Ansgar Graf

Frédéric Piéchon

Berry Curvature and Quantum Metric in N -band Systems -an Eigenprojector Approach

The eigenvalues of a parameter-dependent Hamiltonian matrix form a band structure in parameter space. In such N -band systems, the quantum geometric tensor (QGT), consisting of the Berry curvature and quantum metric tensors, is usually computed from numerically obtained energy eigenstates. Here, an alternative approach to the QGT based on eigenprojectors and (generalized) Bloch vectors is exposed. It offers more analytical insight than the eigenstate approach. In particular, the full QGT of each band can be obtained without computing eigenstates, using only the Hamiltonian matrix and the respective band energy. Most saliently, the well-known two-band formula for the Berry curvature in terms of the Hamiltonian vector is generalized to arbitrary N . The formalism is illustrated using three-and four-band multifold fermion models that have very different geometrical and topological properties despite an identical band structure. From a broader perspective, the methodology used in this work can be applied to compute any physical quantity or to study the quantum dynamics of any observable without the explicit construction of energy eigenstates.

I. INTRODUCTION

In a quantum mechanical system characterized by an N -dimensional Hilbert space, the N × N (Hermitian) Hamiltonian matrix H(x) frequently depends on a set of parameters x = (x 1 , x 2 , ...). In condensed matter physics, such parameters may take the form of a crystal momentum, the intensity of an external field, an applied strain or any mean-field order parameter. A Hamiltonian of this kind has (real) eigenvalues E α (x), and eigenstates |ψ α (x) , where α ∈ {1, ..., N }. The eigenvalues form a band structure, and one may thus speak of an N -band system, or equivalently a (parametric) SU(N ) system.

In this paper, we will be concerned with the quantum geometric properties of N -band systems. More specifically, we will focus on a fundamental quantum geometric object, the (Abelian) quantum geometric tensor (QGT)

[1-3] T α,ij (x) ≡ g α,ij (x) - i 2 Ω α,ij (x). (1)
Here, the quantum metric (tensor) [START_REF] Provost | Riemannian structure on manifolds of quantum states[END_REF] g α,ij (x) = Re T α,ij (x) is symmetric in the indices i, j, and the Berry curvature (tensor) [START_REF] Berry | Quantal Phase Factors Accompanying Adiabatic Changes[END_REF][START_REF] Simon | the Quantum Adiabatic Theorem, and Berry's Phase[END_REF]] Ω α,ij (x) = -2 Im T α,ij (x) is antisymmetric.

The concept of a Berry curvature has a long history. It popped up early on, for example quite explicitly in Blount's work [START_REF] Blount | Formalisms of Band Theory[END_REF], and implicitly in studies on geometric phases [START_REF] Ehrenberg | The Refractive Index in Electron Optics and the Principles of Dynamics, Proceedings of the Physical Society[END_REF][START_REF] Aharonov | Significance of Electromagnetic Potentials in the Quantum Theory[END_REF][START_REF] Pancharatnam | Generalized theory of interference, and its applications[END_REF][START_REF] Stone | Spin-orbit coupling and the intersection of potential energy surfaces in polyatomic molecules[END_REF]. Nowadays, the Berry curvature is recognized as a key quantity for explaining many fundamental physical phenomena [START_REF] Resta | The insulating state of matter: a geometrical theory[END_REF][START_REF] Thouless | Quantized Hall Conductance in a Two-Dimensional Periodic Potential[END_REF][START_REF] Nagaosa | Anomalous Hall effect[END_REF][START_REF] Xiao | Berry phase effects on electronic properties[END_REF][START_REF] Qi | Topological insulators and superconductors[END_REF][START_REF] Sinova | Spin Hall effects[END_REF]. The influence of the quantum metric on physical effects is more subtle but has started to attract growing attention in recent years. On the theory side, many measurable effects influenced by it were identified [START_REF] Zanardi | Information-Theoretic Differential Geometry of Quantum Phase Transitions[END_REF][START_REF] Srivastava | Signatures of Bloch-Band Geometry on Excitons: Nonhydrogenic Spectra in Transition-Metal Dichalcogenides[END_REF][START_REF] Raoux | Orbital magnetism in coupled-bands models[END_REF][START_REF] Claassen | Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators[END_REF][START_REF] Gao | Geometrical effects in orbital magnetic susceptibility[END_REF][START_REF] Piéchon | Geometric orbital susceptibility: Quantum metric without Berry curvature[END_REF][START_REF] Julku | Geometric Origin of Superfluidity in the Lieb-Lattice Flat Band[END_REF][START_REF] Freimuth | Geometrical contributions to the exchange constants: Free electrons with spin-orbit interaction[END_REF][START_REF] Lapa | Semiclassical wave packet dynamics in nonuniform electric fields[END_REF][START_REF] Gao | Nonreciprocal Directional Dichroism Induced by the Quantum Metric Dipole[END_REF], several measurement protocols were developed [START_REF] Neupert | Measuring the quantum geometry of Bloch bands with current noise[END_REF][START_REF] Kolodrubetz | Classifying and measuring geometry of a quantum ground state manifold[END_REF][START_REF] Lim | Geometry of Bloch states probed by Stückelberg interferometry[END_REF][START_REF] Bleu | Measuring the quantum geometric tensor in two-dimensional photonic and exciton-polariton systems[END_REF][START_REF] Ozawa | Extracting the quantum metric tensor through periodic driving[END_REF], and recently the full QGT of a two-level system was measured directly [START_REF] Yu | Experimental measurement of the quantum geometric tensor using coupled qubits in diamond[END_REF][START_REF] Gianfrate | Measurement of the quantum geometric tensor and of the anomalous Hall drift[END_REF]. It is now well established that the entire QGT, not only the Berry curvature part, is essential for understanding the quantum geometric contributions to observables in N -band systems.

The standard perspective on the QGT, following the historical development, is the following (see Fig. 1):

(i) The system is described by the eigenstates |ψ α (x) , and the concept of quantum geometry arises from the nonzero overlap ψ α (x)|ψ α (x + dx) of eigenstates. In this language, the QGT reads [1-3]

T α,ij (x) = ∂ i ψ α | (1 N -|ψ α ψ α |) |∂ j ψ α , (2)
where

∂ i ≡ ∂/∂x i .
Here, our goal is to emphasize the utility of an alternative point of view that can be adopted:

(ii) The system is described by the eigenprojectors P α (x) = |ψ α (x) ψ α (x)|. Thus, quantum geometry can be viewed as being encoded in rates of changes of the eigenprojectors upon variation of the parameters x i . In this language, the QGT reads [START_REF] Berry | The Quantum Phase, Five Years After[END_REF] T α,ij (x) = Tr [(

∂ i P α ) (1 -P α ) (∂ j P α )] . (3)
Note that there exists a one-to-one correspondence between each eigenprojector and the associated (generalized) Bloch vector b α (x) -the SU(N ) analog of the familiar SU(2) Bloch vector on the unit sphere (see Section III). Thus, the QGT can also conveniently be written in terms of Bloch vectors:

T α,ij (x) = 1 4 [∂ i b α • ∂ j b α +ib α • (∂ i b α × ∂ j b α )] . (4)
While point of view (i) is of considerable conceptual value, eigenstates prove rather cumbersome in actual calculations of the QGT for three main reasons. First, the gauge arbitrariness in the parameter-dependent global phase of the eigenstate is problematic. Second, singularities in some of the components of |ψ α (x) may occur at certain points x 0 in the parameter space X ⊆ R d . Third, closed-form expressions for |ψ α (x) that can be used independently of the Hamiltonian of interest are quite complicated already for N = 2 (cf. Appendix A) and essentially useless for N > 2 [34].

Point of view (ii) is adopted more rarely. However, already the simple N = 2 case demonstrates its usefulness for obtaining analytical insight into the relation between the system's Hamiltonian and its quantum geometry. Namely, for two-band systems, it is well known that

P α (x) = 1 2 1 2 + 1 E α (x) H(x) , b α (x) = 1 E α (x) h(x), (5)
with h(x) a vector that decomposes the Hamiltonian into Pauli matrices [cf. Eq. ( 6)]. Inserting Eq. ( 5) into Eq.

(3) or (4) immediately allows to make the following observation [see Eq. ( 14) below]: The QGT of a two-band system can be written in terms of the Hamiltonian vector h(x), the energy eigenvalues as well as their parametric derivatives. This approach circumvents the need to explicitly compute energy eigenstates.

The main contribution of this paper is to clarify and quantify the generalization of this convenient property to arbitrary N . The QGT T α,ij (x) will be written in terms of only the Hamiltonian [matrix H(x) or vector h(x)] and the band energy E α (x), without having to construct energy eigenstates at all.

Over the last few years, some efforts in this spirit have been made, mostly focusing on the N = 3 case [START_REF] Barnett | SU(3) Spin-Orbit Coupling in Systems of Ultracold Atoms[END_REF][START_REF] Lee | Arbitrary Chern Number Generation in the Three-Band Model from Momentum Space[END_REF][START_REF] Bauer | Quantum geometry and stability of the fractional quantum Hall effect in the Hofstadter model[END_REF]. Very recently, Pozo and de Juan [START_REF] Pozo | Computing observables without eigenstates: Applications to Bloch Hamiltonians[END_REF] pointed out that, quite generally, any observable (not only the QGT) can be computed without energy eigenstates if the eigenenergies are known. While similar in spirit, our work is less ambitious in scope, since we essentially restrict ourselves to the QGT. The issue of computing more complicated physical quantities without using eigenstates is deferred to future work [39].

The setup of this paper is as follows. In Section II, the familiar two-band case is reviewed. This sets the stage for the N -band generalization, which is developed in Sections III and IV. First, in Section III, we focus on the eigenprojectors and Bloch vectors, i.e. we derive formulas

P α (x) = P α (E α (x), H(x)) and b α (x) = b α (E α (x), h(x))
that generalize Eq. ( 5) to any N . Second, in Section IV, these results are combined with Eqs. ( 3) & (4), yielding the QGT of N -band systems as a function of the Hamiltonian and its eigenvalues. Writing the Berry curvature in terms of the Hamiltonian vector, we recover the familiar SU(2) result as well as the SU(3) formula found in Ref. [START_REF] Barnett | SU(3) Spin-Orbit Coupling in Systems of Ultracold Atoms[END_REF], and write down new explicit formulas for the SU(4) and SU(5) case. Section V serves to illustrate the formalism using explicit multifold fermion models. In particular, we introduce a three-(four-) band low-energy model that has exactly the same spectrum as a simple pseudospin S = 1 (S = 3/2), but completely different geometrical and topological properties. Finally, we sum up and conclude in Section VI.

II. SHORT REVIEW OF TWO-BAND SYSTEMS A. SU(2) Hamiltonian

Consider a two-band system, described by a 2 × 2 Hermitian matrix H(x) with eigenvalues {E α (x)| α = ±} and (orthonormal) eigenstates {|ψ α (x) }. Without loss of generality, the trivial part of H is set equal to zero, i.e. Tr H = 0. The Hamiltonian can be expanded in the Pauli matrices σ = (σ x , σ y , σ z ) as

H(x) = h(x) • σ, (6)
with the Hamiltonian vector h(x) = Tr{Hσ}/2. The eigenvalues of Eq. ( 6) are

E α (x) = α Tr(H 2 )/2 = α|h|. (7)

B. Eigenprojectors and Bloch vectors

The eigenprojectors P α (x) ≡ |ψ α (x) ψ α (x)| of the Hamiltonian (6) can be expanded in the Pauli matrices as

P α (x) = 1 2 [1 2 + b α (x) • σ] , (8)
which defines a Bloch vector b α (x). It is related to the Hamiltonian vector as

b α (x) = α h |h| . (9)
Note the agreement with the important expression [START_REF] Berry | Quantal Phase Factors Accompanying Adiabatic Changes[END_REF].

In contrast to the eigenstates, eigenprojectors and Bloch vectors are explicitly gauge-independent quantities and well-behaved in parameter space: a singularity can only appear at band touching points in parameter space.

C. Three different SU(2) Bloch spheres

Here, we briefly discuss three different notions of Bloch spheres encountered in the treatment of two-band systems. A precise distinction of these Bloch spheres is necessary to avoid confusion when introducing Bloch vectors for N -band Hamiltonians.

The unit vector u h (x) ≡ h/|h| associated to the Hamiltonian vector can be parametrized as u h = (sin θ h cos φ h , sin θ h sin φ h , cos θ h ). The two parameters (θ h (x), φ h (x)) may be called Hamiltonian's angles, since cos θ h = h z /|h| and tan φ h = h y /h x . The unit vector defines a map

u h : X → B (2) h , x → u h (x) (10)
from the parameter space X to the Hamiltonian's Bloch sphere B

h . It is simply a unit two-sphere, B

h = S 2 , see Fig. 2(a).

The Bloch vector defines a map

b α : X → B (2) Pα , x → b α (x) (11)
from parameter space to a space that may be called the eigenprojector's Bloch sphere B

Pα . Importantly, the peculiarity and simplicity of two-band systems, as compared to the more general N -band case, consists in the fact that the eigenprojector's Bloch sphere is a unit twosphere, B

Pα = S 2 [see Fig. 2(b)], and that the Bloch vector is proportional to the Hamiltonian vector, cf. Eq. [START_REF] Aharonov | Significance of Electromagnetic Potentials in the Quantum Theory[END_REF].

The third type of Bloch sphere arises from the eigenstates. A two-band eigenstate can be parametrized as

|ψ α (x) = e iΓα   cos θ α sin θ α e iφα   , (12)
with a global phase Γ α (x) and two eigenstate's angles (θ α (x), φ α (x)). These angles define a map

|ψ α : X → B (2) α , x → |ψ α (x) (13)
from parameter space to an eigenstate's Bloch sphere B

α . Since (θ α , φ α ) can be interpreted as spherical coordinates, B

α is a unit two-sphere, too, see Fig. 2(c). Note that the eigenstates [START_REF] Thouless | Quantized Hall Conductance in a Two-Dimensional Periodic Potential[END_REF] are explicitly gauge-dependent and potentially suffer from singularities, as discussed in Appendix A.

Although, in the two-band case, all three Bloch spheres happen to correspond to a unit two-sphere, it is important to keep in mind that they are a priori distinct spaces. This distinction is necessary to avoid considerable confusion in the SU(N > 2) case, where the three Bloch spheres are completely different from one another [see Fig. 2 In Eqs. ( 1)-( 4), the explicit dimension and form of the matrices g α and Ω α clearly depend on the number and type of parameters x i contained in the chosen set x. Some examples for different interesting choices of x are given in Appendix B. If the set of parameters is kept completely general, the most convenient formulation of the QGT follows from Eq. (3): [START_REF] Xiao | Berry phase effects on electronic properties[END_REF] with the shorthand notation m i ≡ ∂ i m. For obtaining the expressions in terms of Bloch vectors, we used Eq. ( 8) and the product identity

g α,ij (x) = 1 4 b i α • b j α = 1 4|h| 2 h i • h j - h • h i h • h j |h| 2 , Ω α,ij (x) = - 1 2 b α • (b i α × b j α ) = - α 2|h| 3 h • (h i × h j ),
(m • σ)(n • σ) = m • n 1 2 + i(m × n) • σ. ( 15
)
The expressions in terms of h then follow from Eq. ( 5).

III. GENERALIZATION TO N -BAND SYSTEMS: PART 1 -EIGENPROJECTORS AND BLOCH VECTORS

We now systematically generalize the two-band discussion of the previous section to arbitrary N . More precisely, after introducing the generic N -band Hamiltonian (Section III A), we will first generalize Eq. ( 5), both for the eigenprojectors (Section III B) and Bloch vectors (Section III C). The properties of the SU(N ) Bloch spheres are also briefly addressed (Section III D).

Once the eigenprojectors P α (E α , H) and Bloch vectors b α (E α , h) are found, one can insert them into Eq. ( 3) or (4), to obtain the N -band generalization of the QGT ( 14). This will be done in Section IV.

A. SU(N ) Hamiltonian

Consider an N -band system, described by an N × N Hermitian matrix H(x) with eigenvalues {E α (x)| α = 1, ..., N } and (orthonormal) eigenstates {|ψ α (x) |} [40].

Again, the trivial part is set equal to zero without loss of generality, Tr H = 0. The Hamiltonian can be ex-panded as

H(x) = h(x) • λ, (16)
with the Hamiltonian vector h(x) = Tr{Hλ}/2. Here, λ = (λ 1 , ..., λ N 2 -1 ) is a vector composed of the N 2 -1 elementary generator matrices of the SU(N ) Lie group. Together with the identity matrix 1 N they consitute a basis for the Lie algebra su(N ). Throughout this paper, we will choose the generators λ a as (generalized) Gell-Mann matrices [START_REF] Pfeifer | The Lie algebras su(N )[END_REF][START_REF] Bertlmann | Bloch vectors for qudits[END_REF]. For the reader's convenience, the Gell-Mann matrices for N = 3 and N = 4 are listed in Appendix C.

B. Eigenprojectors as a function of the

Hamiltonian: Pα(Eα, H)

According to the Cayley-Hamilton theorem [START_REF] Cayley | II. A memoir on the theory of matrices[END_REF][START_REF] Horn | Matrix Analysis[END_REF], any function of the N × N Hamiltonian matrix H can be written as a matrix polynomial in H, where the highest power is N -1. In particular, as demonstrated in detail in Appendix D, the eigenprojector P α can be written as a matrix polynomial

P α = N -1 n=0 q N -1-n (E α )H n N -1 n=0 q N -1-n (E α )E n α = N -1 n=0 q N -1-n (E α )H n N -1 n=0 q N -1-n (E α )C n , (17)
where the polynomials q n (z) ≡ n k=0 c k z n-k are closely related to the Hamiltonian's characteristic polynomial. The coefficients c k appearing here are listed in Table I for k ≤ 5, and depend solely on the (classical) Casimir invariants [START_REF] Kusnezov | Exact matrix expansions for group elements of SU(N)[END_REF]

C n ≡ Tr(H n ) = N α=1 E n α , (18)
where obviously

C 0 = N and C 1 = 0. k 0 1 2 3 4 5
c k 1 0 -C 2 2 -C 3 3 C 2 2 8 -C 4 4 C 2 C 3 6 -C 5 5 TABLE I. Coefficients c k determining the polynomial qn(z).
The utility of Eq. ( 17) becomes immediately apparent if we write down the explicit eigenprojectors P α (E α , H) for the cases N = 2 to N = 5:

P α = 1 2E α (E α 1 2 + H), P α = 1 3E 2 α -C2 2 E 2 α - C 2 2 1 3 + E α H + H 2 , P α = 1 4E 3 α -C 2 E α -C3 3 E 3 α - C 2 2 E α - C 3 3 1 4 + E 2 α - C 2 2 H + E α H 2 + H 3 , P α = 1 5E 4 α -3C2 2 E 2 α -2C3 3 E α + C 2 2 -2C4 8 E 4 α - C 2 2 E 2 α - C 3 3 E α + C 2 2 -2C 4 8 1 5 + E 3 α - C 2 2 E α - C 3 3 H + E 2 α - C 2 2 H 2 + E α H 3 + H 4 . (19)
This represents the N -band generalization of the twoband projectors [START_REF] Berry | Quantal Phase Factors Accompanying Adiabatic Changes[END_REF]. Since the Hamiltonian is typically known for any given problem, the only ingredient required for explicitly computing the eigenprojector P α is the corresponding eigenenergy E α .

Beyond the computation of eigenprojectors, Eq. ( 17) has two interesting applications. First, it can be used to rewrite any function f (H) as a polynomial of order N -1 in the Hamiltonian, which corresponds to an alternative version of Sylvester's formula [START_REF] Horn | Matrix Analysis[END_REF]:

f (H) = N α=1 f (E α ) N -1 n=0 q N -1-n (E α )H n N -1 n=0 q N -1-n (E α )E n α . (20)
Second, the eigenprojector is in some sense a more fundamental object than the eigenstate, namely Eq. ( 17) may be employed for constructing energy eigenstates

|ψ α = 1 ψ g | P α (E α , H) |ψ g P α (E α , H) |ψ g , (21)
by projecting onto a gauge freedom state |ψ g that can be chosen arbitrarily (for more details, see Appendix E).

The eigenstates [START_REF] Gao | Geometrical effects in orbital magnetic susceptibility[END_REF] can then be used further to compute matrix elements of observables, or the Berry connection and other quantities of interest.

C. Bloch vectors as a function of the Hamiltonian vector: bα(Eα, h)

The above key results ( 17) & ( 19) on the eigenprojectors are more convenient for some practical purposes when reformulated in a vectorial language. More precisely, just like a Hamiltonian vector was introduced in Eq. ( 16), one may define a (generalized) Bloch vector b α by expanding the eigenprojector as [START_REF] Hioe | N -Level Coherence Vector and Higher Conservation Laws in Quantum Optics and Quantum Mechanics[END_REF]

P α (x) = 1 N 1 N + 1 2 b α (x) • λ (22)
in the SU(N ) generators, where b α ≡ Tr{P α λ}. Again, be reminded that throughout this paper we will always use the generators to be (generalized) Gell-Mann matrices. Importantly, however, the functional form of all SU(N ) vector identities discussed below is independent of the particular choice of generator matrices. Now, by inserting the expansions ( 16) & ( 22) into Eq. ( 17), one obtains the analog of the function P α (E α , H) in the vectorial language, namely b α (E α , h), as explained in Appendix F. The usefulness of this procedure is most apparent when considering the explicit Bloch vector expressions resulting from it, provided here for N = 2 to N = 5:

b α = 1 E α h, b α = 2 3E 2 α -C2 2 (E α h + h ) , b α = 2 4E 3 α -C 2 E α -C3 3 E 2 α - C 2 4 h + E α h + h , b α = 2 5E 4 α -3C2 2 E 2 α -2C3 3 E α + C 2 2 -2C4 8 E 3 α - 3C 2 10 E α - 2C 3 15 h + E 2 α - 3C 2 10 h + E α h + h . (23)
This is the N -band generalization of the two-band Bloch vectors [START_REF] Berry | Quantal Phase Factors Accompanying Adiabatic Changes[END_REF], and contains exactly the same information as Eq. ( 19). Two ingredients of Eq. ( 23) require some more explanation, namely the Casimir invariants C n , and the "star product vectors" h , h , etc. Importantly, both quantities are completely determined by the vector h.

Casimir invariants from the Hamiltonian vector

Since the Casimir invariants are traces of powers of the Hamiltonian matrix, cf. Eq. ( 18), they can be directly expressed in terms of the Hamiltonian vector (cf. Appendix F):

C 2 = 2|h| 2 , C 3 = 2 h • h , C 4 = 4|h| 4 /N + 2|h | 2 . ( 24
)
It is convenient to combine Eq. ( 24) with Eq. ( 23), such that the Bloch vector for a given band α depends only on the single eigenvalue E α .

Star products of the Hamiltonian vector

For given N , each Bloch vector ( 23) is a kind of "vector polynomial" of degree N -1 in h, just like each eigenprojector is a matrix polynomial in H. In other words, the vectors h , h , etc. -though rather unfamiliar objects of dimension Energy 2 , Energy 3 , etc. -are uniquely determined by the Hamiltonian vector h, as explained in the following.

The properties of the Lie algebra su(N ) underlying the N -band Hamiltonian [START_REF] Sinova | Spin Hall effects[END_REF] are determined by the commutation and anticommutation relations [START_REF] Kaplan | Matrix Products and the Explicit 3, 6, 9, and 12-j Coefficients of the Regular Representation of SU(n)[END_REF]

[λ a , λ b ] = 2if abc λ c , {λ a , λ b } = 4 N δ ab 1 N + 2d abc λ c , (25)
where repeated lower indices imply summation (Einstein convention). In particular, these relations define totally antisymmetric and totally symmetric structure constants of su(N ), respectively:

f abc ≡ - i 4 Tr ([λ a , λ b ]λ c ) , d abc ≡ 1 4 Tr ({λ a , λ b }λ c ) . (26)
These are a known set of real numbers once a matrix representation is chosen for the generators λ a . Note that for N = 2, where λ = σ, the d abc vanish identically and f abc = abc , where abc is the Levi-Civita tensor.

From the structure constants, one defines dot, star and cross products of SU(N ) vectors:

m • n ≡ m c n c , (m n) a ≡ d abc m b n c , (m × n) a ≡ f abc m b n c , (27)
where m and n are real and (N 2 -1)-dimensional. The star product is unfamiliar because it does not play any role in N = 2 situations (where d abc = 0), but it is crucial for N > 2 systems. Finally, we may introduce the notation for repeated star products of a vector with itself:

m (0) = m, m (1) = m ≡ m m, m (2) = m ≡ m (m m), m (k+1) = m ... ≡ m m (k) . (28)
The vectors h , h , etc. are thus simply star products of the Hamiltonian vector with itself.

D. Three different SU(N ) Bloch spheres

The Hamiltonian's, eigenprojector's and eigenstate's Bloch spheres introduced in Section II C for the two-band case are here briefly discussed for higher N .

The Hamiltonian vector h has N 2 -1 real-valued components, such that the unit vector u h ≡ h/|h| can be parametrized by N 2 -2 Hamiltonian's angles. In other words, u h (x) defines a map

u h : X → B (N ) h , x → u h (x) (29)
from the parameter space to a Hamiltonian's Bloch sphere

B (N ) h = S N 2 -2
, where S N 2 -2 is the unit (N 2 -2)-sphere, as depicted in Fig. 2(d).

Naively, since b α also has N 2 -1 real-valued components, the map b α (x) (for given α) would similarly seem to define an (N 2 -2)-sphere, but this is prevented by constraints on the Bloch vectors. In particular, the usual orthogonality relation P α P β = δ αβ P α and completeness relation α P α = 1 N translate to the Bloch vector picture as

b α • b β = 2 δ αβ - 1 N , b α b β = δ αβ - 2 N (b α + b β ), b α × b β = 0, α b α = 0. ( 30
)
As a consequence, the vector b α (x) defines a map

b α : X → B (N ) Pα , x → b α (x) (31)
from parameter space not to an (N 2 -2)-sphere but to a 2(N -1)-dimensional subset thereof, which may be called the eigenprojector's Bloch sphere B (N )

Pα , or simply the generalized Bloch sphere, see Fig. 2(e). In contrast to the N = 2 case, the Bloch vector is no longer parallel to the Hamiltonian vector for N > 2 [cf. Eq. ( 23)], such that u h (x) and b α (x) are completely distinct maps. An understanding of the true geometrical structure of the generalized Bloch sphere is not at all easy to acquire. Many efforts have been undertaken to figure out its properties for N > 2, which is a surprisingly nontrivial issue, see Refs. [START_REF] Harriman | Geometry of density matrices. I. Definitions, N matrices and 1 matrices[END_REF][START_REF] Jakóbczyk | Geometry of Bloch vectors in two-qubit system[END_REF][START_REF] Kimura | The Bloch vector for N -level systems[END_REF][START_REF] Zyczkowski | Hilbert-Schmidt volume of the set of mixed quantum states[END_REF][START_REF] Byrd | Characterization of the positivity of the density matrix in terms of the coherence vector representation[END_REF][START_REF] Kimura | The Bloch-Vector Space for N -Level Systems: the Spherical-Coordinate Point of View[END_REF][START_REF] Mendaš | The classification of three-parameter density matrices for a qutrit[END_REF][START_REF] Goyal | Geometry of the generalized Bloch sphere for qutrits[END_REF] and references therein. For the interested reader, we outline the main results in Appendix H.

The third type of Bloch sphere arises from the eigenstates. Each eigenstate ( 21) of an N -band system can be minimally encoded by a global phase Γ α (x) and N -1 pairs of eigenstate's angles (θ i α (x), φ i α (x)) (i = 1, ..., N -1). These angles define a map

|ψ α : X → B (N ) α , x → |ψ α (x) (32)
from the parameter space to a 2(N -1)-dimensional eigenstate's Bloch sphere

B (N )
α , which is depicted, for comparison with the N = 2 case, in Fig. 2(f).

In summary, the Hamiltonian's, the eigenprojector's and the eigenstate's Bloch sphere are all isomorphic spaces for N = 2, but all different spaces for N > 2.

IV. GENERALIZATION TO N -BAND SYSTEMS: PART 2 -QUANTUM GEOMETRIC TENSOR

The QGT in N -band systems is usually defined in terms of N -component eigenstates |ψ α . The standard eigenstate-based QGT formula is given by Eq. ( 2). (For more details on its origin, see Appendix I.) A more popular eigenstate-based QGT expression, which is easily found by treating H(x + dx) -H(x) as a formal perturbation to |ψ α (x) , reads

T α,ij = β =α ψ α | ∂ i H |ψ β ψ β | ∂ j H |ψ α (E α -E β ) 2 . ( 33
)
The advantage of Eq. ( 33), as compared to Eq. ( 2), is that it involves the parametric velocity operators ∂ i H, thus avoiding derivatives of eigenstates. Therefore, it is the most common formula used for computing the QGT. However, this in general still requires numerical construction of the eigenstates.

The main goal here is to develop a deeper analytical understanding of the geometric tensors and their direct relation to the Hamiltonian. This can be achieved by avoiding eigenstates and drawing on the eigenprojectorbased formula [START_REF] Kolodrubetz | Geometry and non-adiabatic response in quantum and classical systems[END_REF]. (For the relation between Eq. ( 2) and Eq. ( 3), see Appendix I.)

First, we will reformulate Eq. ( 3) in terms of generalized Bloch vectors (Section IV A). Then, inserting Eq. ( 23) into the result, we will be able to write the QGT in terms of only the Hamiltonian vector h and the eigenenergy E α (Section IV B).

A. QGT in terms of Bloch vectors

In order to obtain a Bloch vector picture of the geometric tensors, one can simply substitute Eq. ( 22) into Eq. (3). Using the SU(N ) product identity (cf. Appendix F)

(m • λ)(n • λ) = 2 N m • n 1 N + (m n + i m × n) • λ (34)
and the star product [START_REF] Bleu | Measuring the quantum geometric tensor in two-dimensional photonic and exciton-polariton systems[END_REF] of Bloch vectors, one finds

g α,ij = 1 4 b i α • b j α , Ω α,ij = - 1 2 b α • b i α × b j α , (35)
for arbitrary values of N (again with the shorthand notation m i ≡ ∂ i m). This is the same as Eq. ( 4) stated in the introduction. Symbolically, Eq. ( 35) is of the exactly same form as in the N = 2 case, cf. Eq. ( 14). However, be aware that for N > 2 the dimension of the Bloch vectors increases, and the cross product has to be interpreted in the SU(N ) sense, cf. Eq. [START_REF] Neupert | Measuring the quantum geometry of Bloch bands with current noise[END_REF]. For the SU(3) case, an equivalent way of writing Berry curvature and quantum metric was already encountered in Refs. [START_REF] Barnett | SU(3) Spin-Orbit Coupling in Systems of Ultracold Atoms[END_REF][START_REF] Lee | Arbitrary Chern Number Generation in the Three-Band Model from Momentum Space[END_REF] and [START_REF] Bauer | Quantum geometry and stability of the fractional quantum Hall effect in the Hofstadter model[END_REF], respectively; for the SU(N ) case, Pozo and de Juan recently found a similar formula in terms of what they call 1-generators [START_REF] Pozo | Computing observables without eigenstates: Applications to Bloch Hamiltonians[END_REF].

While Eq. ( 35) is conceptually important, a certain disadvantage consists in the presence of explicit parametric derivatives of the Bloch vector; this is potentially complicated, since cumbersome expressions follow from applying the product rule to Eq. ( 23). An alternative form of writing the QGT that does not involve such explicit derivatives of the Bloch vectors is provided in Appendix J.

B. QGT from the Hamiltonian and its eigenvalues

Having established the general expressions [START_REF] Barnett | SU(3) Spin-Orbit Coupling in Systems of Ultracold Atoms[END_REF] for arbitrary values of N , the final step towards obtaining the QGT in terms of only the Hamiltonian vector h and the eigenenergy E α consists in inserting the explicit formula [START_REF] Julku | Geometric Origin of Superfluidity in the Lieb-Lattice Flat Band[END_REF].

For the Berry curvature, we obtain a closed form expression for arbitrary N , which is illustrated here for N = 3, 4, 5 for comparison with the simple N = 2 expression [START_REF] Xiao | Berry phase effects on electronic properties[END_REF]; the discussion for arbitrary N is given in Appendix K.

In the N = 3 case, the Berry curvature tensor is given by

Ω α,ij = - 4(E α h + h ) (3E 2 α -|h| 2 ) 3 • E α h i + h i × E α h j + h j , (36)
with h i ≡ ∂ i h . This expression is equivalent to the result found by Barnett et al. [START_REF] Barnett | SU(3) Spin-Orbit Coupling in Systems of Ultracold Atoms[END_REF] if one inserts a closedform energy parametrization in terms of trigonometric functions, see for example Ref. [START_REF] Rosen | Finite Transformations in Various Representations of SU(3)[END_REF]. Note that Eq. ( 36) can be expressed in a more compact form (cf. Appendix K), which allows to explicitly verify the sum rule

α Ω α,ij = 0. For arbitrary N = 4 systems, we have

Ω α,ij = - 4(Q α h + E α h + h ) (4E α Q α -2 3 h • h ) 3 • Q α h i + E α h i + h i × Q α h j + E α h j + h j , (37)
with

Q α ≡ E 2 α -|h| 2 /2.
Similarly, for arbitrary N = 5 systems, the Berry curvature is given by

Ω α,ij = - 4(R α h + Rα h + E α h + h ) 5E α R α + 3 10 |h| 4 -1 2 |h | 2 3 • (R α h i + Rα h i + E α h i + h i ) × (R α h j + Rα h j + E α h j + h j ) , (38)
where R α ≡ E α Rα - 4 15 h • h and Rα ≡ E 2 α -3 5 |h| 2 . In the same way, one may obtain expressions for N > 5.

For the quantum metric, writing down explicit formulas in terms of h and E α proves too cumbersome for N > 2, due to the absence of appropriate orthogonality relations. The quantum metric can however be straightforwardly computed by a two-step procedure: first calculate the Bloch vectors ( 23), then substitute the result into Eq. ( 35).

V. EXAMPLES: MULTIFOLD FERMIONS

In this section, the utility of the formalism developed above is demonstrated by applying it to concrete Hamiltonians of interest. In particular, we will consider N = 3 and N = 4 models in the class of multifold (Dirac) fermions [START_REF] Flicker | Chiral optical response of multifold fermions[END_REF], i.e. low-energy models H(q) with a linear spectrum and an N -fold degeneracy at q = 0. Here, x = q should be understood as some generic threedimensional quasi-momentum vector, which may take various physical meanings [START_REF] Qi | Topological insulators and superconductors[END_REF][START_REF] Zak | Berry's phase for energy bands in solids[END_REF][START_REF] Dalibard | Colloquium: Artificial gauge potentials for neutral atoms[END_REF][START_REF] Armitage | Weyl and Dirac semimetals in three-dimensional solids[END_REF][START_REF] Cooper | Topological bands for ultracold atoms[END_REF][START_REF] Ozawa | Topological photonics[END_REF][START_REF] Zhang | Topological quantum matter with cold atoms[END_REF].

A. Pseudospin fermions

As a first simple class of models that lends itself to an analysis within the above language, consider (pseudo)spin-S multifold fermion models, where S = (N -1)/2. Such models involve matrices S x,y,z that satisfy a spin algebra

[S i , S j ] = i ijk S k ,
for any triple (i, j, k),

as well as S 2 = S(S + 1)1 N . The corresponding Hamiltonian can be written as

H(q) = q • S = q x S x + q y S y + q z S z . (40)
It may be checked that this Hamiltonian has a (global) charge conjugation symmetry C † H * (q)C = -H(q) [START_REF] Ludwig | Topological phases: classification of topological insulators and superconductors of noninteracting fermions, and beyond[END_REF],

with C = e iπSy and CC * = (-1) 2S 1 2S+1 . The spectrum is symmetric about zero energy, as well as isotropic and linear in the quasi-momentum:

E m = m|q|, (41)
where m = -S, ..., S.

In the conventional eigenstate picture, the QGT of a pseudospin-S fermion can be computed as follows. One may construct the q-dependent spin eigenstates |S, m, q , where m = -S, ..., +S, as rotated eigenstates of the form |S, m, q = e iϕqSz e iθqSy |S, m , with S z |S, m = m|S, m , cos θ q = q z /|q|, and tan ϕ q = q y /q x . One may then use Eq. ( 33) to obtain the QGT T m,ij (q). In particular, defining the three-component pseudo-vector Ω m ≡ (Ω m,yz , Ω m,zx , Ω m,xy ), the Berry curvature takes the form of a topological monopole [START_REF] Berry | Quantal Phase Factors Accompanying Adiabatic Changes[END_REF]:

Ω m (q) = -m q |q| 3 . (42)
This monopole carries a topological charge measured by the first Chern number, C m = -2m. As is well known, C m is odd (even) for half-integer (integer) spin. We now restrict ourselves to the three-and four-band case for simplicity, i.e. to pseudospin-1 and pseudospin-3/2 fermions, respectively. The respective band structures are shown in Fig. 3(a)&(b). Our goal is to obtain the eigenprojectors and Bloch vectors, which will allow us to compute the QGT without constructing spin eigenstates. In particular, we will recover the result [START_REF] Bertlmann | Bloch vectors for qudits[END_REF] and additionally compute the quantum metric. We now separate the treatment for spin-1 and spin-3/2.

Pseudospin-1 fermions

The spin Hamiltonian (40) can be easily rewritten in the general form of Eq. ( 16) by using the relation between spin matrices and Gell-Mann matrices. In the standard representation, the spin-1 matrices are related to the N = 3 Gell-Mann matrices by Eq. (C1). Thus, the Hamiltonian vector reads

h(q) = 1 2 ( √ 2q x , √ 2q y , q z , 0, 0, √ 2q x , √ 2q y , √ 3q z ). ( 43
)
Upon using the spectrum (41) and C 2 = 2|q| 2 [cf. Eq. ( 24)], the eigenprojectors for a pseudospin-1 fermion are obtained from Eq. ( 19) as

P m (q) = 1 3m 2 -1 (m 2 -1)1 3 + m |q| (h • λ) + 1 |q| 2 (h • λ) 2 . ( 44
)
It is readily verified that Tr(P m ) = 1 and m P m = 1 3 as expected. Similarly, the Bloch vectors follow from Eq. ( 23) as

b m (q) = 2 3m 2 -1 m h |q| + h |q| 2 , (45)
where

h (q) = q x q z √ 2 , q y q z √ 2 , 3q 2 z -|q| 2 4 , q 2
x -q 2 y 2 , q x q y , -

q x q z √ 2 , - q y q z √ 2 , |q| 2 -3q 2 z 4 √ 3 . (46)
The Berry curvature may now be obtained from Eq. [START_REF] Barnett | SU(3) Spin-Orbit Coupling in Systems of Ultracold Atoms[END_REF] [or more directly from Eq. ( 36)], and one recovers Eq. ( 42) as expected. Similarly, the quantum metric follows from Eq. ( 35) as

g m,ij (q) = 2 -m 2 2|q| 2 δ ij - q i q j |q| 2 , ( 47
)
in agreement with Ref. [START_REF] Lin | Dual Haldane sphere and quantized band geometry in chiral multifold fermions[END_REF].

Pseudospin-3/2 fermions

In the standard representation, the spin-3/2 matrices are related to the N = 4 Gell-Mann matrices by Eq. (C2). The Hamiltonian vector thus reads

h(q) = 1 2 ( √ 3q x , √ 3q y , q z , 0, 0, 2q x , 2q y , √ 3q z , 0, 0, 0, 0, √ 3q x , √ 3q y , √ 6q z ). ( 48
)
The star product vectors h and h can be computed from Eq. ( 48) using Eq. ( 28). Upon using the spectrum (41), as well as C 2 = 5|q| 2 and C 3 = 0, the eigenprojectors for a spin-3/2 fermion are obtained as

P m (q) = 1 4m m 2 -5 4 m m 2 - 5 2 1 4 + m 2 -5 2 |q| (h • λ) + m |q| 2 (h • λ) 2 + 1 |q| 3 (h • λ) 3 .
(49) Similarly, the Bloch vectors are given by

b m (q) = 1 2m m 2 -5 4 m 2 - 5 4 h |q| +m h |q| 2 + h |q| 3 . ( 50
)
Importantly, be aware that the coefficient of the linear term (in H) of the eigenprojector is in general not the same as the coefficient of the linear term (in h) of the Bloch vector, and similarly for the higher-order terms. This is due to the fact that H n+1 = h (n) • λ for n ≥ 0. The Berry curvature follows from Eq. ( 35) [or more directly from Eq. ( 37)], and again takes the expected form [START_REF] Bertlmann | Bloch vectors for qudits[END_REF]. Similarly, the quantum metric reads

g m,ij (q) = 15/4 -m 2 2|q| 2 δ ij - q i q j |q| 2 , ( 51
)
again in agreement with Ref. [START_REF] Lin | Dual Haldane sphere and quantized band geometry in chiral multifold fermions[END_REF].

B. Beyond pseudospin fermions

Let us now discuss two examples that go beyond the simple pseudospin models. More precisely, we present a three-band (four-band) multifold fermion model with exactly the same energy spectrum as a spin-1 (spin-3/2), cf. Fig. 3, but with completely different symmetries and quantum geometric properties. It is important to realize that the analytical expressions presented below are easily obtained using our approach, while establishing them from an energy eigenstate approach would be rather cumbersome.

Three-band example

Consider the Hamiltonian

H(q) =     0 1 √ 2 (q x -iq y ) -iq z ... 0 1 √ 2 (q x + iq y ) ... ... 0     , (52)
where the lower left matrix elements are obtained by complex conjugating the upper right ones. This model has a global chiral symmetry S † H(q)S = -H(q) [START_REF] Ludwig | Topological phases: classification of topological insulators and superconductors of noninteracting fermions, and beyond[END_REF] with S = antidiag(1, -1, 1). The energy spectrum reads

E α (q) = α|q|, α = 0, ±1, (53)
which is the same as for a pseudospin-1 model. In the language of Eq. ( 16), the model ( 52) is characterized by a Hamiltonian vector h(q) = 1 √ 2 (q x , q y , 0, 0, √ 2q z , q x , q y , 0) [START_REF] Mendaš | The classification of three-parameter density matrices for a qutrit[END_REF] and its star product

h (q) = q y q z √ 2 , q x q z √ 2 , 3q 2 z -|q| 2 4 , q 2
x -q 2 y 2 ,

q x q y , q y q z √ 2 , q x q z √ 2 , |q| 2 -3q 2 z 4 √ 3 . ( 55
)
The Bloch vectors follow from Eq. ( 23) as

b α (q) = 2 3α 2 -1 α h |q| + h |q| 2 . ( 56
)
Notably, they are formally completely equivalent to the spin-1 case, but with slightly different vectors h and h .

The QGT for all bands α = 0, ±1 can be summarized in compact form as

Ω α (q) = (2 -3α 2 )(q • e z ) q |q| 4 , g α,ij (q) = 2 -α 2 2|q| 2 δ ij - q i q j |q| 2 - 2 -α 2 2|q| 2 (q × e z ) i (q × e z ) j , (57)
where we again introduced a Berry curvature pseudovector Ω α and where e z = (0, 0, 1). There are several notable features of Eq. ( 57). First, the Berry curvature is completely different from its spin-1 analog (42) and yields first Chern numbers C α = 0. Second, the quantum metric contains a part that is exactly like for a spin-1 [cf. Eq. ( 47)], plus an additional term. Third, the QGT is anisotropic despite an isotropic band structure.

Four-band example

Consider now the Hamiltonian

H(q) =        0 q y + qz 2 -i q x + qy 2 qx 2 -q z ... 0 qx 2 + q z i q x - qy 2 ... ... 0 q y -qz 2 ... ... ... 0        . ( 58
)
This model has a global charge conjugation symmetry with C = diag(1, -1, 1, -1), but no chiral symmetry, and a band structure

E α1α2 (q) = α 1 1 + α 2 2 |q|, α 1 = ±1, α 2 = ±1
(59) which is exactly the same as for a pseudospin-3/2. Note that a similar model has been studied in Ref. [START_REF] Roy | From Birefringent Electrons to a Marginal or Non-Fermi Liquid of Relativistic Spin-1/2 Fermions: An Emergent Superuniversality[END_REF].

The model ( 58) is described by a Hamiltonian vector h(q) = q y + q z 2 , 0, 0, 0, q x + q y 2 , q x 2 + q z , 0, 0,

q x 2 -q z , 0, 0, -q x + q y 2 , q y - q z 2 , 0, 0 , (60)
from which the vectors h and h are readily obtained. The Bloch vectors follow from Eq. ( 23) as

b α1α2 (q) = α 1 1 + 2α 2 α 2 h |q| +α 1 1 + α 2 2 h |q| 2 + h |q| 3 . (61)
The QGT for all four bands (calculated in the same way as for the spin-3/2 case above) is then given by

Ω α1α2 (q) = -α 1 (1 + α 2 ) q 2|q| 3 , g α1α2,ij (q) = 1 2|q| 2 δ ij - q i q j |q| 2 . (62)
Though Eq. ( 62) is formally similar to the QGT of a spin-3/2 fermion, there are crucial differences. Most importantly, the first Chern numbers obtained from Eq. ( 62) are {2, 0, 0, -2} from the lowest to the highest band, while they are {3, 1, -1, -3} for a spin-3/2. Moreover, the quantum metric is band-independent.

VI. CONCLUSIONS

For a physical system described by a (Hermitian) parametric Hamiltonian matrix H = H(x), there are situations in which it is useful to recall that any eigenstate is more fundamentally encoded in the eigenprojector matrix

P α (x) = |ψ α (x) ψ α (x)| [67].
Consequently, for a given quantity of interest whose standard expression is known in terms of eigenstates, it can prove rewarding to aim for a reformulation in terms of eigenprojectors, which will entirely eliminate the necessity for constructing eigenstates explicitly. Such a reformulation can be done for any physical quantity in principle. The present work demonstrates this by focusing on the case where the quantities of interest are the quantum metric and Berry curvature tensors (forming together what is known as the quantum geometric tensor, QGT). Selecting the QGT is natural for two reasons. First, it is relatively simple (compared to more involved quantities like the orbital magnetic susceptibility), and therefore serves well for illustrating the main features of the eigenprojector approach, which are preserved in the treatment of more complicated observables. Second, the inconvenient features of eigenstates become particularly problematic when one is interested in geometrical quantities; therefore, the QGT is one of the observables for which the projector formalism is the most powerful and beneficial.

The most striking qualitative conclusion that can be drawn from the eigenprojector approach is that any quantity of interest can be written in terms of the Hamiltonian vector h and the eigenvalues E α , without using eigenstates, in agreement with recent results of Pozo and de Juan [START_REF] Pozo | Computing observables without eigenstates: Applications to Bloch Hamiltonians[END_REF] (who used a somewhat different mathematical framework). At the origin of this conclusion is the fact that each eigenprojector is a matrix polynomial of degree N -1 in H, according to the Cayley-Hamilton theorem. This result, though implicitly known for a long time [START_REF] Halmos | Finite-Dimensional Vector Spaces: Second Edition[END_REF], is perhaps most useful to physicists when formulated as an explicit function P α (E α , H) of a single eigenvalue, see Eqs. ( 17) & [START_REF] Raoux | Orbital magnetism in coupled-bands models[END_REF]. These expressions for P α (E α , H) represent our first intermediate result.

The realization that the knowledge of H as well as its eigenenergies immediately yields the eigenprojectors implies that any physical quantity of interest can be computed in the following way:

(1) Express the quantity in terms of eigenprojectors.

(2) Explicitly insert the function P α (E α , H).

In the course of carrying out this procedure, it proves extremely enlightening to switch between two "languages" where appropriate. The first language simply employs the relevant matrices (H, P α , etc.), while the second one expands those matrices in the generators of SU(N ). In this second (vectorial) language the Hamiltonian H is given by a real vector h, and similarly the eigenprojector P α is represented by a real (generalized) Bloch vector b α . The analog of the matrix function P α (E α , H) is the vector function b α (E α , h), see in particular the important result [START_REF] Julku | Geometric Origin of Superfluidity in the Lieb-Lattice Flat Band[END_REF]. Both languages contain exactly the same information, such that the above protocol can be equivalently formulated as:

(1) Express the quantity in terms of Bloch vectors.

(2) Explicitly insert the function b α (E α , h).

In the present paper, starting in Section IV, the above protocol was carried out for the QGT, and the systematic application to other, more complicated observables will be published elsewhere [39]. The known formula (3) for the QGT in terms of eigenprojectors can be written in the vectorial language as described in Section IV A [cf. in particular Eq. ( 35)], which proves quite handy for practical computations and completes step [START_REF] Berry | The Quantum Phase, Five Years After[END_REF]. Step (2) was carried out in Section IV B, where we obtained the Berry curvature tensor Ω α,ij in terms of the vector h and the relevant energy eigenvalue E α only. This generalizes the well-known Berry curvature expression [START_REF] Xiao | Berry phase effects on electronic properties[END_REF] to arbitrary N .

As a concrete illustration of the results obtained for the QGT, Section V served to present models belonging to the class of multifold (Dirac) fermions, characterized by an N -fold degenerate nodal point accompanied by a linear band crossing. First, we found interesting Bloch vector expressions for spin-1 and spin-3/2 fermions that yield a QGT consistent with the literature. Second, we introduced two multifold fermion models, which emphasize that different Hamiltonians can exhibit completely distinct geometrical and topological properties despite being indistinguishable according to their band structure. Such Hamiltonians may serve as interesting platforms for studying effects such as orbital magnetism or Friedel oscillations, similar to the α -T 3 model [START_REF] Raoux | From Dia-to Paramagnetic Orbital Susceptibility of Massless Fermions[END_REF].

To close this paper, we state a few immediate perspectives of our work. First, as already mentioned, it is worthwhile to apply the eigenprojector approach to observables that are not purely of geometric origin. In particular, the analytical insight afforded by this approach will help to unveil and distinguish purely spectral and geometrical contributions to such observables, similar in spirit to what has been done for the orbital magnetic susceptibility of two-band models in Ref. [START_REF] Raoux | Orbital magnetism in coupled-bands models[END_REF], superconducting stiffness or non-linear responses and more generalized thermodynamic stiffnesses.

Second, it would be interesting to establish a better understanding of the geometry and topology of multifold fermion models with linear but also quadratic band crossings. More concretely, how to systematically design models that all share a certain type of energy spectrum but belong to different topological classes according to the tenfold way classification [START_REF] Ludwig | Topological phases: classification of topological insulators and superconductors of noninteracting fermions, and beyond[END_REF]?

Finally, note that the validity of the formalism developed in this paper goes beyond the case of Hermitian Hamiltonian matrices. In particular, the key expressions Eq. ( 19) for the eigenprojector P α (E α , H) and Eq. ( 23) for the Bloch vector b α (E α , h) stay valid for systems where the Hermiticity condition is relaxed [START_REF] Bender | Making sense of non-Hermitian Hamiltoni-ans[END_REF][START_REF] Brody | Biorthogonal quantum mechanics[END_REF].

The N = 3 Gell-Mann matrices [START_REF] Gell-Mann | Symmetries of Baryons and Mesons[END_REF] are given by

λ 1 =     0 1 0 1 0 0 0 0 0     , λ 2 =     0 -i 0 i 0 0 0 0 0     , λ 3 =     1 0 0 0 -1 0 0 0 0     , λ 4 =     0 0 1 0 0 0 1 0 0     , λ 5 =     0 0 -i 0 0 0 i 0 0     , λ 6 =     0 0 0 0 0 1 0 1 0     , λ 7 =     0 0 0 0 0 -i 0 i 0     , λ 8 = 1 √ 3     1 0 0 0 1 0 0 0 -2     .
They can easily be related to spin-1 matrices, which in the standard representation read

S x = 1 √ 2 (λ 1 + λ 6 ), S y = 1 √ 2 (λ 2 + λ 7 ), S z = 1 2 (λ 3 + √ 3λ 8 ). ( C1
)
The N = 4 Gell-Mann matrices [START_REF] Pfeifer | The Lie algebras su(N )[END_REF][START_REF] Bertlmann | Bloch vectors for qudits[END_REF] are given by the extended SU(3) Gell-Mann matrices,

λ 1 =        0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0        , λ 2 =        0 -i 0 0 i 0 0 0 0 0 0 0 0 0 0 0        , λ 3 =        1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0        , λ 4 =        0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0        , λ 5 =        0 0 -i 0 0 0 0 0 i 0 0 0 0 0 0 0        , λ 6 =        0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0        , λ 7 =        0 0 0 0 0 0 -i 0 0 i 0 0 0 0 0 0        , λ 8 = 1 √ 3        1 0 0 0 0 1 0 0 0 0 -2 0 0 0 0 0       
, plus an additional seven matrices

λ 9 =        0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0        , λ 10 =        0 0 0 -i 0 0 0 0 0 0 0 0 i 0 0 0        , λ 11 =        0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0        , λ 12 =        0 0 0 0 0 0 0 -i 0 0 0 0 0 i 0 0        , λ 13 =        0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0        , λ 14 =        0 0 0 0 0 0 0 0 0 0 0 -i 0 0 i 0        , λ 15 = 1 √ 6        1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -3        .
This can easily be used to express the standard representation of spin-3/2 matrices:

S x = 1 2 ( √ 3λ 1 + 2λ 6 + √ 3λ 13 ), S y = 1 2 ( √ 3λ 2 + 2λ 7 + √ 3λ 14 ), S z = 1 2 (λ 3 + √ 3λ 8 + √ 6λ 15
).

(C2)

Appendix D: Derivation of Eq. ( 17)

To find the eigenprojector P α as a polynomial in the Hamiltonian, our starting point is the following textbook formula, involving the set {E β , β = 1, ..., N } of all energy eigenvalues [START_REF] Halmos | Finite-Dimensional Vector Spaces: Second Edition[END_REF]:

P α = β =α H -E β 1 N E α -E β . (D1)
Note that in the language of matrix theory, Eq. (D1) defines the Frobenius covariants of H. Note also that the denominator of Eq. (D1) corresponds to the derivative of the Hamiltonian's characteristic polynomial. More generally, before proceeding it proves useful to compile some more details on the characteristic polynomial p N (z).

Characteristic polynomial

The characteristic polynomial of an N × N matrix A is given by [START_REF] Gantmacher | The Theory of Matrices[END_REF] pN (z) = det(z1

N -A) = N k=0 ck z N -k = N α=1 (z -a α ), ( D2
)
where a α denotes an eigenvalue of A and c0 = 1. According to the Faddeev-Le Verrier algorithm [START_REF] Gantmacher | The Theory of Matrices[END_REF], the coefficients ck may be computed from the traces

s k ≡ Tr A k of powers of A as ck = - 1 k (s k + c1 s k-1 + ... + ck-1 s 1 ) = (-1) k k! Y k (s 1 , ..., (-1) k-1 (k -1)!s k ). (D3)
The second equality involves (exponential) complete Bell polynomials Y k (z 1 , ..., z k ) [START_REF] Bell | Exponential Polynomials[END_REF][START_REF] Comtet | Advanced Combinatorics: The Art of Finite and Infinite Expansions[END_REF], the first few of which read explicitly

Y 0 = 1, Y 1 (z 1 ) = z 1 , Y 2 (z 1 , z 2 ) = z 2 1 + z 2 , Y 3 (z 1 , z 2 , z 3 ) = z 3 1 + 3z 1 z 2 + z 3 , Y 4 (z 1 , z 2 , z 3 , z 4 ) = z 4 1 + 6z 2 1 z 2 + 4z 1 z 3 + 3z 2 2 + z 4 . (D4)
Focusing now on the case where A = H represents an N × N Hamiltonian matrix, we have the Hamiltonian's characteristic polynomial

p N (z) = det(z1 N -H) = N k=0 c k z N -k = N α=1 (z -E α ). (D5)
Using the traceless character of the Hamiltonian, cf. Eq. ( 16), and the Casimir invariants defined in Eq. ( 18), the coefficients are given by

c k = (-1) k k! Y k (0, -C 2 , ..., (-1) k-1 (k -1)!C k ), (D6)
and the first few of them read explicitly

c 0 = 1, c 1 = 0, c 2 = - C 2 2 , c 3 = - C 3 3 , c 4 = C 2 2 8 - C 4 4 , c 5 = C 2 C 3 6 - C 5 5 , c 6 = - C 3 2 48 + C 2 3 18 + C 2 C 4 8 - C 6 6 .
(D7)

where the gauge freedom state |ψ g can be chosen arbitrarily. Each eigenstate (E2) is minimally encoded by a global phase Γ α and N -1 pairs of eigenstate's angles (θ i α , φ i α ) (i = 1, ..., N -1). Two more remarks can be made. First, although this is not immediately obvious, the eigenstate's angles stay unchanged upon varying |ψ g for fixed x. In contrast, the global phase is changing, implying Γ α ≡ Γ α (x, |ψ g ). As a consequence, the possible singular behaviors of the wavefunction in parameter space are gauge-dependent. Second, the state |ψ g must not be orthogonal to the projector P α (x), i.e. one requires P α (x) |ψ g = 0. This constraint implies that it might be necessary to change the state |ψ g when the parameter x is varying because it is never guaranteed that a single |ψ g (meaning a fixed gauge) is sufficient to describe a given eigenstate |ψ α over the entire parameter space x.

Applying the recursion (F6), for a traceless Hamiltonian matrix, we obtain successively up to n = 4 the important identities

C 1 = 0, η 1 = h, C 2 = 2|h| 2 , η 2 = h , C 3 = 2 h • h , η 3 = C 2 N h + h , C 4 = 4|h| 4 N + 2|h | 2 , η 4 = C 3 N h + C 2 N h + h . (F8)
More generally, for a generic N > 1, the form of the vector η n (h) is compactly written as:

η n = 1 N n-1 p=0 C p h (n-1-p) . ( F9
)
The final step required for completing our task of finding the explicit expressions of the Bloch vector b α (h, E α ) consists in substituting Eqs. (F8) & (F9) into Eq. (F2). One can then write down the explicit Bloch vectors for any N , as done explicitly in Eq. ( 23) for N = 2 to N = 5. As a final remark, be aware that, for given N , only C n≤N and η n<N are relevant, and there are only N -1 independent vectors h (k) , k = 0, ..., N -2. For example, for N = 3, all information we need is encoded in C 2 , C 3 , h and h . This again follows from the Cayley-Hamilton theorem, which states that q n=N (H) = p N (H) = 0. From this property it is easy to establish the following useful identities:

N = 3 : h = C 2 6 h, h h = C 3 3 h - C 2 6 h , N = 4 : h = C 3 12 h + C 2 4 h , h h = C 3 3 h, h h = |h | 2 2 h + C 3 12 h . ( F10
)
Appendix G: SU(N ) Jacobi identities

The first Jacobi identity [START_REF] Kaplan | Matrix Products and the Explicit 3, 6, 9, and 12-j Coefficients of the Regular Representation of SU(n)[END_REF][START_REF] Macfarlane | On Gell-Mann's λ-matrices, d-and f-tensors, octets, and parametrizations of SU(3)[END_REF] can be written alternatively for the generator matrices, the antisymmetric structure constants and the SU(N ) vectors as

[[λ i , λ j ], λ k ] + [[λ j , λ k ], λ i ] + [[λ k , λ i ], λ j ] = 0, f ijm f klm + f ikm f ljm + f ilm f jkm = 0, m × (n × o) + n × (o × m) + o × (m × n) = 0, (m × n) • (o × p) + (m × o) • (p × n) + (m × p) • (n × o) = 0, (G1)
where the third and fourth lines are obtained from the second line depending on whether or not one keeps a free index. Similarly, the second Jacobi identity is given by ρα with the large sphere, is the generalized (eigenprojector's) Bloch sphere. For the familiar N = 2 case, these complications are hidden, since all of the relevant sets coincide.

[{λ i , λ j }, λ k ] + [{λ j , λ k }, λ i ] + [{λ k , λ i }, λ j ] =
The first two lines are valid for general N and can be proved using the second Jacobi identity (G2). The proof of the third line requires the Jacobi indentity as well, but additionally the SU(3)-specific identities (F10). Inserting all of these identities into Eq. (K4), and exploiting the characteristic equation E 3 α = C2 2 E α + C3 3 , one obtains the generic SU(3) Berry curvature formula Ω α,ij = -4

(3E 2 α -|h| 2 ) 3 E α |h| 2 h • h i × h j + 3 h • h i × h j + 3E 2 α + |h| 2 h • h i × h j . (K6)
The advantage of this formula, as compared to Eq. ( 36), consists in the fact that it contains only three terms, and that the Berry curvature sum rule α Ω α,ij = 0 is more evident, since α E α (3E 2 α -|h| 2 ) -3 = 0 and α (3E 2 α + |h| 2 )(3E 2 α -|h| 2 ) -3 = 0. In principle, one can continue in this way to obtain the Berry curvature for any N . As a shortcut, it is however useful to realize that, when the Bloch vectors [START_REF] Julku | Geometric Origin of Superfluidity in the Lieb-Lattice Flat Band[END_REF] are substituted into Eq. ( 35), the derivatives effectively do not act on the prefactors but only on the vectors h, h , etc. This is due to the orthogonality relations (F5) & (F7). For example, for the N = 2 case, it suffices to replace b i α → 1 Eα h i in Eq. [START_REF] Barnett | SU(3) Spin-Orbit Coupling in Systems of Ultracold Atoms[END_REF], which allows to directly read off the Berry curvature: Ω α,ij = -1 2 h Eα • ( h i Eα × h j Eα ). Similarly, for N = 3, it suffices to replace b i α → 2 3E 2 α -C2/2 (E α h i + h i ), and so on for higher N . In this way, one directly obtains Eqs. ( 36)- [START_REF] Pozo | Computing observables without eigenstates: Applications to Bloch Hamiltonians[END_REF] in the main text.

FIG. 1 .

 1 FIG.1. Two alternative points of view on quantum geometry.

FIG. 2 .

 2 FIG. 2. Schematic visualization of the different Bloch sphere's appearing in the treatment of SU(N ) Hamiltonians. The familiar SU(2) case (Section II C) is illustrated in the first row, while the second row treats the general case of arbitrary N (Section III D). The Hamiltonian's Bloch sphere B (N ) h is the relevant space for the mapping u h (x), and corresponds to a proper unit sphere for all values of N . The eigenprojector's Bloch sphere B (N ) P α associated to the mapping bα(x) is only a proper sphere for N = 2. In the N > 2 case, where it is often called generalized Bloch sphere, it is of very complicated shape. The eigenstate's Bloch sphere B (N ) α associated to the mapping |ψα(x) is also a proper unit sphere for all N , but of different dimension than the Hamiltonian's Bloch sphere if N > 2.

  (d)-(f) and Section III D].

FIG. 3 .

 3 FIG. 3. (a) Energy spectrum of a pseudospin-1 fermion and of the model (52). (b) Energy spectrum of a pseudospin-3/2 fermion and of the model (58).

FIG. 4 .

 4 FIG. 4. Schematic illustration of the generalized Bloch sphere and other relevant sets, for N = 2, 3, 4. The large (N 2 -2)sphere W N 2 -2 , of radius 2(N -1)/N , corresponds to the highest permissible length of a Bloch vector. In its interior, the Ball B N 2 -1 , there is the space Σ (N ) ρα accessible to a mixed state of type (H1), with topological boundary ∂Σ (N ) ρα , and in the interior of Σ (N ) ρα there is the small (N 2 -2)-sphere V N 2 -2 , of radius 2/[N (N -1)]. The extremal boundary B (N ) Pα of Σ (N ) ρα , which is the intersection of Σ (N )

ACKNOWLEDGEMENTS

We thank Mark-Oliver Goerbig and Andrej Mesaros for fruitful discussions.

Appendix A: Gauge dependency and singularities of eigenstates Here, we describe some issues of eigenstates that are absent in an eigenprojector approach. Consider the two-band eigenstates [START_REF] Thouless | Quantized Hall Conductance in a Two-Dimensional Periodic Potential[END_REF]. The eigenstate for α = + can be written in either of the forms and similarly for |ψ -(x) . The first expression corresponds to the formal parametrization of a complex-valued twocomponent unit vector. It expresses the fact that, at each point x in the parameter space, the eigenstate |ψ + (x) is minimally encoded by a global phase Γ + (x) and two eigenstate's angles (θ + (x), φ + (x)). Thus, the first evident problem of Eq. (A1) consists in the fact that it is gauge-dependent. The second expression in Eq. (A1) illustrates a second problem, namely the possible singular behavior of eigenstates. A singularity is evidently to be expected when h z (x 0 )/E + (x 0 ) = -1 for some point x = x 0 in parameter space. Similarly, if the eigenstate components are written in terms of the angles (θ h , φ h ), the explicit relation tan φ h = h y /h x points towards further possible singular behavior in parameter space when h x (x 0 ) = 0. Finally, the closed-form expression of |ψ + (x) (in terms of the corresponding energy eigenvalue and the components of the Hamiltonian vector h) is rather cumbersome and not very convenient if one is dealing with classes of Hamiltonians beyond a specific Hamiltonian of interest. This issue in particular becomes much more dramatic as soon as N > 2. In this case, similar closed-form expressions are practically useless.

Appendix B: Quantum geometric tensor for different parameter spaces

The explicit dimension and form of the matrices g α and Ω α clearly depends on the parameter space. For example, in a two-band system, taking x = (θ h , φ h ) (d = dim(x) = 2) and inserting the state (A1) into Eq. ( 2), the quantum metric and Berry curvature tensors are 2 × 2 matrices that write as

which is a well-known result [START_REF] Kolodrubetz | Geometry and non-adiabatic response in quantum and classical systems[END_REF]. Taking instead x = (h x , h y , h z ) (d = dim(x) = 3) for the same two-band state, the quantum metric and Berry curvature tensors are now 3 × 3 matrices with matrix elements

where ijk is the Levi-Civita antisymmetric tensor. Note that g -,ij = g +,ij and Ω -,ij = -Ω +,ij for the state |ψ -. More generally, the QGT of interest is often related to the explicit dependency of the Hamiltonian vector h(x) on some vector of external parameters x, with d = dim(x) ≥ 2. For example, in condensed matter physics, one will often have x = k, where k represents crystal momentum. In that situation, the corresponding QGT T α,ij (x) may be obtained from either T α,kl (h) or T α,kl (θ h , φ h ) by a simple composition rule as

with y = h or y = (θ h , φ h ).

Appendix C: Gell-Mann and spin matrices

Here we list the N = 3 (N = 4) Gell-Mann matrices and relate them to spin-1 (spin-3/2) matrices. Note that the generalization to N ≥ 5 Gell-Mann matrices (not listed here explicitly) is straightforward, see for example Refs. [START_REF] Pfeifer | The Lie algebras su(N )[END_REF][START_REF] Bertlmann | Bloch vectors for qudits[END_REF]: There are N (N -1)/2 symmetric matrices (purely real), N (N -1)/2 antisymmetric matrices (purely imaginary), and N -1 diagonal matrices. The goal is now to eliminate all E β =α from Eq. (D1), such that P α becomes a proper polynomial in H with coefficients that depend only on the single eigenvalue E α , i.e. P α = P α (E α , H). Consider first the numerator of Eq. (D1) and note that by explicit multiplication one may write

where e n = e n (E 1 , ..., E α-1 , E α+1 , ..., E N ) are known as elementary symmetric polynomials [START_REF] Macdonald | Symmetric Functions and Hall Polynomials, Oxford classic texts in the physical sciences[END_REF]. One has e 0 = 1 and all higher e n are determined recursively by Newton's identities:

where the C k are the Casimir invariants of Eq. ( 18) and it was exploited that

This may further be rewritten as

where c k are the coefficients (D6) of the characteristic polynomial. If we now define polynomials

is the characteristic polynomial (D5), and q n (E α ) = (-1) n e n . Moreover, inserting into Eq. (D8), we have

Similarly, for the numerator of Eq. (D1), exactly the same procedure as above (where H gets replaced by E α ) leads to

As mentioned in the main text,

From this one may also show that

Combining all of these results, one arrives at Eq. [START_REF] Zanardi | Information-Theoretic Differential Geometry of Quantum Phase Transitions[END_REF].

Appendix E: Eigenstates from eigenprojectors

The eigenprojector P α permits to construct the eigenstate |ψ α from an arbitrary gauge freedom state |ψ g . For example, in the two-band case, using the eigenprojector P + given by Eq. ( 19), a state equivalent to Eq. (A1) is easily constructed as

where the gauge freedom angles (θ g , φ g ) can be chosen at will at any point in parameter space x. For example, if cos θ g = 1 and sin θ g = 0, one exactly recovers the second expression in Eq. (A1). More generally, the N -band eigenstate |ψ α in an arbitrary gauge may be obtained as

Appendix F: Derivation of Bloch vector formula bα(Eα, H)

Here we detail the steps that allow to find b α (h, E α ) from the eigenprojector formula [START_REF] Zanardi | Information-Theoretic Differential Geometry of Quantum Phase Transitions[END_REF]. The only missing ingredient is H n = (h • λ) n in terms of the generators of SU(N ). Defining vectors η n ≡ Tr(H n λ)/2, we can write:

where obviously η 0 = 0 and η 1 = h. Inserting into Eq. ( 17) and using Eq. ( 22), we obtain the intermediate result:

At this point it remains the task to find the explicit form of the vectors η n (h) for n > 1. To accomplish this task, we require a product identity generalizing the familiar SU( 2) identity [START_REF] Qi | Topological insulators and superconductors[END_REF] to any N . This can be readily achieved by using Eq. ( 25) to obtain

Since all generators are traceless, Tr λ c = 0, the convenient trace orthogonality Tr(λ a λ b ) = 2δ ab holds. From Eqs. (F3) & ( 27), one directly obtains the desired product identity in vector form:

When computing H n = (h • λ) n by applying Eq. (F4) repeatedly, it proves useful to introduce the notation [START_REF] Kolodrubetz | Classifying and measuring geometry of a quantum ground state manifold[END_REF] for repeated star products of a vector with itself. The resulting vectors have the following properties (with n 1 , n 2 ∈ N 0 ):

The former identity follows directly from the total symmetry of the structure constants [START_REF] Lapa | Semiclassical wave packet dynamics in nonuniform electric fields[END_REF], and the latter is a consequence of the second Jacobi identity, as can be proved by mathematical induction. More generally, from the generic properties of the algebra, we have established useful generalized vector and scalar Jacobi identities, listed in Appendix G.

With all these prerequisites we may now calculate (h • λ) n i.e. determine C n (h) and η(h). In particular, we obtain the following simple recursion relations:

with initial conditions C 0 = N and η 0 = 0. Making use of the fact that η n and the structure constants f abc are real by definition, we can also establish the identity

Appendix H: The generalized Bloch sphere

We give here a short summary of the concept of a generalized Bloch sphere [i.e. the SU(N ) eigenprojector's Bloch sphere B

(N )

Pα introduced in Section III C], drawing largely on Refs. [START_REF] Harriman | Geometry of density matrices. I. Definitions, N matrices and 1 matrices[END_REF][START_REF] Jakóbczyk | Geometry of Bloch vectors in two-qubit system[END_REF][START_REF] Kimura | The Bloch vector for N -level systems[END_REF][START_REF] Zyczkowski | Hilbert-Schmidt volume of the set of mixed quantum states[END_REF][START_REF] Byrd | Characterization of the positivity of the density matrix in terms of the coherence vector representation[END_REF][START_REF] Kimura | The Bloch-Vector Space for N -Level Systems: the Spherical-Coordinate Point of View[END_REF][START_REF] Mendaš | The classification of three-parameter density matrices for a qutrit[END_REF][START_REF] Goyal | Geometry of the generalized Bloch sphere for qutrits[END_REF]. For an N -dimensional Hilbert space, the three defining properties of a (mixed state) density matrix ρ α representing an N -component (mixed) quantum state |ψ α , where α = 1, ..., N , are hermiticity ρ † α = ρ α , probability conservation Tr ρ α = 1 and positive semidefiniteness, ρ α ≥ 0. Pure states have, in addition, ρ 2 α = ρ α , in which case ρ α = P α is an eigenprojector. An expansion in the basis of SU(N ) generator matrices analogous to Eq. ( 22) can be made for a general (mixed state) density matrix:

where now |b α | can take various values depending on the pureness of the state. In the pure state case, we have |b α | = 2(N -1)/N . Considering the vector space R N 2 -1 , and denoting the (N 2 -1)-dimensional subspace accessible to the (mixed state) Bloch vector b α as Σ (N ) ρα , one needs to distinguish between two kinds of boundaries, namely its (N 2 -2)-dimensional topological boundary ∂Σ Pα . The latter is of dimension 2(N -1), and it comprises the vectors of maximal |b α |, i.e. it is the space of pure state Bloch vectors, or in other words the generalized Bloch sphere, sketched in Fig. 4.

Another set of immediate interest in this discussion is the sphere

that contains all pure state Bloch vectors; in other words, it is the surface of the ball B N 2 -1 , the smallest ball that contains Σ

(N ) ρα . For N = 2, one trivially has ∂Σ

(2)

Pα = W 2 (and W 2 = S 2 , with the unit two-sphere S 2 ), as illustrated in Fig. 4; but for higher N the three spaces are different. One has

i.e. the generalized Bloch sphere is a proper subset of (i) the boundary of the space of mixed states and (ii) the sphere W N 2 -2 ; in fact, it is their intersection:

Pα . In other words, if b α corresponds to a pure state, it lies on W N 2 -2 , but not necessarily the other way around, i.e. not all points on W N 2 -2 represent physically valid pure states.

Analogously, the interior of W N 2 -2 , i.e. the ball B N 2 -1 , is not composed of only physically valid mixed states. In fact, calculations [START_REF] Zyczkowski | Hilbert-Schmidt volume of the set of mixed quantum states[END_REF] of the volume of Σ At the origin of all of these complications is the constraint of positive definiteness, ρ α ≥ 0: for N = 1, this condition is trivially fulfilled. For N = 2, it is equivalent to Tr ρ 2 α ≤ 1, so for a given density matrix there is one additional constraint as compared to N = 1. In the same way, whenever N increases by one, one additional constraint involving traces D α,n ≡ Tr(ρ n α ) needs to be fulfilled in order to have positive definiteness. For example, for N = 3, there are the three constraints [START_REF] Kimura | The Bloch vector for N -level systems[END_REF] D α,1 = 1, D α,2 ≤ 1, and -2D α,3 + 3D α,2 ≤ 1, which can be translated to constraints on the Bloch vectors via Eq. (H1). This restrict the elements of B 8 that represent physically valid states. As a consequence, the relevant sets (including the generalized Bloch sphere) acquire a very nontrivial shape already for N = 3 [START_REF] Goyal | Geometry of the generalized Bloch sphere for qutrits[END_REF]:

where the star product is defined in Eq. ( 27), and the agreement of the first line with Eq. ( 30) is to be noted.

Similarly, more complicated constraints can be obtained for higher N , see Ref. [START_REF] Kimura | The Bloch vector for N -level systems[END_REF]. Finally, note that there exists another sphere

ρα lies between V N 2 -2 and W N 2 -2 . The essence of all of these established facts is visualized schematically in Fig. 4. Note that the figure is only a rough sketch supposed to highlight the different sets involved. A visual insight into the true (very complicated) shape of those sets can be gained by considering two-or three-sections, see for instance Refs. [START_REF] Jakóbczyk | Geometry of Bloch vectors in two-qubit system[END_REF][START_REF] Kimura | The Bloch-Vector Space for N -Level Systems: the Spherical-Coordinate Point of View[END_REF][START_REF] Mendaš | The classification of three-parameter density matrices for a qutrit[END_REF][START_REF] Goyal | Geometry of the generalized Bloch sphere for qutrits[END_REF].

Appendix I: From eigenstate to eigenprojector picture of the QGT The QGT is a tensor quantifying the overlap of two states that are "infinitesimally close" in parameter space [START_REF] Provost | Riemannian structure on manifolds of quantum states[END_REF]: x) , where the modulus F α (x) has come to be known as fidelity [START_REF] Gu | Fidelity approach to quantum phase transitions[END_REF]. The fidelity is related to the quantum metric, while the phase φ α (x) gives rise to the Berry curvature. This can be seen by expanding

Noting that Re ψ α (x)|∂ ij ψ α (x) = -Re ∂ i ψ α (x)|∂ j ψ α (x) and introducing the Berry connection A α,i (x) ≡ i ψ α (x)|∂ i ψ α (x) = -Im ψ α (x)|∂ i ψ α (x) , one immediately finds

Now, the quantum metric tensor g α,ij (x) is defined [START_REF] Provost | Riemannian structure on manifolds of quantum states[END_REF][START_REF] Kolodrubetz | Classifying and measuring geometry of a quantum ground state manifold[END_REF] by ij g α,ij dx i dx j ≡ 1 -F 2 α , such that

Upon calculating the Berry curvature Ω α,ij (x) ≡ ∂ i A α,j -∂ j A α,i , one can observe that g α,ij and Ω α,ij are parts of the single complex tensor (2). An eigenstate-based version of the QGT that is used more frequently than Eq. ( 2) is given by Eq. [START_REF] Gianfrate | Measurement of the quantum geometric tensor and of the anomalous Hall drift[END_REF]. From the latter, Eq. ( 2) is easily recovered upon using the identity

To obtain the projector-based formula (3), Eq. ( 33) may be rewritten in an explicitly gauge-invariant form:

where we used the identity ψ α | O |ψ α = Tr {P α OP α } = Tr {P α O}. From there, by inserting H = γ E γ P γ , it is straightforward to arrive at Eq. (3).

Appendix J: Alternative QGT formula in terms of Bloch vectors A formula equivalent to Eq. ( 35) but without derivatives acting on Bloch vectors can be obtained by inserting Eq. ( 22) into Eq. (I4) and exploiting the Jacobi identity (G3):

(J1)

Here, we have defined

It is straigthforward to check that S ij αβ is symmetric under the exchange of indices i, j (or of α, β), whereas A ij αβ is antisymmetric under such exchange, and as a consequence α Ω α,ij = 0. While apparently much less compact than Eq. ( 35), the expressions (J1) appear more convenient for numerical implementation. Conceptually, they mainly distinguish themselves from Eq. ( 35) in that, on the one hand, they involve solely the parametric derivatives of the Hamiltonian vector, while on the other hand they also illustrate the interband nature of the two geometric tensors, and lastly they also show explicitly the importance of the star product for both the N -band quantum metric and Berry curvature.

q N -1-n (E α )q N -1-m (E α ) τ (i) n × τ (j) m .

(K1)

Here we use the polynomials q n (z) and have defined vectors

With the notation m i ≡ ∂ i m, the first few τ

From Eq. (K1), one can obtain the Berry curvature for any given N . For N = 2, Eq. ( 14) is immediately obtained. For N = 3, analogously, we have

or equivalently Eq. [START_REF] Lee | Arbitrary Chern Number Generation in the Three-Band Model from Momentum Space[END_REF] in the main text. This result can be simplified due to the following identities:

(K5)


Philip Murgatroyd

Kieran Routledge

Samantha Durdy

Michael Gaultois

T Wesley Surta

Matthew Dyer

John Claridge

Stanislav Savvin

D Pelloquin

Sylvie Hébert

Chemically Controllable Magnetic Transition Temperature and Magneto-Elastic Coupling in MnZnSb Compounds

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Marc Dhenain
email: marc.dhenain@cea.fr

Ines Abdeljaoued-Tej
email: ines.abdeljaoued@u-carthage.tn

Ines Abdeljaoued

Keywords: estimated number of cases, mortality, reported prevalence, reported and unreported cases, isolation, quarantine, public closings, epidemic mathematical model

Estimating the number of people affected by COVID-19 is crucial in deciding which public health policies to follow. The authorities in different countries carry out mortality counts. We propose that the mortality reported in each country can be used to create an index of the number of actual cases at a given time. The specificity of whether or not deaths are rapid or not by COVID-19 also affects the number of actual cases. The number of days between the declaration of illness and death varies between 12 and 18 days. For a delay of 18 days, and using an estimated mortality rate of 2%, the number of cases in April 2020 in Tunisia would be 5 580 people. The pessimistic scenario predicts 22 320 infected people, and the most optimistic predicts 744 (which is the number of reported cases on April 12, 2020). Modeling the occurrence of COVID-19 cases is critical to assess the impact of policies to prevent the spread of the virus.

INTRODUCTION

Infection with the Sars CoV-2 coronavirus that causes COVID-19 has spread throughout the world and is causing a significant number of deaths [START_REF] Who | Coronavirus disease 2019 (COVID-19) situation reportâĂŤ43[END_REF]. Every day estimates of the number of people affected and a death toll are provided by authorities in different countries and distributed worldwide. We address the following fundamental issues concerning this epidemic: How will the epidemic evolve in Tunisia for the number of reported cases and unreported cases? How will the number of unreported cases influence the severity of the epidemic? How will public health measures, such as isolation, and public closings, impact the final size of the epidemic? Knowing the number of people affected is important for implementing strategies to protect populations and to end the crisis: In [START_REF] Lauer | The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application[END_REF] the median incubation period of COVID-19 was studied, but publicly reported cases may over-represent severe cases. This number reveals flagrant differences between countries. For example, on April 13 th , a day when the number of worldwide deaths was close to 120 000, Tunisia reported around 707 people affected (60 per million the population) and 31 deaths, compared with many more 136 779 affected persons in France (2 095 per million) for 14 967 deaths. China reported around 82 000 people affected (57 per million) and more than 3 300 deaths. Germany, for its part, reported 130 000 people affected (1 552 per million) and 3 194 deaths. At first glance, differences between countries can be objectified by calculating the mortality rate T on a given day (t 0 ):

T = M t0 C t0 (1)
With M t0 : Number of deaths on that day t 0 ; C t0 : Number of cases reported on t 0 . We follow the reasoning of [START_REF] Dhenain | Estimation du nombre de cas de COVID-19 en France et dans differents pays : Homogeneisation basée sur la mortalité[END_REF] that the fatality reported in each country can be used to create an index of the number of actual cases at a time t 0 . This index is :

C t0 = M t0 T * C t0 C t0-18 (2)
Where T : Estimated mortality rate; C t0-18 : Number of cases reported in the country at time t 0 minus 18 days. The proposed formula also makes it possible to evaluate the impact of policies to prevent the spread of the virus on the appearance of new cases. It is useful to more accurately estimate the number of people affected using a more consistent method for all countries [START_REF] Baumgaertner | How deadly is the new coronavirus? Scientists race to find the answer[END_REF]. We propose to follow a method developed in [START_REF] Dhenain | Estimation du nombre de cas de COVID-19 en France et dans differents pays : Homogeneisation basée sur la mortalité[END_REF], to use the number of deaths reported by each country to estimate and compare the actual rate of affected people. This approach is based on four assumptions:

1. The mortality reported by each country is reliable 2. The fatality rate (T ) is known and is similar in different countries 3. The average time between the onset of symptoms and death is known 4. The increase over time in the number of cases reported in the databases over the average time from symptom onset to death reflects the increase over time in the actual number of cases over the same period. In [START_REF] Li | Early transmission dynamics in Wuhan, China, of novel coronavirusinfected pneumonia[END_REF] the epidemiological characteristics of patients are studied, and all patients who die on a given day have been infected much earlier, so the denominator of the fatality rate should be the total number of patients infected at the same time as those who died [START_REF] Lipsitch | Estimating case fatality rates of COVID-19[END_REF]. This is especially true as the rates of progression of the pandemic evolve differently in different countries: in April 2020, the number of people affected was increasing sharply from day to day in France, while it had stabilized in China. Figure 2 shows the reported mortality rate of COVID-19 in Tunisia.

A better estimation of the fatality rate is therefore available:

T -x = M t0 C t0-x (3)
Where C t0-x is the number of cases reported on day t 0 minus x days, and x is the average time from symptom onset to death. A mean duration of 18 days is reported between the onset of symptoms of COVID-19 and patient death [START_REF] Verity | Estimates of the severity of coronavirus disease 2019: a model-based analysis[END_REF]. It has a 95% credible interval of 16.9 to 19.2 days [START_REF] Flaxman | Report 13: Estimating the number of infections and the impact of nonpharmaceutical interventions on COVID-19 in 11 European countries[END_REF]. An adjusted mortality rate T -18 can, therefore, be calculated by taking into account this average 18-day delay.

We can easily see in Figure 1 that the reported cases of the first two weeks were underestimated. Containment measures in Tunisia start on March 12 and become more restrictive on March 25. The median of T -18 in Figure 3 is equal to 0.67 and its mean is equal to 0.71. They are equal to 0.5 from March 30, 2020. For t 0 equal to 12 April 2020, Table 1 gives the main mortality rates. The computation takes into account the number of cases on March 26 th , 2020. Data is available in [START_REF]Data repository for the 2019 Novel Coronavirus Visual Dashboard[END_REF].

RESULTS

With C t0-18 the number of cases reported on the day t 0 minus 18 days. Values of -18 in Table 1 reveal a great variation compared to T with variations ranging from 1% (Hong Kong, China) to 80% for Algeria.

The slope of this relationship varies greatly between countries, which is consistent with variable T -18 . In [START_REF] Dhenain | Estimation du nombre de cas de COVID-19 en France et dans differents pays : Homogeneisation basée sur la mortalité[END_REF] the values of T -18 based on cases reported by different countries are therefore unreliable. Several highly controlled international studies (mainland Chinese residents, returnees from mainland China, returnees from China, passengers on the Diamond-Princess cruise ship) reported T -18 values ranging from 0.7 to 3.6% [START_REF] Verity | Estimates of the severity of coronavirus disease 2019: a model-based analysis[END_REF]. Based on the latter study, an estimated mortality rate with T = 2% can be used. Knowing the number of deaths on day t 0 , the number of cases C t0-18 eighteen days before is estimated:

C t0-18 = M t0 T . (4)
The progression of the number of cases reported over the last eighteen days P 18 is known in each country. It depends on the rate of containment and its effectiveness. It is equal to:

P 18 = C t0 C t0-18 . (5)
When comparing M t0 and C t0-18 in different countries, [START_REF] Dhenain | Estimation du nombre de cas de COVID-19 en France et dans differents pays : Homogeneisation basée sur la mortalité[END_REF] shows a linear relationship between mortality at t 0 and the number of cases at t 0 -18 days for most of them. The number of cases estimated at t 0 in each country is therefore

C t0 = C t0-18 * P 18 . (6)
Values of T = 0.5% to 4% could be other suitable options. One of the limitations of our model is that this value may vary from country to country, for example depending on the distribution of the population into different age groups that have different sensitivity to COVID-19 [START_REF] Virlogeux | Comparison of incubation period distribution of human infections with MERS-CoV in South Korea and Saudi Arabia[END_REF]. Figure 4 shows the Tunisia cases' estimation, where T = 2% and 4%: The number of reported cases is underestimated. Note that some authors suggest that the real mortality rate for Covid-19 could be 5.6 to 15.6% [START_REF] Baud | Real estimates of mortality following COVID-19 infection[END_REF], which is much higher values than those we used.

The population of Tunisia is equal to 11 818 619. For C t0 = 707 we obtain Table 3, which gives the evaluation of the number of cases in Tunisia as of April 12 th , 2020 by modulating the estimated mortality rate T and taking into account delays of 12, 15 and 18 days between the onset of symptoms (or asymptomatic) and death [START_REF] Zhou | Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study[END_REF]. We obtain P 12 = 1.8, P 15 = 2.5 and P 18 = 3.6.

It is conceivable that some people who died as a result of COVID-19 may not be counted. The number of infected people may be much higher than the number reported. Figure 5 estimates the number of cases in Tunisia as of April 12 th , 2020 by modulating the estimated mortality rate T from 0.5 to 4% and taking into account delays of 12 days between the onset of symptoms (or asymptomatic) and death.

Ines Abdeljaoued-Tej and Marc Dhenain Table 3. Evaluation of the number of COVID-19 cases as of April 12 th , 2020 by where the estimated mortality rate T equals to 0.5%, 2%, 4%, 15%, and taking into account delays of 12, 15 and 18 days between the onset of symptoms (or asymptomatic) and death

DISCUSSION

This work uses an intuitive model, rather than other more complex models, such as [START_REF] Flaxman | Report 13: Estimating the number of infections and the impact of nonpharmaceutical interventions on COVID-19 in 11 European countries[END_REF] based on Bayesian approaches or [START_REF] Chinazzi | The effect of travel restrictions on the spread of the 2019 novel coronavirus (COVID-19) outbreak[END_REF]. As expected, our analysis suggests that the number of cases of COVID-19 in several countries exceeds the number of cases presented in international databases. Our results can contribute to the prevention and control of this epidemic in Tunisia. As a consequence of our study, we note that the number of cases is very sensitive. We can find multiple values of reported and unreported cases that provide a good fit for the data. Currently, there are many unanswered questions about this novel coronavirus, and this work enables us to clarify the effect of the mortality rate on the dynamic of the virus infection.

Public health measures, such as isolation, quarantine, and public closings, greatly reduce the final size of the epidemic and make the turning point much earlier than without these measures. For example, for Tunisia on April 12, P 18 = 3.6, while on March 25, 2020, when regulation of progression rate based on population containment was still not fully efficient, P 18 = 127. We can therefore estimate that the containment made it possible to reduce P 18 from 127 to 3.6. The number of cases estimated on April 12 using P 18 = 175 would have been 26 246 cases (with T = 15%). Thus, the containment from March 25 to April 12 prevented the appearance of 25 539 new cases in Tunisia, as well as associated fatalities.

An epidemic outbreak of a new human coronavirus COVID-19 will occur in Tunisia. For this outbreak, the unreported cases and the disease transmission rate are not identified. In order to recover these values from reported medical data, this simple mathematical model for estimating the COVID-19 cases is used. The knowledge of the cumulative reported symptomatic and asymptomatic infectious cases and assuming the infectious delay until death to be between 12 days and 18 days, this model estimates the number of infected cases. Then numerical simulations of the model are done to predict forward in time the severity of the epidemic, with adjusted data. We find that the most pessimistic number of infected people is 22 320 and the most optimistic is 744 at April 12 th , 2020, and when the mortality rate is equal to 2%, the estimation number is equal to 5 580 (the number of reported cases at this date is 707 people).

This model inflated the number of cases, but it is at most a preliminary estimate. What is significant, in Tunisia and also in France, is that the public authorities are becoming aware of the necessity to properly quantify the number of deaths. This work raises interesting questions about the relation between the fatality rate and the number of infected cases. Our model used a mortality rate of 2% while several international studies reported rates of 0.7 to 3.6% [START_REF] Verity | Estimates of the severity of coronavirus disease 2019: a model-based analysis[END_REF]. Values from 0.5 to 4% could thus be other reasonable options to estimate mortality rate. One of the limitations of our model is that mortality rates can change from one country to another, depending on the distribution of the population in different age groups and on the co-morbidity that have different sensitivities to Covid-19. In the other hand, the ability of the virus to persist in different environments (hot climate) can affect this relation. Hence the importance of comparisons between countries with more and less sunshine, at different seasons and periods.

Ines Abdeljaoued-Tej and Marc Dhenain

The progression rate P 18 shows the increase in the number of reported cases over the past eighteen days. Figure 7 proves the efficiency of confinement measure in Tunisia, where P 18 decreases from March 25.

Figure 1 .

 1 Figure 1. Infected cases and deaths in Tunisia

Figure 2 .Figure 3 .

 23 Figure 2. The mortality rate T in Tunisia varies from 1.67% to 4.68%. The first case of positive COVID-19 was declared in Tunisia on March 2 th , 2020. The first case of death occurred on March 19 th , 2020. The median of the fatality rate is equal to 3.26%

Figure 4 .

 4 Figure 4. Estimation of Ct 0 , with death delay of 18 days, and T = 2% or T = 4%. The number of estimated cases in Tunisia ranges from 2 790 and 5 580 at time t 0 equals to April 12 th , 2020

Figure 5 .

 5 Figure 5. Estimated cases and reported cases of COVID-19 in Tunisia. Simulations using the average time, from first symptom to death, 12 days. The worse case is 11 160 and the optimistic case on April 12 th , 2020 is 1 395

Figure 7 .Figure 8 .

 78 Figure 7. Effect of confinement measures on the progression rate P 18 of COVID-19 in Tunisia. The progression rate P 18 is equal to 173 on March 25 and decreases to 3.6 on April 12, 2020

Table 1 .

 1 Fatality rates in different countries and in the cruise Diamond Princess

		M t0	C t0 C t0-18	T	T -18
	Diamond-Princess	11	712	712	1.6%	1.6%
	Tunisia	31	707	197 4.39% 15.7%
	Hong Kong, China	4	1004	453	0.4%	0.9%
	South Africa	25	2173	927	1.2%	2.7%
	Germany	3022 127854	43938	2.4%	6.9%
	Japan	108	6748	1387	1.6%	7.8%
	France	14393 132591	29155 10.9% 49.4%
	Italy	19899 156363	80589 12.7% 24.7%
	Algeria	293	1914	367 15.3% 79.8%

Table 2 .

 2 Estimation of the number of cases in different countries as of April 12 th , 2020 (t 0 ) by correcting for the mortality rate taking into account a delay of 18 days between the onset of symptoms and death, and using T = 2%

		M t0	C t0-18 C t0-18	C t0 P 18	C t0
	Diamond-Princess	11	550	712	712	1.0	550
	Tunisia	31	1 550	197	707	3.6	5 580
	Hong Kong, China	4	200	453	1 004	2.2	440
	South Africa	25	1 250	927	2 173	2.3	2 875
	Germany	3 022 151 100 43 938 127 854	2.9	438 190
	Japan	108	5 400	1 387	6 748	4.9	26 460
	France	14 393 719 650 29 155 132 591	4.5 3 238 425
	Italy	19 899 994 950 80 589 156 363	1.9 1 890 405
	Algeria	293	14 650	367	1 914	5.2	76 180

  Simulation of cases in Tunisia at April 12 th , 2020

	Delay until death		18 days			15 days			12 days
	T	0.5%	2%	4%	15%	0.5%	2%	4%	0.5%	2%	4%
	C t0	22 320 5 580 2 790 744 15 500 3 875 1 937 11 160 2 790 1 395

ACKNOWLEDGMENTS

We would thank Khaled Abdeljaoued for his remarks following his careful reading of the paper.

APPENDIX

Figure 6. Number of deaths at t 0 as a function of the number of cases eighteen days earlier (t 0 -18 days). These data are from March 2 to April 12 th , 2020. The number of deaths in France, Italy, and Germany is greater than the number of deaths in Tunisia, Algeria, South Africa and Japan. These graphics show the relations between Mt 0 and Ct 0


René É Van Bevern

Gregory Kucherov
email: gregory.kucherov@univ-mlv.fr

event was hosted by Novosibirsk State University and chaired by René van Bevern. The conference proceedings appeared in volume 11532 of Springer's Lecture Notes in Computer Science series. The Program Committee of the conference, chaired by Gregory Kucherov, invited authors of several presentations to submit extended versions of their works to this special issue. All papers underwent a rigorous reviewing process following the usual standards of Theory of Computing Systems. As a result, this special issue contains 10 papers out of the 31 papers presented at the conference. The CSR conference series is devoted to a broad range of topics of theoretical computer science, which is reflected in the selected papers. A short description of the 10 contributions presented in this issue follows.

correct randomized sliding window algorithm can be derandomized without increasing the worst-case space consumption, where a sliding window algorithm is strictly correct if, for every data stream, it outputs, with high probability, a correct value at every time instant.

The paper "On Decidability of Theories of Regular Languages" of Sergey Dudakov and Boris Karlov studies decidability issues for theories of regular languages with the union, concatenation, and Kleene star operations and subsets of those operations. Several new decidability and undecidability results are proved.

The paper "Transition Property for Cube-Free Words" by Elena A. Petrova and Arseny M. Shur contains a series of results about combinatorial properties of cubefree words, both binary and over larger alphabets. In particular, these results answer some open problems raised back in 1985.

The paper "Nearly Linear Time Isomorphism Algorithms for Some Nonabelian Group Classes" by Bireswar Das and Shivdutt Sharma shows a near linear-time algorithm for testing isomorphism of Hamiltonian groups or of groups that are the direct product of an abelian group and a nonabelian group of bounded order. Herein, it is assumed that the groups are given as multiplication tables with constant-time lookup.

The paper "Parameterized Complexity of Conflict-Free Set Cover" by Ashwin Jacob, Diptapriyo Majumdar, and Venkatesh Raman studies the parameterized complexity of a variant of the Set Cover problem where the sets chosen for the cover must form an independent set in some given graph or matroid.

The paper "Belga B-Trees" by Erik D. Demaine, John Iacono, Grigorios Koumoutsos, and Stefan Langerman belongs to the area of competitive analysis of online algorithms. In relation to the "dynamic optimality conjecture" about binary search trees, the authors study the competitiveness of B-trees under the so-called "diskaccess model". The authors propose a construction of a B-tree, called Belga B-tree, and prove several competitiveness bounds.

The paper "The Non-hardness of Approximating Circuit Size" by Eric Allender, Rahul Ilango, and Neekon Vafa highlights obstacles that would have to be overcome by any proof that the Minimum Circuit Size Problem (MCSP) is hard for NP under AC 0 reductions. This may be a step toward confirming a conjecture of Murray and Williams that MCSP is not NP-complete under logtime-uniform AC 0 reductions.

The paper "Unpopularity Factor in the Marriage and Roommates Problems" by Suthee Ruangwises and Toshiya Itoh studies the popular matching problem in bipartite and non-bipartite graphs. The study relies on the so-called unpopularity factor of a matching and proposes new algorithms to compute the unpopularity factor for both settings considered.

The paper "Forward Looking Huffman Coding" by Shmuel T. Klein, Shoham Saadia, and Dana Shapira proposes a new approach to dynamic Huffman encoding which is adapted to the input data model. It is shown, in particular, that the method provably outperforms the static Huffman encoding.

The paper "On Tseitin Formulas, Read-Once Branching Programs and Treewidth" by Ludmila Glinskih and Dmitry Itsykson won the best paper award at the conference. It shows that any nondeterministic read-once branching program that decides a satisfiable Tseitin formula based on an n × n grid graph has size at least 2 (n) . For arbitrary graphs, an exponential lower bound on the branching program size in terms of the treewidth of the graph is given.

In conclusion, we express our deep gratitude to the authors of the papers and to the reviewers for their work. We are also grateful to the Program and Steering Committees of the CSR 2019 conference.

Novosibirsk and Paris, December 2020, R. van Bevern and G. Kucherov.

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