This discovery makes the high-energy physicists' dream come true that one can access relativistic systems in a table-top device in which one can simulate relativistic experiments. One requires much less experimental experimental conditions than the usual ones in traditional high-energy physics. Therefore, high-energy physicists and eld theorists now have a fascinating playground in these low-energy systems where their theory can be applied and veri ed.

where η µν = Diag(1, -1, -1, -1) is the metric tensor and I 4 is the four-by-four identity matrix. This anti-commutation ensures the square root of the operators is possible.

Here, I do not specify an explicit set of matrices for γ µ because there are di erent possibilities due to the Lorentz invariance of Eq. ( . ). One can thus choose a set of γ µ matrices, i.e., a coordinate This is only one of many basis, called the Dirac basis, in which the matrices αi and β are written. This will be further explained in terms of γ matrices in the following. Depending on the context, I will also denote the three Pauli matrices as σi with i = x, y, z.

Here, I call Dirac materials all the materials described by a Dirac-type Bloch Hamiltonian with only linear terms in momentum no matter whether they are gapped or gapless.

Depending on the ratio |w/v|, WSMs are classi ed into two phases [ ]: type-I WSMs if |w/v| < 1 and type-II WSMs if |w/v| > 1. The crucial di erence between the two types of WSMs is that the Fermi surface is ellipsoid and closed for type-I but it is hyperbolic and opened for type-II (see Fig. . ).

The two shapes of the Fermi surface induce distinguishable responses to magnetic eld as I will show in Chapter .

A more elementary proof is given in [ ].

for his fatherly patience, enlightening guidance, and endless vitality. He taught me a lot of beautiful physics that I have dreamt of. He has shared with me his good research taste from which I will bene t from my career. All his insightful comments on my works have always been an invaluable source of inspiration to deepen my understanding. I also would like to thank Mark for his friendship, empathy, and great sense of humor. I will remember forever the comforting message that Mark sent to me during the rst con nement in France. He is always prepared to listen to me and try to resolve my anxiety.

Without Mark, I might have still nished my Ph.D. but I could never end it full of warm memories, happiness, and satisfaction.

My sincere thanks go to the rest of the jury of my thesis: Dr. Dario Bercioux and Pr. Björn Trauzettel as reporters, Pr. Silke Biermann, Dr. Milan Orlita, and Dr. Pascal Simon as examiners, for their acceptance to participate in the defense. Special thanks to Silke, who was my teacher of solid-state physics at Ecole Polytechnique and opened the door for me to the world of condensed matter theory, Milan, with whom I collaborate closely in the NbAs 2 project and who always has refreshing ideas, and Pascal, who taught me the basics of many-body physics and who continues to discuss with me in the lab. I would like to also thank my committee of thesis formed by Dr. Louis-Anne de Vaulchier and Dr. Miguel

Monteverde, who have given me many useful pieces of advice for my future career.

I thank all the collaborators I have worked with on various fascinating projects. In particular, I would like to thank Dr. Dibya Kanti Mukherjee for our brotherhood-like tight friendship and e orts on the project of Volkov-Pankratov plasmons, Jan Wyzula for all the informative exchanges on the NbAs 2 project, David Alspaugh who incited me to rephrase my work in a general mathematical context.

During my three years' stay in Laboratoire de Physique des Solides, I would like to thank all the permanents in the theory group, especially Dr. Frédéric Piechon, from whom I have learned a lot of physics, Pr. Marc Gabay, who is the mentor to initiate me in teaching, and Marie-France Mariotto for all her help in administration stu .

Of course, I will not forget to thank all my young fellow lab-mates of which I cannot exhaust the list here. I am grateful to Dr. Pierre Bruneel, with whom I share the o ce as well as the knowledge in physics and who is the rst listener of my complaints, Dr. Maxime Garnier for his French humor, iii Ansgar Graf who was also my roommate during the workshop in Aussois, Dr. Jonathan Atteia with whom I discuss many papers on graphene, and Brendan Le Pennec who was my classmate during my master and told me recent progress on spin liquids.

Undoubtedly, I deeply thank all my relatives, especially my wife for her warm accompaniment through my thesis and comforting words of encouragement that o er me con dence and courage, and my parents and parents-in-law for their continuous support and care.

Finally, I would like to say something to encourage all other young physicists like me, who are trying to do a good thesis. Ph.D. is never easy. "Doing a Ph.D. is like having a baby", said Mark, my Ph.D.

advisor. One needs to give one's motherly passion and patience to grow the baby, i.e., the thesis. One of my energy sources that I would like to share here was best said by the scienti c giant, Paul Dirac, "Living is worthwhile if one can contribute in some small way to this endless chain of progress". By the way, the time span of the thesis reminds me about Nezha the Third Lotus Prince, a protection deity in Chinese folk religion who was born after three years of pregnancy just like a thesis. I wish that my thesis could be Nezha in the sense that it could inspire other scientists and bring them courage and luck.

Paul Dirac

The Dirac equation is a quantum mechanical wave equation originally proposed by Paul Dirac in [ , ] to describe all spin-1/2 relativistic particles in high-energy physics moving at a speed v close to the speed of light c. It incorporates Einstein's theory of special relativity in the framework of quantum mechanics. The Dirac equation is thus a relativistic version of the Schrödinger equation that governs the motion of non-relativistic particles in quantum mechanics, or conversely, the Schrödinger equation is an approximate version of the Dirac equation when v c. In condensed matter physics, low-energy electrons moving at a speed v/c 1 are soundly seen as non-relativistic particles. One might think that the Schrödinger equation would su ce to describe electrons in solids and the relativistic e ects are a small correction term such as spin-orbit coupling (SOC) [ ]. It seems that the ultra-relativistic, i.e., massless, Dirac equation (also known as the Weyl equation [ ]) was thought to be only applicable for high-energy elementary particles. Surprisingly, by the interplay between the periodic crystal lattice potential and electrons, the massless Dirac equation emerges in the low-energy description of the electronic structure of solids, yet another manifestation of the idea "More is di erent" by Anderson [ ]. This is one of the most spectacular recent ndings in the realm of condensed matter physics. The relativistic massless Dirac equation or variations of it can actually describe the physics around the Fermi level of real materials such as graphene [ , ], Dirac/Weyl semimetals [ , , ], nodal-line semimetals [ ], etc. These materials are named Dirac materials in which the mathematical treatment for the electrons is perfectly identical to that of relativistic highenergy elementary particles.

Reciprocally, the idea that relativistic physics is manifest in condensed matter is not uniquely rewarding for high-energy physicists. It is also enlightening for condensed matter physicists. Many unexpected observations in Dirac materials are thus attributable to the special relativity embedded in the condensed-matter physicists have been interested mostly in the electronic band structure of solids and often forgotten information encoded in the wavefunctions apart from their symmetry properties which matter in optical transitions. The importance of wavefunctions is best illustrated by a brain-refreshing example that two Hamiltonians with exactly the same energy spectrum could have distinguishing electromagnetic responses as shown in the seminal paper of Haldane [ ]! These two Hamiltonian are now classi ed as topologically di erent phases of matter by an index called topological invariant. The idea behind this is that the topology of a Hamiltonian describes how the energy bands are coupled to each other so that only responses involving interband process such as the Hall conductivity [ ] can reveal the topological properties of matter. This is also the reason why people have overlooked the topological aspect of Hamiltonians for a long time while considering only one band alone, especially in semi-classical treatments [ ]. Therefore, a matrix formulation of the Hamiltonian is necessary to study all the relevant bands together in order to capture the topological properties of solids. The scalar Schrödinger Hamiltonian should be thus replaced by the matrix Dirac Hamiltonian. Hence, the Dirac equation and its variants are also suitable to describe the low-energy physics of topological materials (including Dirac materials), namely materials having intriguing properties due to the non-trivial topology of their Hamiltonian.

One of the tantalizing properties of topological materials is the bulk-edge correspondence which means that topologically protected conducting edge states exist when the bulk Hamiltonian is topologically non-trivial. These edge states are believed to enable the fabrication of more energy-e cient microelectronic components, better catalysts, improved thermoelectric converters, new magnetic storage media or even quantum computers [ ]! The mechanism of emergence of the edge states of topological materials is the band inversion meaning that the order of valence and conduction bands in energy is inverted across the interface between materials and vacuum. To explicitly show the presence of the edge states, the Dirac equation, which now becomes spatially dependent, is the unavoidable theoret-Especially, a linear conical band dispersion requires at least two bands where the use of the Pauli matrices are natural. ical formalism. It is actually a unifying general approach to study all the surface states of topological systems related to band inversions.

In summary, the Dirac equation occupies an essential position in the recent progress of condensed matter physics. The reader will also nd the Dirac equation throughout this thesis which is interested in the spectroscopic properties of topological materials using the Dirac equation. In the present introductory chapter, I would like to rst recapitulate the history of the Dirac equation. Then, I will show some examples of Dirac materials of which the Hamiltonians are described by generalized versions of the Dirac equation. As a paradigm of Dirac materials, graphene is the best to illustrate the idea of "doing high-energy physics in low-energy systems" of which I will give three examples here. After that, I will brie y present topological band theory and topological materials using the model proposed by

Haldane [ ], also known as a quantum anomalous Hall insulator. Within this model, the bulk-edge correspondence manifests itself as a bulk insulating phase with topologically protected conducting channels at its edges. This will be explained by a general argument based on the topological band theory. Furthermore, I will derive the energy and the wavefunctions of these topological edge states with the help of the Dirac equation by which the band inversion mechanism is explicitly shown. The stability of the edge states is understood by the Jackiw-Rebbi I cannot resist to sharing a historical perspective on the Dirac equation. In , Max Planck postulated that electromagnetic energy could be emitted in quantized form, later known as photon, based on his experimental observation of black-body radiation. This observation is now seen as the ignition of the following explosion of knowledge on quantum mechanics. Twenty-six years later, to describe a particle as a wave in an explicit way, Erwin Schrödinger [ ] came up with one of the most important equations in the Human history: ) where = h/2π is the reduced Planck constant, m is the mass of the particle of interest, V (r, t) is the energy potential which the particle experiences at a given position r and time t and ψ(r, t) is the wavefunction that fully determines the particle's movement in a probabilistic way. This equation is now given the name of Schrödinger's equation. The right part in the square brackets can be seen as an operator known as the Hamiltonian acting on the wavefunction ψ. The eigenvalue of the Hamiltonian is the total energy of the particle including kinetic and potential energy. Solving the spectrum of a given particle is nding the eigenvalue of its corresponding Hamiltonian. By the principle of correspondence, one can write the Hamiltonian for Eq. ( .)

i ∂ ∂t ψ(r, t) = - 2 2m ∇ 2 + V (r, t) ψ(r, t) ( .
Ĥ = - 2 2m ∇ 2 + V (r, t) = p2 2m + V (r, t) ( . )
where the momentum operator p is introduced. In this form, it is clear that Schrödinger's equation is only valid for non-relativistic particles, e.g., electrons with moderate velocity compared to the speed of light, because the kinetic energy part is p 2 /2m. Then, Eq. ( . ) becomes i ∂ ∂t ψ(r, t) = Ĥψ(r, t).

( . )

However, in Einstein's special relativity [ ], the energy of a relativistic particle is

E 2 = p 2 c 2 + m 2 c 4 ( . )
where E is the energy of the particle, p is the momentum, m is the rest mass of the particle and c is the speed of light. It is the famous Einstein's energy-momentum relation. Furthermore, Einstein suggested in his special relativity theory that the time dimension should play the same role as the spatial dimensions. But, the Schrödinger equation isolates the time dimension as a special one. It has the second derivative in space but only the rst derivative in time. To write a relativistic equation for quantum mechanics, it is tempting to replace E by i ∂ t and p by -i ∇ in Eq. ( . ). By doing so, one has ) which is known as the Klein-Gordon equation [ , ]. It treats indeed the dimension of time and space both with second derivative. However, this equation is more suitable to describe relativistic bosons of integer spin. The use of this equation for fermions such as electrons is rather limited.

-2 ∂ 2 ∂t 2 ψ(r, t) = -2 c 2 ∇ 2 + m 2 c 4 ψ(r, t) ( .
When Paul Dirac was facing the Klein-Gordon and Schrödinger equations, his solution was to transform the spatial and time derivatives to rst order. It turns out that the price to pay to make Eq. ( . ) a rst-order equation is that the equation must be written in matrix form. Paul Dirac introduced a four-As an anecdote, the proposal of the Klein-Gordon equation was in fact before the Schrödinger equation.

component wavefunction and four-by-four matrices that would allow him to deal with the square root of the operators. The nal form in the four-dimensional spacetime world was: . ) in terms of

i ∂ ∂t ψ(r, t) = -i cα • ∇ + βmc 2 ψ(r, t), (
β = σ 0 0 0 -σ 0 , α i = 0 σ i σ i 0 , ( .
)
where σ 0 is the two-by-two identity matrix and σ i (i = 1, 2, 3) are the Pauli matrices

σ 1 = 0 1 -1 0 , σ 2 = 0 -i i 0 , σ 3 = 1 0 0 -1 . ( . )
As for the Schrödinger equation, one can de ne the associated Hamiltonian as

Ĥ = -i cα • ∇ + βmc 2 = cα • p + βmc 2 , ( . )
which is also called the Dirac Hamiltonian, so that Eq. ( .) can be also written as Eq. ( . ). It is sometimes useful to write the original Dirac equation in the covariant form ic γ µ ∂ µ -mc 2 ψ(r, t) = 0.

( . )

which is invariant under Lorentz transformations. Here the notation implies summation over repeated indices µ (Einstein convention). The indices take the values 0, 1, 2, 3 and correspond to the t, x, y, z space-time dimensions, respectively, while ∂ µ is the covariant four-gradient and γ µ are the four-by-four matrices obeying the Cli ord algebra {γ µ , γ ν } = 2η µν I 4 ( . ) system for the space-time dimensions in the convenience of the problem of interest. For example, if the particle is massless, it is worthy to write the Dirac Hamiltonian ( . ) in the Weyl basis in which the γ µ matrices are H D Thanks to the periodicity and space group symmetry of some crystal lattices, the low-energy physics of electrons in solids can be sometimes modeled by the Dirac-type Hamiltonians. In hindsight, the hints for the previous fact are already given in Bloch's theorem which states that the energy spectrum of solids consists of several bands. In particular, if one considers excitations from valence to conduction bands, a matrix formulation is inevitable. It is thus not surprising that the analogy of the notion of particle and anti-particle [ ] with that of electron and hole fascinated physicists in the great epoch of semiconductors in the s [ ]. While the massive Dirac Hamiltonian easily nds its application in condensed matter physics, it is unexpected that the massless Dirac Hamiltonian can also emerge in real materials. Although some theoretical predictions were already made in s [ , ], these studies did not attract much attentions until the synthesis of graphene in [ ].

γ 0 = 0 I 2 I 2 0 , γ i = 0 σ i -σ i 0 (
Then begins the gold rush for Dirac materials. People are searching in the periodic table all the candidates whose low-energy band structure can be modeled by the variations of the Dirac Hamiltonian.

The advent of graphene also inspires eld theorists to verify in materials their theories initially proposed for elementary particles in high-energy physics.

In this section, I will present several Dirac materials which I will investigate further in the following chapters. Some of them are widely surveyed and some of them have been realized only recently. Then, I will take graphene as an example to demonstrate three relativistic phenomena already seen in materials.

Z D

The genesis of Dirac materials starts at the synthesis of graphene thanks to Geim and Novoselov, two Nobel prize laureates in physics of . Graphene is a two-dimensional ( D) semimetal which possesses two in inequivalent band touching points in the rst Brillouin zone at the charge neutral point.

In the vicinity of these band crossing points, the Hamiltonian to rst order in momentum reads ( . )

H = v(ξk x σ x + k y σ y ), (
While the derivation of the original Dirac equation ( . ) or Hamiltonian ( . ) is based on Einstein's theory of special relativity which has to preserve the Lorentz symmetry, there is no such a constraint in the emergent Dirac Hamiltonian of Dirac materials. In particular, the velocity could be anisotropic for in the x, y-plane or the Dirac cones could be tilted such that the Hamiltonian becomes

H = wk x + v(k x σ x + k y σ y ) + ∆σ z ( . )
where the velocity w is the slope of tilt. See The three-dimensional ( D) Weyl Hamiltonian ( . ) can also emerge in materials know as Weyl semimetals (WSMs). The band touching points are called Weyl nodes. As shown by Nielsen and Ninomiya in [ ], later known as fermion doubling theorem or Nielsen-Ninomiya no-go theorem, the number of the Weyl nodes of chirality λ = + must be equal to that of chirality λ = -in the lattice realization of the Weyl equation. Unlike in graphene, the band touching points of WSMs are generic and stable. This can be seen by considering the most general two-by-two Hamiltonian

H = f 0 + f x σ x + f y σ y + f z σ z whose spectrum is E = f 0 + ± f 2 x + f 2 y + f 2 z .
The valence and conduction band touch if f x = f y = f z = 0 simultaneously. In D space, each coe cient f i depends on D momentum (k x , k y , k z ) so that three equations with three variables yields always solutions. Introducing a mass gap like ∆σ z would not gap but only move the Weyl nodes in reciprocal space. However, the band touching in D requires symmetries or ne tuning. So, the term "generic touching point" means that while it is not guaranteed to nd any such touching events in an arbitrary material, one should not be surprised to nd them either. Nevertheless, inversion and time-reversal symmetries play important roles for WSMs. If the inversion symmetry is present and time-reversal symmetry is broken, the minimum model needs only two Weyl nodes. In contrast, if the inversion symmetry is broken and time-reversal symmetry is present, the minimum model needs at least four Weyl nodes. The reason is that one must verify the fermion doubling theorem while the time-reversal operation preserves the chirality of the Weyl nodes and the inversion switches it. If the Hamiltonian commutes with both inversion and time-reversal symmetries, then all bands are Kramers degenerate already and any additional touchings would generate points of fourfold degeneracy known as Dirac points. The associated material is called Dirac semimetal.

The rst theoretical proposal of material realization of WSMs was given by Herring in [ ].

More recently, the book by Volovik published in [ ] has been in uential in re-drawing the attention of theorists to the Weyl physics in materials. Similar to the D massless Dirac Hamiltonian, the anisotropy of velocity and the tilt are also allowed in the D Weyl Hamiltonian. For example, a tilted WSM's Hamiltonian reads

H = wk z + v(k x σ x + k y σ y + k z σ z ).
( . )

Many materials are now theoretically and experimentally identi ed as instances of the Weyl Hamiltonian. For example, as a representative of the transition metal monopnictide, TaAs is a non-magnetic and non-centrosymmetric material and con rmed to be an inversion-broken WSM with twelve pairs In addition to Weyl and Dirac nodes, the band touching points can form a continuous line in reciprocal space in materials called nodal line semimetals [ ] (see Fig. . ). They have been predicted and observed via the angle-resolved photoemission spectroscopy (ARPES), for example, in ZrSiS [ , ].

However, the combination of inversion and time reversal alone cannot protect the nodal line from being gapped out if the SOC is included. The resulting phase of matter is thus called gapped nodal line semimetal. An example that I will explain thoroughly in Chapter and is the transition metal dipnictides NbAs 2 described by a low-energy Hamiltonian

H = wk z + v(k x σ x + k y σ y ) + ∆σ z . ( . )
Its energy pro le is shown in electrons can tunnel through a potential step with perfect transmission. This particular phenomenon has been shown to be relevant in electron scattering in graphene. When electrons pass through an extremely narrow graphene bipolar junctions, the Klein tunneling manifests itself as the phase shift observed in the conductance fringes at low magnetic elds, a signature of the perfect transmission of normally incident electrons [ , ]. A theoretical pedagogical review on the Klein tunneling in graphene can be found in [ ].

Another famous phenomenon in atomic physics to mention is termed "atomic collapse" [ , ]: an atom with an atomic number beyond a critical value (Z = 172) cannot exist because of the leakage to the negative energy spectrum. To experimentally prove this theoretical prediction is extremely di cult.

Looking at the periodic table, the highest atomic number that can be achieved by all the present highenergy techniques is only below , still long way to go. However, atomic collapse is shown to be able to be simulated in graphene with point charge impurities over it [ ]. The experimental demonstration is the observation of a resonant peak in the density of states in the hole band of graphene by scanning tunneling microscope (STM) if atom collapse happens. These two examples already depict a promising prospect of doing high-energy physics in a piece of Dirac materials.

The relativistic renormalization, a famous consequence of the Lorentz transformation, is also predicted to be able to see in graphene in the presence of crossed electric magnetic eld [ ]. One does not need to contemplate the Galaxy using immense astronomical telescopes to see the relativistic renormalization of electromagnetic elds. As I will show in Chapter and , a tiny piece of crystal will su ce.

The manifestation of the relativistic renormalization in graphene is that the Landau level structure collapses into a continuum in the presence of a magnetic eld and an intense electric eld given by a high potential barrier induced on graphene sheet. An indirect experimental signature of the phenomenon is that the oscillations of conductance are found to abruptly disappear when the strength of the magnetic eld is reduced below a certain critical value [ ]. I will also elaborate the theory on the relativistic renormalization in Dirac materials in Chapter and where I will show a more direct and unambiguous experimental signature for it.

Although the projection of relativistic phenomena to Dirac materials is exciting, one must keep in mind that the pseudo-relativity in Dirac materials is purely formal and mathematical. It does not mean that the particles in our treatment are really relativistic. They are indeed in the non-relativistic regime. For example, electrons in graphene move with the Fermi velocity far below the speed of light (v/c ≈ 1/300). For the same reason, the analogy with special relativity is not always guaranteed in Dirac materials. Additional terms, with no known relativistic counterpart, arise naturally in the Hamiltonian describing general Dirac materials as I have shown previously. Although many highenergy phenomena have their corresponding condensed matter representation, one should remember that they must be understood in the language of condensed matter, that is non-relativistic. I will unfold this point in Chapter .

T

Topology is initially a mathematical term to describe the geometrical properties of manifolds which are robust to continuous deformations without any "violent event". In algebraic topology, objects of di erent topology are classi ed by discrete and countable numbers known as topological invariant. A violent event is thus de ned as a transformation on objects switching abruptly the topological invariant from one value to another. At the moment of the violent event, the topological invariant is not wellde ned. For example, D objects can be classi ed by the number of holes, i.e., the genus, though themselves. The number of holes will not change if one just pinches the object without cutting it or pasting together its parts, namely without violent events.

Unlike the Gaussian curvature which is a local property of manifold, topological properties are global geometrical properties. Topology can be thus thought to be the integrated geometry of manifolds. This idea is best illustrated by the Möbius strip which is formed by cutting a trivial strip as a scotch tape, twisting one end by degrees and then pasting the two ends together (see Fig. . ).

Intuitively, the Möbius strip is topologically di erent object from the trivial strip for a Human surviving in D space. However, it is not evident for an ant living on the D strip which seems to only feel the local curvature of the strips. Nevertheless, an ant knowing topology can distinguish the two strips as illustrated in Fig. . . It can mark its starting position and orientation in space before looping on the strip. When it arrives for the rst time back to its initial position, it will know on which strip it has had this journey. If it nds itself on the same side of the strip as the starting point, namely the same orientation, it knows the strip is the trivial one. In contrast, if it nds on the opposite side of the strip, it must live on the Möbius strip. Another equivalent way to classify the two strips is to count the winding number (a topological invariant), namely the number of times that an ant loops on the strip to go back to its initial position on the same side of the strip as that of departure. The ant needs twice for the Möbius strip but only once for the trivial one.

The previous idea can be applied in physics as shown by Berry in [ ]. Given a parameterdependent Hamiltonian H(φ), its corresponding wavefunction |ψ(φ) depends thus also the parameter φ. When one changes adibatically φ along a closed path and back to the initial value in the manifold for the parameter, the energy spectrum of the Hamiltonian H(φ) remains unchanged but the wavefunction |ψ(φ) accumulates a geometric phase exp(iθ) depending on the topology of the Cutting and pasting are the two violent events that change the topology of the strip. Here, the ant is considered as a point moving on the two sides of the strip. Although the strips are D objects, the dimension of the width of the strip is irrelevant in the discussion so that the strip is e ectively considered as a D object. Note that an eigenstate multiplied by a phase factor is still an eigenstate of the same energy. Choosing the phase as a function of φ is choosing a gauge for the eigenstates. The fact that all the observables should be independent of the choice of gauge is one of the fundamental principles of quantum mechanics. However, it does not mean that the gauge phase is not important as I will show in the following by the Aharonov-Bohm e ect and all the topological e ects. closed path. The phase θ is now known as the Berry phase. The revolutionary message from Berry in his seminal paper [ ] is that the Berry phase θ is gauge-independent when the path tracked by the parameter φ in its manifold is closed. In other words, the phase φ is in principle a physical observable.

The notion of the Berry phase may be more understandable if one compares it with the ant on the strip. In that case, the position of the ant on the strip plays the role of the parameter φ, the ant itself is regarded as the wavefunction and the strip is thus the manifold of the parameter-dependent Hamiltonian. Then, a closed path traveled by φ is equivalent to a loop by the ant. The Berry phase associated to the closed path is 0 if the ant goes back to the initial position with the same orientation; the Berry phase is π if the ant nds itself upside down after a loop on the strip. Therefore, the two di erent

Hamiltonians associated to the trivial and Möbius strip are called topologically trivial and non-trivial Hamiltonian, respectively. Sometimes, the topologically non-trivial Hamiltonian is said to be twisted by the analogy with the Möbius strip.

The rst proposal and observation of the Berry phase can be traced back to the paper published in and in [ , ] on the Aharonov-Bohm e ect. In this case, the Berry phase is the phase accumulated by a charged particle in the vector potential along a closed loop so that it is proportional As an anecdote, this result was too shocking to be accepted by the physicists of that time. There was even a paper with the title "Nonexistence of the Aharonov-Bohm e ect" published in [ ].

to the magnetic ux. On the one hand, the Aharonov-Bohm e ect demonstrates the wave nature of electrons by interference. On the other hand, it also reveals the importance of the gauge in modern theoretical physics and inspires the notion of the Berry phase.

In the Aharonov-Bohm e ect, the parameter to change is the position in real space. One can thus transplant this idea in reciprocal space in which the crystal momentum k is a natural choice of the parameter. In this section, I will show how Berry phase arises in electronic band theory of solids while studying Bloch Hamiltonians H(k). Then, I will present one of the important consequences of the Berry phase e ect, i.e., the bulk-edge correspondence. Besides general topological arguments, I will provide a treatment by the Dirac equation to explain the bulk-edge correspondence.

T B

As shown by Bloch in [ ], the energy of electrons in a periodic crystal lattice can be indexed by the crystal momentum k, which yields electronic band structure. Given the Hamiltonian H, the corresponding eigenstates |ψ n (k) of energy E n (k) can be written as

|ψ n (k) = e ik•r |u n (k) ( . )
where |u n (k) is the cell-periodic Bloch state. Via a unitary transformation, the Bloch Hamiltonian

H(k) = e -ik•r He ik•r ( . )
has the same spectrum E n (k) as H and the corresponding eigenstates |u n (k) verify

H(k)|u n (k) = E n (k)|u n (k) . ( . )
The crystal momentum k is thus the parameter to vary in the de nition of Berry phase in topological band theory. The quantity to measure the variation of the phase accumulated by |u n (k) while varying k is called Berry connection de ned as

A n (k) = i u n (k)|∇ k u n (k) ( . )
A pedagogical introduction of Bloch's theorem can be found in [ , ].

where n is the band index. By de nition, the integration of Berry connection along a closed loop is equal to Berry phase. One can also de ne the associated Berry curvature as

B n (k) = ∇ × A n . ( . )
In the context of topological band theory, the closed path in reciprocal space often refers to paths going though the rst Brillouin zone. Therefore, the corresponding Berry phase θ n of a band n is given by

θ n = ∂BZ dk • A n (k) = BZ d 2 k • B n modulo 2π. ( . )
As implied by the notation, the Berry connection and curvature are said to be the vector potential and magnetic eld in reciprocal space, respectively.

The Berry phase is closely related to the topological invariant of materials whose de nition depends on the dimensionality of the parameter space. For example, the topological invariant for a D band of index n is the rst Chern number de ned as

C n = θ n 2π ∈ Z ( . )
which must be an integer on lattice according to the Atiyah-Singer index theorem [ , , ]. The topological properties of a given Hamiltonian are thus determined by the sum of the Chern numbers of all the lled bands. Note that the Chern number is only well-de ned in a gapped system. In the case of degenaracies in multi-band systems, one can generalized the de nition of Berry connection and phase to a tensor form

A mn (k) = i u m (k)|∇ k u n (k) ( . )
as in the Yang-Mills gauge theory where the non-abelian Wilson loop in a matrix form is more suitable [ , ]. In the following discussions, I suppose that all the bands are separated from each other and there exists a gap between the bands. Here, I assume the band n is separated from all other bands.

The reader may wonder that the above expressions [see for example ( . )] give the impression that the topological properties encoded in the Berry curvature in the nth band only depends on a single band contrary to what I have said at the beginning of the present chapter. Actually, this is only a pseudo-paradox if one remembers that the rst derivative of the vector eld is orthogonal to itself, i.e., it can be written as a combination of vectors orthogonal to itself. To explicitly show that the Berry curvature is related to virtual transitions between bands (at xed k), one uses rst-order perturbation theory for wavefunction so that the Berry curvature for a D band reads

B n (k) = i n =n u n |∂ kx H|u n u n |∂ ky H|u n (E n -E n ) 2 + c.c. ( . )
where the energies E n are supposed to be non-degenerate. The reader should now be convinced that topological band theory is a band theory considering the coupling between bands via the virtual interband transitions.

B

The consequence of virtual interband transitions was proposed by Karplus and Luttinger in Here, I adapt this method to the Haldane insulator.

"Anomlaous" because it is a Hall e ect in the absence of magnetic eld.

Quote from Bernevig, one of the founders [ ] of topological band theory [ ].

For the reason of simplicity, I consider an interface between the Haldane insulator and a trivial one in D described by a Hamiltonian of similar structure called Semeno insulator [ , ]. The Hamiltonian for the latter reads

H = v(k x τ z σ x + k y σ y ) + ∆σ z ( . )
where the mass gap remains the same value for two valleys. The spectrum of the Semeno insulator is exactly the same as that of the Haldane one. The Chern number at the charge neutral point for ( . ) is zero meaning that the Semeno insulator is topologically trivial and its Hall conductivity in the absence of a magnetic eld is zero. This is precisely a concrete example of what I have said at the beginning of the chapter: two Hamiltonians of identical spectrum can have di erent response to electric eld.

An interface between two topologically di erent phases is called topological heterojunction [ ]. Suppose that the interface between the two insulators is formed around x = 0 in space. When x → -∞, one nds the Haldane insulator; when x → +∞, one nds the Semeno insulator. To model the topological heterojunction connecting a Haldane and Semeno insulator, one can replace the constant mass gap by a position-dependent one

∆(x) =    ∆ 0 τ z if x → -∞ ∆ 0 if x → +∞. ( . )
The order in energy of valence and conduction bands is inverted in the valley τ z = -1 (band inversion) since the gap change its sign. Note that the mass gap is inverted only for τ z = -1 so that one can focus on the valley τ z = -1. The resulting Hamiltonian reads

H = v(-k x σ x + k y σ y ) + ∆(x)σ z ( . )
where

∆(x) =    -∆ 0 if x → -∞ ∆ 0 if x → +∞ ( . )
and k x should be considered as the derivative -i∂ x . The wavefunction of the eigenstate of the Hamiltonian can be written as ψ = e ikyy χ(x).

( . )

First, considering k y = 0, one can show that E = 0 is also an eigenvalue of the Hamiltonian [ , ] with the two possible wavefunctions

χ ± (x) = exp ±1 v x x 0 dx ∆(x ) 1 ±i ( . )
which is also the eigenstate of the Hamiltonian ( . ) with energy E = ∓ vk y . However, only one of the two wavefunction is square-integrable depeding on the sign of ∆ 0 . Given ∆ 0 > 0, the only physically viable solution is

χ -(x) = exp - 1 v x x 0 dx ∆(x ) 1 -i ( . )
with the energy E = + vk y . Therefore, only one edge mode is present at the surface of the Haldane insulator. Most saliently, it propagates only in the positive k y -direction with a de nite chirality given by the spinorial part of its wavefunction. So, it is also called chiral state. Since there is no counterpropagating partner nearby this edge mode, it is immune to the back-scattering.

The argument of the emergence of edge states using the Dirac equation is rather general. One only relies on the fact that the mass gap ∆(x) changes its sign across the topological heterojunction so that the bands are inverted. The only way to get rid of the edge state E = vk y is to invert the gap of one of the two bulk phases. Since this would cost large amount of energy, the edge state is stable and protected by the gap.

Additional surface states called Volkov-Pankratov states can arise when the interface is smooth enough as shown by Volkov and Pankratov in their work [ , ]. Since the gap changes its sign across the topological heterojunction, one can generally linearize the gap function in the vicinity of the gap closing point, e.g., x = 0. The wavefunction of the chiral state becomes

χ -∼ e -∆ v x 2 for |x| e -∆ v |x| for |x| , ( . )
where is the length scale associated with the smoothness of the topological heterojunction. The smoothness of the surface can be interpreted as a pseudo-magnetic eld [ ] so that the stability of the chiral state is also understood by the argument given by Aharonov and Casher • Quantum eld theory for condensed matter [ ]

• Quantum Hall e ect [ , ],

• Special relativity [ ],

• Topological band theory [ , ],

• Topological insulators and semimetals [ , , , ] • Topology in physics [ ].

S D

The mathematical education of the young physicist [Albert Einstein] was not very solid, which I am in a good position to evaluate since he obtained it from me in Zurich some time ago.

Hermann Minkowski

The present chapter will show in detail how frame-dependent electrodynamics, known in Einstein's special relativity, can be also observed in Dirac materials. On the one hand, the reader will see the full derivation of how relativistic electrodynamics emerges in graphene and how to solve it using so-called hyperbolic transformations, a spinor representation of Lorentz boosts. On the other hand, I will show how to extend the use of hyperbolic transformation to a generic problem in the condensed-matter community: Landau quantization for Dirac Hamiltonians with tilting term.

I will start from the prototypical Dirac material, i.e., graphene, and discuss how an out-of-plane magnetic eld and an in-plane electric eld conspire to reconstruct the low-energy spectrum of graphene. I will solve the problem in two di erent ways: one is to map it to a problem in Einstein's special relativity and the other one is to solve it quantum mechanically using hyperbolic transformations. These two methods are physically equivalent. However, the latter provide a general approach to solve a larger set of problems: the energy spectrum of tilted Dirac materials in the presence of magnetic eld. Besides, it also provides wavefunctions that will be crucial for the next chapter on the spectroscopic properties of Dirac materials. Apart from graphene, I will illustrate the application of hyperbolic transformations to the problem of Landau quantization with three other examples: tilted gapped graphene, tilted

Weyl semimetal and a gapped nodal-line semimetal, NbAs 2 . As for the latter material, I have studied the role of the tilt in magneto-optical spectroscopy within a collaboration with Milan Orlita's group at the Grenoble High-Filed Lab (LNCMI). While previous studies focus on the gapless system like Dirac/Weyl semimetals [ , ], I am mostly interested in systems with gap.

After this more abstract relativistic description of Dirac materials, I will bring the discussion to earth with my proper understanding by interpreting the pseudo-relativity in the language of condensed matter and clarifying their mechanisms. This less fancier way of looking at these system is, in my view, as important as the relativistic description. It emphasizes that the analogy with special relativity can be viewed in simple geometric terms. The two conceptually di erent perspectives are actually the two sides of the same coin.

S G

Being the rst experimental realization of Dirac materials, graphene is also the simplest one to study both experimentally and theoretically. In particular, if one wants to study the energy spectrum of Dirac materials in the presence of crossed electric and magnetic elds [ , ], it is the best candidate to begin with. The system is two-dimensional so that only the out-of-plane components of the magnetic eld is relevant to the spectrum. For the same reason, the electric eld can be supposed to be applied in the plane of graphene because an out-of-plane electric eld cannot induce any current but only dopes

the system changing the chemical potential. Since its low-energy Hamiltonian is isotropic, the electric eld is equivalently applied in each direction. All the traits justify selecting graphene as the starter of the banquet.

In this section, I will rst show how the Dirac equation emerges in graphene after a short introduction to the Dirac equation in 2 + 1 space-time. Before introducing the electric eld, the energy spectrum with only a magnetic eld will be shown to form a series of Landau levels indexed by an integer. Then, I will solve the spectrum when the electric eld is also present by analogy to special relativity.

To be self-contained, a short reminder of electrodynamics in special relativity will be given in advance.

U D

Graphene is an atomically thin monolayer of carbon atoms. Electrons of graphene live in a two-dimensional space. To show the emergence of the Dirac equation in graphene, I will rst derive the Dirac equation in the 2+1 space-time dimensions. Then, I will compare it with the low-energy Hamiltonian of graphene derived from the tight-binding model. The reader will see soon that the low-energy Hamiltonian of graphene has exactly the same mathematical structure as the 2 + 1 Dirac equation.

D

So far, the reader has encountered the Dirac equation in the form of Eq. ( . ) in the introduction for 3 + 1 (three spatial and one temporal) dimensions. Now I derive its 2 + 1 dimensional counterpart.

The equation has the same form as Eq. ( . ) but now µ = 0, 1, 2 ⇔ t, x, y. Furthermore, γ µ only need to be two-by-two matrices to satisfy the respective anti-commutation relation by replacing I 4 by I 2 in Eq. ( . ) and η µν = Diag(1, -1, -1). A possible choice is to set

γ 0 = σ 3 , γ 1 = -iσ 2 , γ 2 = -iσ 1 , ( . ) so that the 2 + 1 Dirac equation reads i ∂ ∂t ψ(r, t) = -i cσ • ∇ + mc 2 σ z ψ(r, t). ( .
)
where I interchangeablely use the notation σ i with i = x, y, z and 1, 2, 3 and σ = (σ x , σ y ). Finally, the corresponding Dirac Hamiltonian is

Ĥ = cσ • p + mc 2 σ z . (
a 1 = a(1, 0), a 2 = a 1 2 , √ 3 2 ( . )
where a is the lattice constant and has a value of around . Å. Each carbon has three nearest neighbors. Given a carbon atom, its nearest neighbors are situated at the positions

δ 1 = d √ 3 2 , 1 2 , δ 2 = d - √ 3 2 , 1 2 , δ 3 = d(0, -1) ( . )
where d = a/ √ 3 is the nearest neighbor distance.

Electron-electron interaction is omitted so that electrons are considered independently in the periodic background potential emanating from the carbon lattice. I am only interested in the low-energy physics of graphene which are governed by the valence electrons of p z orbital of carbon atoms. Since the typical Fermi wavelength λ F is much longer than the length of chemical bond between the nearest

The dimension of γ matrices must be an even number since they obey Cli ord algebra. One can prove that the dimension should be 2 [D/2] where [D/2] is the integer part of half of the space-time dimension [ ]. carbon atoms and the typical amplitude of lattice ripple, the electrons are supposed to be con ned in two-dimensional space. It turns out that a tight-binding model considering only the nearest neighbor hopping is able to capture the low-energy physics of graphene [ ]. In the second quantization, the tight-binding Hamiltonian is

H = -t R c † R,A c R,B + c † R,A c R-a 1 ,B + c † R,A c R+a 2 ,B + H.c. ( . )
where c

K = 4π √ 3a (1, 0), K = 4π √ 3a (-1, 0). ( . )
Since the p z orbitals are occupied by only one electron, the resulting band structure is half-lled and the Fermi energy is situated at the band touching points. To describe the low-energy electronic dynamics, one can therefore expand the Hamiltonian ( . ) to linear order around K and K points, with q = k -K( ). This yields the low energy Hamiltonian of graphene around K and K valleys:

H ξ (q) = v F (ξq x σ x + q y σ y ), ( . )
where ξ = ±1 stands for K and K valley, respectively, and v F = 2ta/2 ≈ c/300 is the Fermi velocity. The spectrum of the Hamiltonian is linear: ) which is isotropic around the Dirac point q = 0 and also electron-hole symmetric.

E(q) = ± v F |q|, ( .
Comparing

p → Π = p + eA(r), ( .
)
where e is the absolute value of elementary charge and A(r) is the vector potential that gives rise to the magnetic eld B = ∇ × A(r). Consider the Hamiltonian for valley K and the results for valley

One should not confound here the speed of light c with the annihilation operator.

K follow naturally once the problem for valley K is solved. Henceforth, I set = 1 to simplify the calculations except when the restoration of is mentioned explicitly. Then, the Hamiltonian reads

H(q) = v F [q + eA(r)] • σ ( . )
where the momentum is equivalent to the wavevector. To explicitly do the calculations, I choose the following Landau gauge for the vector potential

A(r) = B(0, x, 0), ( .
)
where B > 0. The corresponding magnetic eld is thus B = Be z . While q y remains a good quantum number, q x should be treated as a derivative operator with respect to x. Remarkably, the o -diagonal terms can be written in terms of ladder operators as for quantum harmonic oscillator. The two ladder operators are de ned as

a = 1 √ 2 q x B -i x + q y 2 B B , ( .
)

a † = 1 √ 2 q x B + i x + q y 2 B B , ( . )
and

1 = [a, a † ], ( . )
where the magnetic length B = /eB is introduced. Using the ladder operators, the Hamiltonian becomes

H = √ 2v F B 0 a a † 0 ( . )
so that its eigenvectors can be written as

|ψ n = sin α n |n -1 cos α n |n if n ≥ 1, |ψ n = 0 |0 if n = 0 ( . )
where |n is the wavefunction of a one-dimensional quantum harmonic oscillator de ned by the previous ladder operators, a † a|n = n|n . The center of |n is linear in q y

x = -q y 2 B . ( . ) For example, in numerical applications or experimentally relevant formula.

Applying the Hamiltonian to a given |ψ n , the matrix to diagonalize becomes

H n = √ 2v F B 0 √ n √ n 0 ( . )
and its spectrum is

E ± n = ± v F √ 2n B ( . )
where is restored (see Fig. . ).

If a staggered potential of type ∆σ z is introduced in ( . ), the new Hamiltonian describes the low energy band structure of graphitic boron nitride hBN. Adopting the same notation as for graphene, the Hamiltonian of BN fo the valley K in the presence of magnetic eld reads

H = √ 2v F B ∆ a a † -∆ ( . )
with the Landau levels

E ± n = ± ∆ 2 + 2 v 2 F 2n 2 B if n ≥ 1, E 0 = -∆ if n = 0. ( . )
Note that the spectrum is not particle-hole symmetric, i.e., a manifestation of the parity anomaly of the 2 + 1D Dirac equation [ , ]. This pseudo-paradox is solved by considering together two valleys K and K where E 0 = ∆ for K Fig. . . Discrete Landau levels are formed when the magnetic eld is perpendicular to the graphene sheet.

If the magnetic eld is not perpendicular but tilted with an angle θ to the plane of graphene, one only needs to consider the out-of-plane part B sin θ. The in-plane magnetic eld which can be generated by an out-of-plane component of vector-potential does not couple to any in-plane momentum. Thus, it will not show up in a Hamiltonian of two-dimensional system. As a result, the Landau levels for an arbitrary magnetic eld remain the same except that B is replaced by B sin θ in the de nition of the magnetic length.

R

Before using the relativistic analogy to solve our problem, one needs the four-vector formalism used in Einstein's theory of special relativity. The reader can refer to App. A. and A. for a short summary of the formalism.

Here, I recall some useful results. If a frame R moves relative to the lab frame R at velocity v, then the electric and magnetic eld in the two frames transform in the following manner:

E = E , E ⊥ = γ(E ⊥ + v × B), B = B , B ⊥ = γ B ⊥ - 1 c 2 v × E ( . )
where the longitudinal and transverse directions are de ned with respect to v and the Lorentz factor

γ = 1/ 1 -β 2 with β = v/c.
Using the electromagnetic eld tensor, one can write two Lorentz invariants :

F µν F µν = 2 B 2 - E 2 c 2 ( . ) µνρσ F µν F ρσ = 8 c E • B ( . )
where E, B > 0 is the magnitude of eld. The value of the two invariants is universal for all the inertial frames. The rst invariant says that if the magnitude of the magnetic eld is larger or smaller than that of the electric eld (in Gaussian units) in one frame, then the inequality holds in all other frames. One can thus distinguish two di erent regimes: the magnetic regime for B > E/c and the electric regime

for for B > E/c [ ].
The second invariant has a profound topological consequence in quantum eld theory [ , ]. A non-vanishing value of E • B is responsible for axion electrodynamics which is theoretically shown to govern in topological materials [ , , ]. Here, I am only interested in the case where E • B = 0, i.e., the magnetic eld is perpendicular to the electric eld. Notably, depending on the sign of the rst invariant, one can always nd a frame where one of the two elds is absent. For example, if B > E/c (B < E/c), there exists a frame in which E = 0 (B = 0). At the critical value B = E/c, there is no frame in which one of the two elds is zero while the other remains nite.

This is coherent with the fact that light is Lorentz invariant and a transverse electromagnetic wave that requires both an electric and a magnetic component.

L

Now I am ready to solve the spectrum of graphene in the presence of crossed electric and magnetic elds, with an in-plane electric and an out-of-plane magnetic eld, along the lines initially proposed by

Lukose et al. [ ].

This con guration of electric and magnetic eld is exactly the one that I have just discussed. To apply the previous results in Einstein's special relativity in graphene, I just replace the limiting speed, i.e., the speed of light c, by the Fermi velocity v F .

Since the Hamiltonian ( . ) is isotropic, one can suppose without losing generality that the electric eld is applied in the x-direction, E = Ee x (E > 0). The magnetic eld B is always in the z-direction, perpendicular to the graphene sheet. Using the Landau gauge A(r) = B(0, x, 0), the Hamiltonian of graphene in the presence of crossed electric and magnetic elds is ) where the rst term on the right is the potential energy of an electron exposed to an electric eld E.

H(q) = eEx + v F q x σ x + v F (q y + eBx)σ y , ( .
One might want to solve the spectrum by writing the Hamiltonian in terms of ladder operators as Special relativity in Dirac materials in the previous section. However, the rst term would be proportional to i(a -a † ) such that the wavefunction expressed in Eq. ( .) is no longer an eigenvector of the Hamiltonian. This problem cannot be avoided by any unitary transformation on the Hamiltonian since the new term is scalar. I will explain this fact in detail in the following section.

This problem may be solved elegantly by appealing to Lorentz invariance of the Dirac equation

[ ].
Here, I reformulate this method in a more elementary way that leads to the spectrum of the Hamiltonian without invoking explicitly the Dirac equation. Using the analogy with special relativity, one is able to nd a moving frame other than the lab frame where the Hamiltonian is written so that the electric or the magnetic eld is zero. In this moving frame, one can solve the problem that graphene is subjected only to an electric or magnetic eld. Finally, one does an inverse Lorentz transformation to get the energy attached to the lab frame. Almost immediately from the result of the previous section, one knows without any calculations that there are two possible regimes: the electric regime where the electric eld is stronger than the magnetic eld (B < E/v F ) and the magnetic regime for the opposite (B > E/v F ). From now on, I am only interested in the the magnetic regime where the magnetic eld is larger than a critical value set by the electric eld, B c = E/v F . There exists then a frame of reference where one has E = E ⊥ = 0 in Eq. ( . ). As a result, the frame of reference R in question moves in the y-direction with the speed v = -E/Be y with respect to the lab frame. The Lorentz factor is de ned as

β = - E v F B e y = - v D v F e y , γ = 1 1 -β 2 , ( .
)
where ||β|| < 1 is the modulus of the drift velocity v D = E/B divided by v F . Then, the magnetic eld in the moving frame is renormalized to B = B/γe z . In the frame of reference R , electrons are thus subjected to an out-of-plane magnetic eld of magnitude B/γ in the absence of electric eld.

The energy spectrum in R can then simply be read o from Eq. ( . )

E ± n = ± v F B 2n γ ( . )
The reason is that graphene is a (semi-)metal with no gap. An electric eld should induce a current making the system in non-equilibrium whereas the Hamiltonian only makes sense at equilibrium. However, tunneling phenomena such as Klein tunneling are possible in the electric regime where the transmission coe cient and the conductance are renormalized by the magnetic eld [ ].

where the magnetic length is still de ned by B in the lab frame. To get the spectrum in the lab frame, it su ces to do an inverse Lorentz transformation on the four-momentum from p µ to p µ de ned by

Λ -1 =     γ 0 γ||β|| 0 1 0 γ||β|| 0 γ     ( . )
so that p µ = (Λ -1 ) µ ν p ν . Note that one has only 2 + 1 dimensional space-time. Finally, the energy spectrum of graphene in the presence of crossed electric and magnetic elds in the magnetic regime is derived

E ± n = ± v F γ B 2n γ -v F βq y , ( . )
where β = v D /v F . To be complete, one anticipates here that the underlying Lorentz transformation also transforms the Hamiltonian and the spinorial wavefunctions as shown later in this chapter. Nevertheless, the spectrum of graphene is obtained without making hands dirty by solving analytically the Hamiltonian ( . ).

R L

As explicitly shown in Eq. ( . ), the spacing between Landau levels is renormalized down by a factor of γ -3/2 . In addition, the previous calculations show that the exponent of -3/2 is the sum of -1/2

and -1 stemming from the magnetic eld and the four-momentum being renormalized when the frame is changed, respectively. Compared to the completely at Landau levels in the absence of the electric eld [see Eq. ( . )], the spectrum acquires a dispersion in q y with a velocity -v D e y . This is precisely the drift velocity of a classical electron due to the Lorentz force created by crossed electric and magnetic elds.

To conclude, a relativistic renormalization e ect characterized by the Lorentz factor γ can indeed manifest in the energy spectrum of originally non-relativistic electrons!

U

In this section, I will show the techniques to solve analytically general Hamiltonians of the form ( . )

and obtain not only its spectrum but also its eigenstates. The latter will be necessary to discuss the spectroscopic properties of Dirac materials. It turns out that one can solve the problem elegantly thanks to This is also the reason why I call it drift velocity in the rst place. The word classical is here the antonym of quantum mechanical.

the symmetry properties of the Dirac equation (or the Dirac Hamiltonian). One does not have to resort to look into the di erential equation but just two sets of transformations acting on the Hamiltonian.

These two kinds of transformations, i.e., the unitary and hyperbolic transformations, will considerably simplify the calculations. The reason why these transformations work is that the Dirac equation is Lorentz invariant but the spinors transform under the two transformations when the frame of reference is changed. So, solving the problem by the unitary and hyperbolic transformations is totally equivalent to switching back and forth the frame of observation as in the previous section.

Here I adopt the attitude of "reinventing the wheel", that is deriving the results well-known in group theory [ , ] according to the need and trying not to invoking too much group theory. This is a practical approach: I need some tools to do the job so I try to rephrase it from zero.

U

A Dirac Hamiltonian only contains Pauli matrices which can be written as

H(x) = x • σ = x 1 σ 1 + x 2 σ 1 + x 3 σ 3 ( . )
where σ i are the three Pauli matrices. Here, I want to nd a unitary transformation U on H(x) so that given a rotation R

U H(x)U † = H(Rx) ( . )
where H(Rx) reads

H(Rx) = (Rx) • σ = R ij x j σ i = x i (R -1 ) ij σ j ( . )
This result justi es why one can organize the Pauli matrices in the form of vector in the previous Hamiltonian: σ acts like a real vector under rotations.

Since the spectrum of H(Rx) and H(x) are both ±||x||, valid matrices U belong to the group U (2). So, I call transformations de ned by U unitary transformations. One can even suppose matrices U is in the group SU (2) since additional phase factor of a unitary matrix is canceled by applying both U and U † . Given U ∈ SU (2), one can thus write

U = e i θ 2 n•σ = cos θ 2 I 2 + i sin θ 2 n • σ ( . )
where n is the normalized vector of the axis of rotation and θ is the angle of rotation. One can check this for a rotation of θ around the z-axis. The corresponding unitary transformation is de ned as

U = e i θ 2 σ 3 = e i θ 2 0 0 e -i θ 2 . ( . )
This would give

(σ 1 , σ 2 , σ 3 ) → (cos θσ 1 -sin θσ 2 , sin θσ 1 + cos θσ 2 , σ 3 ) H(x 1 , x 2 , x 3 ) → H(cos θx 1 + sin θx 2 , -sin θx 1 + cos θx 2 , x 3 ) ≡ H(x 1 , x 2 , x 3 ). ( . )
which precisely do the rotation on x.

One may be curious why only half of the angle appears in the de nition of U . The reason is encoded in the relation between the group SU (2) and SO(3). In group theory, SU (2) doubly covers SO(3), meaning that for each R(θ n) one can nd two di erent SU (2) matrices doing the same transformation, namely ±U (θ n/2). This is precisely the meaning of half angle that θ/2 → (θ + 2π)/2 add a negative sign in front of U while R(θ) = R(θ + 2π).

H L

In some problems like Eq. ( . ), the Hermitian Hamiltonian matrix can always be written as

H(x) = η µν x µ σ ν = x 0 σ 0 -x 1 σ 1 -x 2 σ 2 -x 3 σ 3 ( . )
where the two-by-two identity σ 0 = I 2 is present compared to above. The position of indices has a meaning: contravariant if it shows as superscript and covariant if it shows as a lower index. So,

{σ 0 , σ 1 , σ 2 , σ 3 } ≡ {I 2 , -σ x , -σ y , -σ z } ( . ) {σ 0 , σ 1 , σ 2 , σ 3 } ≡ {I 2 , σ x , σ y , σ z }. ( .
)
The last equality can be veri ed by expanding the exponential in the series.

In analogy with the above discussion, x is considered as a four-vector in the sense that its components transform under the Λ matrices in the Lorentz group, i.e., SO(3, 1). I want to nd a transformation M on H(x) so that given a Lorentz transformation Λ ) where H(Λx) reads

M H(x)M † = H(Λx), ( .
H(Λx) = (Λx) • σ = η µν Λ µ ρ x ρ σ ν = η µν x µ Λ ρ ν σ ρ = x • (Λ -1 σ). ( . )
This result justi es why one can organize σ µ in the form of four-vector in the previous Hamiltonian.

Indeed, σ acts like a real four-vector under Lorentz transformations.

Concretely, Λ matrices are the matrices for Lorentz boosts and rotations. Since rotations have already been discussed, I focus on how to nd a transformation M on σ µ associated to a Lorentz boost on x µ . Clearly, M cannot be a unitary matrix because a Lorentz boost alters x 0 in front of I 2 which is invariant under any unitary transformations. One has to look for candidates in a larger group namely SL(2, C), i.e., the complex special linear group. This group is actually the optimal choice one can have. On the one hand, the Lorentz transformations preserves the Minkowski metric x µ x µ which is also the determinant of the Hamiltonian matrix. A matrix M preserving the determinant of the Hamiltonian belongs to SL(2, C) × U (1). On the other hand, as before, since both M and M † apply, one can only deal with M with unity determinant, that is precisely the SL(2, C). Given M ∈ SL(2, C), one can write

M (φ nφ , θ nθ ) = e (-φ 2 nφ +i θ 2 nθ) •σ ( . )
where nφ,θ are the normalized vector of the axis of boost and rotation and (φ, θ) are the rapidity of boost and angle of rotation, respectively. Here, σ = (σ x , σ y , σ z ). Focusing on Lorentz boosts, one sets θ = 0 so that

M = e -φ 2 n•σ = cosh φ 2 I 2 -sinh φ 2 n • σ ( . )
where hyperbolic functions appear, that is the reason why M is called hyperbolic transformation. The boost Λ determined by the hyperbolic transformation M shows how a four-vector x of the frame R , moving relative to R with a speed v/c = tanh φe z , is written in terms of x of R.

One can check this for a Lorentz boost of rapidity φ along the z-axis. The corresponding matrix M is exp(-φσ z /2). Applying M HM † would give

(I 2 , σ x , σ y , σ z ) → (cosh φI 2 -sinh φσ z , σ x , σ y sinh φI 2 -cosh φσ z ) (σ 0 , σ 1 , σ 2 , σ 3 ) → (cosh φσ 0 + sinh φσ 3 , σ 1 , σ 2 , sinh φσ 0 + cosh φσ 3 ) H(x 0 , x 1 , x 2 , x 3 ) → H(cosh φx 0 -sinh φx 3 , x 1 , x 2 , -sinh φx 0 + cosh φx 3 ) ≡ H(x 0 , x 1 , x 2 , x 3 ). ( . )
This is exactly what does the given Lorentz boost to a four-vector x.

Similar to unitary transformations, only half rapidity appears in the de nition of M because the SL(2, C) group double covers the Lorentz group SO(3, 1) consisting of all the Lorentz transformation matrix Λ. The de nition of M given here is called (0, 1/2) projective representation, thus two-by-two matrices acting on spinors, of the SO(3, 1) group. In fact, the two possible de nitions of M (±φ nφ , θ nθ ) map a matrix in SL(2, C) to a matrix in SO(3, 1). Unlike the mapping from SU (2) to SO(3), the two spin-1/2 projective representations, (0, 1/2) and (1/2, 0), with opposite sign in front of φ in the de nition of M are inequivalent. However, this fact is irrelevant for the practical usage in the following.

The reader should now understand the relation between unitary transformations and rotations and that between hyperbolic transformations and Lorentz boosts. Within each pair, the two di erent operations are the two di erent representations of the same physical transformation acting on their corresponding mathematical objects. Talking about the transformation of the frame of reference, the four-vector is transformed by the Lorentz boost Λ while the Hamiltonian and the spinorial wavefunctions are transformed by the hyperbolic transformation M . In the jargon of group theory, Hamiltonian, spinorial wavefunctions and four-vectors provide di erent representations of the same group SO(3, 1).

C

To emphasize the di erence between the unitary and hyperbolic transformations, I list the properties of both transformations in this part. This would also facilitate the calculations without always doing matrix product such as T HT † .

U

Unitary transformations are two-dimensional representation of rotations. A matrix U of unitary transformations is de ned as

U = e i θ 2 n•σ = cos θ 2 I 2 + i sin θ 2 n • σ ( . )
with the following properties:

• When one applies a unitary transformation on a Hamiltonian H, it means: U HU † .

• U conserves the spectrum of Hamiltonian.

• U is in general non-hermitian. U -1 = U † represents the inverse transformation.

• U forms a multiplicative group namely SU (2) such that the product of two unitary transformations is also a unitary transformation.

• If U = exp(iθσ i /2), then U σ j U † = σ j if i = j ( . )
where σ i,j are the Pauli matrices.

• Sometimes it is useful to think of unitary transformation U (θ n) as a rotation R(θ n) on σ so that σ = Rσ. This is done by U † σU (be careful to the order of matrices).

H

Hyperbolic transformations are two-dimensional representation of Lorentz boosts. A matrix M of hyperbolic transformations is de ned as

M = e -φ 2 n•σ = cosh φ 2 I 2 -sinh φ 2 n • σ ( . )
with the following properties:

• When one applies a hyperbolic transformation on a Hamiltonian H, it means: M HM † .

• M modi es the spectrum of Hamiltonian as a Lorentz boost does to the component 0 of a four-momentum namely energy.

• M is hermitian so that one can write M HM † = M HM .

• M -1 = M † represents the inverse transformation.

• The pure Lorentz boosts M do not form a multiplicative group so that the product of two hyperbolic transformations are in general not another hyperbolic transformation. This is in line with the fact that the product of two non-collinear Lorentz boosts is the product of another Lorentz boost and a rotation called Thomas-Wigner rotation. Alternatively, if M is de ned by Eq. ( . ), they form a multiplicative group named SO(3, 1) including Lorentz boosts and rotations.

• If M = exp(-φσ i /2), then

M σ j M = σ j if i = j ( . )
where σs are the Pauli matrices.

• Sometimes it is useful to think of a hyperbolic transformation M (φ n) as a Lorentz boost Λ(φ n) on σ ≡ (I 2 , σ x , σ y , σ x ) such that σ = Λσ. This is also done by M σM .

G

In the previous section, I considered only Pauli matrices since the Hamiltonian of interest can always be decomposed into their linear combination. One might think that unitary and hyperbolic transformations are only useful for this form of Hamiltonians, but this is not the case. In this section, I will extend the use of unitary and hyperbolic transformations to all the Dirac Hamiltonians that can always be written as a linear combination of the tensor products of the Pauli matrices, in which the Lorentz invariance is generally absent. Rather than a generalization of Lorentz transformations to general Dirac Hamiltonians, it is more appropriate to think of Lorentz transformations as just one particular manifestation of hyperbolic transformation.

Nevertheless, the "generalization" allows us to solve all the Dirac k • p Hamiltonians in the presence of a crossed magnetic and electric elds and alike with only linear momentum such as

H = d i=1 t i k i + d i,j=1 v ij k i M j ( . )
where M i are tensor products of the Pauli matrices and the two-by-two identity matrix I 2 . In particular, I give a recipe to nd the unitary and hyperbolic transformations according to need. The core of this

This can be seen by the Baker-Campbell-Hausdor formula:

e A e B = e C with C = A + B + 1 2 [A, B] + • • • . ( . )
If A, B are two di erent Pauli matrices so that the commutator of A and B is two times the complex i times the third Pauli matrix where a rotation emerges. One can even generalized further the Hamiltonian to allow the index j to have a di erent range than i. Nevertheless, the present Hamiltonian is su ciently generalized for the discussions in the thesis.

recipe is to reduce the number of matrices in the Hamiltonian of interest by unitary transformations and then solve the problem by a Lorentz boost de ned by a hyperbolic transformation.

Before introducing any external eld, one should render the k • p Hamiltonian as simple as possible. A practical criterion for simplicity is that every non-identity matrix is only accompanied by one component of the momentum. To do so, one factorizes the Hamiltonian by momentum k i and then focuses on those components with two or more matrices associated, v ij M j . Recall that two matrices written in a tensor product of the Pauli matrices and I 2 either commute or anti-commute. To merge two matrices, say M 1 and M 2 , into a simple one by unitary transformation, they must anti-commute so that there exist a matrix R that M 1 , M 2 , R form a Lie algebra like the Pauli matrices. In particular, M 1 M 2 = iR. So the desired unitary transformation is de ned by U = exp(iθR/2) where θ is found by writing the coe cients of M 1 and M 2 in trigonometric functions. For example, given

H = v x σ x + v y σ y and thus R = σ z , one can write v x σ x + v y σ y = v cos θσ x + v sin θσ y with v = v 2 x + v 2 y tan θ = v x v y .
The unitary transformation U = exp(iθσ z /2) can thus simplify H to U HU † = vσ x . One can repeat this step for all the pairs of matrices in factor that anti-commute until the Hamiltonian cannot be simpli ed further.

Once the Hamiltonian is simpli ed, one can introduce electric and magnetic eld by the Peierls substitution choosing a Landau gauge so that only one spatial coordinate appears in the Hamiltonian which does not commute with one of the components of the momentum. Thus, the other components k i remains good quantum numbers. When a real or an e ective electric eld is present, the spatial coordinate appears in front of two matrices, say N 1 and N 2 , that commute. To merge these two matrices, one has to use a hyperbolic transformation T = exp(-φN/2). The choice of matrix

N depends on the coe cient in front of N 1 and N 2 . If the coe cient of N 1 is larger in absolute value, then N = N 1 . Otherwise, N = N 2 .
Here, one also determines the regime of the Hamiltonian of interest, electric or magnetic regime. The rapidity φ is then found by writing the coe cients in hyperbolic functions. Once all the steps are done, the transformed Hamiltonian has only either magnetic or electric eld and it is thus easy to solve. while this recipe might seen horribly abstract for the moment, I will repeatedly illustrate it in the following part of this chapter as well as in Chapter , and .

W H

Let me illustrate the recipe mentioned in the previous section with the example of graphene in crossed electric and magnetic elds. Using the hyperbolic transformations, one can now solve the Hamiltonian ( . ) in the magnetic regime which I recall here

H = eEx + v F q x σ x + v F (q y + eBx)σ y = v F Bx v D v F + σ y + v F (q x σ x + q y σ y ) ( . )
where v D = E/B < v F and the Landau gauge A y = Bx is used so that k y is a good quantum number. Since one knows how to solve the case with only magnetic eld, it is tempting to get rid of eEx (or equivalently v D /v F ) using a hyperbolic transformation. Replacing v D /v F by -tanh φ, the Hamiltonian becomes

H = v F Bx(-tanh φ I + σ y ) + v F (q x σ x + q y σ y ) ( . ) = v F B cosh φ x(-sinh φ I + cosh φ σ y ) + v F (q x σ x + q y σ y ) ( . )
where I emphasize the matrix structure by adding the general identity matrix I which is two-by-two in the context. Here, one nds the structure mentioned in Sec. . . with the two commuting matrices N 1 = I and N 2 = σ y , since cosh φ > sinh φ, the required hyperbolic transformation is generated by σ y ,

T = e φ 2 σy . ( . )
The trick to nd quickly this transformation is to note that σ y is the Pauli matrix in the exponential because the hyperbolic transformation is e ective only when the Pauli matrix in the exponential commutes with the one of interest. The rapidity φ is already given in the form above. Finally, one just needs to set the sign in front of φ to be the opposite of the relative sign between the two terms in study. After the hyperbolic transformation, we have

H T = T HT = v F q y sinh φ + v F q x σ x + v F q y cosh φ + e B cosh φ x σ y . ( . )
Since T does not conserve the spectrum of H, one needs the relation between the spectra of two Hamiltonians. Given |ψ an eigenstate of H, we can write T HT T -1 |ψ = ET 2 T -1 |ψ . Thus, we can de ne |ψ T = N T -1 |ψ with a normalization factor N such that

H T |ψ T = ET 2 |ψ T = E(cosh φ + sinh φσ y )|ψ T ( . ) from which one nds a Hamiltonian H E of eigenstate |ψ T , H E = (H T -E sinh φσ y )/ cosh φ,
having the same spectrum as the original Hamiltonian H

H E = v F q y tanh φ + 1 cosh φ v F q x σ x + v F q y cosh φ + e B cosh φ x - E sinh φ v F σ y .
( . )

Even if the Hamiltonian H E depends itself on the energy E, I will show that the energy-dependent term in H E is irrelevant to the spectrum but encoded in the wavefunction |ψ T . As only the magnetic eld is present, the o -diagonal terms can be also written in terms of energy-dependent ladder operators. We de ne

a E = cosh φ 2 q x B -i x + q y 2 B cosh 2 φ -E v F 2 B sinh φ cosh φ B cosh φ , ( . ) a † E = cosh φ 2 q x B + i x + q y 2 B cosh 2 φ -E v F 2 B sinh φ cosh φ B cosh φ , ( . ) 1 = [a E , a † E ] ( . )
where B = /eB. Using the ladder operators, H E becomes

H E = v F q y tanh φ + v F B cosh φ 2 cosh φ 0 a E a † E 0 ( . )
so that its eigenvectors can be written as

|ψ n = sin α n |n -1 E cos α n |n E if n ≥ 1, |ψ n = 0 |0 E if n = 0 ( . )
where |n E is the wavefunction of the one-dimensional quantum harmonic oscillator de ned by the previous ladder operators, a † E a E |n E = n|n E . The lower index E indicates that the center of |n E is energy-dependent namely

x E = -q y 2 B cosh 2 φ + E v F 2 B sinh φ cosh φ ( . )
where the energy-dependent term in H E is absorbed in the de nition of the ladder operator so that it is irrelevant to the spectrum. However, the center of the wavefunction depends on the energy. In particular, the eigenstates for the same momentum q y for di erent Landau levels are not orthogonal to each other due to the mismatch of its energy-dependent guiding centers. This induces new selection rules in Landau level spectroscopy which I will present in Chapter .

Applying H E on |ψ n , the spectrum is found

E ± n = ± v F B cosh φ 2n cosh φ + v F q y tanh φ ( . )
where is restored. The relation between hyperbolic transformations and Lorentz boosts is clear if one compares this method using hyperbolic transformations with the relativistic argument in the previous section of this chapter. Identifying cosh φ = γ > 0 and tanh φ = β < 0, the spectrum solved with the help of hyperbolic transformations is identical to ( . ). Furthermore, the hyperbolic transformation T can be seen as exp(-(-φ)σ y /2). The physical meaning of T is a Lorentz boost that allows to write a four-vector x of the moving frame in terms of x of the lab frame. x and x are two observations of the same physical quantity associated to the lab frame and the moving frame, respectively.

Mathematically, it means that x = Λ(-φ)x. This interpretation is useful if one works always in the lab frame. However, if one wants to work in the moving frame, as I did in the previous section, one should interpret it as a Lorentz boost that changes the frame of reference from σ ≡ {I 2 , σ x , σ y , σ z } of the lab frame to σ of the moving frame. This means that

σ = Λ(φ)σ. ( . )
After this transformation, one works in the moving frame. The speed of the moving frame relative to the lab frame is v F tanh φe y namely -v D e y . This is precisely the moving frame found in the previous section to cancel the electric eld. Hence, the hyperbolic transformation is the mathematical formalism to incorporate the e ect of a Lorentz boost on the Dirac Hamiltonian and its eigenstates. Another way to see this is to compare ( . ) with ( . ):

q y → q y cosh φ - E sinh φ v F = γq y -γβ E v F . ( . )
This is exactly how q y momentum transforms under a Lorentz boost of rapidity φ in the y-direction, namely relativistic Doppler e ect [ ]. Hence, the two methods presented in this chapter to solve the problem are physically equivalent.

Besides o ering the wavefunction, the advantage of hyperbolic transformation is not limited to a speci c problem such as the one treated in this section. For example, the Lorentz invariance of the Dirac equation describing graphene is not necessary for the usage of hyperbolic transformations. In other words, the problem to solve for Dirac materials is not limited to those having a high-energy cousin as graphene having neutrino. As a mathematical tool, hyperbolic transformation can be extended to solve a larger set of problems that I will show immediately.

H(q) = w • q + 3 i,j=1 v ij q i σ j + ∆σ 3 ( . )
where = 1, the velocity w is the tilted velocity, ∆σ 3 is the Dirac mass term, and v ij is a symmetric velocity tensor. An immediate example of this model is graphene where w = ∆ = 0 and v ij is proportional to the Kronecker symbol δ ij . I would like to emphasize that the generic model serves only as a uni ed mathematical framework. In principle, it does not have a physical meaning until I bestow one upon it in a particular material.

In this section, I discuss the spectrum of the Hamiltonian ( . ) in the presence of a magnetic eld.

By rescaling the Fermi velocity and also the applied magnetic eld, one can always equivalently treat a simpler isotropic model,

H(q) = wq z + vq • σ + ∆(sin θσ x + cos θσ z ) ( . ) or H(q) = w z q z + w x q x + vq • σ + ∆σ z ( . )
where two forms are exchangeable by a unitary transformation after rede ning the axes (see in App.

B). Three models that will be discussed in this section are:

• Three-dimensional type-I and type-II Weyl semimetals (WSM):

H(q) = wq z + vq • σ ( . )
where the energy is E ± = wq z ± v||q||. Remember that for type-I WSMs, one has |w| < |v|, while |w| > |v| for type-II WSMs.

• Two-dimensional tilted gapped graphene:

H(q) = wq x + vq x σ x + vq y σ y + ∆σ z ( . )
where the energy is

E ± = wq x ± v 2 q 2 x + v 2 q 2 y + ∆ 2 . • Gapped dispersive nodal-line semimetals in D: H(q) = wq z + vq x σ x + vq y σ y + ∆σ z , ( . )
where the energy is

E ± = wq z ± v 2 q 2 x + v 2 q 2 y + ∆ 2 .
In these models, I suppose for simplicity w, v, ∆ are strictly positive. In the presence of a magnetic eld, an electric-eld-like term appears in the Hamiltonian due to the tilting term, i.e, the one proportional to the two-by-two identity. For example, by Peierls substitution,

wq x → w(q x -eBy) = wq x + eE e y
where the e ective electric eld E e = -Bw is in the y-direction. If one adopts this point of view, the Hamiltonian to solve is an analogue one to that of graphene in crossed electric and magnetic elds.

Although the tilting term w • q excluding the e ective electric eld term cannot be incorporated into a Lorentz invariant Dirac equation, the mathematical treatment makes no di erence because it is di-agonalized with respect to the matrix structure and also diagonal in q upon the appropriate choice of the Landau gauge.

One can even push the discussion to include also a real electric eld perpendicular to the applied magnetic eld. As a matter of text length, I will not include any real electric eld in this discussion because the Hamiltonian to solve has exactly the same mathematical structure after well-chosen unitary transformations. Combined with unitary transformations, hyperbolic transformations serve to solve the generic problem of the Landau levels in the presence of crossed electric and magnetic elds of a generic Dirac Hamiltonians.

T W

The Landau quantization for tilted WSM was solved by Tchoumakov et. al. in [ ]. Since a WSM is a three-dimensional version of graphene, a magnetic eld can in principle point in any direction of the three-dimensional space. Thanks to the Hamiltonian ( . ) being isotropic in the x, y-plane, one can suppose without losing generality that B = B(cos θe z + sin θe x ) = Be z for 0 < θ < π/2.

Here, a new basis is de ned {e x , e y , e z } = {cos θe x -sin θe z , e y , cos θe z + sin θe x }.

( . )

If one writes the Hamiltonian in this new basis followed by a unitary transformation U = exp(iθσ y /2), it becomes

H(q) = w cos θq z -w sin θq x + q • σ. ( . )
Choosing the Landau gauge A = -Bye x , the nal Hamiltonian to solve is

H = -w sin θ(q x -eBy) + v(q x -eBy)σ x + w cos θq z + v(q y σ y + q z σ z ) = eEy + v(q x -eBy)σ x -w sin θq x + w cos θq z + v(q y σ y + q z σ z ) ( . )
where I remove the prime for q. One identi es an e ective electric eld E = Bw sin θ applied in the y-direction. Compared to ( . ), one manages to map this new problem to a problem that one knows how to solve by hyperbolic transformations. Depending on the drift velocity of the problem v D = w sin θ, the system is in the electric regime with a continuum spectrum if v D > v, or in the magnetic regime described by Landau levels if v D < v. Thus, there are two control parameters to distinguish the regimes. If the WSM is of type-I (w < v), it is in the magnetic regime regardless the orientation of the magnetic eld. However, if the WSM is of type-II (w > v), there exists a critical angle

sin θ c = v w ( . )
beyond which the Landau levels collapse into a continuum. Experimentally, one can thus tweak the system continuously from one regime to another by varying the angle θ if w > v.

In the magnetic regime, the Landau levels are

E ± n = w cos θq z ± v γ q 2 z + 2n γ 2 B for n > 0 E 0 = w cos θq z - vq z γ for n = 0 ( . )
where γ = 1/ 1 -β 2 and β = w sin θ/v. The results are obtained with the help of a hyperbolic transformation M = exp(φσ x /2) where tanh φ = β. As in graphene, the spacing between the Landau levels is renormalized by a factor of γ -3/2 . Since one considers only one Weyl node, the Landau levels are not symmetric with respect to zero energy due to the parity anomaly. As in graphene, the only relevant direction for magnetic eld is perpendicular to the D plane namely in the z-direction. The resulting Hamiltonian is thus

H(q) = w(q x -eBy) + v(q x -eBy)σ x + vq y σ y + ∆σ z ( . )
where one identi es an e ective electric eld E = -Bw applied in the y-direction and thus the drift velocity v D = -we x . The Hamiltonian is in the magnetic regime only if w < v because the magnetic eld is xed to be out-of-plane unlike in three-dimensional Weyl semimetals. Using the hyperbolic transformation M = exp(φσ x /2) where tanh φ = -w/v, the spectrum of the Hamiltonian in the magnetic regime is Only one valley in the gapped graphene is considered. Therefore, the spectrum is not particle-hole symmetric due to the parity anomaly. The spacing between the Landau levels are also renormalized by the same factor of γ -3/2 . Most saliently, the Dirac mass is renormalized by a factor of γ -1 and so does the gap measured by extrapolating the Landau levels to the zero eld limit, 2∆ = 2∆/γ. This is analogue to the de nition of relativistic mass in special relativity. However, as it is well-known in solid state physics, one would measure 2∆ for the direct band gap of ( . ) by spectroscopy. It sounds strange that the magnetic eld modi es the optical gap of the tilted Dirac Hamiltonian. If it was true, one should have questioned the general method of gap measurements in Landau level spectroscopy.

E ± n = ± 1 γ ∆ 2 + 2nv 2 γ 2 B for n > 0 E 0 = - ∆ γ for n = 0 (
As will be explained at the end of this chapter, 2∆/γ is actually the indirect gap of the band structure

G

Now I consider the gapped nodal-line semimetal NbAs 2 , the low-energy Hamiltonian of which is described by Eq. ( . ). Since Eq. ( . ) describes a system that is isotropic in the x, y-plane, one sets with no loss of generality that B = B(cos θe z + sin θe x ) = Be z for 0 < θ < π/2. Here, a new basis is de ned {e x , e y , e z } = {cos θe x -sin θe z , e y , cos θe z + sin θe x }.

( . )

Writing the Hamiltonian in this new basis and choosing the Landau gauge A = -Bye x , the

Hamiltonian becomes

H = w[cos θq z -sin θ(q x -eBy)] + v[cos θ(q x -eBy) + sin θq z ]σ x + vq y σ y + ∆σ z ( . )
where I also remove the prime. One can de ne the e ective electric eld E = Bw sin θ applied in the y-direction and the drift velocity v D = w sin θe x . Since the limiting speed is v cos θ instead of v, one needs to compare w sin θ with v cos θ to determine whether the system is in the magnetic or electric regime. Compared to Weyl semimetals, the angle θ plays a more decisive role in the the gapped nodal-line semimetal since the value of tan θ can take any real number. Although the material-related parameter w/v cannot be varied experimentally, one can always nd a critical angle θ c below which the Hamiltonian is in the magnetic regime. Conversely, the Landau levels can be continuously tuned to extinction by increasing θ to θ c . The critical angle called the angle of extinction is de ned as

tan θ c = v w . ( . )
Using the hyperbolic transformations M = exp(φσ x /2) with tanh φ = w tan θ/v, the spectrum in the magnetic regime reads

E ± n = wq z cos θ ± 1 γ ∆ 2 + 2nv 2 cos θ γ 2 B for n > 0 E 0 = wq z cos θ - ∆ γ for n = 0 ( . )
where γ(θ) = 1/ 1 -β 2 and β(θ) = w tan θ/v. For the same reason as previously shown, the Landau levels are not particle-hole symmetric and the spacing between them is renormalized by γ -3/2 .

More details on NbAs2 in Chapter and App. C where I discuss also the magneto-optics of this gapped nodal-line semimetal.

Moreover, the Dirac mass is renormalized by a factor of γ -1 . The main di erence is that γ now can be tweaked by the angle θ. Therefore, the extrapolated gap depends on the orientation of the magnetic eld. Most saliently, the gap is closed when θ = θ c . The gap closing is an unambiguous signature to demonstrate special relativistic renormalization in a material like NbAs 2 . It is especially easy to experimentally show since the orientation of the magnetic eld is freely con gurable. Concerning the solid-state mechanism of this gap closing, I invite the readers to go to the next section.

In summary, hyperbolic transformations allow one to solve completely the problem of Landau quantization for tilted Dirac Hamiltonians by giving the energy spectrum and wavefunctions. By the Peierls substitution in the Landau gauge, an e ective electric eld is generated due to the presence of tilting term. One can still make an analogy with special relativity as I did for graphene. However, this analogy is only partially valid because the tilting term includes momentum (see, for example, D tilted gapped graphene), which breaks the Lorentz invariance of Dirac equation. In other words, one cannot write a Dirac equation associated with the generalized linear Hamiltonian in a Lorentz invariant form.

Another thing to pay attention to is that one might think the tilting speed w and the Fermi velocity v as the drift velocity and the speed of light c, respectively. This is explicitly proved to be wrong in the calculations for tilted Weyl semimetal where the de nition of speed is angle-dependent. Hence, the analogy with special relativity is only physically valid after the Peierls substitution and the identi cation of an e ective electric eld. All these caveats make hyperbolic transformation even more tempting for its multiple interpretations of calculations and elegance of simplicity.

R

In this section, I will elucidate the solid-state mechanism of the fancy relativistic renormalization in Dirac materials, so far not yet discussed in the community. I will retrieve the energy spectrum of the previous models by the semi-classical method. One can interpret the relativistic renormalization e ect as an anisotropy of the Fermi surface introduced by the tilting. The criterion for electric and magnetic regime is whether the cross section of the Fermi surface perpendicular to the magnetic eld is compact, i.e., nite and closed. Furthermore, I will explain the microscopic mechanism that leads to the renormalization of the optical gap: the energy dispersion is attened by Landau quantization.

A F

Quantum oscillations are the phenomena related to response functions, for example magneto-resistance and magnetic susceptibility, that oscillate as a function of magnetic eld. They o er information on the Fermi surface of a given material by looking at the oscillation pattern since response functions are related to the density of states at the Fermi level. The interpretation of quantum oscillations relies on the formation of Landau levels. Semi-classically, the Onsager relation [ ] or equivalently the Roth-Gao-Niu relation [ , ] provide a quantization condition to derive Landau levels from the geometry of the Fermi surface: ) where S(E F ) is the reciprocal-space area of a closed cyclotron orbit at the Fermi level and γ(E F , B)

S(E F ) = 2πeB[n + γ(E F , B)], ( .
is a phase shift whose Taylor series in B is given by [ ]

γ(E F , B) = 1 2 -M 0 (E F ) - B 2 χ 0 (E F ) - p≥3 B p-1 p! R p (E F ), ( . )
where the rst term is the Maslov index, the second term is the sum of Berry curvature and the derivative of orbital magnetization, the third term is the derivative of magnetic susceptibility and the last term is derivatives of other higher order response functions. In the following, I take γ(E F , B) ≈ 0 since I am interested in Dirac Hamiltonians for which M 0 (E F ) = 1/2. I also neglect further higher order terms of B. Here, it will be shown that the relation Eq. ( .) can accurately give Landau levels in tilted Dirac

Hamiltonians and thus furnishing a complementary understanding of the relativistic renormalization in Dirac materials, namely the induced anisotropy of Fermi surface.

Suppose the Fermi level is above the gap such that

E F = wq x + ∆ 2 + v 2 q 2 x + v 2 q 2 y . ( . )
This equation de nes the trajectory of a cyclotron motion in the reciprocal space. Given that β = w/v and γ = 1/ 1 -β 2 , I transform this equation into a more explicit form

1 = v 2 q x + E F v βγ 2 2 γ 4 E 2 F -∆ 2 γ 2 + v 2 q 2 y γ 2 E 2 F -∆ 2 γ 2 ( . )
This statement is actually true for the Shubnikov-de Haas oscillations of conductivity but not true for the de Haas-van Alphen oscillations of magnetization [ ].

which has the shape of the de ning equation of an ellipse. Several conditions need to be ful lled if this equation actually describes an ellipse:

|β| < 1 and E F > ∆ γ , ( .
)
where the rst condition is precisely that for the magnetic regime, and the second condition says the smallest gap of the system is 2∆/γ. Note that the center of the ellipse is not at q = 0 but shifted in the q x -direction such that 2∆/γ denotes the indirect gap (see Fig. . ). With the area of the ellipse

S = πγ 3 E 2 F -∆ 2 γ 2 v 2 = πγ γ 2 E 2 F -∆ 2 v 2 , ( .
)
the Onsager relation gives ) in agreement with the above result ( . ) for Landau levels.

E F = 1 γ ∆ 2 + 2nv 2 γ 2 B , ( .
If w = 0, the Hamiltonian describes an isotropic gapped graphene. The equation of ellipse becomes

1 = v 2 q 2 x E 2 F -∆ 2 + v 2 q 2 y E 2 F -∆ 2 , ( . )
which actually depicts a circle centered at q = 0. Therefore, the tilting term wq x squeezes the circular Fermi surface (contour) to an elliptic one along the x-direction by a factor of γ. This induces an anisotropy of Fermi surface. Furthermore, the tilting term also modi es the density of states so that the Fermi energy is renormalized by another factor of γ for a xed number of electron. These two e ects give together the factor of γ 3 in the equation above.

T W

To proceed the calculations for Weyl semimetal, one simply has to replace ∆ by vq z in the previous calculations, The trajectory of cyclotron is now the contour of a cross section of the ellipsoid Fermi surface of Weyl semimetal. The ellipsoid Fermi surface is described by

1 = v 2 q x + E F v βγ 2 2 γ 4 E 2 F + v 2 q 2 y γ 2 E 2 F + v 2 q 2 z γ 2 E 2 F ( . )
Here, I replace wqz by wqx in ( . ) which is another valid simple model to describe Weyl semimetals.

where β = w/v and γ = 1/ 1 -β 2 . Condition of an ellipsoid Fermi surface is |β| < 1. But this is not necessary to nd a closed orbit for cyclotron because the orientation of magnetic eld matters. Depending the orientation of the magnetic eld, one has to rewrite the previous equation to an elliptic equation. Here, I show only two particular orientations: the magnetic eld is in the z-or the

x-direction. The general geometric calculation can be found in the literature [ ].

For a magnetic eld in the z-direction, one just needs to replace ∆ by vq z in the results for twodimensional titled gapped graphene

E F = 1 γ v 2 q 2 z + 2nv 2 γ 2 B . ( . )
The condition for a closed orbit is |β| < 1.

For a magnetic eld in the x-direction parallel to the tilting term wq x , q x remains a good quantum number and thus wq x can be treated as an additive constant to the energy. The resulting spectrum is given by Landau levels in a non-tilted WSM adding wq x . The calculations using the Onsager relations are more involved since one need rewrite the ellipsoid equation considering q x as constant. This leads to

1 = v 2 q 2 y γ 2 E 2 F 1 - v 2 qx+ E F v βγ 2 2 γ 4 E 2 F + v 2 q 2 z γ 2 E 2 F 1 - v 2 qx+ E F v βγ 2 2 γ 4 E 2 F ( . )
with no conditions on w/v. The area enclosed by a cyclotron orbit in reciprocal space is

S = πγ 2 E 2 F v 2   1 - v 2 q x + E F v βγ 2 2 γ 4 E 2 F   .
( . )

Finally, the Onsager relation gives the correct Landau levels

E F = wq x + v 2 q 2 x + 2nv 2 2 B . ( . )

G

Starting from the Hamiltonian discussed previously

H = w(cos θq z -sin θq x ) + v(cos θq x + sin θq z )σ x + vq y σ y + ∆σ z , ( . )
I choose the magnetic eld to be oriented in the z-direction. The cyclotron trajectory is described by

1 = v 2 q2 x cos 2 θ γ 4 E F -vβqz sin θ 2 -γ 2 ∆ 2 + v 2 q 2 y γ 2 E F -vβqz sin θ 2 -∆ 2 ( . )
where

β(θ) = w v tan θ γ(θ) = 1 1 -β(θ) 2 ( . ) qx = q x + (v 2 + w 2 )q z sin θ cos θ -wE F sin θ v 2 cos 2 θ -w 2 sin 2 θ . ( . )
This equation describes an ellipse when |β(θ)| < 1 and E F > ∆/γ +wq z / cos θ. The rst condition gives rise to the critical angle beyond which Landau levels collapse. The second condition states that the Fermi energy should be above the renormalized gap at a given momentum q z so that the carrier density is non-zero. Invoking the Onsager relation, one gets the spectrum

E F = w cos θ q z + 1 γ ∆ 2 + 2nv 2 cos θ γ 2 B ( . )
which is perfectly in line with the results obtained quantum mechanically.

To conclude, the renormalization e ect in tilted Dirac materials in the presence of a magnetic eld is a relativistic interpretation of the consequence of an anisotropic Fermi surface. The Lorentz factor γ actually quanti es the degree of anisotropy of Fermi surface induced by the tilting term. To some extent, the relativistic renormalization should be measurable in quantum oscillations experiments which would probe a squeezed Fermi surface due to the tilting term. The Lorentz factor γ would emerge as a multiplicative factor on the periodicity of quantum oscillations. However, in the language shared by experimentalists, it is more understandable to talk about the anisotropy of Fermi surface rather than the relativistic renormalization.

D

Anisotropic Fermi surfaces are ubiquitous in materials. However, when refering to anisotropy, di erent velocities or e ective masses along the crystalline axis are considered. In parabolic bands, the tilting has no e ect on the shape of Fermi surface because it is absorbed in the quadratic term by rede ning the origin of momentum q. However, this is impossible in linear bands describing the low-energy physics of Dirac materials. Most saliently, unlike the anisotropy in the Fermi velocity, the tilting term does not only make the longitudinal and transverse directions inequivalent but can also modify the gap of the system. To see this, one takes the tilted gapped graphene as a starting example [ ].

In the absence of magnetic eld, one can easily calculate the energy spectrum of ( . )

E ± = wq x ± v 2 q 2 x + v 2 q 2 y + ∆ ( . )
where the direct gap 2∆ is formed at q = 0. However, the smallest energy di erence between valence and conduction band is not the direct gap but the indirect gap. Direct calculation of the extrema of the two bands shows the band edges of the valence and conduction band is at momenta vq x = ±γβ∆ and vq y = 0 where β = w/v and γ = 1/ 1 -β 2 . The energy di erence between these two band edges, i.e., the indirect gap, reads 2∆/γ. Without appealing to relativity one notices that the indirect gap equals the direct gap renormalized by the Lorentz factor γ de ned in the language of special relativity. The relativistic gap renormalization is just a smaller indirect gap induced by the tilting term! The optical gap measured in the absence of magnetic eld is to nd the smallest energy di erence between valence and conduction band conserving the same momentum q. However, in the presence of magnetic eld in the z-direction, q x and q y are no longer good quantum numbers so that it makes no sense to de ne a direct gap preserving the momentum q x and q y . In other words, the role of magnetic eld is to atten the energy dispersion in the transverse direction q x and q y . Since the optical gap measured in the presence of magnetic eld is obtained by extrapolating the Landau levels to zero, the measured gap is now an indirect gap.

Generalizing the argument from two to three dimensions, the gap measured by extrapolating the Landau levels to zero eld limit is the lowest energy di erence between the two bands when varying the two momenta perpendicular to the direction of the magnetic eld. The longitudinal momentum along the magnetic eld is untouched by the Peierls substitution and the two transverse momenta are no longer good quantum numbers. Therefore, the transverse dispersion is atten by the magnetic eld and turns into Landau levels while the longitudinal dispersion is intact. For example, in the gapped nodal-line semi-metal, the spectrum in the absence of magnetic eld reads

E ± = w(cos θq z -sin θq x ) ± v 2 q 2 y + ∆ 2 + v 2 (cos θq x + sin θq z ) 2 ( . ) = w q z cos θ -w sin θ qx ± v 2 q 2 y + ∆ 2 + v 2 cos 2 θ q2 x ( . )
where q z is the longitudinal momentum, (q x , q y ) are transverse ones and qx = q x + q z tan θ is the recentered q x . While minimizing the energy di erence between E + and E -, q z is irrelevant and q y must be zero. Since I want to nd the smallest gap no matter whether it is direct or indirect, one can separately maximize E -and minimize E + . Taking derivative of E ± , the condition of zero derivative

Special relativity in Dirac materials

gives precisely the condition of the magnetic regime, i.e, |β| < 1. Most prominently, the minimal gap is again found to be 2∆/γ, in line with the previous quantum mechanical calculations. In this example, one realizes once more that the renormalized gap is merely the minimal gap of the energy dispersion transverse to the magnetic eld. This is the one probed by Landau level spectroscopy. Nevertheless, it is still surprising that one can tune the indirect gap of a material by the orientation of magnetic eld in this particular model. In the absence of magnetic eld, only direct optical transitions are possible since the photon momentum is negligible. As shown in the next chapter, where I will discuss Landau level spectroscopy, the magnetic eld induces mismatch between wavefunctions of di erent energy so that dipole transitions from one Landau level to another do not respect the usual selection rules. In other words, photons probe the indirect gap.

T

A theoretical physicist can spend his entire lifetime missing the intellectual challenge of experimental work, experiencing none of the thrills and dangers -the overhead crane with its ten-ton load, the flashing skull and crossbones and danger, radioactivity signs. A theorist's only real hazard is stabbing himself with a pencil while attacking a bug that crawls out of his calculations.

Leon M. Lederman

Optical properties of solids are tangible in daily life. For example, the color of materials is a manifestation of all the complex interactions between light and matter. In electromagnetism, light is considered as an electromagnetic wave described by the Maxwell's equations. As it is well-known from the basic electrodynamics classes, the wide-ranging optical properties observed in materials are encoded in the constitutive equations

D = 0 r E B = µ 0 µ r H j = σE
where the elds D, E, B, H are displacement, electric, magnetic and magnetizing eld, respectively; j is charge current density; the coe cients 0 , r , µ 0 , µ r , σ are vacuum electric permittivity, relative electric permittivity, vacuum magnetic permeability, relative magnetic permeability, conductivity, respectively. These equations are valid when an incident light is not so intense that the response of material to an incident electromagnetic wave is linear. However, these optical coe cients are not measured directly by spectroscopy but can be derived from other measurable optical constants such as refractive index, extinction coe cient, transmission and re ectivity.

Theoretical approach to spectroscopy

The frequency-dependent complex dielectric function (ω) or the complex conductivity σ(ω) are directly related to the electronic energy band structure of solids. Therefore, light is an excellent probe to resolve the band structure of materials. In fact, one can also use electrons in the spectroscopy encoded also in the complex dielectric function. However, the momentum of electron is no longer negligible with respect to the size of Brillouin zone, the dielectric function must be momentum-dependent, i.e., (q, ω).

In this chapter, I present succinctly the theoretician's way to study the spectroscopic properties of materials, namely how to calculate dielectric function and conductivity knowing the Hamiltonian.

Especially, I elaborate some known results in magneto-optics, which will be useful in Chapter . As an illustration, I will apply the theoretical formalism in a real material that I have been recently working on with experimentalists at LNCMI (Grenoble). Then, I will talk about the physics of plasmon and how to measure it by spectroscopy.

Q

In this section, I introduce the tool that theoreticians would use to study the optical properties of materials, namely the optical conductivity σ(ω). The goal is not to develop the full construction of the theory but to emphasize on key ingredients of the theory and mention the approximations used in the theory.

In a complete quantum mechanical treatment, light is quantized in bosonic elds known as photons. The light-matter interaction is then described by a vector potential A(r, t) including the bosonic ladder operators for the photons. This treatment is necessary when the number of photons is very small such as in a single-photon source used in quantum optics [ , , ]. In most cases in condensed matter physics, one uses classical light sources so that light can be treated as an electromagnetic wave because light is used as a probe of the electronic properties of materials rather than the object of a physical study itself. When the light source is not an intense laser, the light-matter interaction is modeled as a small perturbation to the matter by the electromagnetic eld. This approach is semi-classical in the sense that light is modeled by an electromagnetic wave and matter is described by a fully quantum mechanical Hamiltonian.

Generally, a crystal coupled to an electromagnetic eld is described by the following Hamiltonian

H = (p + eA) 2 2m 0 + V (r) ( . )
For the construction of theory from zero, I suggest the following references [ , , ] from which I am inspired to formulate this section.

where V (r) is a periodic lattice potential, e > 0 is the elementary charge and m 0 is the rest mass of the electron. The above Hamiltonian can be written as a sum of the Hamiltonian in the absence of A denoted H 0 , and a perturbation term H . To rst order in A, H reads

H = e m 0 p • A ( . )
where the Coulomb gauge ∇ • A = 0 is used so that p = -i ∇ and A commute. As an electromagnetic wave, A is written in the following form

A(r, t) = A 0 exp(ik • r -iωt) ( . )
where A 0 is in the direction of electric eld indicating the polarization of light. In time-dependent perturbation theory, Fermi's golden rule tells that the transition rate from the state 1 to the state 2 is

W 1→2 = 2π |M 12 | 2 g( ω) ( . )
where g( ω) is the density of states (DOS) and M 12 is the matrix element

M 12 = 2|H |1 = d 3 rψ * 2 (r)H (r)ψ 1 (r). ( . )
Note that the time-dependence in exp(-iωt) is already taken into account in Fermi's golden rule.

E

Since the frequency domain that I am interested in is the visible-infrared regime where the wavelength of light is much larger than the periodicity of lattice potential, one can legitimately omit the spatial dependence in A, namely exp(ik • r) ∼ 1. In other words, the momentum of photon is negligible in front of the lattice momentum of electron. In this approximation, the matrix element reads

M 12 = e m 0 2|p • A 0 |1 . ( . )
When the two involved states are two discrete levels, g( ω) represents the DOS of photon. If one considers the transition between k-dependent bands of solids, one should use the joint density of states summing over k.

This is called electric dipole approximation. The name for this approximation is justi ed by writing the equation of motion of the position operator r. Note that p = m 0 dr/dt, the equation of motion reads

p = m 0 dr dt = m 0 i [H 0 , r] ( . ) so that M 12 = ie E 2 -E 1 2|r • A 0 |1 ( . )
where E 1,2 are the energy of two states. Since E 0 = iωA 0 and, if ω = E 2 -E 1 , the matrix element becomes

M 12 = 2|er • E 0 |1 ( . )
where the perturbation term H becomes the energy of an electric dipole in an electric eld E 0 . The corresponding transition is called electric dipole transition.

If one pushes the Taylor expansion of exp(ik • r) to higher orders, other types of transition are possible. For example, ik • r yields magnetic dipole transition and (k • r) 2 yields electric quadrupole transition. These higher order corrections are 10 -3 weaker than electric dipole transition in most cases thanks to large wavelength of light. However, they are essential to explain the optical activity, when the time-reversal symmetry is not broken, and the circular dichroism of chiral materials, which possesses neither an inversion center nor a mirror plane [ , ]. This can be understood by looking the rst order correction to the electric dipole approximation in Fermi's golden rule

2|er • E 0 |1 1|(er × p m 0 ) • B 0 |2 ( . )
in which the second term is the magnetic dipole transition. If the system possesses an inversion center, the eigenstates are either even or odd under inversion operation. Since the electric dipole is odd under inversion, a non-zero electric dipole matrix element has to involve two states of di erent parity. However, this would kill the magnetic dipole matrix element due to the angular momentum r × p being invariant under inversion. If the system possesses a mirror plane, the eigenstates are either even or odd under re ection with respect to the mirror plane. Similarly, a polar vector as electric dipole and an axial Optical activity is the capability of a material to rotate the linear polarization of transmitted light. Circular dichroism is the property of materials whose absorption spectrum is di erent for left and right handed circularly polarized light.

vector as magnetic dipole imposes incompatible symmetry constrictions on the eigenstates involved in the transitions so that the rst order correction to the electric dipole approximation is forbidden.

In organic chemistry, the optical activity is due to the chiral molecules. However, the optical activity of crystalline quartz arises from the crystal structure. The unit cell of quartz belongs to the trigonal crystal class labeled 32, which has no mirror symmetry, and is therefore chiral [ ].

Nevertheless, the higher order corrections to electric dipole approximation is beyond the scope of the thesis. In all the following discussions, I consider only electric dipole transitions in optical properties.

L K

Light-matter interaction is considered as a perturbation to the system. If the electric eld of light is not too strong, which is the case I consider here, then the conductivity is independent of the electric eld.

It only depends on the material properties in the absence of electric eld (at equilibrium). This type of response is called the linear response. In general, one should consider the momentum dependence in the conductivity σ(q, ω). In the electric dipole approximation, the momentum of photon is omitted so that the optical conductivity becomes local (q = 0) and depends only on ω. All information on the optical properties of solids is encoded in the optical conductivity. Most saliently, the optical conductivity can be easily retrieved from the transmission coe cient and/or re ectivity that are directly measurable by experiments.

Here, the formulation of the perturbation is di erent from that in the time-dependent perturbation theory. In real space

H = -d 3 r j(r) • A(r, t).
The current density has two parts, paramagnetic and diamagnetic. The latter plays an important role in superconductors but not in the present discussions. Nevertheless, this formulation is physically equivalent to the previous one. Equivalent to the electric dipole approximation, this is justi ed by the fact that the speed of light is typically several orders of magnitude larger than the electrons' velocity in materials.

The transition rate given by Fermi's golden rule is of course related to the optical conductivity. See details in [ ].

While the absorption method is viable for D thin lms, the re ectivity method is more suitable for D bulk materials.

Theoretical approach to spectroscopy

The Kubo formula of the optical conductivity reads

σ ij (ω) = i e 2 V n,n ,k f D (E n (k)) -f D (E n (k)) E n (k) -E n (k) × n |v i |n n|v j |n E n (k) -E n (k) -ω + i0 + ( . )
where V is the volume of system for normalization; n, n are the band index; i, j represent the direction of polarization; vi is the velocity operator; f D (E) is the Fermi-Dirac distribution, and 0 + ensures the causality of the response in the clean limit. In general, one can replace 0 + by /τ with a phenomenological relaxation time τ to encounter the peak broadening due to disorders. Note that this expression includes both intra-and inter-band processes. When n = n , the numerator and denominator of the rst fraction on the right hand side has a nite limit. One would obtain the same results in the classical Drude's theory for metals [ ].

In the following, I focus on the interband transition in solids caused by an incident light of energy ω. The absorption process is related to the real part of the optical conductivity thinking of the dissipative Joule's e ect. In the clean limit, the real part of the diagonal elements of the optical conductivity

for ω = 0 reads [σ ii (ω = 0)] = πe 2 ω n =n d d k (2π) d (f D (E n ) -f D (E n ))| n |v i |n | 2 δ[ ω -(E n -E n )] ( . )
where the dependence on k is not explicitly shown, and the subscript d is the dimension of the system of study.

The formula ( . ) consists of three key ingredients of optical transitions:

. The term Here, I use the relation:

f D (E n )-f D (E n ) is
1 x + i0 + = P.V.( 1 x ) -iπδ(x)
where P.V. is the Cauchy principle value and δ is the Dirac distribution.

strictions on the optical transitions dictated by group theory. Exactly like in atomic physics, these constrictions are called selection rules. By de nition, the velocity operator is given by

v = i [r, H] ( . )
where r is the position operator. In practice, it is easier to work in the basis of the cell-periodic part of Bloch wavefunction to evaluate the matrix element. A crystal Hamiltonian

H = p 2 2m 0 + V (r) ( . )
with a periodic lattice potential V (r) and the energy is E n (k) for the Bloch wavefunction |ψ n,k , the k-dependent e ective Hamiltonian, i.e., the Bloch Hamiltonian, reads

H k = e -ik•r He ik•r = (p + k) 2 2m 0 + V (r) ( . )
with the same energy E n (k) for the eigenstate |u n,k . The velocity operator in this basis reads

vk = e -ik•r ve ik•r = 1 ∇ k H k . ( . )
When working on k • p Hamiltonians [ ], this is the most suitable way to calculate the matrix elements.

. However, the inverse statement is not true because the optical activity is also present in chiral materials in the absence of magnetic eld. A key di erence between the optical activity in the presence and in the absence of magnetic eld is whether the TRS is broken. For example, preserving the TRS, the optical activity manifests itself only in the transmitted light but not in the re ected light, namely

The term k δ[ ω -(E n -E n )] is
The convention of handedness is de ned from the point of view of the source. One should not confound the Pauli matrix σy with the conductivity tensor σij

Theoretical approach to spectroscopy

no Kerr e ect is observed. When the direction of propagation is reversed, the rotation angle of linear polarization does not invert its sign in the absence of TRS. But it does so in the presence of a magnetic eld, namely the TRS is broken. Therefore, only the optical activity induced by TRS-breaking is called Faraday e ect. One does not have to invoke the magnetic dipole moment to explain the optical activity in the presence of a magnetic eld, which is already present in the electric dipole approximation. Kubo formula manages to recover the two steps in the optical conductivity. Furthermore, one can identify the cause of the two-step feature, namely the onsets of the dispersive and the at part of the nodal line with di erent gap, respectively. The ratio between the optical conductivities for two polarizations is also accurately retrieved. This allows one to evaluate the anisotropy of Fermi velocity.

Here, I only consider the interband transitions so the Drude peak below 0.1 eV is not captured in Fig.

. (a). Nevertheless, one should be convinced by the predictive power of the Kubo formula ( . ) used in these rather simple low-energy models.

S

To understand the obscure magneto-optical observations for the dispersive part of the nodal line, one must go back to the Hamiltonian ( . ) where I omit the anisotropy of the Fermi velocity. Since the Landau fan structure is still present, the system is in the magnetic regime (see Chapter ). The angle More details are shown in App. C. θ de ned in ( . ) is thus smaller than the critical angle θ c de ned by ( . ). In the presence of a magnetic eld in the z-direction, the Hamiltonian reads H = w(cos θq z -sin θ(q x -eBy)) + v(cos θ(q x -eBy) + sin θq z )σ x + vq y σ y + ∆σ z ( . )

where = 1 (also in the following discussions) and the Landau gauge A = (0, -eBy, 0) is used. The Hamiltonian H can be solved using the hyperbolic transformation M = exp(φσ x /2) with tanh φ = w tan θ/v following the same procedure in Chapter for graphene in a crossed electric and magnetic eld. The transformed Hamiltonian reads:

H T = M HM = wv v * q z + w 2 + v 2 v * q z sin θ cos θσ x + v * (q x -eBy)σ x + vq y σ y + ∆σ z ( . )
where a new velocity v * is de ned as

v * 2 = v 2 cos 2 θ -w 2 sin 2 θ with γ = cosh φ = v cos θ v * ( . )
where γ > 1 is the relativistic Lorentz factor. Note that H T does not have the same spectrum as H.

In the basis of |ψ T = γ -1/2 M -1 |ψ given the eigenstate |ψ of H, one can construct from H T another Hamiltonian H E = (H T -E sinh φσ x )/ cosh φ with the same spectrum as H

H E = wq z cos θ + 1 γ ∆ √ 2vv * B a E √ 2vv * B a † E -∆ ( . )
where a pair of energy-dependent ladder operators is de ned

a E = - B √ 2vv * v * B (y -y E ) + ivq y ( . ) a † E = - B √ 2vv * v * B (y -y E ) -ivq y ( .
)

y E = 2 B v * v * q x + q z (w 2 + v 2 ) sin θ cos θ -E w sin θ v * ( . ) with [a E , a † E ] = 1.
The subscript E indicates the dependence on energy E. In particular, the center of cyclotron y E shifts with energy.

Since the energy-dependent term in H E is absorbed in the de nition of the ladder operator, the energy spectrum is self-consistently found in the eigenstates of energy E λ n [see ( . )] in the form

|ψ T,n,λ = sin α n,λ |n -1, E λ n cos α n,λ |n, E λ n ( . )
where α n,λ is an angle depending on n and the sign of energy λ = ±; |n , E λ n is the wavefunction of one-dimensional quantum harmonic oscillator de ned by the previous ladder operators. Given the Landau level index n, n = n or n -1. As shown in Fig. . (b), the gap measured by magneto-optics is indeed renormalized by γ, which depends on the orientation of the magnetic eld.

To study the magneto-optical properties of NbAs 2 , one evaluates the matrix element

ψ n |v k |ψ m .
Indeed, it is more practical with the basis |ψ T,n already at hands by remarking that ) thanks to the fact that M is k-independent. So, in the basis |ψ T , the velocity operators for

ψ n |∇ k H|ψ n = γ ψ T,n |∇ k H T |ψ T,n ( .
H T are vT,x = v * σ x ( . )
vT,y = vσ y ( . )

vT,z = wv v * + w 2 + v 2 v * sin θ cos θ ( . )
where one notice an emergent anisotropy of Fermi velocity induced by the magnetic eld. For the dispersive part in NbAs 2 , the x, y-directions in the model ( . ) correspond to the a, b crystalline directions (see in App. C). The anisotropy of Fermi velocity yields a signi cant discrepancy in the optical conductivity

σ aa σ bb = sin 2 θ σ xx σ yy ∝ v * 2 sin 2 θ v 2 = sin 2 θ cos 2 θ γ 2 . ( . )
Using the parameters in [ ], the angle θ is ∼ π/3 and thus γ = 1.17. Taking into account the inherent Fermi velocity v a /v b ∼ 0.5, the optical conductivity for photon polarized along the a-axis is more than times smaller than that of a polarization along b-axis! Using the same scale of plots, the Landau fan is thus apparent for σ bb but di cult to see for σ aa .

If one follows the same calculations on the at part, one would nd the ratio between the conductivities is cos 2 θ. The relativistic renormalization is again demonstrated by the additional Lorentz factor γ. Then, they would be pushed back in the opposite direction by a restoring force. This process can lead to oscillatory motion called plasma oscillations. Like any other oscillation, the plasma oscillations can be quantized into quasi-particles called plasmon.

Plasmons can be measured by electron energy loss spectroscopy (EELS), in which the incident electron scatters with the electrons in metals and thus loses a quantum amount of energy ω p , energy of the plasmon. Details of technique can be found in [ , ]. Recently, even the momentum dispersion of plasmon energy has been resolved [ ], which further boosts the study of plasmons in solids.

The physics of plasmons is encrypted in the polarization function χ(q, ω), i.e., the charge-charge correlation function in linear response theory, which is related to the dielectric function (q, ω). The theoretical way to study plasmon is then to calculate χ(q, ω) or (q, ω). However, this is di cult to do because the origin of plasmon is due to the Coulomb interaction between electrons, which is notoriously intractable. Some approximations must be used.

The simplest approximation to demonstrate the existence of plasmons is the random phase approximation (RPA) [ ] applied on the dielectric function . It is exact if the electronic density is high.

The main idea behind the RPA is to suppose that the local charge uctuation is induced as a response of non-interacting electrons to a local potential, which is self-consistently determined by the sum of the external and induced potential (see Fig. . ). This yields the following equations

φ ind = V C χφ ext ( . ) φ ind = V C χ (0) φ loc ( .
)

φ loc = φ ext + φ ind ( . )
where φ represent electric potentials, χ (0) is the (non-)interacting charge susceptibility and V C is the Coulomb potential. One thus nds the RPA charge susceptibility

χ RPA = χ (0) 1 -V C χ (0) ( . )
By de nition,

φ ext = RPA φ loc ( . )
from which one retrieves the RPA dielectric function

RPA = 1 -V C χ (0) . ( . )
The zero of RPA is thus the plasmon mode, namely a long-lived charge oscillation (and thus a persistent As shown in Chapter , the powerful theoretical tools also helps one to unequivocally identify the nature of states and guides the search of new optical and plasmonic applications.

T V P

Every theoretical physicist who is any good knows six or seven different theoretical representations for exactly the same physics. He knows that they are all equivalent, and that nobody is ever going to be able to decide which one is right at that level, but he keeps them in his head, hoping that they will give him different ideas for guessing.

Richard P. Feynman

In the introduction, the reader has seen that topologically protected edge states emerge at the surface of topological materials by means of gap closing. The wavefunction of topological edge states can be explicitly shown within the Dirac equation including a spatially dependent mass gap, for example ∆(x). Functional analysis indicates that the existence of an exponentially localized topological edge states only requires an inverted gap across the interface. This is in line with general arguments in topological band theory that do not rely on a particular form of ∆(x). Therefore, most studies on topological edge states are justi ed for using a sharp step-like ∆(x) for simplicity. However, the surface of real materials is never perfectly cleaved so that the mass gap ∆(x) has to be a smooth function rather than Heaviside function. For example, the surface of prototypical two-dimensional topological insulator (TI), HgTe/(Hg,Cd)Te quantum well, is notoriously di cult to handle [ , ]. This solicits a study on the e ect of smoothness on the surface of topological materials, which is another work in my thesis.

As I will show in this chapter, the study of smooth surfaces and interfaces is particularly rich. In addition to the topologically protected chiral states, massive surface states called Volkov-Pankratov (VP)

states [ , , ] emerge at the surface of topological materials when the surface is smooth enough.

H ξ=+1 = ∆(x)σ z + v(k x σ x + k y σ y ) ( . )
In some literature [ ], the topological surface state are sometimes called the topological/massless VP state.

where σ α (α = x, y, z) are Pauli matrices that represent the sublattice (A and B), ξ = +1 denotes the K valley [see Eq. ( .)]. The interface is described by a position-dependent gap ∆(x), with ∆ < 0 for x < 0 and ∆ > 0 for x > 0. In the other valley K , the Hamiltonian is similar to that in the K valley,

H ξ=-1 = ∆(x)σ z + v(-k x σ x + k y σ y ) ( . )
where ∆(x) remains positive for both sides of the interface, even if its value may vary across the interface, and ξ = -1 denotes the K valley. So, I will only consider the Hamiltonian for K valley in the following calculations.

The interface Hamiltonian has to match the bulk Hamiltonian when one is situated at a place far away from the position where the gap is closed. In this example, the Hamiltonian becomes the bulk Hamiltonian of Semeno insulator in the limit x → -∞ and that of Haldane insulator when x → +∞. Concretely, ∆(x) veri es

∆(x) =    -∆ 0 if x → -∞ ∆ 0 if x → +∞ ( . )
where ∆ 0 > 0. To emphasize the fact that two topologically di erent phases at x → -∞ and

x → +∞ cannot be distinguished from their spectrum, I have assumed additionally the gap of the two insulating phases to be the same. I will also make the same assumption in the following models.

For interested readers, the case for di erent gap is discussed in the references [ , ]. In the vicinity of

x ∼ 0 where the gap varies, k x is no longer a good quantum number so that Inspired by this model, one can construct a Hamiltonian to describe a D topological heterojunction

H 0 = ∆(z)τ z + vk z τ y + vτ x (k y σ x -k x σ y ) ( . )
where the Pauli matrices τ and σ represent orbital and spin degrees of freedom, respectively. As in the D model, one forms a D topological heterojunction by replacing the constant bulk gap by a spatially varying one which changes its sign across the interface along the z-axis.

For a D TI in Z 2 classi cation as Bi 2 Se 3 , the sign of ∆ is also the topological invariant: topological if the gap is negative and trivial if positive [ ]. Suppose the half-space z < 0 is lled by topological phase and the other half-space z > 0 lled by trivial phase. As before, I assume a symmetric topological heterojunction such that

∆(z) =    -∆ 0 if z → -∞ ∆ 0 if z → +∞ ( . )
where ∆ 0 > 0.

T W

The way to construct a Hamiltonian describing the smooth interface between a WSM and a trivial insulator is slightly di erent from that for TIs. One starts by the simplest Hamiltonian of a timereversal-broken two-node WSM [ ]

H = v(k x σ x + k y σ y ) + k 2 z 2m -∆ 0 σ z ( . )
where ∆ 0 , m > 0 and two Weyl nodes are placed at k = (0, 0, ± √ 2m∆ 0 ) between which the gap is inverted in reciprocal space. A topological transition occurs when two Weyl nodes of opposite chirality are annihilated by merging them together [ ]. A trivial insulator is thus formed beyond the merge of two nodes by tuning the gap to zero and then to negative values. The desired Hamiltonian is modeled as

H = v(k x σ x + k y σ y ) + k 2 z 2m -∆ 0 + ∆(x) σ z ( . )
where ∆ 0 > 0 and

∆(x) =    0 if x → -∞ 2∆ 0 if x → +∞.
( . )

The bulk Hamiltonian of the trivial insulating phase interfacing with WSM is then

H = v(k x σ x + k y σ y ) + k 2 z 2m + ∆ 0 σ z ( . )
L Until now, the only arti cial hypothesis for the gap parameter is that the two sides of a topological heterojunction have the same band gap. The condition on the gap parameter given above is to make the interface Hamiltonian match correctly the bulk Hamiltonian of two phases at x → ±∞. None of the models imposes any explicit form for the position-dependent gap parameters. Yet, the above condition implicitly requires a valid function of a gap parameter describing a topological heterojunction, for example ∆(x) in the D toy model, to have the property that ∆(x) is vanishing somewhere at nite

x only an odd number of times. Otherwise, the neighboring gap closing points can be annihilated pairwise by continuous transformation (see green dots in Fig. . ). The gap can reopens consuming a nite amount of energy, that is not topologically protected. Therefore, from the topological point of view, one is allowed to suppose without losing generality that the gap is closed only once at, for example, x = 0. A plausible choice for the gap parameter could be ∆(x) = ∆ 0 tanh(x/ ) where a length scale for smoothness describing the size of the interface is introduced (see blue line in Fig. . ).

Since the gap changes its sign only once at x = 0, one can always do a Taylor expansion of ∆(x) in the vicinity of the gap closing point where ∆(x) can be fairly supposed to be analytical at x = 0. This amounts to write ∆(x) = ∆ 0 x/ + O((x/ ) 2 ) where the typical size of the interface serves as a cut-o . Therefore, one can always use a linearized gap parameter at the gap closing point whenever one focuses on the surface states localized in a topological heterojunction. In other words, the linearization of model is universally possible in any topological heterojunction. Every valid functional form of gap parameters is permitted to do such a linear expansion. For example, tanh(x/ ) is linearized to x/ as expected.

Introducing a length scale characterizing the smoothness of topological heterojunction, the linearized version of the three models are

• D toy model:

H = ∆ 0 x σ z + v F (k x σ x + k y σ y ) ( . )
where the gap is closed at x = 0 and only one valley is considered.

• DTI:

H = ∆ 0 z τ z + v F k z τ y + v F τ x (k y σ x -k x σ y ) ( . )
where the gap is closed at z = 0. • Two-node WSM:

H = v(k x σ x + k y σ y ) + k 2 z 2m -∆ 0 + 2∆ 0 x σ z ( . )
where the inverted gap of WSM is closed at x = .

In all the models above, I consider that two topologically di erent phases are semi-in nite so that only one interface is considered. This is valid when the typical size of two materials L is much larger than the size of the interface . The nite-size e ect will be discussed in the following sections of the chapter.

O V P

Besides massless topological states, massive surface states also emerge in topological heterojunction thanks to a smooth interface. These states are called Volkov-Pankratov (VP) states, being rst theoretically predicted by Volkov and Pankratov in s [ , ]. The VP states have been rediscovered

recently by our group and collaborators [ , ] with the advent of topological material.

In this section, the Hamiltonians of topological heterojunctions consisting of TIs are solved both analytically and numerically. Various methods are used and each of them re ects a conceptually di erent perspective on the origin of VP states. I will discuss the energy spectrum of topological heterojunctions made by TIs. The analysis for the surface states of WSM are postponed to the next chapter.

S

I start with the simplest model, i.e., the D toy model, which bears all relevant phenomena that are all present in the more complete models. Linearization of the gap parameter permits to solve analytically the Hamiltonian with the help of unitary transformations presented in Chapter . After the unitary transformation T = exp(-iπσ x /4), the linearized Hamiltonian of the D toy model ( . ) reads:

H T = T HT † = v k y k x + i x ξ k x -i x ξ -k y ( . )
where a characteristic length ξ = v/∆ 0 is de ned. This length scale is intrinsic because its value depends only on the Fermi velocity v and the gap parameter ∆ 0 given by the bulk material. The e ect of the unitary transformation is to interchange σ z and σ y Pauli matrices so that the x-dependence of the linearized gap function is now placed in the o -diagonal elements with k x . The Hamiltonian written in the new basis is reminiscent of the Hamiltonian of gapped graphene in the presence of magnetic eld where the gap is replaced by vk y which is a good quantum number. However, the linear term due to the spatially varying gap parameter along with k x quantizes the motion in the x-direction in the same way as a magnetic eld does by the Peierls substitution. Hence, the smoothness of topological heterojunction can be seen as a pseudo-magnetic eld. As a real magnetic eld yields Landau quantization, the pseudo-magnetic eld should also induce energy levels indexed by an integer n.

To explicitly show this surprising analogy, one can de ne a pair of ladder operators

â † = S √ 2 k x + i x 2 S ( . a) â = S √ 2 k x -i x 2 S ( . b)
that satisfy [â, â † ] = 1 and S = √ ξ which plays the role of a pseudo-magnetic length associated with the pseudo-magnetic eld, i.e., the smoothness. This is mostly evocative by comparing the Hamiltonian

H T = √ 2 v S ky S √ 2 â † â - ky S √ 2 ( . )
with ( . ). To solve this Hamiltonian, one thus proceeds in the same way as for graphene in a magnetic eld using harmonic oscillator basis de ned by the ladder operators (see Chapter ). The center of harmonic oscillators x is precisely at the gap closing point x = 0. This is actually an artifact of using an odd function of ∆(x). Nevertheless, the wavefunction should be localized at x = 0. Unlike usual Landau level wavefunction, x is independent of k y making the resulting energy levels nondegenerate.

The spectrum of H T reads

E ± n = ± v k 2 y + 2n 2 S if n ≥ 1 E 0 = vk y if n = 0 ( . )
where n is a positive integer. As shown in the left panel of Fig.

. , instead of discrete Landau levels that one would nd for a gapped graphene in the presence of a magnetic eld, the spectrum of the D topological heterojunction in the presence of a pseudo-magnetic eld created by the interface smoothness consists of dispersing Landau bands indexed by an integer n. I will explain in detail in the next part that the analogy of smoothness with magnetic eld is partial and formal. They are actually fundamentally di erent quantizing eld. This di erence is encoded in the de nition of the ladder operators. The dispersion in k y is an appreciable characteristic of VP states which distinguishes them from other disorder-induced surface states. This is the reason why spectroscopy could be a good tool to unambiguously reveal VP states as shown in the next chapter.

Nevertheless, the analogy of smoothness with magnetic eld is both convincing and inspiring. On the one hand, the massless topological state, which emerges as the n = 0 Landau band, is soundly included in the model describing a smooth junction. Most saliently, the massless state with E 0 = vk y does not depend on any details of the interface, in particular the smoothness , as it should do by general topological argument. The massless state is chiral because it propagates only in the positive y-direction. This is due to the fact that one considers only one side of a topological Haldane insulator. The chiral state with opposite dispersion can be found if one takes into account the other side of a nite size TI. On the other hand, the emergence of massive VP states are explicitly shown by the states n ≥ 1 with n being any large value. At rst sight, it might seem that there is no bound for n. It might be tempting to think that the existence of the VP states would be universal on the surface of every topological materials because the surface smoothness is at least limited from below by the atomic length scale. This is however not the case and simply an artifact of the linearized model in which only surface states are considered. The presence of the bulk system prohibits the existence of the VP states on any surface of real samples above a certain value of n (see the left panel of Fig. . ). Here, the spectrum of the bulk system is ±

2 v 2 (k 2 x + k 2 y ) + ∆ 2 0 .
If the VP states with large n are immersed in the bulk continuum, they will not be localized on the surface due to the coupling with the bulk. Alternatively, only the VP states present in the gap is physically allowed and experimentally observable. Following this logic, one can de ne a critical length for smoothness only beyond which the interface is said to be smooth and the n = 1 VP state emerges in the gap. This critical length is precisely the intrinsic length scale ξ. The condition of the emergence of VP states in a topological heterojunction is thus ξ. This condition is ful lled when the surface is very smooth (large ) or the bulk gap is large enough (small ξ). For some large gap DTI, the intrinsic length scale is about 1 nm while the quintuple layer is also around 1 nm [ , ]. The realization of a smooth topological heterojunction is promising. The above argument that the VP states of energy should be smaller that the bulk gap, √ 2n v/ S < ∆ 0 , yields an estimate of the maximum number n max of VP states that a given topological heterojunction can display,

n max ≈ ξ . ( . )
Another mathematical understanding of the limit on n is to solve the model with ∆(x) for a full pro le of space in which both the surface and bulk are considered [ ]. Small value of admits only a

Topological heterojunction: emergence of Volkov-Pankratov states nite number of localized solutions to the di erential equation. In comparison, one would prefer the analysis using linearized models since the argument is universally applicable and the physical picture is more intuitive.

It has been shown that the energy levels indexed by an integer n resemble Landau levels with dispersion justifying the similarity between smoothness and magnetic eld. It is thus natural to ask how this pseudo-magnetic eld and a real magnetic eld interplay on the surface of topological materials. I invite the readers to go to the next chapter for this topic.

S D

In this part, I will elaborate a point of view on the origin of VP states that is conceptually di erent from the analogy with magnetic eld. I have already mentioned several attributes of VP states distinct from Landau levels induced by a real magnetic eld. The discrepancy is encoded in the de nition of the ladder operators, which are independent of k y . The VP states are thus non-degenerate, (d -1)dimensional bands, indexed by n, if one considers a d-dimensional bulk system. For the same reason, the energy spectrum is a function of k y but independent of k x . The most prominent distinction is thus the dimensionality of the energy spectrum. Only one direction is quantized due to the positiondependent gap parameter. The dimension of system is thus is reduced from d to d -1, as one would expect for the relation between surface and bulk. However, a real magnetic eld quantizes the motion in the plane perpendicular to itself so that the dimension is reduced from d to d -2. In this sense, the smoothness a ects the spectrum in a similar way as a quantum con ning e ect in the direction along which the gap varies.

Quantum con nement needs a potential quantum well which is not explicitly present in the model of the topological heterojunction. However, the Dirac Hamiltonian written in matrices can be decoded into a pair of conventional Schrödinger equations in a con ning potential that arises from the (linearly) varying gap parameter for the components of the spinors. I will illustrate this point by a topological heterojunction formed by DTI ( . ). Working in the so-called Weyl basis T = exp(iπτ y /4),

the Hilbert space can be decomposed into an orthogonal direct sum of two subspaces with opposite chiralities. The eigenstates |ψ are four-component spinors, where χ ± are themselves two-component spinors of chirality ±. One obtains thus a set of two decoupled di erential equations:

|ψ = χ + (z) χ -(z) , ( . )
E 2 -2 v 2 k 2 χ λ = [∆(z) + λ v∂ z ][∆(z) -λ v∂ z ]χ λ , ( . )
where k 2 = k 2 x +k 2 y and λ = ± represents the chirality. Most saliently, the equations can be rewritten as

E 2 -2 v 2 k 2 χ λ = Ẽ2 λ χ λ = -2 v 2 ∂ 2 z + U λ (z) χ λ , ( . )
the right hand side of which shows now a second-order derivative in z, as it is the case for a D Schrödinger equation with a con ning potential In this treatment, solving E for the Dirac Hamiltonian ( . ) is equivalent to solving

U λ (z) = ∆(z) 2 + λ v∂ z ∆(z), (
Ẽ2 λ ≡ E 2 -2 v 2 k 2 ( . )
for this Schrödinger equation whose spectrum Ẽ2 λ must be non-negative. Be careful about the fact that the spectrum and the potential in the Schrödinger-type equation ( . ) have the physical dimension of a squared energy. To emphasize that I am working with such auxiliary quantities that do not have the dimension of energy (but its square), I invent the term virtual energies in the context of the Schrödinger equation ( . ) to refer to the quantity Ẽ2 λ . The conversion relation from a virtual energy to a physical energy reads

E ± λ (k ) = ± Ẽ2 λ + 2 v 2 k 2 . ( . )
With the help of the conversion relation, one can thus get the spectrum of the original Dirac Hamiltonian after solving the virtual energy of the corresponding the system of two Schrödinger equations.

A given value of virtual energy corresponds to two opposite physical energies making the spectrum particle-hole symmetric. This is a distinguishable characteristic of VP states from other surface states.

Though the particle-hole symmetry is an artifact of the Hamiltonian containing only linear terms in k, it still applies for the low-energy spectrum especially for k = 0.

To illustrate a single Dirac QW, it is instructive to study the linearized form of the gap parameter

∆(z) =          -∆ 0 if z < - ∆ 0 z if z ∈ [-, ] ∆ 0 if z > , ( . )
where the bulk is also included [see Fig. A particularly interesting character of this choice could be that the gap parameter coincides with the linearized model ( . ) for z ∈ [-, ] where the gap is closed and the range of interface is de ned.

In this region, the Schrödinger Hamiltonian has the form of the Hamiltonian for a D quantum harmonic oscillator. This can be seen by formally substituting ∆ 0 /v 2 → 2m and v/l → ω c /2 (or equivalently ∆ 0 /l 2 → mω 2 c /2) in ( . ), such that the e ective Schrödinger Hamiltonian reads

Ẽ2 λ ∆ 0 χ λ = - 2 2m ∂ 2 z + 1 2 mω 2 c z 2 + λ ω c 2 χ λ ( . )
where an energy shift depending on the chirality due to a vertical shift between U -and U + is determined by the interface width . Recall that the wavefunction of Landau levels in the Landau gauge is a D quantum harmonic oscillator as it is for the VP states. So, the form ( . ) suggests an alternative way to understand the analogy of smoothness with magnetic eld. The spectrum of this Hamiltonian of the linearized model is given by

Ẽ2 λ ∆ 0 = ω c n + 1 + λ 2 ( . ) or Ẽλ = 2 ξ l n + 1 + λ 2 ∆ 0 ( . )
where n ≥ 0 is an integer and λ = ±. Only one zero mode exists with a de nite chirality λ = -while all other levels possess two chiralities. Using the conversion relation ( . ), the zero mode corresponds to the famed topologically protected single Dirac cone on the surface of DTI. The other Dirac cone is presented on the other side of DTI which is not considered in this single boundary model so that only one zero of one chirality is found here. The massive VP states manifest themselves as the non-zero energy levels of a D harmonic oscillator. They are thus not protected by topology from back-scattering

[ ].
The above solution is only correct if ∆(z) = ∆ 0 z/ for all over the space along the z-direction.

Nevertheless, the reader already sees how a spatially variant gap parameter in Dirac Hamiltonian becomes a con ning potential in the transformed Schrödinger Hamiltonian for the components of spinor.

One can thus easily complete the full pro le of the con ning potential induced by the gap parameter ( . ). This is shown by ]. An undergraduate level physics course [ ] on D quantum mechanics tells one that there must be a bound state for λ = -, but not necessarily for λ = +. The virtual energy of bound states should be smaller than ∆ 2 0 . Using ( . ), one estimates the number of bound state in the gap that a topological heterojunction can host, which is exactly the thumb rule ( . ).

Since the virtual energy potential U λ is parabolic inside z ∈ [-, ] and constant outside, the wavefunction χ λ for n = 0 behaves as a Gaussian within the interface and decays exponentially in the bulk.

The standard deviation of the Gaussian part is thus described by a length scale S = √ ξ which depends on the well width and the bulk gap, the same I have shown in the previous part. The width and depth of Dirac QW are both determined by the smoothness of the surface, which is encoded in ∆(z) 2 and ∂ z ∆(z) [see the orange and green lines in Fig. . ]. This is essentially di erent from the conventional (square) QW of which one can independently engineer its depth and width. Most saliently, a smoother interface gives rise to a wider but shallower Dirac QW which can nevertheless host more bound states. This can be understood in terms of supersymmetric quantum mechanics as will be shown in the next part.

It is satisfying to nd the same results on the bound states using a full pro le ∆(z) as that on the localized states in the linearized model. Using ∆(z) = ∆ 0 tanh(z/l) will of course provide the same conclusions [ , , ]. This puts in evidence the argument that the band inversion mechanism allows one to linearize the spatially varying gap at the interface. Indeed, the validity of the linearized model to discuss the localized states is con rmed in the framework of Dirac QW as well as the statement that massive VP states can in principle emerge in any topological heterojunction when the interface is su ciently smooth. In the language of quantum well physics, the Dirac QW can host more bound states if it is su ciently wide.

C D

Knowing the spectrum of a single Dirac QW, it is appealing to study the coupling between two Dirac QWs. The con guration of two adjacent Dirac QWs arises naturally when a DTI is sandwiched between two trivial insulators, as depicted in Fig. . (b). A confusion of jargon could be that the sandwich heterostructure is conventionally said to be a topological QW but with two interfaces so that it hosts two Dirac QWs. The thin lm is thought to be a promising way to get rid of the metallic bulk states in TIs, which are undesirable for applications that rely on conduction from only the surface states [ ]. In this part, I will show how the coupling strength between the states localized at di erent sides of a topological material depends on the thickness of the sample and also the smoothness of sur-face. As shown below in a concrete example, the double Dirac QW is intrinsically asymmetric re ecting the topological protection of the topological surface state which is little a ected by the presence of an adjacent Dirac QW. In contrast, the (virtual) energy of massive VP states are considerably shifted.

One chooses the following ∆(z) in the Hamiltonians ( . ) to model the above heterostructure:

∆(z) =                      ∆ 0 if z < -L 2 - -∆ 0 (z + L 2 ) if z ∈ [-L 2 -, -L 2 + ] -∆ 0 if z ∈ [-L 2 + , L 2 -] ∆ 0 (z -L 2 ) if z ∈ [ L 2 -, L 2 + ] ∆ 0 if z > L 2 + ( . )
where L is the thickness of the DTI as indicated in is no longer valid. One can thus already anticipate that the zero mode is not protected anymore and acquires a non-zero energy. In terms of Dirac QW, it is understood as the possible tunneling between two adjacent Dirac QWs separated by a nite virtual energy potential barrier of height ∆ 2 0 . For each U λ , the Schrödinger Hamiltonian can be solved analytically by the usual method used in wave mechanics. Just like the problem of particle in a box, it su ces to transform the eigenvalue problem in an energy-dependent di erential equation whose general solutions are analytically known. The boundary conditions then give rise to a secular equation that determines the spectrum. Here, I am only interested in localized states and the details of calculations can be found in App. D.

Since the presence of the second Dirac QW can be seen as a perturbation term to a single Dirac QW, especially for L , the energy spectrum of double Dirac QW will not deviate too much from that of single Dirac QW. Therefore, only the energy shift due to the double QW con guration is worth a careful analysis which I present here. Denote the deviation of the energy at k for each index n as ±∆E n . Indeed, this deviation can be calculated with the help of the virtual energies Ẽ2 ,

∆E n = || Ẽ| -|E 0 n (k = 0)|| ( . )
where the energies at k = 0 of a single Dirac QW are represented henceforth by the superscript 0, E 0 n (k = 0). For the zero mode n = 0 which is no longer topologically protected, surface Dirac cones acquire a mass gap such that

E ± n=0 (k ) = ± ∆E 2 0 + 2 v 2 k 2 . ( . )
In contrast to the n = 0 states, the massive VP states are "split" in energy by ±∆E n with

E ± n (k ) = ± E 0 n (k = 0) ± ∆E n 2 + 2 v 2 k 2 , ( . )
as a consequence of quantum tunneling between the two Dirac QWs and the resulting hybridization of the QW states.

Two possible regimes are discussed separately.

S

When ξ, only the n = 0 modes exist at two sides of DTI and the presence of the other respective side yields a small mass gap. They live in two decoupled potential U λ so that they cannot hybridize directly by quantum tunneling since their respective spinorial wavefunctions are orthogonal (see Fig.

. ). One can only imagine an indirect tunneling with the help of the plane-wave bulk states that open a gap. However, this is not the dominant mechanism that opens the mass gap. As shown in the smooth surface regime where tunneling is even enhanced in the presence of VP states, it is more suitable to understand it in perturbation theory due to the exponential tail of the wavefunction probing an adjacent Dirac QW nearby. In the sharp interface limit, one obtains by Taylor expansions of the secular equation the mass gap for the n = 0 modes

∆E 0 = ∆ 0 e -L ξ 1 + 4l 2 3ξ 2 , ( .
)
where the ratio L/ξ in the exponential illustrates the idea of topological protection by gap. Besides the exponential decay with increasing thickness which benchmarks the result with those in literature [ , , , ], a correction arising from the smoothness of the surface is added. This formula gives a good estimation in order of magnitude for, for example, a thin lm of Bi 2 Se 3 . For Bi 2 Se 3 , taking ξ as . nm [ , ], one nd a mass gap 2∆E 0 = 0.03 eV close to the experimentally measured gap 0.05 eV [ ]. In addition, this formula allows a quick estimation of the smoothness of the surface of topological material from the measured mass gap.

S

When > ξ, massive VP states are also present in the region of con ning potential wells. The calculations show that the energy splitting ∆E n depends non-trivially on /ξ, L/ξ and the VP state index n xing other material-related parameter such as v and ∆ 0 . I de ne a reduced energy ω r with dimension of unity used in the following plots

ω r = E 2 -2 v 2 k 2 ∆ 2 0 ∈ [0, 1].
( . )

For n = 0, ω r = ∆E 0 /∆ 0 [see Eq. Fixing L/ξ = 20, the evolution of the energy splitting of the VP states as function of /ξ is shown in Fig. . . The spectrum of massive VP states follows as predicted to a small relative error the behavior ω r = 2nξ/ derived Eq. ( . ). Increasing the smoothness of the surface permits VP states with large n to emerge in the gap, or equivalently, be localized in the con ning potential. The larger the index n is, the more prominent is the energy splitting.

The /ξ-dependence of the energy splitting of the VP states and the topological state is given in In contrast to VP states, the zero mode of a given chirality in one Dirac QW is little a ected because the adjacent Dirac QW does not host a zero mode. This explains the huge di erence in the order of magnitude of the energy splitting between the VP states and the topological states.

Pursuing the idea of tunneling, one can in principle estimate the energy splitting in order of magnitude using a heuristic formula. This formula is the result of a pedestrian exercise of undergraduate level that the energy splitting due to the tunneling between adjacent nite symmetric square QWs reads

[ ] 2∆E = 2 π 2 4m 2 4e -K(L-2 ) 2K , ( .
)
where 2 is the width of a square QW, L the separation between the centers of the two square QWs and K = √ 2mV 0 / with the e ective depth of the square QW, V 0 . The heuristic formula for double Dirac QW is thus derived after replacing 2m by 1/v 2 and V 0 by (1 -ω 2 r )∆ 2 0 for the e ective potential depth for a surface state of reduced energy ω r . Since the energy splitting is that of the virtual energies in the Dirac QW approach, one obtains actually a formula for ∆(E 2 n ) instead of ∆E n . One has to do an expansion of the virtual energies to linear order around the VP energies by writing ∆(E 2 n ) ≈ 2E n ∆E n to translate the splitting of virtual energy into that of physical energy. The wavevector that describes the exponential suppression of the hybridization is given by K = 1 -ω 2 r /ξ. Since the linearized version for the energy of the VP states matches the correct one to great accuracy, one can use it for ω r so that the nal formula reads

∆E n = π 2 4 ∆ 0 √ 2n ξ 5/2 e -1-2nξ (L-2 )/ξ 1 -2nξ ( . )
for the VP states. As shown by dots and crosses in Figs. . (a) and . (a), the heuristic formula gives a good approximation on the order of magnitude of energy splitting, especially for n = 1 and 2. Most saliently, the non-trivial dependence on /ξ is captured by the formula ( . ). The non-monotonic behavior of energy splitting as a function of /ξ can be thus understood in terms of the antagonistic interplay between the e ective separation of two Dirac QWs L -2 and the single QW energy dependence on . A smooth interface, i.e. large stabilizes the massive VP states so that the tunneling process should get through a higher potential barrier which prohibits the hybridization. However, it also reduces the e ective distance between the two adjacent Dirac QWs so that the tunneling becomes more probable. The competition between the two e ects manifests itself as the existence of a minimal value of the energy splitting while varying . Since the formula ( . ) itself is only valid for the states of small n, the heuristic formula does not give a good approximation for the states of large n (here n = 4). These states have an virtual energy close to the virtual energy edge of Dirac QW. In other words, these VP states just emerge from the bulk continuum. In this case, the energy splitting of higher VP states increases with . The tunneling between two adjacent Dirac QWs is thus the reason to lift the degeneracy of the VP states. The formalism of Dirac QWs explicitly put in evidence the intuition on the mechanism of energy splitting and captures the essence of physics.

One might want to follow the same argument to derive a heuristic formula for the state n = 0.

However, as I show in this paragraph, this approach yields an erroneous scaling of the energy splitting.

Since the unperturbed n = 0 state has zero virtual energy, one must push to the second order in ∆E 0 unlike the n > 0 states, i.e.,∆(E 2 0 ) = (∆E 0 ) 2 . This would give a heuristic estimation -(L -2 )/2ξ) predicted by the heuristic formula. Actually, the tunneling on which the heuristic formula is based is not the predominant factor that opens a mass gap for the topological state. The n = 0 state has no partner in the adjacent Dirac QW with the same chirality so that tunneling is repressed. One has to nd another mechanism than the resonant hybridization to account for the splitting for the n = 0 state. Indeed, a treatment in perturbation

∆E 0 = π 2 ∆ 0 ξ 3/2 e -(L-2 )/2ξ (
∆E 0 = ∆ 0 2π 1/4 ξ 5/4 e -L-1.5 ξ . ( .
)
where the e ective separation is L -1.5 . The calculations are reported in App. D. . To the leading order, the energy of the n = 0 state is a ected by the deviation in U λ (z) induced by the second adjacent Dirac QW in the exponential tail of the wavefunction.

The absence of the resonant hybridization of wavefunctions explains why the the splitting for n = 0 states is by several orders of magnitude smaller than that for the n > 0 states. This is because a double Dirac QW is inherently asymmetry in contrast to the symmetric double square QW. The asymmetry in the double Dirac QW can be thus seen as another interpretation of topological protection.

S D

As for thin lm of topological material, there are both experimental [ , , , ]

[ ] e orts on topological superlattices which consists in alternating layers of topological and normal insulators. Forming a superlattice from topological materials is a strategy to enhance the surface response with respect to the bulk response in magneto-optics [ , ]. In the framework of Dirac QW, the modeling of topological superlattice, which is just a periodic series of QWs as shown in Suppose the layer thickness of TI is L TI and that of normal insulator (NI) is L NI . Two topologically di erent bulk phases are assumed to have the same bulk gap. The smoothness of interface is set to be ξ with 2 L TI , L NI . When L TI = L NI , the periodicity of the superlattice is L = L TI = L NI . The coupling between VP states at two surfaces of a NI is thus the same as that between VP states at two surfaces of a TI. Given n, one thus realizes a D chain in which each interface represents a site and the hopping amplitude is t n = ∆E n (L) where ∆E n (L) is the expression ( . ). So, the splitting for VP states can be estimated

∆E n = 2t n cos(k z L).
( . )

If L TI = L NI , the periodicity of the superlattice is L = L TI + L NI so that each unit-cell contains two interfaces TI/NI. The coupling between VP states at two surfaces of a NI is di erent from that between VP states at two surfaces of a TI. Given n, the hopping amplitude between two surfaces of a TI is t n = ∆E n (L TI ) and that between two surfaces of a NI is t n = ∆E n (L NI ). One thus realizes a D chain which can be described by the SSH model! Most saliently, the ratio between the hopping terms t n and t n are experimentally tunable simply by changing the thickness of TI and NI layer. Consequently, the splitting for VP states reads

∆E n = ± t 2 n + t 2 n + 2t n t n cos(k z L). ( . )
The estimation for the n = 0 state is more subtle. On the one hand, the n = 0 state has no partner at the nearest neighbor Dirac QW but at the second nearest neighbor Dirac QW. The expression ( . ) is thus valid by replacing L by L TI + L NI . On the other hand, the dominant mechanism to open a mass gap is still described by the rst-order perturbation theory, i.e., Eq. ( . ). Consequently, the oscillating part due to the tunneling process is negligible compared to the term given by the rst-order perturbation theory. Thus, the mass gap of the n = 0 state should be still almost dispersionless.

In summary, the idea shown by the Dirac QW is as follows. The emergence of VP states is a hid- The topological protection is encoded in the inherently asymmetric double Dirac QW.

den

S D

One might be astonished by the fact there exists always a localized solution at zero energy in Eq. ( . ).

In the linearized model, i.e, a D quantum harmonic oscillator, the zero mode is magically stipulated by a chirality-dependent shift in the con ning potential so that the energy of the zero-point movement is compensated. As rst remarked by Volkov and Pankratov et al. in their papers [ , , ], the reason is given in terms of supersymmetric quantum mechanics. In Eq. ( .), the con ning potential is a linear combination of ∆(z) 2 and ∂ z ∆(z) is called the Witten equation [ ] in the literature for supersymmetric quantum mechanics.

To brie y illustrate the idea of supersymmetry, I consider only the band extrema where k = 0 in Eq. ( .). The Hamiltonian in the Weyl basis becomes:

H s = -∆(z)τ x + vk z τ y . ( . )
In the context of supersymmetric quantum mechanics [ ], H s plays role of the supercharge operator which connects linearly the subspaces of fermions and bosons (here two subspaces of chirality), and H = H 2 s is thus the supersymmetric Hamiltonian. Containing only o -diagonal Pauli matrices, H s maps χ + to χ -and vice versa when it acts on χ -. A functional analysis stipulates that only χ λ gets a zero-energy mode with a de nite chirality λ while χ -λ does not whenever ∆(z) has opposite sign at z = ±∞. The massive VP states are thus the excited states in the boson-fermion correspondence.

More mathematically, they are all the other solutions of the di erential equation Eq. ( . ) that veri es the boundary condition of being vanishing at the in nity.

There is also a quasi-topological invariant called Witten index [ ]:

I W := dim kerH s | V --dim kerH s | V + ( . )
where dim kerH| V is the dimension of the kernel of a linear operator H acting on a subspace V and V λ are the two Hilbert subspaces of opposite chirality. Thus, I W must be an integer and invariant under continuous changes of ∆(z). It dictates also the number of zero mode, zero or one, at the interface.

The reader has just seen another beautiful example how a high-energy formalism, i.e., supersymmetry, emerges naturally in a condensed matter system such as topological materials.

Supersymmetric quantum mechanics is not only a beautiful mathematical formalism but also helps to understand something practical in Dirac QW formulation of the VP states. Recall that a smoother interface induces a wider but shallower Dirac QW which can nevertheless host more bound states. This can be understood in the language of supersymmetric quantum mechanics. If a non-zero mode χ n,- exists in V -, one can nd a non-zero mode of same energy in V + by χ n-1,+ = H s χ n,-because H s ] so that a localized solution becomes viable. However, this is a necessary but not su cient condition for the existence of bound states because the zero-point energy is nite. Yet, a smoother interface can host more bound states.

L

Until now, all the discussions are restricted to low-energy continuum models. Especially, the role played by the bulk system in a topological heterojunction is illusive. In this part, I will show how the VP states emerge in a lattice model.

Here, a Chern insulator on a square lattice [ ] is modeled within the tight-binding approach. Since the low-energy surface physics are governed by the same chirality-dependent Schrödinger equation as Eq. ( .), the discussions in this model can be extended to general consideration of topological heterojunction. The D system is cut along the y-direction so that k y remains a good quantum number. the splitting of the massive VP states is underestimated in the continuum model. A likely explanation for the discrepancy could be due to the higher order terms in k in the lattice model which are not taken into account in the continuum model. Actually, the non-linear terms in k a ects the value of the mass gap of the topological states due to nite-size e ects [ , , , ]. Since the energy splitting of the topological state is too weak to be captured in the numerical precision limited by a lattice cut-o , the mass gap for n = 0 is not visible in the spectrum of the lattice model.

Nevertheless, the lattice model not only con rms many conclusions from the continuum model but also clari es how VP states descend from the bulk. As shown in Fig. . (a), the n = 1, 2 VP states in the gap connect adiabatically to bulk states by varying k y . They are actually the consequence of the bulk states pervading to the surface. They are localized when the surface acquires a nite-size. One can continuously push the VP states into the bulk continuum by varying /ξ. This is essentially di erent from the topological states that the previous action is impossible. Since the VP states are the leakage of the bulk states to the surface, they are bulk states in nature so that the bulk-edge correspondence does not apply for them. This is also the reason why they are not topologically protected. But, this does not mean that they are not topological in a general sense. Their topological properties are embedded in, for example, the π Berry phase given by its massive Dirac fermion's spectrum and wavefunction leading to a quantized surface Hall conductivity [ ].

S

In this chapter, I have shown several di erent perspectives on the origin of the VP states that arise in smooth topological heterojunctions. Each perspective reveals an aspect on the nature of the VP states. First, smoothness can be seen as a source of a pseudo-magnetic eld that gives rise to bands indexed by n similar to Landau levels in a real magnetic eld. This encourages a study of magnetooptics on the VP states as I will shown in the next chapter. In the formulation of the topological heterojunction in terms of the Dirac QW that stems from squaring the original Dirac Hamiltonian, one interprets the topological protection of the topological zero modes in two parallel interfaces as the absence of a resonant hybridization due to the inherently asymmetric double Dirac QW. Meanwhile, the VP states are shown to be not topological. The fact that the two components of opposite chiralities are simultaneously non-zero enables the resonant hybridization. An estimation of the energy splitting is derived using the analogy to the tunneling problem for a symmetric square double QW. Furthermore, the Dirac equation encoded in the VP states is the one studied in supersymmetric quantum mechanics so that the VP states can be understood as excited states in the boson-fermion supersymmetric space. Therefore, one must focus on unique features embedded in the wavefunction of VP states. An immediate proposal would be the particle-hole symmetry dictated by the duality between Dirac and Schrödinger equation, i.e., the supersymmetry. This is not the case for the surface states induced by the previous mechanisms. If the non-linear terms in k are considered, the particle-hole symmetry has a generalized version that VP states always emerge in pairs with opposite e ective mass out of the bulk valence and conduction bands. The observation of this feature requires a very smooth surface or a large bulk gap so that none of the two VP states in pair is immersed in the bulk spectrum.

The symmetry of the wavefunction of VP states provides an even stronger way to identify VP states.

The matrix element in the de nition of optical conductivity encodes precisely the symmetry of wave-However, the simultaneous appearance of conduction and valence subbands is still possible by surface band bending mechanism when the bulk bandwidth is small [ ]. function. Since the smoothness is interpreted as a pseudo-magnetic eld, some optical selection rules similar to those for the Landau levels in a real magnetic eld should exist for the VP states. Furthermore, the interplay between smoothness and real magnetic eld gives more de ning features in the magneto-optical response of VP states that can further substantiate their presence. Therefore, the magneto-optical spectroscopy is believed to o er a clear-cut signature of VP states. This is indeed my original motivation to study the magneto-optical response from VP states.

In this section, I present rst the calculated optical conductivity for the D toy model ( . ) in the presence and absence of a magnetic eld. Only the real part of the diagonal terms in the optical conductivity tensor is given. While the calculation of the optical conductivity of edge states in the D model may be of limited experimental interest, it allows us to illustrate the basic e ects. Then, the same calculations are also done for the more realistic, yet more involved DTI model ( . ). The use of the linearized models is justi ed since only optical transition between the surface states are considered and the bulk system is gapped. As I will strengthen the argument by calculations in the following, the transition between surface and bulk states will not wash out the key signature of the transition between surface states. Finally, I will brie y discuss the e ect of an electric eld perpendicular to the surface.

The temperature is set to be zero (T = 0). For simplicity, I also suppose the chemical potential to coincide with the charge neutral point (µ = 0).

T

Consider a very smooth surface ξ for the topological heterojunction modeled by the linearized D toy model ( . ) whose form after transformation I recall below:

H T = T HT † = v k y k x + i x ξ k x -i x ξ -k y ( . )
where the same notation as in Chapter is adopted and k y is a good quantum number. In the following calculations, I x the characteristic length of the interface smoothness to be = 5ξ for illustrations.

The optical conductivity in the clean limit for this Hamiltonian is thus

σ ij (ω) = i e 2 m,n∈N λ,λ dk y 2π f D (E λ n ) -f D (E λ m ) E λ m -E λ n -ω + i0 + ψ λ m |v i |ψ λ n ψ λ m |v j |ψ λ n * E λ m -E λ n ( . )
where λ = ± indicates if a band belongs to valence or conduction bands; the index i, j = x, y represents the linear polarization of the photon; the two positive integers m, n are the band index and |ψ λ n is the corresponding Bloch wavefunction;. Taking the real part of the optical conductivity, the diagonal term reads

[σ ii (ω = 0)] = πe 2 ω m,n∈N λ,λ dk y 2π f D (E λ n ) -f D (E λ m ) | ψ λ m |v i |ψ λ n | 2 × δ ω -(E λ m -E λ n ) ( . )
where δ is the Dirac distribution. As explained in Chapter , the matrix elements encode the selection rules and δ ω -

(E λ m -E λ n ) is
vk = 1 ∇ k H k ( . )
One should not confound it with the chirality λ in Chapter .

One should not confound it with the complex i at the beginning of the formula.

where H k is the Bloch k • p Hamiltonian in the basis |u λ n . The analytical expressions of the optical conductivity are given in App. E.

In the following plots, I suppose a very smooth interface = 5ξ and I set the magnetic eld (if present) to be B = 1.5 S which corresponds to B = 20.8 T if 2∆ 0 = 0.3 eV and v = 2.5 eV • Å.

The parameters are of typical magnitude for Bi 2 Se 3 [ , ]. I choose a rather strong magnetic eld for illustration, to render the shift of the peaks visible in the plots.

I

When the magnetic eld is absent, the spectrum of ( . ) reads the interband transition between VP states at k y = 0 has a diverging JDOS. They have di erent heights because of nite mesh points in the plot. The rst regular peak originates from the transition between the massless n = 0 topological state and the n = 1 VP state whose JDOS is not diverging thanks to the chiral linear dispersion of the topological state. This is also the reason why its decay rate of the tail

E λ n = λ v k 2 y + 2n 2 S if n ≥ 1 E 0 = vk y if n = 0, (

I

The application of a real magnetic eld o ers richer features of VP states helping to understand further the similarities and di erences between smoothness and magnetic eld. Since the Hamiltonian ( . ) describes a D system, only an out-of-plane magnetic eld is relevant. By the Peierls substitution, one chooses the Landau gauge A = (0, Bx, 0) with (B > 0) to preserve the translational invariance along the y-axis. After replacing k y by k y + x/ 2 B , where B = /eB is the magnetic length, one clearly notices here the similarity between the smoothness via the linear variation of the gap and the magnetic eld: both couple linearly to the space coordinate of the x-direction, but through di erent Pauli matrices. The Hamiltonian ( . ) then becomes

H = v x 2 T (sin θσ z -cos θσ y ) + v(k x σ x + k y σ z ) ( . )
where a characteristic length T and an angle θ are de ned by

1 4 T = 1 4 S + 1 4 B ⇐⇒ B 2 tot = B 2 pseudo + B 2 real ( . a) cos θ = 2 T 2 S ( . b) sin θ = 2 T 2 B . ( . c)
With this form, the recipe given in Chapter shows that the unitary transformation T = exp(iθσ x /2) simpli es further the Hamiltonian to:

H T = v   k y cos θ k x + i x 2 T + k y sin θ k x -i x 2 T + k y sin θ -k y cos θ   . ( . )
such that one can diagonalize the Hamiltonian again with the help of a pair of ladder operators,

â † = T √ 2 k x + i x 2 T + k y sin θ ( . a) â = T √ 2 k x -i x 2 T + k y sin θ ( . b)
that satisfy [â, â † ] = 1. The spectrum of this Hamiltonian thus reads

E λ n = λ v k 2 y cos 2 θ + 2n 2 T if n ≥ 1 E 0 = vk y cos θ if n = 0 ( . )
where λ = ±. Comparing this result to that in the absence of a magnetic eld ( . ), the velocity of the massless n = 0 state is contracted by a factor of cos θ and the spacing between the bands is ampli ed because the characteristic length T , which assembles the in uence from smoothness and magnetic eld, is strictly smaller than S . The de nition of an angle θ suggests that the smoothness and the magnetic eld are two perpendicular components of a total magnetic eld in the direction de ned by θ with respect to the plane. This is compatible with what is suggested before. The total magnetic eld is geometrically composed of a pseudo-magnetic eld in the y-direction and a real magnetic eld in the z-direction.

However, the statement is not true in all the aspects because the selection rules Polarization along x: n → n ± 1

Polarization along y: n → n and n → n ± 1 disprove it. A magnetic eld along the direction neither parallel nor perpendicular to the polarization of photon should not only allow the transition n → n ± 1. The result can be understood in the following manner. The velocity operators associated with the Hamiltonian ( . ) are, vx = vσ x and vy = v cos θσ z -v sin θσ y , ( . )

and the unitary transformation T thus mixes the σ y -and σ z -components of the Hamiltonian so that vy acquires an o -diagonal part while vx is untouched. The selection rules for the polarization in the

x-direction are therefore una ected by the magnetic eld while those in the y-direction acquires an additional dipolar component n → n ± 1. For the same reason, the conductivity tensor is a priori not diagonal.

The real part of the optical conductivity in the presence of a magnetic eld is represented by orange lines in Fig. . . Comparing with those in the absence of a magnetic eld, the peaks in the presence of a magnetic eld have the same shape but the spacing between the two nearest peaks is larger. This re ects the enhanced spacing at k y = 0 of the surface bands [see Eq. ( . )]. Furthermore, one notices that the optical conductivity is larger when the magnetic eld is switched on. This is due to the band dispersion that is contracted by cos θ, in other words, the JDOS is enhanced by the same amount. As a consequence of the supplementary n → n ± 1 selection rules due to the unitary transformation, additional peaks appear in [σ yy ] at the same energies of the peaks in [σ xx ]. Theses peaks are all the more intense for a stronger magnetic eld. Indeed, one can retrieve the selection rules n → n ± 1 for both polarization, x and y, matched with those in the Faraday geometry by taking B → ∞ for which the quantization by smoothness is totally washed out.

In spite of the simplicity of the D toy model, it is already informative about the key signatures of the optical responses from VP states, i.e., the speci c selection rules similar to those for Landau levels in a real magnetic eld. Applying a real magnetic eld does not change radically the shape of the signals but mixes the selection rules and shifts the peaks. This is due to the alliance between the smoothness and the magnetic eld, which is encoded in T . Therefore, the intricate interplay between interface smoothness and magnetic eld allows in principle for a detailed experimental identi cation and investigation of VP states. As shown later, the results of the D model can be easily extended to the realistic DTI model.

T

Proceeding exactly the same way as for D toy model, the matrix form of the Hamiltonian ( . ) after a unitary transformation T = exp(iπτ y /4) reads

H = v       0 k y + ik x √ 2 S â 0 k y -ik x 0 0 √ 2 S â √ 2 S â † 0 0 -k y -ik x 0 √ 2 S â † -k y + ik x 0       ( . )
where the wavevector components k x,y remain good quantum numbers and the ladder operators â and â † are de ned as

â † = - S √ 2 ( z 2 S -ik z ) ( . a) â = - S √ 2 ( z 2 S + ik z ) ( . b) with â, â † = 1.
The optical conductivity in the clean limit for this Hamiltonian is

σ ij (ω) = i e 2 m,n∈N λ,λ dk x dk y 4π 2 f D (E λ n ) -f D (E λ m ) E λ m -E λ n -ω + i0 + ψ λ m |v i |ψ λ n ψ λ m |v j |ψ λ n * E λ m -E λ n ( . )
where the index for the polarization of photon now runs over the three components i, j = x, y, z, and the other notations have the same meaning as for the D toy model. Taking the real part of the diagonal term of the conductivity tensor, one has

[σ ii (ω = 0)] = πe 2 ω m,n∈N λ,λ dk x dk y 4π 2 f D (E λ n ) -f D (E λ m ) | ψ λ m |v i |ψ λ n | 2 × δ ω -(E λ m -E λ n ) . ( . )
For the reasons explained previously for D toy model, one should replace the ket |ψ λ n by its periodic part |u λ n and the velocity operators are just k-derivatives of the Bloch Hamiltonian. The analytic results of the optical conductivity are given in the supplementary material of our work [ ].

As for the D toy model, the following gures are plotted with a very smooth interface = 5ξ.

I

The spectrum of ( . ) is obtained by diagonalizing it in the harmonic oscillator wavefunctions:

E λ n = λ v k 2 x + k 2 y + 2n 2 S if n ≥ 1 E λ 0 = λ v k 2 x + k 2 y if n = 0 ( . )
where in the Kubo formula thus yields a constant optical conductivity, as it is known for graphene, where the frequency-independent absorption is given in terms of the ne-structure constant α 1/137 of quantum electrodynamics [ , ]. In fact, this is the signature of the optical transition in the D Dirac cone. This constant optical conductivity can be expressed only by the intrinsic physical constant:

σ c = π 8 σ 0 ( . )
for a single cone, with σ 0 = e 2 /h.

I

Suppose now that a magnetic eld is applied parallel to the interface, for example, in the x-direction.

For illustration purposes, I have chosen a eld strength so that B = 1.5 S . Choosing the gauge A = (0, -Bz, 0), the Hamiltonian ( . ) after the Peierls substitution becomes:

H = -v z 2 S τ x + z 2 B τ z σ x + vk z τ y + vτ z (k y σ x -k x σ y ). ( . )
After a unitary transformation T = exp(-iθτ y σ x /2) found by the recipe given in Chapter , the Hamiltonian reads:

H Bx = v       0 k y cos θ + ik x √ 2 T â 0 k y cos θ -ik x 0 0 √ 2 T â √ 2 T â † 0 0 -k y cos θ -ik x 0 √ 2 T â † -k y cos θ + ik x 0       ( . )
where the angle θ is the same as the one introduced in Eq. ( . ) and the ladder operators are de ned as which satisfy [â, â † ] = 1. The spectrum of this Hamiltonian is

â † = T √ 2 k y sin θ - z 2 T + ik z ( . a) â = T √ 2 k y sin θ - z 2 T -ik z ( . b)
E λ n = λ v k 2 x + k 2 y cos 2 θ + 2n 2 T if n ≥ 1 E λ 0 = λ v k 2 x + k 2 y cos 2 θ if n = 0 ( . )
where λ = ±. The impact of a parallel magnetic eld in the D model is similar to that in the D model. The velocity of the massless topological state is contracted by a factor of cos θ along k y while the dispersion in the k x -direction (that is the direction of the magnetic eld) is una ected. The equienergy pro le of the Dirac cones becomes oval and the spacing between the Landau bands increases since the characteristic length T is shorter than S as discussed in the D toy model.

The e ect of a parallel magnetic eld in the D model on the selection rules is also similar to the D model as shown below:

Polarization along z: n → n ± 1 Polarization along x: n → n Polarization along y: n → n and n → n ± 1.
It mixes the rules n → n ± 1 and n → n for the photon polarized along the y-axis, while those in the x, z-directions remain the same as in the absence of a magnetic eld. This also can be understood

σ cB = σ c cos θ = 2 S 2 T σ c . ( . )
Furthermore, [σ yy ] acquires additional peaks as a consequence of the unitary transformation that

I

Since the system is D, it is physically relevant to apply a magnetic eld perpendicular to the interface, i.e., in the z-direction. For example, I set here B = 2.15 S . The situation changes drastically. Taking a gauge A = (0, Bx, 0) to apply a magnetic eld B = Be z (B > 0), the Hamiltonian ( . )

becomes

H Bz = √ 2 v       0 b B â S 0 b † B 0 0 â S â † S 0 0 - b B 0 â † S - b † B 0       ( . )
where â and â † are de ned in ( . ) and the applied magnetic eld yields now a second pair of ladder operators:

b † = B √ 2 k y -ik x + x 2 B ( . a) b = B √ 2 (k y + ik x + x 2 B ), ( . b)
which satisfy [ b, b † ] = 1. Note that the operators â and b act on the two orthogonal directions so they commute with each other. In other words, they yield two independent quantum numbers n and m.

The velocity operators are still given by Eq. ( . ), and the spectrum of this Hamiltonian reads

E λ n,m = λ v F 2n 2 S + 2m 2 B ( . )
where λ = ± and n, m ∈ N. One notices that the spectrum is now completely quantized by the two e ects, smoothness and perpendicular magnetic eld, and labeled by the two decoupled quantum numbers, n and m, respectively. As explained in Chapter , smoothness as a quantum con nement e ect quantizes the motion only in the z-direction while magnetic eld quantizes the motion in the

x, y-directions. Comparing ( . ) with ( . ), one can already anticipate that the optical conductivity for a dispersionless energy spectrum consists of a series of Dirac peaks with no tail. To calculate the optical conductivity, one should replace the integral of k x,y in Eq. ( .) by the density of magnetic ux quanta n B = eB/h.

This choice for the magnitude of the magnetic eld allows the readers to clearly associate the optical response with their corresponding interband transition.

It is instructive to juxtapose the selection rules for the two quantum numbers n and m according to the polarization of photon as shown below:

Polarization along z: m → m, n → n ± 1 Polarization along x, y: n → n, m → m ± 1.
One nds the usual selection rules for transitions between the Landau levels under magnetic eld for the quantum number m, while the quantum number n describing the smoothness as a pseudomagnetic eld respects a di erent but complementary set of selection rules. This underlines that smoothness and magnetic eld are two distinct quantization elds. Since the selection rules are incompatible for the two sets of polarization, the tensor σ ij is diagonal.

The results of our analytical calculations are shown in which is identical to [σ xx ] because the perpendicular magnetic eld does not break the isotropy in the x -y plane. Since the spectrum ( . ) is now composed of discrete energy levels, the optical conductivity consists of a set of Dirac peaks that are represented by Lorentzian functions of nite width for visibility. The width of the Lorentzian function encodes phenomenologically the peak broadening by disorders. For a magnetic eld such that B S , one should identify di erent groups among all the Dirac peaks. To illustrate this phenomenon, I plot separately di erent groups of peaks in di erent colors so that the whole optical conductivity should be the sum of all peaks. For [σ zz ] [see Fig.

. (b)], the blue peaks correspond to the transition between the n = 0 and n = 1 levels and the orange peaks to the transition between n = 1 and n = 2. Each of these transitions which are associated with smoothness is now split into additional m → m transitions that are separated by a characteristic energy of v/ B due to Landau level quantization by the physical magnetic eld. The two groups of peaks are interleaved with each other like two hair combs. Likewise, for [σ zz ] [see Fig. . (a)], the blue peaks correspond to the transition between the n = 0 levels, the orange peaks to the transition between the n = 1 levels and the green ones to the transition between the n = 2 levels. More and more peaks will overlap if n is increased.

In this part, I have discussed the e ect of a parallel and perpendicular magnetic eld to the interface. If magnetic eld is along an arbitrary direction, it can always be decomposed to a component perpendicular to the interface and the other component parallel to the interface. Suppose that the magnetic eld forms an angle φ with respect to the interface plane. By using the gauge A = (0, Bx sin φ -Bz cos φ, 0), the problem is divided into the two problems whose solution I have already provided.

Besides the in-gap features shared also by the D toy model, the optical conductivity in the DTI model depend crucially on the orientation of the magnetic eld. A magnetic eld parallel to the inter- face simply changes the position of the peaks and slightly modi es the optical selection rules, whereas a magnetic eld perpendicular to the interface fully quantizes the surface bands and thus yields prominent peaks. The orientation of magnetic eld is therefore an excellent probe of both massless and massive surface states. Therefore, the other perspicuous way to reveal the VP states by magneto-optical spectroscopy is to monitor the spectra of absorption while rotating the magnetic eld from parallel to perpendicular direction. The shape of the optical conductivity will evolve continuously from a step-like function of photon energy into a series of Lorentzian peaks during this rotation. This transformation of the shape of the optical conductivity by rotating the magnetic eld is a smoking gun to distinguish the VP states from other in-gap surface states.

B

The surface-to-surface transitions are not the only contribution to the optical response below the gap.

Optical transitions are also possible between delocalized bulk states and localized surface states. These transitions also yield the sub-gap signal in the optical conductivity. Thus, it is necessary to consider the relative magnitude of the bulk-surface transitions compared to the surface-surface transitions and their modi cation to the shape of the optical conductivity. Here, I numerically calculate the optical conductivity of the bulk-surface transitions for a topological heterojunction modeled by ( . ) in the absence of a magnetic eld. The conclusions are general for other systems hosting VP states.

To treat this problem, I model the bulk states as plane waves [ ] considering an interface in the zdirection at z = 0 between two very large systems of size L so that one can still model it by a single topological heterojunction. Although the bulk spectrum of these two systems is exactly the same, their eigenstates which encode their topological properties are certainly di erent. So, one has to separate the space into two parts, z > 0 and z < 0 to calculate the overlap between the wave-functions. The Kubo formula reads:

σ ij (ω) = i e 2 L n∈N (n,λ,λ ) dk x dk y dk z 8π 3 f D (E λ n ) -f D (E B,λ ) E B,λ -E λ n -ω + i0 + ψ λ B |v i |ψ λ n ψ λ B |v j |ψ λ n * E B,λ -E λ n ( . )
where the superscript B for energy means the energy of the bulk. Note that the optical conductivity is still for two dimensions even though one considers here a D system. Since the bulk states are inherently D and the surface states are essentially D, one should limit the de nition of the optical conductivity in D in order to have compatibility in the thermodynamic limit. Therefore, L in the prefactor is crucial because the matrix elements have another factor of 1/L from the normalization of the bulk states. As a result, the nal expression of the optical conductivity for the bulk-surface transitions is in the unit of e 2 /h and independent of the size of the bulk. This also allows one to compare its magnitude with the magnitude of the surface-surface transitions.

Suppose as before that the chemical potential is at the charge neutral point, the temperature is zero and = 5ξ. As shown in Fig. . , the contribution from bulk-surface transitions starts to be visible at the half band gap (ω ≈ ∆ 0 ) and it is a continuous function of photon energy with three kink spots.

Each of the kink spots is the onset of one kind of transitions. For example, the rst one is due to the transition between the bulk states and the chiral states and the second one stems from the transition between the bulk states and the n = 1 massive VP states. Only three kinks exist because only one massless state and two massive VP states are present in the band gap. Fortunately, even though the magnitude of the bulk-surface transition is not negligible as compared to that of the surface-surface transition especially for energy near the gap, the total optical conductivity still has step-like feature thanks to the continuity of the bulk-surface contribution as a function of photon energy.

In the optical conductivity from the transition between two states with dispersion of di erent dimensions, the singularities from lower-dimensional states are integrated out due to higher dimensional state. This is due to the fact that the JDOS, which determines the functional shape of the optical conductivity, has a higher-dimensional behavior. For example, when one considers a transition from a D surface state to a D bulk one, the JDOS is δ Similar arguments should also apply to the case in the presence of a perpendicular magnetic eld where the surface states are D, i.e., dispersionless, and the bulk states are D. The total optical conductivity should have additional one-dimensional band contributions with onsets at energies corresponding to the bulk-surface transitions. However, the hair-comb-like shape should still be visible and thus remains a valid signature for the identi cation of VP states. More interestingly, in the presence of a parallel magnetic eld, the surface states are D whereas the bulks states are D. As a result, the contribution from the bulk-surface transition is a series of steps. Therefore, additional steps will appear in the total conductivity. This is indeed another spectroscopic signature of the VP states.

[ ω -E D (k D ) + E D (k D )] so that
To conclude, the features of the VP states in magneto-optical spectroscopy due to the surface-surface transition are still palpable even the bulk-surface transitions are considered. E I brie y discuss in this part the case of an electric eld perpendicular to the interface. In a topological heterojunction modeled by DTI, the Hamiltonian including an electric eld E in the z-direction reads

H = eEzI + vk z τ y + vτ z (k y σ x -k x σ y ) -∆ 0 z τ x ( . )
where I is the identity matrix. Here, I de ne not only S = √ ξ characterizing the smoothness but also a length characterizing the electric eld E = v/eE. The Hamiltonian can be rewritten as

H = -v z 2 S τ x - z 2 E I + vk z τ y + vτ z (k y σ x -k x σ y ). ( . )
. Faraday and Kerr rotation This is the form suitable for a hyperbolic transformation which can merge I into τ x if the electric eld is weak (see Chapter ). The critical electric eld is de ned by S = E . One nds again the electric and magnetic regimes as for NbAs 2 in Chapter and as for graphene in a crossed electric and magnetic elds in Chapter . If the electric eld is too strong, the VP states no longer exist and the spectrum becomes a continuum due to the electric breakdown. For a subcritical electric eld, the mismatch of the centers around which the wavefunctions are localized yields the proliferation of peaks exactly as for NbAs 2 in Chapter . The presence of a magnetic eld perpendicular to the interface will not change the argument since the corresponding ladder operators commute with all the z-dependent terms. If the applied magnetic eld is parallel to the interface, one just needs to use the same argument after replacing S by T , the length scale encoding both smoothness and magnetic eld.

Before going to the next part, I want to mention that the study of the optical conductivity has also been done for WSMs [ ]. As a metal, the chemical potential turns out to be another tunable parameter that gives even richer feature to characterize the VP states in WSMs. The e ect of an electric eld is also discussed for WSMs in [ ].

F K

In this section, I brie y discuss other magneto-optical e ects, namely Faraday and Kerr rotations, due to the presence of VP states. Using the results already known in the literature on DTI [ , , , ], I would like to simply point out another possible way to identify VP states in magneto-optical spectroscopy.

When time-reversal symmetry is broken by an out-of-plane magnetic eld, a conventional D electron gas yields quantized magnetic Faraday and Kerr e ects due to the Landau quantization [ ].

DTI lms have also been shown to exhibit interesting magneto-optical Faraday and Kerr e ect [ , ]. Thanks to surface Hall conductivity arising from surface Dirac cone, the magnetic Faraday rotation angle is quantized as multiples of the ne-structure constant α = 1/137 and the magnetic Kerr rotation angle is surprisingly large and close to π/2 for normal incidence. These e ects are observable in experiments if:

• The thickness of DTI lm is smaller than the wavelength so that the electric components in an electromagnetic wave are nearly constant through the whole sample;

• The system is in the quantum Hall regime and thus the longitudinal conductivity is vanishing if the chemical potential is in the gap;

• The frequency of incident wave is smaller than any gap including Zeeman gap and the cyclotron gap ω c so that no interband transitions are allowed.

Spectroscopic properties of Volkov-Pankratov states

Alternatively, the wavelength of the incident wave is the largest one of all the length scales of the system.

It turns out that the wavelength is in the far infrared regime for typical parameters in practice [ ].

Here, it is also important to neglect any residual bulk conductivity and the coupling between two surfaces. In other words, the DTI lm is just a stacking of two surface Dirac cones quantized by an out-of-plane magnetic eld.

When the above conditions are ful lled, the Faraday and Kerr angles read One can perfectly extend the arguments above to DTI thin lms with smooth surfaces ( > ξ)

θ F = tan -1 [(ν T + ν B )α] ≈ (ν T + ν B )α ( . ) θ K = -tan -1 1 (ν T + ν B )α ≈ - π 2 (
when VP states are present in the bulk gap. The smoothness should be much smaller than the thickness of the sample so that the coupling between the two surfaces of the lm can be neglected. Faraday and Kerr rotations should yield interesting behavior varying the surface chemical potential [ , ]. From this perspective, the topological properties of VP states are encoded in the spinor structure of wavefunctions as for D massive Dirac fermion.

Although an elaborate discussion on the Faraday and Kerr rotations of topological heterojunction is out of the scope of this part, I just want to point out that this could be a promising direction for future research.

S W

Until now, I have only discussed the single-particle physics of VP states. In this section, I will discuss how electron-electron interactions enrich the spectroscopic properties of VP states, one of which is the surface plasmon. Two reasons have encouraged me to study plasmon physics. On the one hand, a plasmon is measurable by EELS and it could be useful for future plasmonic applications. On the other hand, the calculations of the dielectric function by which plasmon is theoretically predicted is in principle analytically tractable using the RPA.

It has been well-known in the community that the FA states can induce a chiral linear FA surface plasmon with total non-reciprocity , , , , , , ], i.e., it propagates only in one direction determined by the chirality of the FA dispersion. Linear dispersion and total non-reciprocity are highly desirable for further plasmonic applications. This is why WSMs are chosen to be the system to study surface plasmons arising from surface states. The study of surface plasmons in WSMs in the presence of VP states are thus valuable both from a fundamental and from an applied point of view.

In this part, I will rst show how to solve the Hamiltonian of a topological heterojunction modeled by WSM. The analogy of smoothness with magnetic eld is more plausible in WSMs than TIs. Then, I will apply the RPA to the Hamiltonian of interest. In particular, I will prove that in the long wavelength limit the RPA leads to a scalar dielectric function while considering only the FA state and the rst VP states. Finally, I will bestow the physical meaning upon the numerical results on plasmon modes using the analytical expressions.

H

Consider a single boundary problem, i.e., a smooth interface in the x-direction between a time-reversal breaking two-node WSM and a trivial insulator modeled by the Hamiltonian ( . ) in Chapter , which I recall here

H = v(k x σ x + k y σ y ) + k 2 z 2m -∆ 0 + 2∆ 0 x σ z ( . )
where the inverted gap of WSM is closed at x = . The Hamiltonian ( . ) after a unitary transformation T = exp(iπσ x /4) reads:

H T = v -k y √ 2 S â √ 2 S â † k y ( . )
where the characteristic length S = v/2∆ 0 . The creation and annihilation operators, a † and a, are constructed from linear combinations of the k x -and the x-dependent terms and read

â † = S √ 2 k x + i x -x 2 S , ( . a) â = S √ 2 k x -i x -x 2 S , ( . b) with x = ∆ -k 2 z /2m 2∆ ( . c)
where x determines the average position of the surface state. Compared to the ladder operators dened for TIs, for example, Eq. ( .), the ladder operators de ned here depend on the surface momentum k z through x . Therefore, the analogy of smoothness with magnetic eld is more appropriate in WSMs. From Eq. ( . ), one can already anticipate that the spectrum is only dispersive in k y . The The eigenstates are thus of the form as The energy spectrum is thus

|ψ λ n = 1 √ 2 u n,λ (k y )|n -1, k z λv n,λ (k y )|n, k z if n ≥ 1 |ψ 0 = 0 |0, k z if n = 0 (
E λ n = λ ( vk y ) 2 + ne 2 0 if n ≥ 1 E 0 = vk y if n = 0 ( . )
where the VP band gap e 0 = √ 2 v/ S , which is the separation between the n = 0 and n = 1 bands at k = 0, sets the characteristic energy scale of this surface model. On the one hand, the FA state (n = 0) has linear dispersion in k y breaking the parity symmetry k y → -k y , while its counterpart with opposite sign of the dispersion is localized at the other surface of the WSM that I do not consider here. The VP states has parity symmetry. On the other hand, the FA state is independent of the surface details such as its smoothness, i.e., the band dispersion does not depend on , indicating its topological nature. In contrast, the n ≥ 1 VP bands depend strongly on the surface modeling. In the sharpsurface limit ( → 0), the VP bands rise up in energy and eventually merge with the bulk states when e 0 ∼ ∆ 0 , while the FA state survives. These two facts re ect the non-topological nature of VP states.

N

The details of full derivation of the RPA is given in the supplementary material of [ ]. Here, I brie y summarize the results. As shown in Chapter , the non-interacting dynamical polarization, i.e., noninteracting charge susceptibility is the building block for the RPA. For a multi-band system, the noninteracting dynamical polarization denoted by χ

(0) i,j is [ , ] χ (0) i,j (q, ω) = 1 V k f D (E i (k)) -f D (E j (k + q)) ω + E i (k) -E j (k + q) + i0 + |F i,j (k, k + q)| 2 , ( .
)
where the i, j indices are shorthand notations for both band labels n and λ. In general, the overlap matrix F i,j is not diagonal because of the aforementioned k z -dependence of the eigenstates so that χ

(0) i,j
is generally a tensor (see in the supplementary material of [ ]). However, for q z = 0, the particlehole excitations are also D and F i,j becomes diagonal meaning that only excitations from n to n are possible. χ

i,j can be thus treated as a scalar. Although the contribution from the bulk states is not explicitly taken into account, the VP states interpreted as leakage of bulk states include implicitly this contribution.

For the same reason, if q z = 0, the Coulomb interacting matrix becomes a scalar. The RPA dielectric function then retrieves its usual form RPA (q y , ω) = 1 -V D (q y )χ (0) (q y , ω), ( . )

where a D Coulomb interaction V D (q y ) = e 2 /2 0 r |q y | is used with vacuum permittivity 0 , and the static screening is encoded in the environmental dielectric constant r . The reader may be curious why the application of the RPA to a D system uses a D Coulomb interaction. This is because I discuss here surface states that are localized at the interface. Their spatial extension is cut o by S . For smooth interfaces with S , the dominant contribution to the Coulomb matrix comes from the region where q( x -x ) 1. The Coulomb potential is e ectively D in the long wavelength limit, i.e., q S 1. This is the reason why I qualify the RPA used in the section quasi-two-dimensional.

When q z = 0, the overlap matrix element F i,j (k, k + q) is mathematically more involved. However, in the long-wavelength limit, the o -diagonal term F i,j is proportional to q |n i -n j | z so that the n → n excitations still remain the leading contributions to the charge susceptibility. Nevertheless, due to the complicated form of the o -diagonal terms, the Coulomb interaction is no longer a scalar even in the long-wavelength limit when several VP bands are present. However, if one considers only the chiral FA and the two n = 1 VP bands (three-band model), the RPA dielectric function for q z = 0 is simpli ed to RPA (q, ω) = 1 -V D (q)χ (0) (q, ω), ( . )

which is again a scalar equation. E q z = 0

Setting q z = 0, the pro le of -Im(χ (0) ) in the (q y > 0, ω > 0)-plane, for di erent values of the chemical potential µ > 0 and a given disorder amplitude by replacing 0 + with 0.001 in Eq. When the chemical potential is below the band edge of the rst VP band, two domains of particlehole continua exist. One of them extends linearly in the (q y > 0, ω > 0)-plane, i.e., the signature of FA particle-hole continuum. Due to the linear dispersion of the FA state, its particle-hole continuum is independent of µ. The second particle-hole spectrum is delimited from below by ω > 4e 2 0 + 2 v 2 q 2 y due to interband excitations involving the two n = 1 VP conduction and valence bands. However, it vanishes at small momenta because the eigenstates associated with the VP conduction and valence bands are orthogonal at q = 0. Since only the interband excitations are involved, the particle-hole continuum for VP states is also invariant of µ as long as the chemical potential is in the VP band gap.

As one increases µ above the VP conduction band, the particle-hole continuum of FA excitations remains unchanged due to the linearity of the FA dispersion whereas that of VP states gets heavily modi ed due to Pauli blocking at the conduction band minima. The particle-hole continuum shifts to higher frequency and larger momentum. Most saliently, the intraband excitations of the VP conduction band induce a third particle-hole continuum at low frequencies. With µ > e 0 just above the conduction band minimum, the VP band is approximately parabolic, and its quasi-D character is apparent in the form of the particle-hole intraband spectrum with its typical exclusion dome for q S < 2k F with k F de ned as µ = 2 v 2 k 2 F + e 2 0 . In the q z = 0 limit, two plasmon modes are present for µ < e 0 as shown by red dashed lines in Fig. . . The rst one is the linearly dispersing FA plasmon with a gap at q y = 0, in agreement with theoretical approaches using classical electrodynamics [ ], hydrodynamic description [ ], or quantum-mechanical calculations [ , , , ]. From the zeros of the real part of the equation RPA (q y , ω) = 0, the FA-plasmon dispersion reads

ω ≈ sgn(q y ) k 0 e 2 4π 2 0 r +   1 + 2k F k 2 F + 2 2 S δ n F ,1   vq y ( . )
where 2k 0 = 2 √ 2m∆ 0 is the separation between two Weyl nodes in the bulk, δ i,j is the Kronecker δfunction and n F is the integer part of the ratio between µ and e 0 . If n F = 0 (µ < e 0 ), for positive ω, the FA plasmon is allowed to propagate only in the direction of positive q y , due to the FA being chiral.

For the usual D electron gas or graphene, recall that their plasmon dispersion is square-root due to the Coulomb potential being D [ , , ]. In spite of the the quasi-D nature of the FA, the Coulomb potential remains D here, and one might naively expect a square-root plasmon dispersion. Surprisingly, this is not the case, and one nds a linear gapped plasmon mode thanks to its chiral nature.

As simple as it is, Eq. ( . ) accurately describes the mode found numerically in Fig. . . The result is the same if considering the FA states alone because the in uence from the interband transition between VP states is prohibited for small q y . I emphasize that the experimentally measurable FA plasmon gap,

γ FA = k 0 e 2 4π 2 0 r , ( . )
yields direct information about the separation 2k 0 between the bulk Weyl nodes knowing the dielectric constant of the substrate r .

VP intraband excitations signi cantly modify the dispersion of the FA plasmon when n F = 1

(µ > e 0 ). The plasmon gap sticks to the same value γ FA . The dispersion of FA plasmon at large q y remains almost linear with the same velocity v. At small q y , the FA plasmon acquires an enhanced velocity that can be further boosted by increasing the chemical potential [see Eq. ( .)]. However, the change of velocity for nite q y , as seen in Fig. . , is not captured by Eq. ( . ) since the long wavelength limit is approaching its limit.

The second plasmon mode in Fig. . , is the VP interband plasmon, which stems mostly from the VP interband excitations. It is also gapped and starts at a nite momentum for the same reason that the spectral weight of the particle-hole continuum is vanishingly small at q y ∼ 0, which makes sustained plasmonic oscillations impossible. The interband VP plasmon mode lies in the VP interband particlehole region and is thus Landau-damped. However, since the amplitude of -Im(χ (0) ) drops at high energy, this plasmon may be visible as an additional bump in EELS as shown later.

Here, what I mean large is to approach qy S ∼ 1 while assuming that the long wavelength limit is still valid.

The third plasmon mode emerges when µ > e 0 . This mode exists in a region delimited by the particle-hole continua of the FA and the VP conduction bands. It starts at small but nite momentum and its energy disperses along with the upper boundary of the intraband continuum and eventually gets merged in it at larger momentum. One may naively think that this VP intraband plasmon originates only from intraband excitations and has a square-root dispersion at small momenta (see yellow dashed lines in Fig. . ). However, the numerical calculations invalidate this picture. This is because one needs to take into account the other particle-hole continua, namely the linear one associated with the FA, which prohibits such a square-root dependence of an undamped plasmon (see in the supplementary material of our work [ ]). Moreover, when the chemical potential crosses a VP conduction band, remote VP interband excitations cannot simply be accounted in r because of the diverging density of states. This signi cantly modi es the dynamical screening. Thus, the VP intraband plasmon acquires positive energy only at non-zero nite momentum and disperses linearly with a velocity smaller than v. The terminating point of the plasmon dispersion is indicated by red dots in Fig. . . Increasing µ from 1.01 to 1.3e 0 , the exclusion dome at low frequencies becomes wider. The available phase space for the VP intraband plasmon between the FA and the VP intraband continua is reduced even further making this plasmon less visible at larger values of µ.

E q z = 0
Since the bands are e ectively D, the particle-hole continua are independent of q z if the coupling between FA and VP bands is omitted. It is indeed legitimate to neglect this coupling in the longwavelength limit where it scales as ∼ (q z S ) 2 . As shown in Fig. . , most of the conclusions for the case q z = 0 are applicable for q z = 0 except that the dispersion of FA plasmon gets strongly modi ed for q y < q z due to the q z -dependence of the Coulomb interaction.

Neglecting a small hybridization between the VP bands and the FA state, the FA-plasmon gap at small momenta becomes

γ FA = γ FA q y q 2 y + q 2 z . ( . )
When q z = 0, Eqs. ( . ) and ( . ) coincide, and the FA plasmon is gapped as shown earlier. However, when q z = 0, the gap vanishes at q y = 0 as a consequence of the strong anisotropy of the FA state, which is only quasi-D but embedded in a D manifold. When q y q z , the FA plasmon disperses again linearly with slope v. This is then validated by the numerical calculations shown in Fig. . . The FA plasmon gap vanishes when q = q z e z . This singular behavior of the gap at q = 0 is also reported in the literature [ , , , , , ]. Therefore, one would always measure a gapless plasmon mode stemming from the FA plasmon in experiments since it is di cult to limit q z = 0. Notwithstanding, the FA-plasmon gap can be retrieved by extrapolating the linear dispersion with slope v at q y S ∼ 1 to q y = 0 so that the intercept gives the value of γ FA .

N

To show what one can see in experiments, the electron loss function -[1/ RPA ], directly measurable by EELS, is plotted in Fig. . in the (q y , ω > 0)-plane with intensity indicated by colorbar. One of the intriguing properties of FA is the non-reciprocity of the FA plasmon, re ecting the chiral nature of the FA state. Therefore, one should also study -[1/ RPA ] for (q y < 0, ω > 0). The result for ω < 0 can be easily retrieved by reversing simultaneously the sign of ω and q in known results.

As shown in Fig where µ = 1.01e 0 , the FA plasmon is completely absent when q y < 0 as well as the corresponding particle-hole continuum. Being non-reciprocal, the FA plasmon only propagates in one direction with xed velocity, highly desirable for applications. Strikingly, the VP intraband plasmon is also non-reciprocal even if it involves the k y ↔ -k y symmetry of the VP bands [see Eq. ( .)]: it has a di erent dispersion for q y < 0, which can be calculated analytically

ω ≈ sgn(-q y )v   |q y | + 2k F k 2 F + 2n 2 S q 2 y + q 2 z   . ( . )
Contrary to q y > 0, it starts from the origin of (q y , ω) and disperses with a velocity larger than v which can be enhanced further by increasing µ. This non-reciprocity is a consequence of the hybridization with the FA mode and particle-hole continuum, which is in close vicinity of the intraband VP plasmon for q y > 0 but further well separated in energy for q y < 0 (see in the supplementary material of our

work [ ]). The chirality of the FA modes thus induces non-reciprocity in other excitations due to their mutual coupling. This can also be seen in the VP interband plasmon, where the starting point moves to higher frequency and larger momentum. As anticipated above, the VP interband plasmon is submerged amid the particle-hole continuum but nevertheless visible on EELS.

In this section, the e ect of surface smoothness on the charge oscillation spectrum of a WSM surface has been investigated. Within quasi-D RPA, I have shown the emergence of two collective modes stabilized by the inter-and intra-VP band excitations, in addition to the FA plasmon. The plasmons exhibit anisotropy and non-reciprocity inherited from the underlying surface model. The ndings could be veri ed experimentally in EELS, which could probe the chirality of the FA and o er an alternative way to prove the presence of the VP states. Furthermore, the FA-plasmon gap gives us a direct experimental measure of the separation between the Weyl nodes.

. Summary

S

In this chapter, I have shown all the theoretical results on the spectroscopic properties of VP states that I have obtained during my PhD.

First, I have calculated the optical conductivity of VP states. Playing with the polarization of incident photon and the orientation of magnetic eld, the optical response from VP states has rich features.

First, additional absorption peaks appear below the gap aside from those due to topological states. Furthermore, the shape of peaks, which is intimately determined by the dimensionality-sensitive DOS, are tunable by the orientation of the magnetic eld which modi es the dimensionality of VP bands. In particular, the in-plane magnetic eld and the smoothness quantization conspire to form a composed magnetic eld which keeps the shape of peaks but yields new peaks for photons of certain polarization. Therefore, these speci c features from VP states in the optical conductivity are a smoking gun to identify the existence of VP states in TIs.

C

The important thing is to never stop questioning [or learning].

Albert Einstein

The Dirac equation, originally derived for particle physics, has been the guideline of this thesis whose There are many other research topics that I have conceived for future. One possible branch is to pursue the idea of "doing high-energy physics in low-energy systems". An interesting branch is to ap-ply not only special relativity but also general relativity to materials. There are already some pioneering works on Weyl semimetal going in this direction. Another possible branch is to further investigate the possibilities given by Volkov-Pankratov states. For example, the at bands in twisted bilayer graphene can be interpreted as pseudo-Landau levels [ , ]. In the continuum model for twisted bilayer graphene [ ], the emergence of the at bands is the consequence of the twisted boundary condition, which inherently constitutes a smooth boundary between di erent domains in real space. is the same to all observers at rest in inertial frames of references and the laws of nature are identical in all inertial frames of reference. An important consequence of his statement is that the spatial and temporal dimensions must be interwoven. This four dimensional space where time is merely another dimension just like space is called space-time. A complete introduction on special relativity could be found in [ , ].

A L I rst show a direct consequence of the two assumptions above. Say a ash of light is emitted at t = 0 from the spatial origin in the point-of-view of the frame R. The trajectory of the light signal propagating spherically at the speed of light ( rst axiom) should verify

c 2 t 2 -x 2 -y 2 -z 2 = 0 (A. )
where (t, x, y, z) is the space-time coordinate of the front of the light wave in R. Therefor, the trajectory of the light in another inertial frame R is characterized by the same equation (second axiom) . ) By the fact that the space is homogeneous and isotropic, one can prove that the transformation law between the space-time coordinates of di erent inertial frames of reference is linear known as the Lorentz boost or transformation. For example, if the frame R moves at the speed v < c in the x-direction relative to R, the coordinate (t , x , y , z ) is written as the linear combination of (t, x, y, z) as

c 2 t 2 -x 2 -y 2 -z 2 = 0. ( A
      ct x y z       =       γ -γβ 0 0 -γβ γ 0 0 0 0 1 0 0 0 0 1             ct x y z       (A. ) A M ' L
One can write Maxwell's equations in a form of four-vector (or more precisely, four-tensor). In terms of electric and magnetic eld (E, B), Maxwell's equations are written as

∇ • E = ρ 0 (A. ) ∇ • B = 0 (A. ) ∇ × E = - ∂ ∂t B (A. ) ∇ × B = µ 0 J + µ 0 ∂ ∂t E (A. )
and the electric and magnetic eld is gradient of potential: . ) It turns out that both elds can be gathered in an anti-symmetric four-tensor called the electromagnetic eld tensor: . ) With the help of the electromagnetic eld tensor and other four-vectors given previously, Maxwell's equations are written in a Lorentz covariant form

E = - ∂ ∂t A -∇φ (A. ) B = ∇ × A. ( A
F µν = ∂ µ A ν -∂ ν A µ =       0 -Ex c - Ey c -Ez c Ex c 0 -B z B y Ey c B z 0 -B x Ez c -B y B x 0       . ( A
∂ µ F µν = µ 0 J ν (A. ) ∂ µ µνρσ F ρσ = 0 (A. )
where µναβ is the Levi-Civita anti-symmetric tensor. Using this Lorentz covariant form, one can derive how electric and magnetic eld transforms under the Lorentz transformations. For example, a frame R moves relative to the lab frame R at velocity v, then

E = E , E ⊥ = γ(E ⊥ + v × B) (A. ) B = B , B ⊥ = γ B ⊥ - 1 c 2 v × E . (A. )
The signi cation of this Lorentz covariant form is that Maxwell's equations are exactly the same for every inertial frame of reference using the four-vectors attached to itself. In other words, Maxwell's equations are Lorentz invariant.

B S

The goal of the present appendix is to simplify the generic model ( . ), which I recall below

H(q) = w • q + 3 i,j=1 v ij q i σ j + ∆σ 3 . (B. )
As shown below, one can legitimately use on an isotropic model to discuss the Landau quantization of the generic Hamiltonian ( . ). The anisotropy of velocity is erased by rescaling the wavevectors and the magnetic eld.

B D

The rst step is to nd the principle axes de ned by the eigenvectors of the real symmetric tensor v ij , which can be diagonalized as a matrix

v = Rv * R T (B. )
where R ∈ SO(3) is an orthogonal rotation matrix and the diagonal matrix v * contains the three eigenvalues v * i . Therefore, in the new basis of space de ned by R, the velocity tensor v ij is diagonal with the coe cients v * i and the wavevector q is written as

q * = R T q (B. ) so that 3 i=1 v ij q i = v * j q * j . (B. )

B Simplification of the generic model

The Hamiltonian becomes

H(q * ) = w * • q * + 3 i=1 v * i q * i σ i + ∆σ 3 (B. )
where

w * = R T w. (B. )
In the following, I will use the basis R as the starting Cartesian coordinate system.

B R

Now getting rid of the star * for simplicity and replacing i = 1, 2, 3 by i = x, y, z, the Hamiltonian to be rescaled becomes

H(q) = w • q + 3 i=x,y,z v i q i σ i + ∆σ z . (B. )
After the rescaling vq i = v i q i , w i q i = w i q i and thus . ) the Hamiltonian becomes

w i = w i v v i , ( B
H(q ) = w • q + vq • σ + ∆σ z . (B. )
Note that one is free to choose the value of v which would not alter the specturm of the Hamiltonian.

When a magnetic eld is applied to the system, the Peierls substitution with the

q i → q i + eA i (B. ) becomes q i → q i + eA i with A i = A i v i v (B. )
where the vector potential A is also rescaled by the rescaling of the wavevectors. The magnetic eld B is rescaled by

B k = ijk ∂ i A j → B k = ijk ∂ i A j (B. )
where . ) In this way, I transform an anisotropic model in the presence of a magnetic eld B to an isotropic model in the presence of a rescaled magnetic eld B . The relation between B and B is

∂ i = v i v ∂ i . ( B
B k = v i v j v 2 ijk ∂ i A j = v i v j v 2 B k (B. )
where i, j are two other indices than k and i = j. Note that this derivation is gauge-independent.

B D

One can simplify further the Hamiltonian (B. ) by writing

w • q = w qz (B. )
where the prime is omitted and w is the modulus of w. This de nes a new basis in which q is written as q and they are related by q = T q with T ∈ SO(3). After a unitary transformation de ned by T , the Hamiltonian (B. ) becomes

H(q) = w qz + vq • σ + ∆(cos θσ z + sin θσ x ) (B. )
where the angle θ is also given by T .

Finally, removing the tilde, the simpli ed Hamiltonian reads

H(q) = wq z + vq • σ + ∆(cos θσ z + sin θσ x ) (B. )
or sometimes another form is also useful in practice [see ( . ) where found by mirror and inversion symmetry of the crystal. In our modeling, each nodal line is divided into four parts: ZA, AB, BC and CY segments. Among the four segments, ZA and CY are at while AB and BC are dispersive.

w x = 0] H(q) = w x q x + w z q z + vq • σ + ∆σ z . ( B
Nevertheless, the energy pro le of all the four segments can be described by the Hamiltonian below ( = 1)

H(k) = (k α )I 2 + ∆(k α )σ z + v(ξk β σ x + k γ σ y ) (C. )
where σ x,y,z are Pauli matrices, ξ = ±1 is the valley index and (k α ) is de ned with respect to the chemical potential. In the following, I consider only one line ξ = +1 and the result for the line ξ = -1

H(k) = E A + E B -E A k B -k A (k α -k A )I 2 + ∆ A + ∆ B -∆ A k B -k A (k α -k A ) σ z + v(ξk β σ x + k γ σ y )
where k A/B are the coordinates of the points A/B along the nodal line, respectively. Here, I consider an isotropic model where v β = v γ = v. The anisotropy of the Fermi velocity can be added a posteriori as a tuning parameter in the t of the optical conductivity.

C Gapped nodal line semimetal niobium di-arsenide

C O

In this section, I will give the analytical expression of the optical conductivity in the absence of a magnetic eld. Since the nodal lines consist of the four segments in our model, one can separately and analytically calculate the optical conductivity for each segment and sum them all to get the nal total optical conductivity.

One only needs to calculate [σ ββ (ω)] and [σ αα (ω)] and [σ ββ (ω)] = [σ γγ (ω)] by isotropy.

In the absence of a magnetic eld,

[σ ββ (ω)] = e 2 4π dk α 4∆ 2 + ω 2 8ω 2 [f ( (k α ) -ω/2) -f ( (k α ) + ω/2)] × Θ(ω -2∆) (C. ) [σ αα (ω)] = e 2 4πv 2 dk α ∂∆ ∂k α 2 ω 2 -∆ 2 4ω 2 [f ( (k α ) -ω/2) -f ( (k α ) + ω/2)] × Θ(ω -2∆) (C. )
where f is the Fermi-Dirac distribution and Θ is the Heaviside function. Since the dispersion in ∆(k α )

is small, the contribution from σ αα is negligible in practice. To retrieve the optical conductivity in the SI units, one only has to multiply -1 . In practice, one calculates the optical conductivity only for the half of one of the two nodal lines as shown in Fig. C. and then multiply by a factor of 4 as the consequence of the mirror and inversion symmetries.

In the re ectivity measurements, incident photons are linearly polarized to the a-and b-axis so that σ aa = cos 2 θ a σ αα + sin 2 θ a σ γγ (C. )

σ bb = cos 2 θ b σ αα + sin 2 θ b σ ββ ( C

C S

Suppose that the gapped nodal line is modeled by the following isotropic Hamiltonian Hamiltonian becomes

H(k) = wk α I 2 + ∆σ z + v(k β σ x + k γ σ y ) ( C
H = w[cos θk z -sin θ(k x -eBy)] + v[cos θ(k x -eBy) + sin θk z ]σ x -vk y σ y + ∆σ z . (C. )
The Hamiltonian H can be solved using the hyperbolic transformation M = exp(φσ x /2) with the rapidity β = tanh φ = w tan θ/v. The transformed Hamiltonian reads

H T = M HM = wv v * k z + w 2 + v 2 v * k z sin θ cos θσ x + v * (k x -eBy)σ x -vk y σ y + ∆σ z (C. )
where a new velocity v * is de ned as

v * 2 = v 2 cos 2 θ -w 2 sin 2 θ with γ = 1 1 -β 2 = cosh φ = v cos θ v * (C. )
where γ > 1 is the relativistic Lorentz factor. Note that H T does not have the same spectrum as H.

In the basis of |ψ

T = γ -1/2 M -
E = wk z cos θ + 1 γ ∆ √ 2vv * B a † E √ 2vv * B a E -∆ (C. )
Compared to the Hamiltonian in the main text, the additional negative sign in front of the ky term is to keep the triad x, y, z to be a right-hand system. There is no in uence for the nal conclusions if the two mirror-symmetric lines are simultaneously considered.

where a pair of energy-dependent ladder operators is de ned

a E = - B √ 2vv * v * B (y -y E ) + ivk y a † E = - B √ 2vv * v * B (y -y E ) -ivk y y E = 2 B v * v * k x + k z (w 2 + v 2 ) sin θ cos θ -E w sin θ v * (C. ) with [a E , a † E ] = 1.
The subscript E indicates the dependence on energy E. In particular, the center of cyclotron y E shifts with energy.

Since the energy-dependent term in H E is absorbed in the de nition of the ladder operators, the energy spectrum is self-consistently found in the eigenstates of energy E λ n in the form

|ψ T,n,λ = cos α n,λ |n, E λ n sin α n,λ |n -1, E λ n (C. )
where α n,λ is an angle depending on n and the sign of energy λ = ±; |n , E λ n is the wavefunction of the one-dimensional quantum harmonic oscillator de ned by the previous ladder operators. Given the Landau level index n, n = n or n -1. The energy spectrum is thus

E ± n = wk z cos θ ± 1 γ ∆ 2 + 2nv 2 cos θ γ 2 B for n > 0 E 0 = wk z cos θ + ∆ γ for n = 0 (C. ) C V
To study the magneto-optical properties of NbAs 2 , one evaluates the matrix element ψ n |v k |ψ m .

Indeed, it is more practical with the basis |ψ T,n already in hands by remarking that

ψ n |∇ k H|ψ n = γ ψ T,n |∇ k H T |ψ T,n (C. )
thanks to the fact that M is k-independent. So, in the basis |ψ T , the velocity operators for

H T are vT,x = v * σ x (C. ) vT,y = -vσ y (C. ) vT,z = wv v * + w 2 + v 2 v * sin θ cos θ (C. ) D C D
This appendix shows the details of calculations of how to derive the spectrum of the double Dirac quantum well (QW) modeled in Chapter . In particular, I derive the energy splitting for the topological n = 0 states for sharp and smooth surfaces.

D D D

In this section, I solve Eq. ( . ) for ∆(z) described by Eq. ( . ). In all the following discussions, the subscript L and R mean the left and right Dirac QW. One should not confound L in the subscript with the separation between two adjacent Dirac QWs L.

If z < -L/2 -or z > L/2 + or z ∈ [-L 2 + , L 2 -], the equation reads ∂ 2 z χ λ -K 2 χ λ = 0 (D. )
where

K 2 = (1 -ω 2 r )/ξ 2 . Recall that ξ = v/∆ 0 .
The solutions are a linear combination of exp (Kz) and exp (-Kz).

If z ∈ [-L 2 -, -L 2 + ],
one can carry out a change of variable z+L/2 = αt L and α 2 = ξ/2. The equation then reads

∂ 2 t L χ λ - 1 4 t L 2 + a L,λ χ λ = 0 (D. )
where

a L,λ = - λ 2 - 2ξ ω 2 r . ( D
u S (a L,λ ; t L ) = e -t 2 L 4 M 1 2 a L,λ + 1 4 ; 1 2 ; t 2 L 2 u A (a L,λ ; t L ) = t L e -t 2 L 4 M 1 2 a L,λ + 3 4 ; 3 2 ; t 2 L 2 (D. )
where S and A mean symmetric and anti-symmetric, respectively.

If z ∈ [ L 2 -, L 2
+ ], one can solve the di erential equation and represent the solutions in the similar way. After a change of variable z -L/2 = αt R ,

∂ 2 t R χ λ - 1 4 t R 2 + a R,λ χ λ = 0 (D. ) where a R,λ = λ 2 - 2ξ ω 2 r . (D. )
Similarly, the solutions for Eq. (D. ) are

u S (a R,λ ; t R ) = e -t 2 R 4 M 1 2 a R,λ + 1 4 ; 1 2 ; t 2 R 2 u A (a R,λ ; t R ) = t R e -t 2 R 4 M 1 2 a R,λ + 3 4 ; 3 2 ; t 2 R 2 (D. )
Using the fact that the wavefunction is vanishing at in nity and it is continuous as well as its derivative, one can match the solution in di erent regions at their common point along the z-direction. For simplicity, I note Since λ = ± are equivalent when one considers double Dirac QW, I will omit λ in the following discussion. The nal secular equation reads which cannot be true except when the surface is sharp ( ξ) and the distance between two QWs is large (L , ξ). In fact, when → 0, there are only three domains along the z-direction:

(1 -ω 2 r ) ξ u S,L + v S,L × (1 -ω 2 r ) ξ u A,L + v A,L × (1 -ω 2 r ) ξ u S,R + v S,R × (1 -ω 2 r ) ξ u A,R + v A,R = e -2 √ 1-ω 2 r ξ (L-2 ) (1 -ω 2 r ) ξ u S,R u A,R -v S,R v A,R (
z < -L/2, z ∈] -L/2, L/2[ and z > L/2. So, there are only two continuity relations for four coe cients, which means two degenerate solutions for ω r = 0. Another interesting value for ω r is ω r = 1. One can check that ω r = 1 is always a solution of Eq. (D. ) for any parameters. So, one also retrieves automatically the bulk spectrum, E = ± 2 v 2 k 2 + ∆ 2 0 , within the model.

D E

I consider rst a sharp surface when ξ and derive a formula to evaluate the mass gap of the chiral mode. To do so, one can develop Eq. (D. ) in terms /ξ and suppose in the rst approximation that ω r is at most of same order of /ξ. After some algebra, one has

2∆E = 2∆ 0 e -L ξ 1 + 4 2 3ξ 2 . (D. )
Next, I calculate the energy splitting for the n = 0 state for a smooth surface ( > ξ) to the rstorder in perturbation theory. As explained in the main text, the leading order of perturbation to open a gap is given by the process that the chiral state is weakly a ected by the deviation of the potential U λ (z) from ∆ 2 0 in the exponential tail. For notational simplicity, = 1. The temperature is set to be zero and the chemical potential is at the charge neutrality point.

E W D E I

The Hamiltonian to study is

H T = v k y √ 2 S â † √ 2 S â -k y (E. )
The eigenstates have the shape of

|ψ λ n = a λ 1,n |n a λ 2,n |n -1 if n ≥ 1 |ψ 0 = a 1,0 |0 0 if n = 0 (E. )
with λ = ±. In this basis, the Hamiltonian becomes

ψ 0 = 1 0 if n = 0 (E. ) E I
The Hamiltonian is

H T = v k y cos θ √ 2 T â † √ 2 T â -k y cos θ (E. )
where the de nition of T is given by ( . ). The eigenstates have the form of

|ψ λ n = a λ 1,n |n -1 a λ 2,n |n if n ≥ 1 |ψ 0 = 0 a 2,0 |0 if n = 0 (E. )
The Hamiltonian in this basis reads

v 2 (a + 1,m a - 2,n δ m,n-1 + a + 2,m a - 1,n δ m-1,n ) 2 δ ω -(E + m -E - n ) = m,n≥1 (n,-),(m,+) v 2 (cos α m 2 cos α n 2 δ m,n-1 -sin α m 2 sin α n 2 δ m-1,n ) 2 δ ω -(E + m -E - n ) = m≥1 v 2 cos 2 α m 2 cos 2 α m+1 2 δ ω -(E + m + E + m+1 ) + n≥1 v 2 sin 2 α n 2 sin 2 α n+1 2 δ ω -(E + n + E + n+1 ) =
dk y 2π (f D (E λ n ) -f D (E λ m ))| ψ λ m |v y |ψ λ n | 2 × δ ω -(E λ m -E λ n ) = πe 2 ω n∈N * +∞ -∞ dk y 2π | ψ + n |v y |ψ - n | 2 δ(ω -2E + n ) (E. )
The matrix element reads

| ψ + n |v y |ψ - n | 2 = v 2 (2 cos

Henri Poincaré

C'est l'une des découvertes récentes les plus spectaculaires dans le domaine de la physique de la matière condensée : l'équation de Dirac relativiste pour des particules sans masse ou ses variantes peuvent décrire la physique autour du niveau de Fermi de matériaux réels tels que le graphène, les semimétaux de Dirac/Weyl, les semi-métaux de lignes nodales, etc. Ces matériaux sont appelés matériaux de Dirac dans lesquels le traitement mathématique des électrons est parfaitement identique à celui des particules élémentaires relativistes de la physique des hautes énergies.

L'application de l'équation de Dirac aux matériaux n'est pas seulement béné que pour la physique de la matière condensée, mais féconde aussi la physique des hautes énergies. Grâce à l'interaction entre les électrons et leur réseau cristallin sous-jacent, l'équation de Dirac, même ultrarelativiste, émerge de manière inattendue dans la description à basse énergie des matériaux de Dirac. Des expériences fondamentales de la physique des particules, qui ne peuvent pas être réalisées en raison des conditions expérimentales inaccessibles ou des coûts astronomiques, deviennent alors concevables dans des matériaux dans des conditions plus pratiques et économiques. Ainsi des phénomènes qui n'ont pas encore été découverts en physique des hautes énergies, comme la supersymétrie, peuvent être réalisés en physique de la matière condensée. En outre, l'hamiltonien de basse énergie des matériaux de Dirac n'est pas limité par la symétrie de Lorentz contrairement aux particules élémentaires. L'imagination des théoriciens est ainsi davantage libérée. On peut légitimement considérer des variantes de l'équation originale de Dirac sans que celles-ci correspondent à des particules élémentaires existantes. Cependant, rien n'empêche leur apparition en physique de la matière condensée.

Réciproquement, la communauté de la physique de la matière condensée pro te du formalisme

  would like to express my sincerest gratitude to my Ph.D. advisor, Dr. Mark-Oliver Goerbig,

  a nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of my work is just playing with equations and seeing what they give.

  Dirac equation. Described by the Dirac instead of Schrödinger equation, Dirac electrons must have peculiar transport and spectroscopic properties di erent from the usual Schrödinger ones. Especially, the arsenal of theoretical tools to study them are already well-established thanks to huge e orts made by high-energy physicists. Working in condensed matter theory, high-energy formalism refreshes the understanding of Dirac materials. The discovery of Dirac materials accompanies also the development of the revolutionary concept of topology encrypted in wavefunctions. For long time since the proposal of Bloch's theorem [ ],

  argument [ ] or by the Aharonov-Casher argument [ ]. D * The historical aspect on the Dirac equation is entirely credited to Austin Kcon Cheng. I learned this fascinating history totally from his thesis [ ] and rephrase it in this section.

  . ) where i = 1, 2, 3 and I 2 is the two-by-two identity matrix. The Weyl basis has the advantage that the four-component spinor is divided into two decoupled two-component spinors with opposite handedness or chirality. The four-by-four Dirac Hamiltonian decomposes into two two-by-two Weyl Hamiltonians Ĥλ = λ cσ • p ( . ) where σ = (σ 1 , σ 2 , σ 3 ) and λ is the chirality of the particle [ ]. The corresponding equation is called the Weyl equation [ ].

  . ) where ξ = ± indicates two inequivalent valleys. The low-energy spectrum is known as the Dirac cones (see Fig. . ). A full derivation of this Hamiltonian from the tight-binding model is given in Chapter . Here, one retrieves the D massless Dirac Hamiltonian in which the two sublattices of graphene's honeycomb lattice play the role of the pseudo-spin. The energy degeneracy of the band touching points is simultaneously protected by the inversion and time-reversal symmetries. For example, graphene is easily gapped by introducing a staggered potential on two sublattices whose material realization is the hexagonal boron nitride h-BN with the Hamiltonian [ ] H = v(ξk x σ x + k y σ y ) + ∆σ z .

  Fig. . for its energy spectrum. The massless version of the D tilted Dirac Hamiltonian can be realized in 8 -P mmn borophene [ ], quinoid-type graphene [ , ], the organic conductor α -(BEDT TTF) 2 I 3 [ ]. Therefore, the physics in Dirac materials are even richer than the original Dirac Hamiltonian. There exist of course many other phenomena having a condensed matter representation, which are well-known in quantum eld theory such as parity anomaly [ ] and chiral anomaly [ ].

Figure . :

 . Figure . : Energy spectra of several types of Dirac materials. For D materials, the third momentum is set to be zero. The blue plane represents the chemical potential which I choose to intersect with the conduction bands. The shape of the Fermi surface is depicted by the red dashed lines.

  Weyl nodes (multiple of four) near the Fermi surface [ ]. The examples of Dirac semimetals are Cd 3 As 2 [ ] and Na 3 Bi [ ]. In spite of the fact the time-reversal broken magnetic WSMs has the simplest theoretical modeling [ ], the material realization has not been recognized yet. Possible candidates would be pyrochlore iridates [ ].

  Fig. . . As I have shown in this section, without the constraint of the Lorentz symmetry, a wide range of the variations of the Dirac Hamiltonian is unleashed in the context of condensed matter. The freedom of adding new terms in the Dirac Hamiltonian have blazoned the Dirac physics in materials. In Chapter , I will talk about the Landau quantization in most of Dirac materials mentioned above. In particular, the spectroscopic properties of the gapped nodal line semimetal NbAs 2 are discussed in detail in Chapter . R Nowadays, many relativistic phenomena have already been observed in Dirac materials via methods that are well-known in the condensed matter community. Since the Hamiltonian of graphene matches perfectly the original D massless Dirac Hamiltonian, it is the rst material lab of high-energy experiments. One of such examples is the Klein tunneling paradox according to which relativistic massless

Figure

  Figure . : D illustration of the trivial and Möbius strip. The ant living on the strips is represented by the blue arrow. The trajectories by the ant are represented by the red lines: dashed lines if the trajectory is hidden behind the strip and solid lines if not. When the ant travels for one loop on the trivial strip, it coincides with itself. However, when it travels for one loop on the Möbius strip, it nds itself on the opposite side of the strip.

  the paper by Berry [ ]. They noticed that the velocity in ferromagnets acquires an additional term later know as anomalous velocity, which is the origin of the anomalous Hall e ect [ ]. Nevertheless, a true understanding of the topology in band theory should wait until the discovery of the integer quantum Hall e ect by von Kliltzing et al. in [ ] and its topological interpretation by Thouless et al. in [ ]. The integer quantum Hall e ect is also seen as the paradigm of the bulk-edge correspondence of topological phases. The integer quantum Hall e ect is that the Hall conductivity of a D electron gas in the presence of a magnetic eld is quantized into integer times of the conductivity quantum e 2 /h. As shown by Thouless et al. in [ ], the integer quantum Hall e ect is present when the chemical potential is between Landau levels so that the Hall conductivity can be written as is the Chern number for the band n. This formula yields a topological invariant known as T.K.N.N. invariant for D integer quantum Hall phases. As the manifestation of the bulk-edge correspondence, the number of the conducting edge channels are determined by the topology of the bulk system characterized by the topological invariant. The emergence of conducting edge states in an insulating bulk phase does not require necessarily a magnetic eld, as shown by Haldane in [ ]. One only requires that the time-reversal symmetry is broken in an insulator known as quantum anomalous Hall insulator or Haldane insulator [ ]. The Hamiltonian of the Haldane insulator built from a honeycomb lattice reads H = v(k x τ z σ x + k y σ y ) + ∆τ z σ z ( . ) where the third Pauli matrix τ z is the valley degrees of freedom. The time-reversal symmetry is broken by a valley-resolved mass and the total Chern number C of lled bands equals thus to at the charge neutral point. Most saliently, C does not change unless the gap is closed. This re ects the fact that the topological properties are robust to continuous deformation without violent events, which are the gap closing events in this context. Since the vacuum can be seen as a trivial insulator with large gap, the associated Chern number is zero. Therefore, the gap must be closed at the interface between the Haldane insulator and vacuum because the Chern number has to change its value from one to zero while moving out of the Haldane insulator into the vacuum. Gapless conducting edge states emerge precisely at the position where the gap is closed. The Haldane insulator possesses one conducting channel at its edge (C = 1) and the Hall conductivity in the absence of a magnetic eld is e 2 /h, yet another example of the bulk-edge correspondence. As long as the gap of the Haldane insulator is not closed, the edge state is present. Here comes the notion of the protection by the gap. Note that the above argument of the emergence of edge states is general and independent how to model the edge. "You cannot get rid of the surface state. You take a hammer, you hit the sample, and the surface state still remains." The robustness of edge states is precisely the reason why topological materials attract so much attention recently. T D Although the general topological argument dictates already the presence of robust edge states at the surface of topological insulators, it gives no information how it looks like. As rst proposed by Jackiw and Rebbi [ ], one can use the Dirac equation with a spatially variant mass gap to explicitly show the presence of edge states. This approach was rst used in the Su-Schrie er-Heeger model for polyacetylene [ ].

  [ ]. In their work, they showed an universal presence of zero mode for the D Dirac Hamiltonian in the presence of a magnetic eld. More details on the smooth topological junction and additional surface states can be found in Chapter and . O The present thesis is thus organized as follows. In the rst part (Chapter and ), I elaborate how the relativistic renormalization arises in the electrodynamics of Dirac materials. I also interpret the relativistic phenomena using the language of condensed matter at the end of Chapter . In particular, I show in Chapter how to reveal the relativistic renormalization by magneto-optics in a gapped nodal line semimetal NbAs 2 . In the second part of the thesis (Chapter and ), I discuss Volkov-Pankratov states in smooth topological heterojunctions. I give various explications on the origin of Volkov-Pankratov states in Chapter . In particular, I am interested in the spectroscopic properties of these states shown in Chapter , which o er a smoking gun for their identi cation by experiments. Throughout the thesis, the Dirac equation appears everywhere. It is THE equation of my PhD. For those who are interested in topics mentioned in the present chapter, I suggest the following references on • Berry phase [ , , ],

  . ) This is the Dirac Hamiltonian for a massive fermion living in the 2 + 1 space-time dimensions. L The carbon atoms of graphene are arranged in a honeycomb-like two-dimensional lattice. Each unitcell has two atoms which form two hexagonal Bravais lattice, i.e., sublattice A and B. In a given unitcell, A = (0, 0) and B = (0, -d) [see Fig. . (a)]. The Bravais lattice vectors are:

Figure

  Figure . : (a) Honeycomb lattice of graphene. (b) Spectrum of the tight-binding model ( . ) with a zoom-in at one of the K points to show the linear Dirac cone structure at low energy. Figures adapted from [ ].

H

  is the annihilation (creation) operator of an electron in a p z orbital situated in the unitcell at a Bravais lattice point of R, t is the hopping amplitude, and H.c. stands for the Hermitian conjugate taking into account the reversed hopping. By the lattice Fourier transformation, we can write the Hamiltonian in reciprocal space in the basis of {c k,A , c k,B } Fig. . (b), the two bands of H(k) touch each other at two inequivalent points in the rst Brillouin Zone dubbed K and K points:

Figure . :

 . Figure . : Schematics of Landau levels for graphene and hBN. The zero energy is marked by blue dashed lines in the plots for hBN. For illustration purpose only, I use a rather strong magnetic eld so that the Landau level spacing is even larger than the band gap of hBN (∼ 6 eV).

  One can derive hyperbolic transformation from calculating how Lorentz boost changes the eigenstates to keep Dirac equation Lorentz invariant [ ].

  generically in the low-energy theory of Dirac materials such as Dirac and Weyl semimetals. Unlike graphene or relativistic particles in high-energy physics, most low-energy Hamiltonians of Dirac materials are anisotropic, meaning that the Fermi velocity depends on crystal axis. Some acquire also a mass term. The following generic Hamiltonian can cover all the Dirac materials I treated during my Ph.D. studies:

T

  Apart from the tilting, one also adds a gap to graphene's two-dimensional Hamiltonian. This model describes the low energy bands of an organic compound α -(BEDT-TTF) 2 I 3 . The Landau quantization in this material has been discussed bySári et al. [ ].

Figure

  Figure . : (a) Schematic of the spectrum for D tilted gapped graphene at k y = 0 in the absence of magnetic eld. The direct band gap marked by red is 2∆ while the indirect band gap by green is 2∆ = 2∆/γ. (b) Landau levels of D tilted gapped graphene where I generalize the de nition of n to include the sign of energy.

  . ) where γ = 1/ 1 -β 2 and β = -w/v [see Fig. . (b)].

[

  see Fig. . (a)].

S

  In this chapter, I have shown by several examples that the renormalization e ect of relativistic electrodynamics can manifest itself in Dirac materials. Using unitary and hyperbolic transformations inspired by the analogy to the special relativity, the Landau quantization in tilted Dirac materials is completed solved. Traditional experimental techniques in condensed matter can thus demonstrate the relativistic renormalization. For example, the gap renormalization is measurable by magneto-optics as I will show in detail in the next chapter.

  Linear response theory is developed through the Kubo formula [ ]. It gives the expression of the retarded correlation function (conductivity σ) that relates the perturbation Hamiltonian H to the observable of interest (current density j). I am interested in the conductivity at nite frequencies, the Fourier transform of which is also called optical or dynamical conductivity. The full derivation of the Kubo formula for σ can be found in [ ].

  the mathematical formulation of Pauli blocking: only the transitions from an occupied to an empty state are possible. Thus, optical transitions are sensitive to the position of the chemical potential. . The term n |v i |n is the same matrix element as in Fermi's golden rule remarking that v = p/m 0 . It encodes the symmetry properties of the wavefunction and thus it stipulates the con-

  Fig. . in which the optical conductivities for di erent dispersion and dimensions are shown. Note that the optical conductivity for parabolic bands in D and D is not divergent in the vicinity of ω = E g thanks to the regular functional form of the DOS in D and D for parabolic bands.

Figure . :

 . Figure . : Illustration on how the dimension of bands a ects the optical conductivity for linear and parabolic bands. Figure reprinted from [ ].

Figure . :

 . Figure . : Schematic to illustrate the magnetic Faraday rotation and ellipticity, magnetic Kerr rotation and ellipticity. Magnetic eld or magnetization of materials is in the z-direction. Figure reprinted from [ ].

L

  Intense magnetic elds induce Landau quantization of the electronic bands in D, i.e., quantization into separated levels indexed by n as elaborately shown in Chapter for graphene. With a xed magnetic eld, the optical conductivity has the shape of a series of peaks and each peak corresponds to one possible transition between two Landau levels. The plot of the evolution of these peaks in the optical conductivity upon variation of the photon energy and the magnetic eld is called Landau fan diagram. An example for graphene is shown in [ ]. The trajectory of peaks depicts the energy of Landau levels as a function of Landau level index and the magnetic eld. Suppose the magnetic eld is applied in the z-direction, the selection rules between Landau levels are [ , ]: Polarization Selection rules circular: clockwise propagating in the z-direction n → n + 1 circular: anti-clockwise propagating in the z-direction n → n -1 linear: in the x, y-directions n → n ± 1 linear: in the z-direction n → n Somewhat surprisingly, the selection rules are the same for parabolic and linear bands. This is because the selection rules inherently originate from the symmetry of the Landau level wavefunctions. In other words, it is a consequence of the conservation of angular momentum. The latter point of view can be elaborated by incorporating the magnetic eld though the vector potential in the symmetric gauge. Besides the index n which indicates Landau levels of di erent energy, another quantum number m emerges and counts the angular momentum L z of the eigenstates if the rotation symmetry is present. However, n and m are not decoupled. In a vertical transition, ∆m = ∆n. So, a circular polarized photon carrying an angular momentum + (clockwise) can only excite an electron from an initial state of angular momentum m to a nal state of angular momentum m + 1 by the conservation of angular momentum. The same reasoning is applied for an anticlockwise circularly polarized photon Bands are quantized in Landau bands in D. If one uses the ordering rule of Landau levels, the obtained selection rules are di erent. In certain literature [ ], people name the zeroth Landau level in my convention n = 1.

Figure . :

 . Figure . : Schematic of Faraday and Voigt geometry for a transverse electromagnetic wave propagating in the z-direction.

Figure . :

 . Figure . : Energy pro le of NbAs 2 tracing one of the two nodal lines. For incident photons of energy around meV, only two parts are optically relevant as indicated in the gure. Figure credited to my collaborator, Jan Wyzula, of the LNCMI in Grenoble.

Figure . :

 . Figure . : Coupling strength between the eigenstates |ψ T,n,λ and |ψ T,m,λ where I consider only transitions from valence (λ = -1) to conduction bands (λ = +1). (a) When θ = 0, the usual selection rules n → n ± 1 are given. (b) When θ = π/3, many other transitions are also allowed.

  oscillation in φ loc ) induced by a vanishingly small external potential [see ( . )]. In Chapter , I will use the RPA to study surface plasmons arsing from Volkov-Pankratov states as well as their spectroscopic signatures.

Figure . :

 . Figure . : Illustration of the self-consistent approximations behind the RPA. ρ ind is the average value of induced charge density. Since all other quantum uctuations are neglected by taking the mean value, the RPA is indeed a mean-eld theory.

  Indeed, they have been shown to arise not only in TIs[ , , , ], but also in surfaces of Weyl semimetals (WSMs)[ , ], topological graphene nanoribbons [ ] and topological superconduc-Topological heterojunction: emergence of Volkov-Pankratov states tors [ ]. Here, I only call the massive surface states other than the topological ones the VP states to underline the particularities of the massive states. The experimental realization of a topological heterojunction could be done by smoothly varying the chemical composition of material in space. For example, one can tune a DTI Bi 2 Se 3 to a trivial one In 2 Se 3 by continuously substituting Bi by In [ ]. In this chapter, I will rst present the three theoretical models I will constantly use in this chapter and the next one: a D toy model, a model for a realistic DTI and a model for a two-node WSM. The smoothness of the surface (or interface) is incorporated in Dirac Hamiltonian by a gap parameter that varies over a certain width across the interface. I will show that the exact functional form of ∆(x) is unimportant and can be linearized if one is interested in localized states in the vicinity of surface. Then, based on my work during my Ph. D., I will show by explicit calculations the emergence of VP states and topological states from di erent perspectives adopting certain well-chosen functional forms of the gap parameter to facilitate the calculations. Each perspective gives a conceptually di erent understanding on the origin of VP states. Furthermore, these perspectives shed light on the properties of VP states. T Since the smooth (inter)surface of a topological material acquires a nite size, it may be more appropriate to call it a smooth junction connecting topological material and vacuum, which can be seen as a trivial insulator with large band gap. The two phases being topologically di erent, such junction is called a topological heterojunction [ ]. More generally, a topological heterojunction is a smooth interface between two materials characterized by di erent values of a topological invariant. In this section, I present three models of topological heterojunctions. T One of the simplest models of topological heterojunctions that one can propose consists of an interface between a trivial Semeno [ ] and a topological Haldane insulator [ ], both stemming from the lowenergy spinless model of graphene. They display both a band structure with two massive Dirac points, in other words two valleys. Suppose the half-space x < 0 is lled by a Semeno insulator and the other half-space x > 0 by a Haldane insulator. The gap changes only its sign in one of the two valleys when passing across the topological heterojunction [ ]. Say, the sign change occurs at the K point. The Dirac-type Hamiltonian for the valley K reads

  [x, k x ] = i. One already anticipates the quantization of the D bands into D bands. Though this D toy model is simplistic, the present picture can however be generalized to the Bernevig-Hughes-Zhang model of D time-reversal-symmetric TI [ ] with spin degree of freedom, s z = ±. One just replaces the valley index by s z and considers the model near the Γ point retaining only linear terms. T A D TI can be described within several di erent Hamiltonians. For example, Bi 2 Se 3 [ ], known as a prototype material of D Z 2 TIs, can be described by a low-energy model thanks to Zhang et al. [ ].

Figure . :

 . Figure . : Schematics of two gap functions ∆(x): one is described by a tangent hyperbolic function (blue line) and another vanishes seven times across the interface (red line). Nevertheless, the neighboring gap closing points (green dots) can be continuously merged together for annihilation so that only one is left (red dot). If one focus on the physics of the interface, one can zoom in in the vicinity of the (last) gap closing point and suppose a linear ∆(x) by introducing the size of interface .

Figure . :

 . Figure . : Spectrum for the D (left) and D (right) model including the chiral mode (red) the modes n = 1...4 (orange). The energy is measured in units of ∆ 0 , i.e. half of the bulk gap. The modes n = 3 and n = 4 are immersed in the purple area which represents the bulk spectrum. So, they cannot give visible signal in experiments.

Figure

  Figure . : (a) Illustration of a single Dirac QW. (b) Illustration of a double Dirac QW. The spatially varying gap ∆(z) changes from positive sign to negative when one goes from the blue area (trivial phase) to the red area (topological phase) and the gap is closed somewhere in between. l, l 1 and l 2 characterize the smoothness of domain wall between phases. Figures adapted from the reference [ ].

  . ) which itself depends on the chirality λ. One therefore has to deal e ectively with the conventional quantum mechanical problem of a particle in a D quantum well, which I call henceforth Dirac quantum well (QW) [ ]. A topological heterojunction is equivalent to two decoupled single Dirac QWs with opposite chirality λ [see Fig. . (a)].

  .

  (a)]. Within this choice of gap parameter, one considers integrally a topological heterojunction consisting two semi-in nite bulk phases and their smooth interface. This choice of ∆(z) is by no means the best to describe the real experiments. But, it is the one that allows a simple analytical solution of the problem of a single Dirac QW. It is also the building block of the model for double Dirac QW. Most importantly, the functional behavior for all the parameters is independent of the choice of a ∆(z) verifying general features discussed above. The spectrum is thus quantitatively relevant (at least in the order of magnitude) compared to real situation in experiments.

Figure

  Figure . : (a) Interface pro les described by a spatially varying gap ∆(z) for two values of characteristic interface width /ξ = 0.9, 2.3. (b) Pro les of Dirac QWs for its corresponding ∆(z) and chirality λ = ±. U -is represented by solid lines and U + by dashed lines. Figures adapted from the reference [ ].

  Fig. . for two di erent values of the smoothness parameter /ξ. Remember that I consider an odd function ∆(z), and ∆(z) 2 and ∂ z ∆(z) are consequently even functions so that U λ (z) is a symmetric QW potential [see Fig. . (b)]. When the interface is abrupt ( /ξ 1), only the D QW for the fermions with λ = -is really con ning [see the solid orange line in Fig. . (b)]. In contrast, fermions with chirality λ = + cannot be con ned in the region z ∈ [-, ] because they can tunnel out of z ∈ [-, ] where the potential is no longer con ning [see the dashed orange lines in Fig. . (b)

  Fig. . (b). For a well-de ned nite size sample, one should impose 2 < L. The gap parameter ∆(z) and its corresponding Schrödinger potential U λ (z) are given in Fig. . . Since the Schrödinger potential is invariant under simultaneous inversion of the chirality and spatial coordinate z, i.e, U λ (z) = U -λ (-z), U + hosts a zero mode of chirality λ = + only at the right QW and U -hosts that of chirality λ = -only at the left QW. However, since the gap parameter has the same sign at in nity, the Jackiw-Rebbi argument given in Chapter

Figure

  Figure . : (a) Pro le of the spatially varying gap ∆(z) for a DTI sandwiched between two trivial insulators with /ξ = 1 and L/ξ = 4. (b) Pro les of two adjacent Dirac QWs for two chiralities. The two dashed lines, blue and red, indicate the energy level (close to zero) of the n = 0 states of λ = ±, respectively. Figures adapted from the reference [ ].

  ( . )]. For n ≥ 1, ω r = |∆E 0 n (k = 0) ± ∆E n |/∆ 0 [see Eq. ( . )]. I de ne also ∆ω r xed, the energy splitting of the topological state and the VP states depends on the thickness of DTI L is shown in Fig. . [see solid lines in (a) and blue line in (b)]. The mass gap of the n = 0 state decays exponentially with L as for sharp surfaces. The splitting of the VP states also is exponentially small with increasing L/ξ showing no particular non-intuitive behavior. Also, the energy splitting of the n = 0 state is 10 2 times weaker than that of the massive VP states with given parameters. See in App. D.

Figure . :

 . Figure . : = 6ξ is xed. (a) Reduced energy ω r as a function of the distance between two Dirac QWs, L, for the massive VP states n = 1 and 2. The results are obtained by solving the secular equation given in App. D. . Inset in (a): the reduced energy splitting ∆ω r decays exponentially with increasing L, only shown for the massive VP states n = 1 and . The solid lines indicate the splitting from the secular equation, while the crosses represent the results based on Eq. ( . ). The dotted lines show results based on the same formula, where one has used the exact energies for the VP states of a single Dirac QW instead of the approximate ones given in Eq. ( . ). (b) log 10 ω r as a function of L/ξ for the n = 0 topological state. The numerical results from the secular equation are represented by blue line, that using Eq. ( . ) by orange line. Figures adapted from the reference [ ].

Fig. . .

 . Fig. . . In the range of /ξ with L = 20ξ xed, the splitting of the topological state is 10 4 times smaller than that of the VP states. The mass gap of the topological state is increasingly enhanced by the smoothness of the surface. The increase is likely to be exponential [see blue line in Fig. . (b)]. This observation is consistent with the intuition that the e ective separation of two Dirac QWs is L -2 therefore giving an exponential behavior in /ξ. However, this argument cannot explain the non-trivial behavior of the energy splitting of the VP states [see solid lines in Fig. . (a)]. For the n = 1, 2 and 3, the energy splitting is not a monotonically increasing function of /ξ. Most saliently, it admits a minimum at di erent /ξ for each n. Larger values of n yield larger /ξ as the position of minimal splitting. When n = 4, the dependence on /ξ of the energy splitting becomes again monotonic. One can explain the observations in the framework of the asymmetric double QW for a given chirality as shown in Fig. . by a thought experiment. Consider initially a thick sample of DTI so that the localized states in two complementary QWs feel little each other. Remember that all the energies in terms of Dirac QW has the dimension of the square of a physical energy and thus are called virtual

Figure . :

 . Figure . : Reduced energy ω r as a function of smoothness for the massive VP states n = 1 to 4 by solving the secular equation (solid lines). The -dependence of ω r is well described by Eq. ( . ) (crosses). Inset: zoom-in to show the splitting for the VP state n = 1. Figures adapted from the reference [ ].

Figure

  Figure . : L = 20ξ is xed. (a) Reduced energy splitting ∆ω r as a function of smoothness for the massive VP states n = 1 to . The solid lines show the splitting numerically obtained from the secular equation and the crosses indicate the values obtained from the heuristic formula ( . ). The dotted lines show results based on the same formula, where one has used the exact energies for the VP states of a single Dirac QW instead of the approximate ones given in Eq. ( . ) (b)log 10 ω r vs. for the n = 0 state and L/ξ = 20. The results from the secular equation are represented by blue line, that using Eq. ( . ) by orange lines and that using the rst-order perturbation theory by green line. Figures adapted from the reference [ ].

Figure . :

 . Figure . : Pro les of two in nitely separated Dirac QWs for two chiralities with /ξ = 3: (a) for λ = + and (b) for λ = -. The dashed lines, blue and red, indicate the energy level of chiral states and n = 1 VP states for λ = ±, respectively. Figures adapted from the reference [ ].

  . ) which would describe the dependence of the mass gap for the topological state both on the smoothness /ξ and the thickness L/ξ. As shown in Figs. . (b) and . (b), the mass gap scales approximately as exp(-(L -)/ξ) instead of exp(

Figure . :

 . Figure . : Illustration of a topological superlattice and its corresponding a series of periodic Dirac QWs. The parameters for the plot of U λ are: L NI /ξ = 5, L TI /ξ = 4 and /ξ = 1.5.

  Fig. . , is conceptually intuitive and natural. One can use the knowledge of double Dirac QW to estimate the spectrum of topological superlattice. If a topological superlattice is periodic and composed of many layers, Bloch's theorem is applied and the spectrum acquires also a dispersion in the z-direction. The problem is thus converted to a D tight-binding model formed by Dirac QWs.

  quantum con nement e ect which is explicitly revealed by the Dirac QW. Squaring the Dirac Hamiltonian yields two decoupled Schrödinger equations for an e ective quantum well given by the chirality-dependent potential U λ (z). The motion parallel to the surface remains plane-wave-like by translation symmetry so that the problem to solve becomes a textbook D Schrödinger problem in wave mechanics. Once one has solved the secular equation derived from imposing boundary conditions on the wavefunction, one can retrieve the spectrum of the original Dirac Hamiltonian using ( . ). A single interface in the form of a topological heterojunction is physically equivalent to a single Dirac QW. Most saliently, the framework of Dirac QW is practical to treat the tunneling e ect between two complementary sides of a nite-size topological material and of topological superlattice.

Figure

  Figure . : (a) Spectrum of the tight-binding model of a Chern insulator on a square lattice with /ξ = 4 and L/ξ = 20. (b) Zoomed-in spectrum for the massive VP states n = 1 (left) and n = 2 (right).

  Two edges are set to have the same smoothness /ξ = 4 and the thickness of the system is L/ξ = 20. The spectrum is shown in Fig. . (a). Two topological states from di erent edges cross at zero energy. Besides the topological states, only two massive VP states appear in the bulk gap. Their energy is consistent with the results shown in Fig. . . The continuum model is thus validated by the lattice model. Especially, it shows that the speci c form of ∆(z) is not important as long as one can linearize it at the band inversion point. In Fig. . (b), the splitting of the massive VP states n = 1 and n = 2 is evaluated. The numerical simulations shows a splitting of 0.001∆ 0 for n = 1 and 0.004∆ 0 for n = 2. Compared to Fig. . (a),

Finally

  , I corroborate the existence of the VP states by a lattice model from which the VP states descends from the bulk spectrum as a leakage of spectral ow from bulk to surface. Since this spectral ow can be continuously deformed into the bulk continuum, the VP states are not topological as already shown in the other continuum approaches. My results on the double Dirac QW have been published in be productive in various ways. One is having the ability to plan and carry out experiments, but the other is having the ability to formulate new ideas, which can be about what experiments can be carried out . . . by making [the] proper calculations. Individual scientists who are successful in their work are successful for different reasons.Linus PaulingIn this chapter, I will present my studies on the spectroscopic properties of Volkov-Pankratov (VP) states. Two reasons pushed me to conduct my research in this direction. On the one hand, the spectroscopic properties are fundamentally appealing, because they allow one to understand the nature of VP states by comparing directly with those of the topological surface states. On the other hand, the calculated physical quantities related to spectroscopic properties are experimentally relevant. Every result of theoretical calculations has in principle its corresponding experiment to observe it. Sometimes, the spectroscopic study opens a route to new potential applications in optics and plasmonics.Three theoretical tools are used in this chapter to study the spectroscopic properties of VP states.One is to calculate the optical conductivity of the VP states of topological heterojunctions in D and D. Then, I give a succinct description on the Faraday and Kerr rotation through a thin lm of D topological insulator (TI) in the presence of VP states. Furthermore, I discuss surface plasmon modes in Weyl semimetals (WSMs) stemming from VP states that are measurable by electron energy loss spectroscopy (EELS). This is done by calculating the dielectric function in the random phase approximations (RPA).This chapter is thus organized as follows. First, I will show the optical conductivity of a single topological heterojunction modeled by the Hamiltonians ( . ) and ( . ) for D and D, respectively, both in the absence and presence of a real magnetic eld. Most saliently, the speci c features in the optical response from VP states provide a smoking gun for their identi cation. Then, I will brie y Similar calculations have also been performed for WSMs within our group [ ].mention possible Faraday and Kerr e ects for a thin lm of DTI subject to an out-of-plane magnetic eld. Finally, I will show how previously unknown plasmon modes arise on the surface of WSMs. The particular band structure of VP states in WSMs yields additional plasmon modes other than the one from the Fermi-arc (FA) states. of the universal presence of VP states on the smooth surface of topological material motivates an experimental hunting for these states. In the high mobility strained heterostructure HgTe/CdHgTe ( DTI), our collaborators at LPENS (Paris) [ ] observed additional conductivity peaks and compressibility bumps in the bulk gap besides the ones attributed to the topological massless states [see Fig.. (a)]. However, the results from transport measurement are only a partial proof for the existence of VP states. Strictly speaking, it only proves the existence of surface states other than the topological one. Surface states can generally arise at the interface of semiconductor heterostructures[ , ]. Due to the mismatch of the valence and conduction bands between two semiconductors, the bulk states of the one with smaller gap induce surface states in the gap of the other semiconductor with larger gap. Another possible related mechanism is due to surface charges induced by the formation of defects on the surface [ ]. Surface charges bend the bulk bands resulting in a con ning potential at the surface and thus surface subbands. This e ect was observed already in Si [ ]. These could lead to false-alarms for the identi cation of VP states since the in-gap subbands also has a dispersion in the direction perpendicular to the surface so that not every spectroscopic method, for example, angle-resolved photoemission spectroscopy (ARPES) cannot provide an unambiguous proof [ , , ] [see Fig. . (b)].

Figure

  Figure . : (a) Quantum capacitance and conductance data (blue-red data points in the heterostructure HgTe/CdHgTe as a function of the surface electronic density. The latter is varied with the help of a backgate. VP states manifest as additional signals that cannot be attributed to the topological massless states. (b) ARPES data on the surface of Bi 2 Se 3 several hours after cleavage reveal a D massive electron gas in addition to the topological D Dirac cone. Reminiscent of VP states formed due to a smoothening of the surface during the aging process, the data is also well tted by surface states induced by surface charges. Two gures adapted from the references [ , ], respectively.

  the joint density of states (JDOS), which determines the shape of the optical conductivity σ by the thumb rule Hamiltonian H T ( . ) is in practice derived by k • p perturbation theory [ ]. One should replace the ket |ψ λ n by its cell-periodic part |u λ n . In the basis of |u λ n and the velocity operator (see Chapter ) thus reads

  Fig. . (a)], except the rst step-like peak, all the peaks represent singularities in the clean limit because

Figure

  Figure . : (a) [σ xx ] and (b) [σ yy ] for the D model, in the absence (blue) and presence (orange) of a magnetic eld, measured in units of σ 0 = e 2 /h, while the excitation energy is represented in units of ∆ 0 . The sharp peaks are singularities and their corresponding optical transitions are labeled above. The dashed perpendicular line denotes the onset of absorption by bulk states, for ω ≥ 2∆ 0 . The peaks in the purple shaded area are immersed in the bulk spectrum and thus cannot be seen experimentally.

  is slower. Similar to [σ xx ], [σ yy ] [see blue lines in Fig. . (b)] has the same shape of peaks placed at di erent frequencies because of the di erent selection rules. Since optical transitions involving the massless state are forbidden by the selection rule n → n, all the peaks are singular for the D diverging JDOS of the VP bands. While the diverging peaks at high frequency are immersed in the bulk continuum represented by the purple shaded area, the peaks in the gap are supposed to be observable. Especially, thanks to the polarization-dependent selection rules, one can continuously measure the optical conductivity, for example from [σ xx ] to [σ yy ], by rotating the linear polarization of photon. The selection rules are a manifestation of the pseudo-magnetic eld interpretation of the origin of VP states, unlike any other defect-induced trivial in-gap surface states. Therefore, this can serve as a smoking gun for the unequivocal identi cation of VP states.

  the n = 0 state represents precisely the famed single Dirac cone on the surface of a DTI and the n ≥ 1 states are the massive VP states which are twice spin-degenerate. The velocity operators are vz = vτ y , vx = -vτ z σ y and vy = vτ z σ x . ( . ) Since τ y is o -diagonal and τ z diagonal, the selection rules for the polarization in the z-direction are di erent from those for the polarization in the x, y-directions as shown below Polarization along z: n → n ± 1 Polarization along x, y: n → n. The selection rules in the absence of a magnetic eld in the DTI model are similar to the D model where n → n ± 1 for the polarization perpendicular to the interface and n → n for that parallel to the interface. The selection rules for the polarization in the x-and y-directions must be identical due to the rotational symmetry of the Hamiltonian around the z-axis. Unlike the D toy model, the polarizationdependent selection rules cannot t in any con guration of a real magnetic eld, neither Faraday nor Voigt, since the two polarizations have the n → n rule. It again shows the intrinsic di erence between smoothness and magnetic eld.

Fig. . shows

  Fig. . shows the optical conductivity in the absence of a magnetic eld for di erent polarizations. The singularities in the D toy model become nite step-like peaks in the DTI model. This is due to the change in the dimensionality of the surface bands. Their D character now yields an essentially at JDOS and thus steps in the conductivity that decays as ω -1 . In [σ zz ] [see Fig. . (c)], the response from the 0 → 1 and -1 → 0 transition has di erent forms from others due to the DOS of the

Figure

  Figure . : (a) [σ xx ], (b) [σ yy ] and (c) [σ zz ] for the DTI model, in the absence (blue) and presence (orange) of a parallel magnetic eld given B = 1.5 S , measured by unity of σ 0 = e 2 /h and excitation energy by the half bulk gap. The corresponding optical transitions of each peak are labeled above. The blue and orange peaks in the light-purple shaded area, where the bulk responses are plotted as purple dot-dashed line, are immersed in the bulk responses and thus cannot be seen experimentally.

  in terms of the unitary transformation T = exp(-iθτ y σ x /2) that adds a new Pauli matrix in vy . Indeed, the velocity operators now read vz = vτ y , vx = -vτ z σ y and vy = v cos θτ z σ x + v sin θτ x . ( . ) As shown in Fig. . by orange lines, the enhanced level spacing yields a shift of the the optical conductivity to higher energy. The plateau in [σ xx ] [see Fig. . (a)] before the rst peak is also enhanced in a magnetic eld with

  admixes a σ zz component to [σ yy ] [see Fig. . (b)]. As in the D toy model, a parallel magnetic eld does not change radically the shape of the optical conductivity but mixes the selection rules and shifts the peaks.

  Fig. . . One does not have to depict [σ yy ],

Figure . :

 . Figure . : Di erent groups of signal of (a) [σ xx ] and (b) [σ zz ] for D model in the presence of a perpendicular magnetic eld given B = 2.15 S , measured by unity of σ 0 = e 2 h and excitation energy by half of the bulk gap ∆ 0 . I make Dirac distributions visible by replacing them by Lorentzian functions with width 0.001 which can be seen as broadening e ects by nite temperature or disorders. Di erent groups of peaks are colored and their corresponding optical transitions are labeled. The bulk signals are plotted as purple dot-dashed line. The surface signal which are immersed in the bulk spectrum cannot be seen experimentally.

  the JDOS is essentially D. Comparing the contribution of bulk-surface transitions and that of surface-surface ones in Fig. . , the step-like feature from D surface states are smeared out by integrating k z , a parameter for D bulk states [see Eq. ( . )]. Thus, the bulk-surface contribution is a continued function of energy but with kinks with no jumps (see orange lines in Fig. . ).

Figure . :

 . Figure . : Di erent contributions in (a) [σ xx ] and (b) [σ zz ] for D model for = 5ξ: surface-surface (blue), bulk-surface (orange) and total contribution (green), measured by unity of σ 0 = e 2 h and excitation energy by the half bulk gap. The bulk signals are plotted as purple dot-dashed line. The signal which are immersed in the bulk spectrum cannot be seen experimentally.

  . ) where ν T,B are the lling factor of the top and bottom surface and α = 1/137 is the ne structure constant. The details of the derivation can be found in [ ]. Note that ν T,B can be di erent if two surfaces di erently doped or gated. Recent experiments have demonstrated the possibility of controlling separately the chemical potential of two surfaces of DTIs [ ].

  As shown by Eq. ( . ), Landau levels are indexed by two quantum numbers n and m associated with the smoothness and the magnetic eld, respectively. So, the lling factor ν counts at the same time n and m. An interesting situation arises when B S . Tuning continuously the chemical potential from the charge neutral point to positive values, one would nd increments of the Faraday angle θ F given only by n when the chemical potential is still smaller than v/ B . This is yet another possible way to corroborate the existence of VP states.Note that time-reversal symmetry can be also broken by an exchange eld, i.e., a Zeeman gap. The single Dirac cone would open a gap and the spin degeneracy of VP states is lifted. Since all the states are now non-degenerate D massive Dirac fermions which are known to have / Chern number [ ],

D

  dispersive energy spectrum of WSMs is quantized to be D, exactly like what a true magnetic eld would do. One can identify here a Weyl cone chirality-resolved pseudo-magnetic eld in the y-direction with B p = ±2∆/ev e y [ ]. This is further con rmed by analyzing the selection rules [ ]. The selection rules for light polarized along the y-axis are n → n (Voigt geometry) while those for light polarized along the x, z-axes are n → n ± 1 (Faraday geometry).

  . )where λ indicating valence (λ = -) or conduction (λ = +) band; u n,λ (k y ) and v n,λ (k y ) are two k y -dependent spinor coe cients; |n, k z are the eigenstates of quantum harmonic oscillator de ned by â † (k z ) and â(k z ). I explicitly mark k z in the notations to emphasize the dependence on k z in the ladder operators and the spinor components |n, k z . The n = 0 state is precisely the famed FA state, which is chiral also in the sense that one of the spinor components is zero. The n ≥ 1 states are the VP states emerging on the smooth surface of WSMs.

Figure . :

 . Figure . : Comparison of density of states of the VP states with that of the bulk. The dotted vertical line denotes the onset of the bulk continuum. Figure adapted from [ ].

  Compared to the spectrum of a topological heterojunction modeled by DTI [see Eq. ( . )], although the eigenstates live on a two-dimensional manifold (k y , k z ), their energies disperse only in the k y -direction, i.e., the direction perpendicular to the interface and the line connecting the two Weyl nodes at k z = ± √ 2m∆ 0 in the reciprocal space. The bands are thus quasi-D and the DOS of the VP states diverges at the band extremum (see Fig..). The dependence on k z is only encoded in the wavefunctions, more precisely the cyclotron center of |n, k z . Indeed, this strong anisotropy in the energy dispersion induces the peculiar properties of plasmon due to the edge states as shown immediately.

P

  Now I investigate in detail the simpli ed three-band model where I consider the two n = 1 VP bands in addition to the chiral state (n = 0). The zero of the real part of RPA in the (q, ω) phase space determines the dispersion relation of the plasmon. All the analytical results for the three-band model are reported in the supplementary material of our work [ ] in which we show that the analytical calculations result in a complicated transcendental equation even in the long wavelength limit. The main di culty comes from the system being multi-band, leading to some features that one cannot account for by considering separately the FA and VP states. So, I numerically do the calculations to nd the plasmon dispersion and analyze the results to interpret them with the help of analytical expression. Because of the log-divergence at the plasma edge in the dynamical polarization, the numerical evaluation of the analytical expression requires a ne mesh in the vicinity of the position of divergence, which requires long execution time and large amount of memory. The trick is to do a linear interpolation in the integral that keeps the ngerprint of the log-divergence and largely accelerates the calculations [ ]. The details on analytical calculations are shown in the supplementary material of our work [ ].

  ( . ), is shown in Fig. . . The imaginary part of χ (0) bears important information about possible electronic excitations and therefore Laudau damping region for the plasmon modes. The dispersion of plasmon is indicated by the red dashed lines (zeros of the real part of RPA ). The plasmon modes are only longlived and undamped in the black regions where Im(χ (0) ) = 0. Within the three-band model, one obtains three particle-hole continua, with -Im(χ (0) ) = 0, shown in Fig. . .

Figure . :

 . Figure . : Pro le of the imaginary part of the non-interacting dynamical polarization -Im(χ (0) ) in the (q y , ω) phase space at µ = 0.01e 0 , 0.90e 0 , 1.01e 0 and 1.30e 0 for q z = 0. The zeros of the real part of RPA (red dashed lines) indicate the plasmon modes. The xed parameters are chosen to be v = 1 eV • Å, = 10 Å, ∆ = 1 eV and k 0 = 10 Å -1 .

Figure . :

 . Figure . : Pro le of the imaginary part of the non-interacting dynamical polarization -Im(χ (0) ) in the (q y , ω) phase space. The parameters are kept to be the same as Fig. . except for q z S = 0.2. The zeros of the real part of RPA (red dashed lines) indicate the plasmon modes.

Figure . :

 . Figure . : Electron loss function at µ = 1.01e 0 for q y S ∈ [-2.0, 2.0]: (a) for q z = 0 and (b) for q z S = 0.2. Yellow dashed lines show the symmetric VP intraband plasmon in the absence of the FA state. Three plasmon modes are all non-reciprocal and visible.

  topic belongs to condensed matter theory. The bene t of the application of the Dirac equation in condensed matter is mutual. Thanks to the interplay between lattice potential and electrons, the Dirac equation, even the ultra-relativistic one, emerges unexpectedly in the low-energy description of Dirac materials. Many theoretically predicted high-energy phenomena, which cannot be tested in particle physics experiments due to inaccessible experimental conditions and astronomical costs, are conceivable in materials with convenient tunability and a ordable costs. Even those which have not yet been discovered in high-energy physics, such as supersymmetry [ ], nd their representation in condensed matter physics. Most saliently, the low-energy Hamiltonian of Dirac materials is not restricted by the Lorentz symmetry in contrast to elementary particles. The imagination of theorists is thus further freed. One can legitimately consider the variations of the original Dirac equation with no associated elementary particles that exist in nature. However, they may exist in condensed matter physics.Reciprocally, the condensed matter community bene ts the formalism that are well established in the context of particle physics. It o ers a rejuvenating and complementary understanding for the electronic properties of materials. Various unusual observations in Dirac materials, which cannot be explained by the Schrödinger equation, are elegantly explained by the Dirac equation and its inherent symmetries. The application of the formalism further boosts the discovery of various Dirac materials.Most saliently, it is one of the momenta to give birth to topological band theory, a ground-breaking progress in solid-state physics. Many surprising properties of matter such as the bulk-edge correspondence in topological materials are also captured by the Dirac equation. Therefore, my thesis has been also unfolded in these two aspects. In the rst part, I have shown how Einstein's theory of special relativity applies on the electrodynamics in Dirac materials. In Chapter , I have reviewed previous works on the magnetic-eld-induced relativistic renormalization in graphene subject to crossed electric eld and magnetic eld. Based on this simple example, I have introduced theConclusions and perspectivesunitary and hyperbolic transformations which can be further extended to other Dirac materials such as Weyl semimetals, tilted gapped graphene and gapped nodal line semimetals. While previous works have been concerned with the gapless system, my contribution is to reveal the gap renormalization in the gapped system. Furthermore, I also provide an interpretation of the relativistic renormalization by the language of condensed matter physics, which completes the understanding of the phenomena of interest.In Chapter where I have included a short introduction to theoretical approaches to spectroscopy, I have elaborated the theoretical study on the magneto-optics a gapped nodal line semimetal, NbAs 2 . This is a project in collaboration with my colleagues of the LNCMI in Grenoble who have made industrious e orts and conducted wonderful magneto-optical measurements on NbAs 2 . The most exciting moment of this project has come recently when the gap renormalization in NbAs 2 was clearly observed by our collaborators and the results t extremely well with our theory.In the second part, I have shown that the Dirac equation is a unifying theoretical framework to study the surface states of topological materials. Especially, additional massive Volkov-Pankratov states emerge in smooth topological heterojunctions. In Chapter , I have o ered di erent perspectives to understand the origin of Volkov-Pankratov states. I have started by showing a point of view given by previous work by Sergueï Tchoumakov, who initiated the work on surface states in our group during his thesis, that the smoothness plays the role of pseudo-magnetic eld quantifying surface bands to Landau bands. Then, I have challenged this perspective by the dimensionality reduction, which has inspired me to interpret the origin of Volkov-Pankratov states in terms of quantum well physics denoted as Dirac quantum well. One of the advantages of this complementary perspective is to permit me to discuss the tunneling between Volkov-Pankratov states of two sides of topological materials in an intuitive way. Especially, the topological protection of topological states is interpreted as the absence of resonant tunneling. To be complete, I have also shared the point of view of mathematician-physicist who may treat the problem by identifying a di erential equation in supersymmetry. Finally, a tightbinding approach has been given to further con rm the presence of Volkov-Pankratov states.In Chapter , I have discussed the spectroscopy properties of Volkov-Pankratov states using the techniques presented in Chapter . The key signature in (magneto-)optical conductivity revealing the presence of Volkov-Pankratov states has been shown to be a smoking gun to distinguish them from trivial surface states. I have also studied plasmonics on the surface of Weyl semimetal. In addition to the well-know Fermi-arc plasmon, Volkov-Pankratov states yield inter-and intra-band plasmons.Unexpectedly, the hybridization between the Fermi-arc and Volkov-Pankratov plasmons attributes the non-reciprocity to Volkov-Pankratov plasmons, which are supposed to be reciprocal.

  It seems thus plausible to re-interpret the at bands in terms of Volkov-Pankratov states, which are generalizable to other twisted systems. Here, I have discussed only the spectroscopic properties. The transport properties of Volkov-Pankratov states are worth to be further investigated. Finally, I have given rst hints on the promising physics induced by electron-electron interactions by looking at the physics of plasmon. Nevertheless, the interaction e ects in the presence of Volkov-Pankratov states remain also unclear and wait for future investigations. A S In , Einstein postulated in his seminal work on the theory of special relativity that the speed of light

  Fig. C. , the unit-cell has four copies of NbAs 2 . By convention in crystallography, the b-axis is de ned as the axis that is perpendicular to the two other axis, a and c, between which the angle β is . °[ ]. This is also the convention used in [ ]. The lattice parameters are [ ]:

Figure

  Figure C. : Crystal structure of NbAs 2 : B and B represent As atoms and A is Nb atom. Picture reprinted from [ ].

Figure

  Figure C. : (A) Brillouin zone of NbAs 2 expanded by k a , k b and k c vectors; (B) Band structure of NbAs 2 calculated by density functional theory (DFT) with and without spin-orbit coupling (SOC). Figures reprinted from [ ].

Figure

  Figure C. : Schematic illustration of the position of the nodal lines in k-space. The upper panel shows the position of the nodal lines with respect to the real space crystal axes a, b, c and their corresponding reciprocal space axes k a , k b , k c . The lower panel shows the local axes along the nodal line: e α is parallel to the line and e β,γ is orthogonal to the line. In the rst approximation, e β is parallel to the b-axis and e α,γ is orthogonal to the b-axis.

Figure

  Figure C. : Energy pro le along the nodal line after tting. k 0 indicates the increment following the nodal line in k-space. The subscript ± indicates the two nodal lines according to their position in the the coordinate system (k x , k y , k z ). The positions of Z ± and Y ± are given by [ ].

  . ) where θ a,b is the angle between e α and the a, b-axes, respectively. The angles θ a,b are determined by the position of A, B, C in reciprocal space (see Fig. C. ).To take into account the anisotropy of the Fermi velocity, one can add a multiplicative constant r such that σ ββ = rσ γγ . The ration r depends on the segments of the nodal line. For example, r AB = r BC .C FAfter tting, the position of the nodal line in reciprocal space is shown below.

Figure

  Figure C. : Position of the nodal line in k-space after tting. k x,y,z are given in Fig. C.

Figure

  Figure C. : Optical conductivities measured by experiments σ 1,aa/bb and those by analytical calculations [σ aa/bb ].

  . ) where w < v and the subscripts α, β, γ indicate the local triad shown in Fig. C. . C Gapped nodal line semimetal niobium di-arsenide C L Since Eq. (C. ) is isotropic in the β, γ-plane, one sets with no loss of generality that B = B(cos θe α + sin θe γ ) = Be z for 0 < θ < π/2. Here, a new basis is de ned (see the lower panel of Fig. C. ) {e x , e y , e z } = {cos θe γ -sin θe α , -e β , cos θe α + sin θe γ }. (C. ) Writing the Hamiltonian in this new basis and choosing the Landau gauge A = -Bye x , the

  . ) D Calculations for double Dirac quantum well Eq. (D. ) is the standard form of the Weber di erential equation whose solution is parabolic cylinder function [ , ]. By concern for symmetry of the wavefunction, I represent the solution in terms of con uent hypergeometric function M (a; b; z) [ , ]. The even and odd solutions read

u

  S/A,L/R,λ = u S/A a L/R,λ ; 2 ξ v S/A,L/R,λ = ∂ ∂t L/R u S/A a L/R,λ ; t L/R )

  1 -ω 2 r ) ξ u S,L u A,L -v S,L v A,L (D. )One can try several particular solution to check the validity of the model. Suppose now ω r = 0 which it is in principle impossible to be a solution for nite L and non-zero . Eq. (D. ) would become 2ξ e -2(L-2 ) ξ (. . . )(. . . ) = 0 (D. )

  Consider, for example, the chirality λ = +, in which the n = 0 state is located in the left QW. The wavefunction is centered at z = -L/2 of formχ 0 + ∼ e -(z+L/2) 2 /2ξ for |z + L/2| < e -|z+L/2|/ξ for |z + L/2| > , (D. )This wavefunction represents the exact zero-energy state when the QW potential is constant when z > -L/2 + so that U + (z > -L/2 + ) = ∆ 2 0 , i.e., when there is no second QW. The other QW at z = L/2 therefore gives rise to a deviation ∆U + (z) in energy of the zero mode can be calculated as is now L -1.5 . One can easily remark that this formula captures the exponential decay of E 0 as a function of L/ξ. In the other hand, the formula (D. ), though limited at the rst order of perturbation, gives a rather good approximation to the result by the secular equation Eq. (D. ) especially when /ξ is not too large [green line in Fig.. (b)]. The reason for the discrepancy is that higher order contributions in perturbation theory are non-negligible when L e becomes smaller. In a tunneling point of view, since the energy spacing between the n = 1 VP states and the chiral state is a decreasing function of /ξ, the hybridization between them is thus stronger with increasing /ξ. , I show the details of calculations of the optical conductivity of the D toy model given in Chapter in the absence and the presence of a magnetic eld. The idea is to show how the machinery of the Kubo formula ( . ) works in a concrete example. The calculations for D heterojunctions have been exactly performed in the same way as shown in detail in the supplementary material of [ ].

  vx = ∇ kx H T = vσ x (E. ) vy = ∇ ky H T = vσ z (vx = ∇ kx H T = vσ x (E. ) vy = ∇ ky H T = v cos θ σ z -v sin θ σ y (E. )The matrix elements then have the form ofψ λ m |v x |ψ λ n = v(a λ 1,m a λ 2,n δ m-1,n + a λ 2,m a λ 1,n δ m,n-1 ) (E. ) ψ λ m |v y |ψ λ n = v[cos θ(a λ 1,m a λ 1,n -a λ 2,m a λ 2,n )δ m,n + i sin θ(a λ 1,m a λ 2,n δ m-1,n + a λ 2,m a λ 1,n δ m,n-1 )] an even function of ω, one can thus suppose ω > 0 and focus on the absorption (E λ m > E λ n ). Since the temperature T = 0, the Fermi-Dirac distribution f D is the Heaviside function. The sum can be divided into three parts:

×S

  n cos α n+1 + 1)δ ω -Using the identity δ(f (x)) = i δ(x-x i ) |f (x i )| where the x i are the simple roots of the function f (x), one can do the integrals. I note heref n (r) = ωr n,+ )| = |f n (r n,-)| = vr n,+ ifand only if the photon energy can overcome the gap. Finally, one has n cos α n+1 + 1)δ(k y -r n,+ ) + δ(k y -r n,-) |f n (r n,+ )| n (r n,+ )| [(cos α n (r n,+ ) cos α n+1 (r n,+ ) + 1) + (cos α n (r n,-) cos α n+1 (r n,-) + 1)] × Θ ωn (r n,+ )| (cos α n (r n,+ ) cos α n+1 (r n,+ ) + 1)Similar calculations lead to the two other contributions to the total optical conductivity.a 2,0 δ m,-1 + a + 2,m a 1,0 δ m-1,0 ) 2 × δ ω -(E + my -r 0,+ ) + δ(k y -r 0,-) |f 0 y -r 0,+ ) + δ(k y -r 0,-) |f 0 (r 0,+ )| × Θ ω -The nal expression for [σ xx ] becomes [σ xx (ω > 0)] = e 2 v 2 2ωn∈N (cos α n (r n,+ ) cos α n+1 (r n,+ ) + 1)

TS

  r n,+ )| = |g n (r n,-)| = 2v 1n |v y |ψ - n | 2 δ(ω -2E + n ) = +∞ -∞ dk y 2π v 2 sin 2 α n δ(k y -r n,+ ) + δ(k y -r n,-) |g n (r n,+ )| × Θ ω -The Kubo formula has exactly the same form as (E. ). Similarly, one can divide the sum into three parts:δ m-1,n + a + 2,m a - 1,n δ m,n-1 ) 2 δ(ω -(E + mω 4 -4(2n + 1) 2 T ω 2 v 2 + 4v 4 (E. )and|p n (r n,+ )| = |p n (r n,-)| =vr n,+ a change of variable k y cos θ → k y in the integral, the integral has exactly the same form by a multiplicative factor as that without a magnetic eld (E. ). However, one has to use a n (r n,+ ) cos α n+1 (r n,+ ) + 1) 1/ cos θ is the consequence of the change of variable k y cos θ → k y in the integral.E P yThe matrix element reads| ψ λ m |v y |ψ λ n | 2 = v 2 [cos 2 θ(a λ 1,m a λ 1,n -a λ 2,m a λ 2,n ) 2 δ m,n + sin 2 θ(a λ 1,m a λ 2,n δ m-1,n + a λ 2,m a λ 1,n δ m,n-1 ) 2 ].(E. )The second term in the matrix element yields σ xx with a prefactor sin 2 θ in σ yy . Thus, one only needs to focus on the rst term in the matrix element which gives rise to a contribution in the 2 θ sin 2 β n δ(ω-2E + n ) Denote q n (r) = ω -2E + n = ω -2v r 2 r n,+ )| = |q n (r n,-)| = 2v 1the same way as before. One can easily retrieve [σ yy ] in the presence of a magnetic eld using the expression in the absence of a magnetic eld after the change of variable k y cos θ → k y . Compared to (E. ), the contribution of the rst term is 2 θ sin 2 β n δ(ω -2E + n ) = cos 2 θ × e 2 v 2 ω cos θ n∈N * sin 2 α n (r n,+ ) real part of σ yy reads [σ yy (ω > 0)] = e 2 v2 sin 2 θ 2ω cos θ n∈N (cos α n (r n,+ ) cos α n+1 (r n,+ ) + 1) (ω > 0)] = sin 2 θ × [σ xx (ω > 0La pensée n'est qu'un éclair au milieu d'une longue nuit, mais c'est cet éclair qui est tout.

  mathématique déjà établi en physique des particules. Il o re une compréhension rajeunissante et complémentaire des propriétés électroniques des matériaux. Diverses observations inhabituelles dans les matériaux de Dirac, qui ne peuvent pas être expliquées dans le cadre de l'équation de Schrödinger, trou-vent une explication naturelle et concise par l'équation de Dirac et ses symétries inhérentes. L'application du formalisme stimule davantage la découverte de divers matériaux de Dirac. En particulier, c'est l'une des forces motrices pour donner naissance à la théorie des bandes topologiques, un progrès révolutionnaire en physique du solide. De nombreuses propriétés surprenantes, par exemple la correspondance volume-bord dans les matériaux topologiques, sont également capturées par l'équation de Dirac.Ces deux aspects de l'équation de Dirac -l'apparition de nouvelles particules et la correspondance volume-bord dans des matériaux topologiques -constituent l'objet de cette thèse dont l'équation de Dirac est donc le l rouge. Dans la première partie, je montre comment la théorie de la relativité restreinte s'applique à l'électrodynamique dans les matériaux de Dirac. Dans le chapitre après une introduction à l'aspect historique de l'équation de Dirac, je passe en revue les travaux antérieurs sur la renormalisation relativiste dans le graphène soumis à un champ électrique et un champ magnétique croisés [ ]. Sur la base de cet exemple simple, j'introduis les transformations unitaires et hyperboliques qui se généralisent à d'autres matériaux de Dirac tels que les semi-métaux de Weyl, le graphène avec gap et inclinaison et les semi-métaux de lignes nodales dispersives avec gap. Alors que des travaux antérieurs se sont intéressés au système sans gap, cette thèse se concentre sur la renormalisation des énergies dans le système avec gap. On trouve en utilisant correctement le formalisme mathématique que le gap du système semble renormalisé en présence d'un champ magnétique par le facteur de Lorentz, γ, qui est déterminé par l'inclinaison dans le hamiltonien. Paradoxalement, le gap est une des propriétés intrinsèques du matériau qui ne devrait pas être modi ée par une perturbation extérieure faible. Surtout, dans certains semi-métaux de lignes nodales dispersives avec gap, la renormalisation du gap dépend de l'orientation du champ magnétique par rapport aux lignes nodales. Autrement dit, le gap du système est contrôlable par le champ magnétique. Pour comprendre ce paradoxe, j'explique la renormalisation relativiste dans le langage de la physique de la matière condensée, ce qui complète la compréhension des phénomènes d'intérêt. D'une part, quand on applique un champ magnétique, le spectre du système est quanti é sous la forme des niveaux de Landau. Ces derniers peuvent être retrouvés semiclassiquement en utilisant la relation d'Onsager qui relie la quanti cation en orbites des cyclotrons à la surface de Fermi. La renormalisation des énergies est donc interprétée comme la retouche de la forme de la surface de Fermi par le terme de l'inclinaison dans le hamiltonien des matériaux de Dirac. D'autre part, la renormalisation du gap est en e et le résultat de l'existence des deux types de gap, direct et indirect, induite par l'inclinaison. Par exemple, pour un cône de Dirac D gappé et incliné [voir Fig. (a)], le gap direct est 2∆ mais la séparation minimale entre les bandes de valence et de conductions est 2∆/γ, i.e., le gap indirect. Quand le champ magnétique est présent, les bandes deviennent les niveaux de Landau. Après avoir extrapolé le champ magnétique à zéro, le gap extrait est donc le gap indirect, d'où la renormalisation du gap [voir Fig. (b)]. Expérimentalement, le gap direct est sondé par la spec-troscopie infra-rouge sans champ magnétique. En revanche, le gap indirect n'est que mesurable par la spectroscopie magnéto-optique, d'où la motivation de l'étude des propriétés spectroscopiques.

Figure

  Figure : (a) Spectre du graphène gappé et incliné pour k y = 0 sans champ magnétique. Le gap direct marqué en rouge est 2∆ et le gap indirect en vert est 2∆ = 2∆/γ. (b) Niveaux de Landau du graphène gappé et incliné.

Figure : i

 : Figure : Conductivité optique de la surface douce d'un isolant topologique D en présence d'un champ magnétique perpendiculaire à la surface.

  These results have been published in Europhysics Letters [ ]. I have also suggested that interesting phenomena in Faraday and Kerr rotations could occur in the presence of VP states. Finally, I have given a rst taste of how Coulomb interactions could enrich the physics of VP states by showing multiple plasmon modes in WSMs. This work has been recently published in Physical Review B [ ]. All these spectroscopic properties of VP states would inspire both fundamental and applied research on VP states in the future.

where β and γ called Lorentz factor are de ned

Generally, one can write

where the Λ matrix represents the Lorentz transformation in the four-dimensional vector space.

The four-dimensional vector space is a pseudo-Euclidean space called the Minkowski space-time because the Minkowski inner product is not positive-de nite. The Minkowski inner product is de ned as

where u, v are two four-vectors and η = Diag(1, -1, -1, -1). As the rotations are isometries of three-dimensional Euclidean space that preserve the Euclidean scalar product, the Lorentz transformations are isometries of the Minkowski four-dimensional space-time that preserve the Minkowski inner product. Note that rotations acting on the three spatial dimensions of four-vector are also isometries Practically, all the quantities that can be written as a total contraction of upper and low indices such as u αβγ v αβγ are invariant under the Lorentz transformations. Such quantities are called Lorentz invariant. For example, c 2 t 2 -r 2 for a given event of coordinate (ct, r) in the frame R is independent of the choice of frames. Depending its sign, the event is time-like if positive, space-like if negative and light-like if zero.

Pursuing the logic in this four-dimensional vector space, other physical quantities can also be arranged in four-vector form. The physical quantities that can be expressed in four-vector, say in group theory, furnish a four-vector representation of the Lorentz group and transforms by Lorentz transformations when changing the frame of reference. Remarkably, these physical quantities follows the Lorentz transformation while changing the frame of reference: coordinates of di erent frame are reηµν = η µν = Diag(1, -1, -1, -1)

A. Maxwell's equations are Lorentz invariant lated to each other by a linear transformation Λ. I list a set of physical quantities, that will be useful in the following, in the contravariant form:

r µ = (ct, r) four-position (A. )

)

, p four-momentum (A. )

, k four-wavevector (A. )

J µ = (ρc, J ) four-current (A. )

where µ = {0, 1, 2, 3} in which index 0 corresponds to the temporal dimension and and indices 1, 2, 3 are the spatial ones. . S. Zhang, X. Lu and J. Liu, "Correlated insulators, density wave states, and their nonlinear optical response in magic-angle twisted bilayer graphene", arXiv: .

List of Symbols