Keywords: Superconductivity, local defects, bound states, scanning tunneling microscopy, transition metal dichalcogenides

Condensed matter systems are sensitive to single local imperfections of the crystalline lattice. In conventional superconductors, a magnetic impurity is one of the most simple pair-breaking defects. It induces spatially-localized in-gap excitations -called Yu-Shiba-Rusinov (YSR) bound states-routinely probed by scanning tunneling microscopy (STM). The spectral properties of YSR states and their spatial structure are complex as they blend the information about the band structure of the host, the superconducting pairing function, and the impurity-substrate coupling. Characterizing these features is crucial to exploit the potential of YSR states as a local probing tool and to design collective impurity states hosting exotic phases of matter.

In view of this ongoing research effort, this thesis investigates various aspects of the interplay between single local impurities and the superconducting state. First, motivated by the improved capability for tuning the Fermi level in novel ma-Along the way, many other people taught me many other things. In a certainly incomplete and unjust list, I thank Fred for staying long hours at the office, and his untiring (but necessary) criticism, and his unfathomable bucket of ideas, and Freek and Tristan for explaining to me some of the arcane mysteries of STM, and Marco and Hervé for the helpful remarks, and Andrzej for his questions that questioned my understanding of things. I also thank the jury of this thesis, Falko Pientka, Miguel Cazalilla, Annica Black-Schaffer and Cristina Bena for taking the time to read these pages and their well-meaning and interesting comments.

Many cheerful souvenirs are shared with my first-colleagues-and-now-friends Ansgar

terials, e.g. monolayer transition metal dichalcogenides (TMDs) and moiré heterostructures, we study how Van Hove singularities in the density of states of the host manifest in YSR states. Second, we unveil a simple analytical relationship between the real-space anisotropy of YSR states and the momentum-space anisotropy of the band structure and gap function of the substrate with which we analyze STM measurements on NbSe 2 . Third, we explore how to infer the existence of a subdominant triplet component in the superconducting order parameter relevant to TMDs from the local response to point defects. Finally, motivated by recent STM measurements on Fe(Se, Te), we analyze multiple energy-tunable in-gap bound states on Fe impurities. To account for the experimental observations, we consider an Anderson impurity model that reflects the adatoms' multi-orbital nature, and we propose a new type of impurity-driven quantum phase transition resulting from Hund's coupling.

Titre : Modélisation théorique et analyse des états liés aux impuretés dans les supraconducteurs. Mots clés : Supraconductivité, défauts ponctuels, états liés, microscopie à effet tunnel, dichalcogénures de métaux de transition.

Résumé : Les systèmes de matière condensée sont sensibles aux imperfections locales du réseau cristallin. Dans les supraconducteurs conventionnels, une impureté magnétique est l'un des défauts les plus simples entraînant la rupture de paires. Elle induit des excitations intra-gap spatialement localisées -appelées états liés de Yu-Shiba-Rusinov (YSR)-qui sont régulièrement sondées par microscopie à effet tunnel (STM, de l'acronyme en anglais). Les propriétés spectrales des états de YSR et leur structure spatiale sont complexes car elles combinent de l'information sur la structure de bandes du substrat, la fonction du gap supraconducteur et le couplage impureté-substrat. Caractériser ces particularités est cruciale pour exploiter le potentiel des états de YSR en tant qu'outil de sondage local, ainsi que pour concevoir des états d'impureté collectifs hébergeant des phases exotiques de la matière.

Dans ce contexte de recherche, cette thèse étudie divers aspects des impuretés locales au sein d'un supraconducteur. Tout d'abord, inspirés par l'amélioration des possibilités de réglage du niveau de Fermi dans des nouveaux matériaux, e.g. les monocouches des dichalcogénures de métaux de transition (TMDs de l'acronyme en anglais) et les hétérostructures moiré, nous étudions comment les singularités de Van Hove dans la densité d'états du subtrat se manifestent dans les états de YSR. Deuxièmement, nous dévoilons une relation analytique simple entre l'anisotropie dans l'espace réel des états de YSR et l'anisotropie dans l'espace réciproque de la structure de bandes et de la fonction du gap supraconducteur du substrat, que nous l'utilisons ensuite pour analyser des mesures STM sur de NbSe 2 . Troisièmement, nous explorons comment détecter l'existence d'une composante triplet sous-dominante dans le paramètre d'ordre supraconducteur, pertinent pour les TMDs, à partir de la réponse locale aux défauts ponctuels. Enfin, en lien avec des mesures récentes de STM sur de Fe(Se, Te), nous analysons des états intra-gap multiples dont l'energie est modifiable sur des impuretés de Fe. Afin d'expliquer les observations expérimentales, nous considérons un modèle d'impureté d'Anderson qui reflète la nature multi-orbitale des adatomes, et proposons un nouveau type de transition de phase quantique induite par l'impureté qui résulte du couplage de Hund.

Invaluable is also the time I spent (and continue to spend) with Joaquín, Aila, Marina, Irmina, Ainhoa, Pablo, Anna, Stefan, and Mireia, exceptionally caring individuals who represented a second home throughout these years. I especially thank Marina for her unconditional support and affection and the laughs that accompanied me both at the LPS and on the magical streets of this city. The final word of gratitude goes to my parents, who supported and encouraged me through the phone line and at all the previous stages that brought me to this very moment, and always wished me and enabled a happy life.

As I turn my gaze to the past to conclude these lines, I can confidently say that even though there were times of hardship, it was a joyful journey, and this is, undoubtedly, thanks to all the human interactions that will always transcend Hamiltonians and wavefunctions. i

General overview

Our current understanding of the physical properties of matter is deeply anchored in P.W. Anderson's emergentist stance: macroscopic aggregates of elementary particles behave qualitatively differently to its individual constituents [1]. Superconductivity is not an exception. When in 1911 G. Holst, a student of K. Onnes, cooled mercury below 4.23 K, a fundamental change in its properties unsued; as we understand it today, the observed sudden drop of three orders of magnitude in the electrical resistivity resulted from the emergent collective behavior of the electrons in the sample [2,3].

More than a hundred years have passed between the first observation of superconductivity and current research efforts addressing unsolved puzzles in the field. Over this time, the description of this phenomenon has substantially evolved. Early theories provided a phenomenological account of the electromagnetic properties of superconductors; later, V. Ginzburg and L. Landau conceptualized superconductivity as a symmetry-broken state of matter by introducing the notion of local order parameter; in 1957, J. Bardeen, L. Cooper, and J.R. Schrieffer (BCS) proposed the first microscopic theory of superconductivity [4], that was later reformulated by L.P. Gorkov in the language of quantum field theory [5]; recent works have reinterpreted superconductivity as a topologically ordered state [START_REF] Hansson | Superconductors are topologically ordered[END_REF]; etc. These various viewpoints are largely compatible and describe more or less naturally different facets of superconductivity, which are many and diverse. This thesis is ultimately concerned with the microscopic behavior of electrons in the superconducting state in the presence of single impurities; therefore, it is most appropriate to treat superconductivity within the BCS framework, where the electronic degrees of freedom are explicit.

Impurities are only one of many imperfections with which crystalline solids are often riddled in nature. Other defects include, for instance, dislocations, grain boundaries, or vacancies. Their presence can fundamentally alter the macroscopic properties of the host, and thus, it constitutes a central topic of study in condensed matter physics. For example, electrical conductivity decreases with increasing impurity concentration, missing strings of ions in a layer originate the malleability of metals, and dopant atoms modify the band structure in semiconductors. The foundations of the study of impurities in superconductors lie in two seminal works that followed soon after the publication of the BCS theory. The first one is due to P.W. Anderson, who showed that the thermodynamic properties of conventional superconductors, e.g. the critical temperature and the magnitude of the superconducting gap, are essentially unaffected by "reasonable" amounts of non-magnetic disorder, where "reasonable" usually means that the substance remains a metal above the critical temperature [START_REF] Anderson | Theory of dirty superconductors[END_REF]. The second chief result is due to iii A.A. Abrikosov and L.P. Gor'kov, who proved that, in contrast, an increasing concentration of magnetic impurities diminishes the excitation gap and eventually destroys the superconducting state [START_REF] Abrikosov | Contribution to the theory of superconducting alloys with paramagnetic impurities[END_REF][START_REF] Skalski | Properties of Superconducting Alloys Containing Paramagnetic Impurities[END_REF]. But solid-state systems are also sensitive to single defects. This fact of nature is epitomized in what is perhaps one of the most beautiful experiments of all time: In 1993, M.F. Crommie and coworkers probed employing a Scanning Tunneling Microscope (STM) the response to point-like contaminants on a copper surface and directly observed a standing-wave pattern that revealed the ondulatory character of electrons in solids [START_REF] Crommie | Imaging standing waves in a two-dimensional electron gas[END_REF]. The developments in local probes and atomic functionalization techniques pioneered at IBM opened a new research era, where impurities and defects in general are studied at nanometer lengthscales. This is precisely the framework that circumscribes this thesis.

In superconductors, the study of local defects started with a theoretical prediction. In the late sixties, L. Yu, H. Shiba, and A.I. Rusinov independently analyzed the effect of a single magnetic impurity under the assumption that it behaves as a localized static magnetic moment [START_REF] Yu | Bound state in superconductors with paramagnetic impurities[END_REF][START_REF] Shiba | Classical spins in superconductors[END_REF][START_REF] Rusinov | Superconductivity near a paramagnetic impurity[END_REF]; they showed that it gives rise to an excited state inside the superconducting energy gap which is spatially localized on the impurity -the so-called Yu-Shiba-Rusinov (YSR) states. This result aligns with the previous works on a finite concentration of scatterers: the exchange interaction between the classical spin and the electrons in the substrate breaks time-reversal symmetry locally; therefore, non-trivial (local) effects should be expected in conventional superconductors. The situation is more dramatic, however: a sufficiently large exchange interaction triggers a quantum phase transition in which the ground state of the system transmutes from a BCS configuration with all the electronic states paired at zero temperature into a slightly degraded BCS state where one Cooper pair is broken, and the impurity binds one quasiparticle and becomes partially screened [START_REF] Sakurai | Comments on Superconductors with Magnetic Impurities[END_REF][START_REF] Salkola | Spectral properties of quasiparticle excitations induced by magnetic moments in superconductors[END_REF]. The original treatment of the problem can be extended to incorporate the spin dynamics of the impurity, either by considering the s-d exchange Hamiltonian [START_REF] Soda | sd exchange interaction in a superconductor[END_REF][START_REF] Nam | Ground-state energy shift in superconductor due to s-d interaction[END_REF] or more general Anderson impurity models [START_REF] Bauer | Spectral properties of locally correlated electrons in a BardeenCooperSchrieffer superconductor[END_REF][START_REF] Zitko | Superconducting quantum dot and the sub-gap states[END_REF]. In this case, the formation of a Kondo singlet where the Fermi-level electrons screen the (dynamic) magnetic moment competes with the collapse of the Fermi surface into a coherent state of Cooper pairs. The complete quantum picture offers a more accurate description of the ground state and can even provide quantitative predictions; however, it is a much more complex problem, and in practice, one needs to resort to other approximations or numerical schemes. At the same time, the semi-classical description proposed by Yu, Shiba, and Rusinov often captures the structure of the in-gap spectrum and the spatial features of the bound states despite its formal simplicity [START_REF] Heinrich | Single magnetic adsorbates on s-wave superconductors[END_REF].

On the experimental side, the story of YSR states started in 1997, when A. Yazdani and coworkers1 first observed in-gap excitations bound to magnetic adatoms on superconducting niobium [START_REF] Yazdani | Probing the local effects of magnetic impurities on superconductivity[END_REF]. Ever since then, the field has immensely evolved: YSR states are nowadays routinely probed in conventional superconductors and also in more exotic substrates, such as graphene [START_REF] Cortés-Del Río | Observation of Yu-Shiba-Rusinov states in superconducting graphene[END_REF], iron-based superconductors [START_REF] Chatzopoulos | Spatially dispersing Yu-Shiba-Rusinov states in the unconventional superconductor FeTe0.55Se0.45[END_REF] or transition metal dichalcogenides (TMDs) [START_REF] Gerbold | Coherent long-range magnetic bound states in a superconductor[END_REF][START_REF] Liebhaber | YuShiba-Rusinov states in the Charge-Density Modulated superconductor NbSe 2[END_REF][START_REF] Yang | Observation of short-range Yu-Shiba-Rusinov states with threefold symmetry in layered superconductor 2H-NbSe 2[END_REF][START_REF] Kim | Long-range focusing of magnetic bound states in superconducting lanthanum[END_REF]; the low-dimensional character of the latter endows YSR states with an extent of tens of nanometers, allowing to resolve their spatial features better; lower operation temperatures and the use of superconducting STM tips offer an improved energy resolution that permits to observe multiple in-gap resonances [START_REF] Ji | High-Resolution Scanning Tunneling Spectroscopy of Magnetic Impurity Induced Bound States in the Superconducting Gap of Pb Thin Films[END_REF][START_REF] Ruby | Orbital Picture of Yu-Shiba-Rusinov Multiplets[END_REF]; and many experiments have succeeded in tuning the energy of YSR states [START_REF] Farinacci | Tuning the Coupling of an Individual Magnetic Impurity to a Superconductor: Quantum Phase Transition and Transport[END_REF][START_REF] Hatter | Magnetic anisotropy in Shiba bound states across a quantum phase transition[END_REF][START_REF] Sujoy Karan | Tracking a Spin-Polarized Superconducting Bound State across a Quantum Phase Transition[END_REF]. This renewed interest in YSR states is partly fuelled by the advancements in STM, which facilitate investigating their properties, but there are also slightly more utilitarian reasons.

One of them is that YSR states are a useful probing tool. This is a general virtue of defects: analyzing the system's response to single impurities is in fact, a widespread strategy to characterize the properties of the host. At the most basic level, observing a bound state on a non-magnetic impurity is very likely a signature of unconventional superconductivity, but there exist more sophisticated schemes; for instance, information about the band structure and the gap function of the substrate can be extracted from the standing-wave pattern around point-like defects probed by STM [START_REF] Hoffman | Imaging quasiparticle interference in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF]. To name a few examples, YSR states can serve to identify a spintriplet order parameter [START_REF] Kaladzhyan | Characterizing p-wave superconductivity using the spin structure of Shiba states[END_REF][START_REF] Kaladzhyan | Asymptotic behavior of impurity-induced bound states in low-dimensional topological superconductors[END_REF] or odd-frequency pairing [START_REF] Perrin | Unveiling Odd-Frequency Pairing around a Magnetic Impurity in a Superconductor[END_REF], and YSR states have been used to examine the magnetic properties of the adsorbates [START_REF] Choi | Influence of Magnetic Ordering between Cr Adatoms on the Yu-Shiba-Rusinov states of the βBi 2 Pd superconductor[END_REF][START_REF] Ruby | Orbital Picture of Yu-Shiba-Rusinov Multiplets[END_REF]. In addition, over the past few years, YSR states have been successfully functionalized in STM tips [START_REF] Huang | Tunnelling dynamics between superconducting bound states at the atomic limit[END_REF][START_REF] Huang | Spin-dependent tunneling between individual superconducting bound states[END_REF][START_REF] Schneider | Atomic-scale spin-polarization maps using functionalized superconducting probes[END_REF]. This breakthrough provides an unprecedented level of energy resolution and paves the way for investigating spindependent phenomena at the atomic scale.

The other major front of research in the field originated from the theoretical proposal that chains of magnetic adatoms could be a route to build topological phases of matter [START_REF] Nadj-Perge | Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor[END_REF]. In the dilute limit, the superconducting substrate mediates the hybridization of YSR states stemming from the magnetic impurities, and an in-gap band arises. If the atoms are helically ordered, p-wave pairing is induced within the emergent sub-band, and a one-dimensional topological superconductor hosting Majorana zero modes (MZMs) on the edges is effectively realized [START_REF] Pientka | Topological superconducting phase in helical Shiba chains[END_REF]. Much experimental progress has been achieved in this direction [START_REF] Yazdani | Hunting for Majoranas[END_REF]; however, at the time of writing, there is still no definitive signature of the elusive MZMs. More intricate structures involving two-dimensional lattices of static magnetic moments have also been proposed to host topological superconductivity [START_REF] Li | Two-dimensional chiral topological superconductivity in Shiba lattices[END_REF][START_REF] Soldini | Building crystalline topological superconductors from Shiba lattices[END_REF].

Understanding and characterizing the properties of impurity-bound states is indispensable to developing new local probing schemes and also, for bottom-up engineering of novel phases of matter. This thesis is expressly motivated by this ongoing research effort. As for the modus operandi, we proceed by constructing models of magnetic impurities in superconductors that are sufficiently simple to be explored with analytical tools, yet, realistic enough to be relevant to specific experiments. Indeed, although the work developed in this thesis is fundamentally theoretical, many of the questions we address originated from experimental observations, and at times, we analyze and test the theory against actual STM data. v

Aperçu général

Notre compréhension actuelle des propriétés physiques de la matière est profondément ancrée dans la vision émergentiste de P.W. Anderson : les agrégats macroscopiques de particules élémentaires se comportent qualitativement différemment de leurs constituants individuels [1]. La supraconductivité n'y fait pas exception. Lorsqu'en 1911 G. Holst, un étudiant de K. Onnes, refroidit le mercure en dessous de 4,23 K, un changement fondamental de ses propriétés se produisit ; tel que nous le comprenons aujourd'hui, la chute subite de trois ordres de grandeur observée dans la résistivité électrique résulte du comportement collectif émergent des électrons dans l'échantillon [2,3]. Plus de cent ans se sont écoulés entre la première observation de la supraconductivité et les efforts de recherche actuels visant à résoudre des énigmes dans ce domaine. Au cours de cette période, la description de ce phénomène a considérablement évoluée. Les premières théories fournissaient une explication phénoménologique des propriétés électromagnétiques des supraconducteurs ; plus tard, V. Ginzburg et L. Landau conceptualisèrent la supraconductivité comme un état de la matière à symétrie brisée en introduisant la notion de paramètre d'ordre local ; en 1957, J. Bardeen, L. Cooper, et J.R. Schrieffer (BCS) proposèrent la première théorie microscopique de la supraconductivité [4], qui fut ensuite reformulée par L.P. Gorkov dans le langage de la théorie quantique des champs [5] ; des travaux récents ont réinterprété la supraconductivité comme un état topologiquement ordonné [START_REF] Hansson | Superconductors are topologically ordered[END_REF] ; etc. Ces différents points de vue sont largement compatibles et décrivent plus ou moins naturellement différentes facettes de la supraconductivité, qui sont nombreuses et diverses. Cette thèse s'intéresse fondamentalement au comportement microscopique des électrons dans l'état supraconducteur en présence d'impuretés ponctuelles ; il est donc plus approprié de traiter la supraconductivité dans le cadre de la théorie BCS, où les degrés de liberté électroniques sont explicites.

Les impuretés ne sont qu'une des nombreuses imperfections dont les solides cristallins dans la nature sont souvent criblés. D'autres défauts comprennent, par exemple, les dislocations, les joints de grains ou les lacunes. Leur présence peut modifier fondamentalement les propriétés macroscopiques de l'hôte et constitue donc un sujet d'étude central en physique de la matière condensée. Par exemple, la conductivité électrique diminue avec l'augmentation de la concentration d'impuretés, le manque de chaînes d'ions dans une couche du réseau crystallin est à l'origine de la malléabilité des métaux, et les atomes dopants modifient la structure des bandes dans les semi-conducteurs. Les fondements de l'étude des impuretés dans les supraconducteurs reposent sur deux travaux précurseurs qui suivirent de peu la publication de la théorie BCS. Le premier est dû à P.W. Anderson, qui montra que les propriétés vii thermodynamiques des supraconducteurs conventionnels, par exemple la température critique et l'ampleur du gap supraconducteur, ne sont essentiellement pas affectées par des quantités "raisonnables" de désordre non magnétique, où "raisonnable" signifie généralement que la substance reste un métal au-dessus de la température critique [START_REF] Anderson | Theory of dirty superconductors[END_REF]. Le deuxième résultat principal est dû à A.A. Abrikosov et L.P. Gor'kov, qui prouvèrent qu'à l'inverse, une concentration croissante d'impuretés magnétiques diminue l'ampleur du gap supraconducteur et finit par détruire l'état supraconducteur [START_REF] Abrikosov | Contribution to the theory of superconducting alloys with paramagnetic impurities[END_REF][START_REF] Skalski | Properties of Superconducting Alloys Containing Paramagnetic Impurities[END_REF]. Mais les systèmes à l'état solide sont également sensibles aux défauts uniques. Ce fait naturel est illustré par ce qui est peut-être l'une des plus belles expériences de tous les temps : En 1993, M.F. Crommie et ses collègues sondèrent, à l'aide d'un microscope à effet tunnel (STM de l'acronyme en anglais), la réponse à des contaminants ponctuels sur une surface de cuivre et observèrent directement un motif d'ondes stationnaires qui révéla le caractère ondulatoire des électrons dans les solides [START_REF] Crommie | Imaging standing waves in a two-dimensional electron gas[END_REF]. Les développements des sondes locales et des techniques de fonctionnalisation atomique mis au point par IBM ouvrirent ensuite une nouvelle ère de recherche, où les impuretés et les défauts en général sont étudiés à des échelles de longueur nanométrique. Ceci est précisément le cadre dans lequel s'inscrit cette thèse.

Dans les supraconducteurs, l'étude des défauts locaux commença par une prédiction théorique. Dans la fin des années soixante, L. Yu, H. Shiba et A.I. Rusinov analysèrent indépendamment l'effet d'une impureté magnétique unique en supposant qu'elle se comporte comme un moment magnétique statique localisé [START_REF] Yu | Bound state in superconductors with paramagnetic impurities[END_REF][START_REF] Shiba | Classical spins in superconductors[END_REF][START_REF] Rusinov | Superconductivity near a paramagnetic impurity[END_REF] ; ils montrèrent qu'elle donne lieu à un état excité à l'intérieur du gap d'énergie supraconducteur qui est spatialement localisé sur l'impureté -les états dits de Yu-Shiba-Rusinov (YSR). Ce résultat est conforme aux travaux antérieurs sur une concentration finie de défauts : l'interaction d'échange entre le spin classique et les électrons dans le substrat brise localement la symétrie de renversement du temps ; il faut donc s'attendre à des effets (locaux) non triviaux dans les supraconducteurs conventionnels. La situation est cependant plus dramatique : une interaction d'échange suffisamment importante déclenche une transition de phase quantique dans laquelle l'état fondamental du système passe d'une configuration BCS avec tous les états électroniques appariés à température zéro à un état BCS légèrement dégradé dans lequel une paire de Cooper est brisée et où l'impureté lie une quasiparticule et devient partiellement écrantée [START_REF] Sakurai | Comments on Superconductors with Magnetic Impurities[END_REF][START_REF] Salkola | Spectral properties of quasiparticle excitations induced by magnetic moments in superconductors[END_REF]. Le traitement original du problème peut être étendu pour intégrer la dynamique de spin de l'impureté, soit en considérant le hamiltonien d'échange s-d [START_REF] Soda | sd exchange interaction in a superconductor[END_REF][START_REF] Nam | Ground-state energy shift in superconductor due to s-d interaction[END_REF], soit des modèles d'impureté d'Anderson plus généraux [START_REF] Bauer | Spectral properties of locally correlated electrons in a BardeenCooperSchrieffer superconductor[END_REF][START_REF] Zitko | Superconducting quantum dot and the sub-gap states[END_REF]. Dans ce cas, la formation d'un singulet de Kondo où les électrons du niveau de Fermi font écran au moment magnétique (dynamique) est en concurrence avec l'effondrement de la surface de Fermi dans un état cohérent de paires de Cooper. Le traitement quantique de l'impureté offre une description plus précise de l'état fondamental et peut même fournir des prédictions quantitatives ; cependant, il s'agit d'un problème beaucoup plus complexe et, dans la pratique, il faut recourir à d'autres approximations ou procédés numériques. Dans le même temps, la description semi-classique proposée par Yu, Shiba et Rusinov capture souvent la structure du spectre à l'intérieur du gap supraconducteur et les caractéristiques spatiales des états liés malgré sa simplicité formelle [START_REF] Heinrich | Single magnetic adsorbates on s-wave superconductors[END_REF].

Sur le plan expérimental, l'histoire des états YSR commença en 1997, lorsque A. Yazdani et ses collègues observèrent pour la première fois des excitations dans le gap supraconducteursett viii 0. Acknowledgements liées à des atomes magnétiques sur du niobium supraconducteur2 . Depuis lors, le domaine a énormément évolué : Les états YSR sont aujourd'hui régulièrement étudiés dans les supraconducteurs conventionnels, mais aussi dans des substrats plus exotiques, comme le graphène [START_REF] Cortés-Del Río | Observation of Yu-Shiba-Rusinov states in superconducting graphene[END_REF], les supraconducteurs à base de fer [START_REF] Chatzopoulos | Spatially dispersing Yu-Shiba-Rusinov states in the unconventional superconductor FeTe0.55Se0.45[END_REF] ou les dichalcogénures de métaux de transition (TMD de l'acronyme en anglais) [START_REF] Gerbold | Coherent long-range magnetic bound states in a superconductor[END_REF][START_REF] Liebhaber | YuShiba-Rusinov states in the Charge-Density Modulated superconductor NbSe 2[END_REF][START_REF] Yang | Observation of short-range Yu-Shiba-Rusinov states with threefold symmetry in layered superconductor 2H-NbSe 2[END_REF][START_REF] Kim | Long-range focusing of magnetic bound states in superconducting lanthanum[END_REF] ; la faible dimensionnalité de ces derniers confère aux états YSR une étendue de plusieurs dizaines de nanomètres, ce qui permet de mieux discerner leurs proprietés spatiales ; des températures de fonctionnement plus basses et l'utilisation de pointes STM supraconductrices offrent une meilleure résolution en énergie qui permet d'observer de multiples résonances dans le gap supraconducteur [START_REF] Ji | High-Resolution Scanning Tunneling Spectroscopy of Magnetic Impurity Induced Bound States in the Superconducting Gap of Pb Thin Films[END_REF][START_REF] Ruby | Orbital Picture of Yu-Shiba-Rusinov Multiplets[END_REF] ; et de nombreuses expériences ont réussi à modifier l'énergie des états YSR [START_REF] Farinacci | Tuning the Coupling of an Individual Magnetic Impurity to a Superconductor: Quantum Phase Transition and Transport[END_REF][START_REF] Hatter | Magnetic anisotropy in Shiba bound states across a quantum phase transition[END_REF][START_REF] Sujoy Karan | Tracking a Spin-Polarized Superconducting Bound State across a Quantum Phase Transition[END_REF]. Ce regain d'intérêt pour les états YSR est en partie alimenté par le progrès du STM, qui facilite l'étude de leurs propriétés, mais il y a aussi des raisons plus utilitaires.

L'une d'entre elles est que les états YSR constituent un outil de sondage. Il s'agit d'une vertu générale des défauts : l'analyse de la réponse du système à des impuretés locales est en fait une stratégie très répandue pour caractériser les propriétés de l'hôte. Au niveau le plus élémentaire, l'observation d'un état lié sur une impureté non magnétique est très probablement une signature de supraconductivité non conventionnelle, mais il existe des stratégies plus sophistiqués ; par exemple, des informations sur la structure de bande et le paramètre d'ordre supraconducteur du substrat peuvent être extraites du motif d'ondes stationnaires autour de défauts ponctuels sondés par STM [START_REF] Hoffman | Imaging quasiparticle interference in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF]. Pour citer quelques exemples, les états YSR peuvent servir à identifier un paramètre d'ordre triplet de spin [START_REF] Kaladzhyan | Characterizing p-wave superconductivity using the spin structure of Shiba states[END_REF][START_REF] Kaladzhyan | Asymptotic behavior of impurity-induced bound states in low-dimensional topological superconductors[END_REF] ou un appariement à fréquence impaire [START_REF] Perrin | Unveiling Odd-Frequency Pairing around a Magnetic Impurity in a Superconductor[END_REF], et les états YSR ont été utilisés pour examiner les propriétés magnétiques des adsorbats [START_REF] Choi | Influence of Magnetic Ordering between Cr Adatoms on the Yu-Shiba-Rusinov states of the βBi 2 Pd superconductor[END_REF][START_REF] Ruby | Orbital Picture of Yu-Shiba-Rusinov Multiplets[END_REF]. En outre, au cours des dernières années, les états YSR ont été fonctionnalisés avec succès dans des pointes STM [START_REF] Huang | Tunnelling dynamics between superconducting bound states at the atomic limit[END_REF][START_REF] Huang | Spin-dependent tunneling between individual superconducting bound states[END_REF][START_REF] Schneider | Atomic-scale spin-polarization maps using functionalized superconducting probes[END_REF]. Cette avancée permet d'atteindre un niveau de résolution énergétique sans précédent et ouvre la voie à l'étude des phénomènes dépendant du spin à l'échelle atomique. L'autre grand axe de recherche dans ce domaine naquit de la proposition théorique selon laquelle les chaînes d'atomes magnétiques pourraient être un moyen de construire des phases topologiques de la matière [START_REF] Nadj-Perge | Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor[END_REF]. Dans la limite diluée, le substrat supraconducteur sert de médiateur à l'hybridation des états YSR provenant des impuretés magnétiques, ce qui donne lieu à l'émergence d'une bande dans le gap supraconducteur. Si les atomes sont ordonnés de manière hélicoïdale, l'appariement des ondes p est induit dans la sous-bande émergente, et un supraconducteur topologique unidimensionnel hébergeant des modes zéro de Majorana (MZM) sur les bords est effectivement réalisé [START_REF] Pientka | Topological superconducting phase in helical Shiba chains[END_REF]. De nombreux progrès expérimentaux ont été réalisés dans cette direction [START_REF] Yazdani | Hunting for Majoranas[END_REF] ; cependant, au moment de la rédaction de ce document, il n'y a toujours pas de signature définitive des MZM. Des structures plus complexes impliquant des réseaux bidimensionnels de moments magnétiques statiques ont également été proposées pour héberger la supraconductivité topologique [START_REF] Li | Two-dimensional chiral topological superconductivity in Shiba lattices[END_REF][START_REF] Soldini | Building crystalline topological superconductors from Shiba lattices[END_REF].

La compréhension et la caractérisation des propriétés des états liés aux impuretés sont indispensables pour développer de nouveaux procédés de sondage local et également pour l'ingénierie ascendante de nouvelles phases de la matière. Cette thèse est expressément motivée par cet effort de recherche en cours. En ce qui concerne le modus operandi, nous procédons en construisant des modèles d'impuretés magnétiques dans les supraconducteurs qui sont suffisamment simples pour être explorés avec des outils analytiques, tout en étant suffisamment réalistes pour être pertinents pour des expériences spécifiques. En effet, bien que le travail développé dans cette thèse soit fondamentalement théorique, de nombreuses questions que nous abordons proviennent d'observations expérimentales et, parfois, nous analysons et testons la théorie par rapport à des données de STM.

x

Organization of this thesis

This thesis is organized into five chapters. We summarize next the content of each of them.

In Chapter 1, we introduce the formalism underpinning the two building blocks of this thesis. We start by laying out the key ideas of the BCS theory and the mean-field treatment, which is the starting point of the subsequent chapters. Particular emphasis is put on explaining the concepts that will be later connected to the YSR states, namely the superconducting gap and the excitation spectrum. For the most part, we follow the books by J.R. Schrieffer [START_REF] Schrieffer | Theory of superconductivity. Advanced book classics. Advanced Book Program[END_REF] and M. Tinkham [START_REF] Tinkham | Introduction to superconductivity[END_REF]. Then, we review the T-matrix approximation and show how to calculate the change in the local density response due to single impurities devoid of quantum degrees of freedom. We also present the general phenomenology of YSR states in a simple setup and discuss some crucial assumptions that will be adopted in the rest of the manuscript. This second part is primarily based on the review by A.V. Balatsky et al. [START_REF] Balatsky | Impurity-induced states in conventional and unconventional superconductors[END_REF].

In Chapter 2, we consider a classical magnetic impurity on a substrate whose normal density of states (DOS) exhibits a logarithmic or a power-law divergence at the Fermi level due to Van Hove singularities (VHSs) and higher-order Van Hove singularities (HOVHSs), respectively. We find that the energy of the YSR states has the same functional form as in the constant DOS scenario, with the effect of the singularity being an enhancement of the effective coupling constants. Interestingly, the YSR energy curve slope at the YSR transition is independent of the impurity parameters and purely reflects the band structure. Our main conclusions remain valid even when the Fermi level is not precisely tuned to the Van Hove singularity but instead lies in an energy range of the order of the superconducting gap. These results show that tuning a superconducting material towards a VHS or HOVHS enhances the possibilities for engineering YSR states and characterizing the host. The material presented in this chapter is original work, and nearly a verbatim adaptation of [START_REF] Uldemolins | Effect of Van Hove singularities on Shiba states in two-dimensional s-wave superconductors[END_REF] with an extended discussion.

In Chapter 3, we study the scattering processes giving rise to YSR states in gapped, twodimensional superconductors employing a saddle-point approximation on the real-space clean Green function. We develop a theory that relates an arbitrary energy dispersion of normal electrons in a two-dimensional host to the spatial features of the YSR states through a simple analytical expression. Namely, we find that flatter segments of the Fermi surface with large Fermi velocity enhance the local density of states (LDOS) around the impurity. Our analytical approximation is quantitatively accurate against tight-binding calculations on various lattices with different Fermi surfaces, and it qualitatively captures certain traits of YSR states observed in scanning tunneling spectroscopy experiments. The material presented in this chapter is original work and mostly reproduced from [START_REF] Uldemolins | Quasiparticle focusing of bound states in two-dimensional s-wave superconductors[END_REF] except for Sec. 3.3.4, where we present an inxi depth analysis of STM data on NbSe 2 that we performed after the publication.

In Chapter 4, we explore how the local response to defects can be used to determine the existence of subdominant triplet correlations in non-centrosymmetric superconductors. In particular, we consider a toy model for singlet-triplet admixture relevant to transition metal dichalcogenides, and we study the quasiparticle interference (QPI) spectrum, which is readily obtained by Fourier-transforming a spatial STM measurement. We show that the substrates inversion symmetry-breaking manifests in the spin-polarized QPI signal, and we provide a detailed analysis of the wave vector q-pattern in systems with lifted spin-degeneracy, opening the way for subsequent studies on QPI in TMDs.

In Chapter 5, we scrutinize recent STM data on Fe(Se, Te) showcasing multiple in-gap bound states on Fe impurities whose energy can be tuned by varying the tip-sample distance. The observation of negative differential conductance suggests that Coulomb interactions play a decisive role in the impurity system; therefore, we consider an Anderson impurity model that reflects the adatoms' multi-orbital nature and goes beyond the classical description adopted in the previous chapters. We propose that the experimental measurements reflect a new type of quantum phase transition characterized by a change of the impurity's mean occupation and total spin, and driven by Hund's coupling. The material presented in this chapter is original work and mostly adapted from [START_REF] Uldemolins | Interaction-driven quantum phase transition of a single magnetic impurity in fe(se,te)[END_REF] with an extended discussion.

Organisation de cette thèse

Cette thèse est organisée en cinq chapitres. On résume ci-après le contenu de chacun d'entre eux.

Dans le chapitre 1, on introduit le formalisme qui sous-tend les deux éléments constitutifs de cette thèse. On commence par exposer les idées clés de la théorie BCS et du traitement du champ moyen, qui est le point de départ des chapitres suivants. Un accent particulier est mis sur l'explication des concepts qui seront ultérieurement liés aux états YSR, à savoir, le gap supraconducteur et le spectre d'excitation. Pour l'essentiel, on suit les livres de J.R. Schrieffer [START_REF] Schrieffer | Theory of superconductivity. Advanced book classics. Advanced Book Program[END_REF] et de M. Tinkham [START_REF] Tinkham | Introduction to superconductivity[END_REF]. Ensuite, on rappelle l'approximation de T-matrix et montre comment calculer le changement dans la réponse de la densité locale due à des impuretés locales dépourvues de degrés de liberté quantiques. On présente également la phénoménologie générale des états YSR dans le cas le plus simple et on discute les hypothèses adoptées sur l'impureté magnétique qui seront pertinentes pour le reste du manuscrit. Cette deuxième partie est principalement basée sur la revue de A.V. Balatsky et al. [START_REF] Balatsky | Impurity-induced states in conventional and unconventional superconductors[END_REF].

Dans le chapitre 2, on considère une impureté magnétique classique sur un substrat dont la densité normale d'états (DOS de l'acronyme en anglais) présente une divergence logarithmique ou en loi de puissance au niveau de Fermi résultant de singularités de Van Hove (VHS de l'acronyme en anglais) et de singularités de Van Hove d'ordre supérieur (HOVHS de l'acronyme en anglais), respectivement. On constate que l'énergie des états YSR a la même forme fonctionnelle que dans le scénario d'une DOS constante, l'effet de la singularité étant une augmentation des constantes de couplage effectives. On note que la pente de la courbe d'énergie à la transition YSR est indépendante des paramètres de l'impureté et reflète purement la structure de bande. Nos principales conclusions restent valables même lorsque le niveau de Fermi n'est pas précisément accordé à la singularité de Van Hove mais se situe plutôt dans une gamme d'énergie de l'ordre du gap supraconducteur. Ces résultats montrent que l'ajustement d'un matériau supraconducteur vers une VHS ou une HOVHS augmente les possibilités d'ingénierie des états YSR et de caractérisation de l'hôte. Le matériel présenté dans ce chapitre est un travail original, et presque une adaptation verbatim de [START_REF] Uldemolins | Effect of Van Hove singularities on Shiba states in two-dimensional s-wave superconductors[END_REF] avec une discussion étendue.

Dans le chapitre 3, on étudie les processus de diffusion donnant lieu à des états YSR dans des supraconducteurs bidimensionnels gappés, en utilisant une approximation du point de selle sur la fonction de Green à l'espace réel. On développe une théorie qui relie une relation de dispersion de l'énergie des électrons normaux dans un hôte bidimensionnel arbitraire aux caractéristiques spatiales des états YSR par le biais d'une expression analytique simple. On constate notamment que les segments plus plats de la surface de Fermi avec une grande vitesse de Fermi augmentent xiii la densité locale d'états (LDOS de l'acronyme en anglais) autour de l'impureté. Notre approximation analytique est quantitativement précise par rapport aux calculs de liaison forte sur divers réseaux avec différentes surfaces de Fermi, et elle capture qualitativement certains traits des états YSR observés dans les expériences de spectroscopie à effet tunnel. Le matériel présenté dans ce chapitre est un travail original et est en grande partie reproduit de [START_REF] Uldemolins | Quasiparticle focusing of bound states in two-dimensional s-wave superconductors[END_REF] à l'exception de la Sec. 3.3.4, où l'on présente une analyse approfondie des données STM sur NbSe 2 que l'on a effectuée après la publication.

Dans le chapitre 4, on explore comment la réponse locale aux défauts peut être utilisée pour déterminer l'existence de corrélations triplet sous-dominantes dans les supraconducteurs non centrosymétriques. En particulier, on considère un modèle-jouet pour le mélange singlet-triplet pertinent pour les dichalcogénures de métaux de transition, et on étudie le spectre d'interférence des quasiparticules (QPI), qui est facilement obtenu par transformation de Fourier d'une mesure STM spatiale. On montre que la brisure de la symétrie d'inversion du substrat se manifeste dans le signal QPI polarisé en spin, et on fournit une analyse détaillée du vecteur d'onde q dans les systèmes avec une levée dégénérescence de spin, ouvrant la voie à des études ultérieures sur le QPI dans les TMDs.

Dans le chapitre 5, on examine les données STM récentes sur Fe(Se, Te) qui montrent de multiples états liés dans le gap supraconducteur sur les impuretés de Fe dont l'énergie peut être réglée en faisant varier la distance pointe-échantillon. L'observation d'une conductance différentielle négative suggère que les interactions de Coulomb jouent un rôle décisif dans le système d'impureté ; par conséquent, on considère un modèle d'impureté d'Anderson qui reflète la nature multi-orbitale des atomes et va au-delà de la description classique adoptée dans les chapitres précédents. On propose que les mesures expérimentales reflètent un nouveau type de transition de phase quantique caractérisée par un changement de l'occupation moyenne et du spin total de l'impureté, et entraînée par le couplage de Hund. Le matériel présenté dans ce chapitre est un travail original et principalement adapté de [START_REF] Uldemolins | Interaction-driven quantum phase transition of a single magnetic impurity in fe(se,te)[END_REF] avec une discussion étendue.

Chapter 1

Introduction

Microscopic theory of superconductivity

In the BCS theory, the onset of superconductivity is understood as the collapse of the Fermi surface into a macroscopic coherent state with lower energy. Below a certain material-dependent critical temperature, the random motion of electrons gives way to a correlated behavior, where pairs of states with close-to-Fermi energy are effectively bound. This idea lies on three fundamental insights that we summarize next:

1. A pair of electrons above an occupied Fermi sea subject to an arbitrarily small attractive force forms a stable bound state called Cooper pair1 . We shall not derive this result here, however, it is worth mentioning a few properties of Cooper the pairs. The two-particle wavefunction of the pair can be written as a product of the orbital and spin parts. For a system with translational invariance, it reads

Ψ(r 1 , σ 1 , r 2 , σ 2 ) = e iq•R ϕ(r)φ σ 1 ,σ 2 , (1.1)
with R = r 1 +r2 2 the center of mass of the pair, r = r 1r 2 , and q the total momentum of the pair. In general, it is expected that pairs at rest will be energetically favored at equilibrium, hence we set q = 0, but of course, this is not the case if there is a net current flow. In addition, under a strong magnetic field, it is sometimes possible to stabilize exotic phases (FFLO state) where Cooper pairs have a finite center-of-mass momentum [START_REF] Matsuda | FuldeFerrellLarkinOvchinnikov State in Heavy Fermion Superconductors[END_REF].

Under the assumption that the interaction is spin-rotation invariant, the total spin of the pair is a conserved quantity; therefore, since electrons are spin 1/2 particles, the spin part of the pair wavefunction is either a singlet (total spin S = 0)2

φ singlet σ 1 ,σ 2 = 1 √ 2 (| ↑↓ -| ↓↑ ) = 1 √ 2 0 1 -1 0 , (1.2)
or a linear combination of triplet states (S = 1)

φ triplet σ 1 ,σ 2 =                            | ↑↑ =   1 0 0 0   , 1 √ 2 (| ↑↓ + | ↓↑ ) = 1 √ 2   0 1 1 0   , | ↓↓ =   0 0 0 1   . (1.3)
Owing to the fermionic nature of electrons, the total pair wavefunction must be antisymmetric under particle exchange; therefore, the orbital part is even and odd for singlet and triplet states, respectively.

2. Despite the Coulomb repulsion, electrons in solids can experience an effective attractive force. In the original BCS treatment, this interaction emanates from local distortions of the crystal lattice. This effect can be rationalized with a classical picture: an electron polarizes the medium by attracting positive ions; since the relaxation time is much larger than the characteristic time of the electron's motion, the positive charge excess remains longer and attracts a second electron, effectively binding the pair. Crucially, it is assumed that the net attractive interaction is only meaningful for states in a shell around the Fermi surface of the order the Debye energy ( ω D ), which is the energy scale of the lattice excitations. This effective interaction conserves the total momentum of the pair (which was assumed to be zero in this discussion); however, it does not conserve the momentum of each electron separately. In momentum space, pairs of electrons in the state (k , -k ) are scattered into another state (k, -k). In particular, the matrix element reads

V k,k ≡ dre -i(k-k )r V (r) =    -|g| if |ε k -ε F | < ω D , for k, k , 0 otherwise, (1.4)
where we implicitly assumed a point contact interaction as well, V (r) = -|g|δ(r). Within this simplifying choice for the interaction potential, the Cooper-pair wavefunction is kindependent, too. In other words, only singlet s-wave pairing states are allowed. This scenario seems to be predominant in real materials, however, as we shall briefly discuss in Sec. 1.1.2, one could expect a more general interaction.

3. The Fermi surface is unstable against the formation of bound Cooper pairs, therefore, pairs will form until an equilibrium is reached. The true many-body ground state describing this situation is an intractable wavefunction. However, in the BCS theory, the ground state of the superconducting state is approximated by a coherent state of the Cooper-pair operator, a many-body wavefunction representing a coherent superposition of pair states whose occupation is treated statistically,

|Ψ BCS = k u k + v k c † k,↑ c † -k,↓ |0 , (1.5)
where |u k | 2 +|v k | 2 = 1, and |0 represents the vacuum. Here, v k and u k are the probability amplitudes of the pair state (k ↑, -k ↓) being occupied and empty respectively. This state is not an eigenstate of the number-of-particles operator, and therefore, the number of particles N is not conserved. Nevertheless, the uncertainty in the number of particles goes as √ N , which becomes negligible compared to N in a macroscopic system.

These ideas are synthesized in the reduced BCS Hamiltonian for singlet s-wave superconductors:

H BCS = k,σ ε k c † k,σ c k,σ -|g| k,k c † k,↑ c † -k,↓ c -k ,↓ c k ,↑ . (1.6)
The first term corresponds to the kinetic energy of the electrons in the substrate, with ε k the single-particle normal-state energy relative to the Fermi energy 3 . The second term represents the Cooper-pair scattering, where |g| is the strength of the effective interaction as defined in Eq. (1.4). In the original formulation of the BCS theory, the coefficients u k , v k in Eq. (1.5) are variational parameters obtained by minimizing

E GS = Ψ BCS |H BCS |Ψ BCS .
Here, we skip this calculation, and instead, continue by presenting the mean-field approach, which is more suitable for studying the excitation spectrum.

Before moving to the next section, we mention that in the presence of impurities that break translational invariance, crystal momentum is not a conserved quantity; therefore, it is not possible to consider pair states (k ↑, -k ↓). However, if the disorder is non-magnetic, the eigenfunctions of the dirty Hamiltonian and their time-reversed partners are degenerate in energy, and one can build an equivalent BCS theory for these states, where the new effective interaction is simply the average interaction between the Bloch wavefunctions making the impurity-scattered states. This averaging procedure juts out the constant part of the interaction, and incidentally, the original BCS approximation becomes more accurate for dirty superconductors. This is the essence of Anderson's theorem, and it justifies the theory's early success in explaining experimental measurements on the prevalently disordered samples of the time. Naturally, the argument does not hold for magnetic impurities, which break time-reversal symmetry, and indeed, they are detrimental to superconductivity.

Mean-field approximation

The need to perform a mean-field decoupling of the reduced Hamiltonian stems from the two-body interacting term in the BCS Hamiltonian (1.6). We proceed by assuming that the pair operator c † k,↑ c † k,↓ has a finite expectation value in the ground state. This choice intuitively follows from the structure of the BCS wavefunction [Eq. (1.5)], which is a coherent superposition of pairs of Bloch states. In addition, since the ground state involves a macroscopic number of particles, fluctuations around the mean value of this operator should be small. These observations lead to choosing the average b k = c -k,↓ c k,↑ , and subsequently neglect quadratic terms in (c -k,↓ c k,↑b k ). The mean-field BCS Hamiltonian then reads

H MF = k,σ ε k c † k,σ c k,σ - k ∆c † k,↑ c † -k,↓ + ∆ * c -k,↓ c k,↑ -∆b * k , (1.7)
where * denotes complex conjugation and

∆ = |g| k c -k ,↓ c k ,↑ . (1.8)
The quantity ∆ is the central object of the BCS theory: the superconducting order parameter. Indeed, ∆ is proportional to the anomalous average c -k ,↓ c k ,↑ , which vanishes if the ground state is the Fermi sea (normal state) and becomes non-zero if the ground state is the BCS wavefunction (onset of superconductivity). Without going into much detail, we mention that the superconducting state spontaneously breaks global U (1) symmetry in the Landau sense, and this is reflected in ∆, which is not invariant under global U (1) transformations, c → c e iφ/2 . The order parameter is in general a complex number, ∆ = |∆|e iφ , and its phase, while not directly observable, is indispensable to describe many of the electromagnetic properties of the superconducting state, e.g. the Meissner effect (or perfect diamagnetism), flux quantization, the Josephson effect, etc. In the remaining of the manuscript, we do not consider orbital magnetic fields nor supercurrents; therefore, we shall treat ∆ as a real variable, which can be enforced by the appropriate gauge transformation 4 .

In addition, it should be noted that the order parameter is defined through the average of an operator: calculating the right-hand side of the equality requires the knowledge of the spectrum of H MF , so that Eq. (1.8) is, in fact, a self-consistent relation. Further, this average should be understood as the thermal average, implying that the superconducting order parameter depends on the temperature. Indeed, the BCS theory allows us to express the critical temperature T c signaling the phase transition as a (non-analytic) function of the interaction strength |g|. We shall not present these results here since they are not crucial for this thesis. In the rest of the manuscript, we will assume that we are deep inside the superconducting phase, and therefore, we will treat |∆| as a free parameter of the theory. As we will see shortly, this parameter is experimentally accessible.

In order to gain further insight, we proceed to diagonalize the Hamiltonian in Eq. (1.7). We note that H MF is bilinear in c † and c, therefore, it is effectively a non-interacting Hamiltonian. It can be diagonalized by introducing the operators

γ † k,+ = u k c † k,↑ + v k c -k,↓ , (1.9)
γ † k,-= u k c † -k,↓ -v k c k,↑ . (1.10)
In addition, we impose

|u k | 2 + |v k | 2 = 1, so that γ † k,+ , γ † k,
-satisfy the anticonmutation relations and effectively describe fermionic quasiparticles. Setting

|u k | 2 = 1 2 1 + ε k E k , (1.11
)

|v k | 2 = 1 2 1 - ε k E k . (1.12)
renders the mean-field Hamiltonian diagonal,

H MF = E 0 + k E k γ † k,+ γ k,+ + γ † k,-γ k,-, (1.13)
where

E 0 = k (ε k -E k + ∆b k ) and
E k = ε 2 k + |∆| 2 . (1.14)
The transformed mean-field Hamiltonian is illuminating in several ways. First of all, we note that the BCS wavefunction defined in Eq. (1.5) is, in fact, the vacuum of the fermionic operators defined in Eqs. (1.9), (1.10),

γ k,+ |Ψ BCS = γ k,-|Ψ BCS = 0.
(1.15)

Then, it becomes evident that |Ψ BCS is the ground state of the mean-field Hamiltonian,

H MF |Ψ BCS = E 0 |Ψ BCS , (1.16)
with E 0 the ground-state energy. At this stage, one clarification is in order. To prove Eq. (1.15), one needs to identify the variational parameters in the BCS wavefunction u k , v k with the coefficients of the linear transformation [we had surreptitiously used the same labelling in Eqs. (1.9), (1.10)]. This connection is not trivial a priori, however, the definitions for u k , v k in Eqs. (1.11), (1.12) are actually the solutions to the variational problem of minimizing Ψ BCS |H BCS |Ψ BCS . Nevertheless, one should not forget that |Ψ BCS is not the ground state of H BCS (which is, after all, an interacting Hamiltonian albeit effective), but "only" an ansatz which turns out to be a good approximation 5 . At the same time, it is not completely unexpected that |Ψ BCS is the ground state of the mean-field Hamiltonian: it was precisely its structure (a coherent superposition of Cooper pairs) the motivation to choose the anomalous average c -k ,↓ c k ,↑ as the order parameter for the mean-field decoupling.

After recognizing that E 0 is the ground state, it becomes apparent that γ † k,+ and γ † k,-represent fermionic single-particle excitations of energy E k over the many-body BCS ground state, often referred to as Bogoliubov quasiparticles in the literature. As the form of the Hamiltonian in Eq. (1.13) indicates, the fundamental excitations in a superconductor form a free fermion gas, but contrary to the standard free electron gas, here the "Fermi sea" is completely empty in the ground state. The lowest-lying excited states read

γ † k,+ |Ψ BCS = c † k,↑ k =k u k + v k c † k ,↑ c † -k ,↓ |0 , (1.17) γ † k,-|Ψ BCS = c † -k,↓ k =k u k + v k c † k ,↑ c † -k ,↓ |0 , (1.18)
and they can be interpreted as localizing an electron into one of the states of the pair [(k, ↑) or (-k, ↓) for the excitations γ † k,+ and γ † k,-respectively], thereby precluding the pair state from entering the coherent wavefunction and increasing the energy of the system. It follows from the form of E k that the minimal energy of these excitations is E k F = |∆|. This is why the order parameter ∆ is commonly referred to as superconducting gap. This gap is manifested in the excitation density of states (DOS),

ρ(E) = k δ(E -E k ) ρ 0 E √ E 2 -∆ 2 Θ(E -∆), (1.19)
where ρ 0 is the DOS of the normal metal at the Fermi level, and Θ(x) is the Heaviside step function. As shown in Fig. 1.1 the spectral weight is pushed above the gap so that the corresponding excitation DOS diverges at ∆. Furthermore, it should be noted that the Bogoliubov quasiparticles are a linear combination 1. Introduction of c † k,↑ (electrons) and c -k,↓ (holes). This fact implies that excited states can be accessed by either adding electrons or removing electrons from the BCS ground state,

c † k,↑ |Ψ BCS = u k γ † k,+ |Ψ BCS , (1.20) c -k,↓ |Ψ BCS = -v k γ † k,+ |Ψ BCS , (1.21)
with analogous equations for the other excitation flavour (γ † k,-). This particle-hole mixing is evidenced, for instance, in scanning tunneling microscopy (STM) experiments, where the differential conductance -which is a priori directly proportional to the local spectral functionis symmetric around zero-bias in superconductors.

A word on unconventional superconductivity

Broadly speaking, unconventional superconductivity is a catchall label for compounds that exhibit vanishing DC resistivity and the Meissner effect below a specific critical temperature but also have properties that do not comply with the standard BCS theory. Some well-known examples are heavy-fermion materials, cuprates, or iron-based superconductors. Here, we precisely mean that the pairing state differs from the original singlet s-wave proposal. In the Landau sense, the order parameter in unconventional superconductors spontaneously breaks additional point group symmetries of the lattice (or even translation or time-reversal) in addition to global U (1). These exotic forms of the order parameter (or equivalently, of the Cooper pair wavefunction) arise from a non-trivial momentum structure of the scattering potential [START_REF] Sigrist | Introduction to Unconventional Superconductivity[END_REF][START_REF] Sigrist | Phenomenological theory of unconventional superconductivity[END_REF][START_REF] Mineev | Introduction to unconventional superconductivity[END_REF]. This thesis mostly deals with conventional s-wave superconductors, but in Chapter 4 we shall consider a more intricate pairing. To prepare the field, we present here the generalization of the BCS order parameter. The most general BCS Hamiltonian describing Cooper pairs with vanishing center-of-mass momentum reads as follows,

H BCS = k,σ ε k c † k,σ c k,σ + k,k {σ i } V σ 1 ,σ 2 ;σ 3 ,σ 4 k,k c † k,σ 1 c † -k,σ 2 c -k ,σ 3 c k ,σ 4 . (1.22)
Owing to the anticonmmutation relations of the fermionic operators, the matrix element of the scattering interaction must satisfy

V σ 1 ,σ 2 ;σ 3 ,σ 4 k,k = -V σ 2 ,σ 1 ;σ 3 ,σ 4 -k,k = V σ 2 ,σ 1 ;σ 4 ,σ 3 -k,-k . (1.23)
In addition, the invariance of the Hamiltonian under the elements of its symmetry group (crystallographic point group operations, spin rotations, etc.) imposes a restriction on the interaction matrix element; namely, it can be decomposed as

V σ 1 ,σ 2 ;σ 3 ,σ 4 k,k = Γ,i v Γ D Γ,i (k; σ 1 , σ 2 )D Γ,i (k ; σ 3 , σ 4 ) † , (1.24)
where D Γ,i (k; σ 1 , σ 2 ) is a basis function in the antisymmetric restriction 6 of the

[k ⊗ σ 1 ⊗ σ 2 ]
space that transforms according to the irreducible representation (irrep) Γ of the symmetry group of H BCS . The sum in Γ goes over all the irreps of said group and the sum in i over the dimension of each irrep. Crucially, the amplitude of the interaction in each channel v Γ is not imposed by symmetry principles but depends on the material.

The mean-field decoupling is analogous to that presented in Section 1.1.1,

H MF = k,σ ε k c † k,σ c k,σ - k σ 1 ,σ 2 ∆ σ 1 ,σ 2 (k)c † k,σ 1 c † -k,σ 2 + ∆ * σ 1 ,σ 2 (k)c -k,σ 2 c k,σ 1 , (1.25)
where we omitted the operator-independent terms that only affect the ground-state energy. Now, the order parameter has a momentum dependence and a spin structure,

∆ σ 1 ,σ 2 (k) = - k V σ 1 ,σ 2 ;σ 3 ,σ 4 k,k c -k ,σ 3 c k ,σ 4 . (1.26)
The gap function has the symmetry properties of the Cooper pairs -which themselves inherit their symmetry from the pairing interaction-, therefore, ∆ σ 1 ,σ 2 (k) can also be expressed very generally as a linear combination of antisymmetric basis functions. In principle, there is one dominating interaction channel (the most negative v Γ ) that gives the highest T c , and triggers the superconducting instability; therefore, the superconducting gap transforms according to one irreducible representation of the symmetry group of the Hamiltonian. It becomes apparent now that unless that is the identity irrep, as it is the case for conventional superconductors, the gap function "spontaneously" breaks certain symmetries of the lattice [START_REF] Annett | Symmetry of the order parameter for high-temperature superconductivity[END_REF][START_REF] Annett | Unconventional superconductivity[END_REF].

It is useful to separate the gap function into an even-k part, which is naturally antisymmetric in spin (singlet), and an odd-k part, which is symmetric in spin (triplet). In particular, we have

∆ ↑↓ (k) -∆ ↑↓ (k) √ 2 -→ singlet OP, ∆ ↑↑ (k) -→ triplet OP (m = 1), ∆ ↑↓ (k) + ∆ ↑↓ (k) √ 2 -→ triplet OP (m = 0), ∆ ↓↓ (k) -→ triplet OP (m = -1).
The order parameter is usually expressed in a vector form in spin space,

∆(k) = ∆ ↑↑ (k) ∆ ↑↓ (k) ∆ ↓↑ (k) ∆ ↓↓ (k) = -d x (k) + id y (k) ψ(k) + d z (k) -ψ(k) + d z (k) d x (k) + id y (k) = (ψ(k)σ 0 + d(k) • σ)(iσ y ),
(1.27) where ψ(k) and d(k) are the singlet and triplet gap functions respectively, and σ the vector of Pauli matrices. We emphasize that the singlet component of the OP is even in k while the triplet counterpart is odd,

ψ(k) = ψ(-k),
(1.28)

d(k) = -d(-k), (1.29)
relations in Eq. (1.23) and further, the ∆(k) may transform non-trivially under other symmetry operations. The various possible pairing states are labeled according to the symmetry of the gap function, e.g. swave, p-wave, d-wave, etc. in analogy with the traditional nomenclature in atomic physics. This notation can be intuitively connected with the total angular momentum of the Cooper pair, although strictly speaking, angular momentum is not a good quantum number in a solid that breaks full rotational symmetry. We make one additional remark: if the normal state is inversion-symmetric, mixed parity states are not allowed; the even-k singlet and odd-k triplet gap functions transform according to different irreps of the symmetry group; therefore, they cannot coexist unless an accidental degeneracy occurs. This is not the case in non-centrosymmetric materials, where the Hamiltonian is not invariant under inversion, and thus, the basis functions of a given irrep may have an ill-defined parity. This scenario will be the focus of Chapter 4.

Finally, there remains the question of what are the possible microscopic mechanisms that induce a more complicated pairing mechanism. This issue is beyond the scope of this introduction, but we mention that an attractive interaction mediated by spin fluctuations is a widespread proposal for d-wave superconductors.

Bogoliubov-de Gennes -Nambu representation

Even though up to this point we have discussed the BCS theory in the language of second quantization, it is possible to construct a first-quantization version of the BCS mean-field Hamiltonian because it is a quadratic object. For that purpose, we introduce the Nambu spinor,

Ψ † k = c † k,↑ , c † k,↓ , c -k,↑ , c -k,↓ (1.30)
which comprises both spin-up and spin-down electron creation and annihilation. The full meanfield Hamiltonian7 reads

H MF = 1 2 k Ψ † k ĤBdG (k)Ψ k + 1 2 k Tr ĤN (k) , (1.31)
where

ĤBdG (k) = ĤN (k) ∆(k) ∆ † (k) -Ĥ N (-k) , (1.32)
with ĤN (k) the normal-state Hamiltonian and ∆(k) the pairing matrix defined in Eq. (1.27). We note that in the present discussion, the normal-state Hamiltonian is simply proportional to the identity matrix, but in general it takes the form ĤN (k) = ε k σ 0 + h(k) • σ with h(k) containing k-symmetric (e.g. Zeeman field) and k-antisymmetric (e.g. spin-orbit coupling) terms.

The diagonalization of the four by four BdG Hamiltonian ĤBdG (k) immediately yields the energy spectrum of the single-particle excitations over the BCS ground state. There is, however, one subtlety that we discuss now. Since the pairing term in the mean-field Hamiltonian is bilinear in c, c † , we had to introduce a first-quantization basis that effectively doubles the degrees of freedom. Indeed, it should be noted that the Nambu spinors Ψ † k and Ψ k are not independent, 1.1. Microscopic theory of superconductivity but fulfill the following relation,

Ψ k = τ x σ 0 Ψ † -k = PΨ -k , (1.33)
with P = τ x σ 0 K an anti-unitary operator. Here and in the rest of this manuscript τ α and σ α are the usual Pauli matrices in particle-hole and spin space, respectively, and K is the complex conjugation operator. This redundancy imposes a constraint on the first-quantized BdG Hamiltonian in the form of a particle-hole symmetry,

P ĤBdG (k)P -1 = -ĤBdG (-k), ( 1
H MF = 1 2 k Υ † k ĤD (k)Υ k + 1 2 k Tr ĤN (k) = = k,α=± E k,α γ † k,α γ k,α + 1 2 k -(E k,+ + E k,-) + Tr ĤN (k) , (1.35)
where the second equality follows from applying the anticommutation relations of the fermionic operators γ † k,± . We can immediately recognize the diagonal form presented in Eq. (1.13), where the operator-independent terms give the ground-state energy. We emphasize that while the BdG Hamiltonian has four eigenvalues that come in particle-hole pairs, the true (single-particle) excitations over the ground state have positive energy. Finally, we note that if the BdG Hamiltonian has time-reversal or inversion symmetry, then E k,+ = E -k,-and E k,α = E -k,α , respectively. In the previous section, we had E k,+ = E k,-because we had considered a spin-degenerate system.

We conclude the discussion by noting that sometimes a different Nambu basis is used in the literature,

Ψ † k = c † k,↑ , c † k,↓ , c -k,↓ , -c -k,↑ . (1.36)
In this basis, the BdG Hamiltonian in Eq. (1.31) reads

Ĥ BdG (k) = ĤN (k) ∆ (k) ∆ † (k) -T ĤN (k)T -1 , , (1.37)
where T = iσ y K, k → -k is the time reversal operator and ∆ (k

) = ∆(k)(iσ y ) † = ψ(k) + d(k) • σ.
The interest of this basis is that the s-wave order parameter becomes proportional to the unit matrix. If the normal Hamiltonian ĤN (k) has spin rotation symmetry, the four-by-four BdG Hamiltonian for s-wave superconductors is diagonal in spin-space and can be effectively reduced to a two-by-two problem. In this basis, the particle-hole constraint in Eq. (1.34) takes the form P = τ y σ y K.

Introduction to Yu-Shiba-Rusinov states 1.2.1 T-matrix formalism

Understanding the properties of Yu-Shiba-Rusinov states is an example of the much more general problem of studying a local defect embedded in an otherwise homogenous substrate. This is, in its very essence, a perturbation problem. In very general terms, it is assumed that one can solve the Hamiltonian describing the pristine system H 0 , while the total Hamiltonian,

H = H 0 + V imp , (1.38)
does not have a straightforward solution because V imp breaks translation invariance. In the problem originally considered by Yu, Shiba, and Rusinov, the substrate is a non-interacting system -an s-wave BCS superconductor at the mean-field level-, and further, it is assumed that the impurity Hamiltonian does not have any internal quantum degrees of freedom. Therefore, the problem can be efficiently addressed within the T-matrix formalism, whereby the singleparticle Green's function of the full system G(ω) ≡ (ω -H) -1 , can be expressed as an infinite series involving the Green's function of the bare Hamiltonian G 0 (ω) and the impurity potential [START_REF] Economou | Green's functions in quantum physics[END_REF]. In momentum representation, and assuming that the impurity scattering is elastic, this relationship (often called the Dyson Equation) between the "dressed" and the "bare" Green functions reads,

G(k, k ; ω) = G 0 (k; ω) + G 0 (k; ω)V imp (k, k )G 0 (k ; ω) + k G 0 (k; ω)V imp (k, k )G 0 (k ; ω)V imp (k , k )G 0 (k ; ω) + . . . = G 0 (k; ω) + G 0 (k; ω)T k,k (ω)G 0 (k ; ω), (1.39)
where T k,k (ω) comprises the infinite summation. Crucially, in order to find bound-state solutions, one has to resummate in Eq. 1.39 to all orders. In Chapter 4, we shall see, however, that it is possible to gain some insight into the effect of weak scatterers by considering a finite number of terms only. Under the assumption of a fully local and isotropic impurity potential, V imp (k, k ) ≡ const., the T-matrix becomes momentum independent,

T (ω) = V imp [1 -G 0 (0, ω)V imp ] -1 , (1.40)
where G 0 (0, ω) is the bare Green's function evaluated at the origin 8 . Expressing Eq. (1.39) in real space, G(r, r ; ω) = G 0 (rr ; ω) + G 0 (rr 0 ; ω) T (ω)G 0 (r 0r ; ω).

(1.41) renders the physical interpretation of the formalism very transparent. The amplitude of probability of a quasiparticle to propagate between two separate points in space r and r in the presence of a local impurity at r 0 can be understood as the linear superposition of the ampli- 8 We recall that r|k = e ik•r √ Ω , so that going to the continuum limit k → Ω dk (2π) d , the real-space representation reads G(r, r ) = dk (2π) d dk (2π) d e i(kr-k r ) G(k, k ). Naturally, translational-invariant objects such as G0(k) are diagonal in momentum space, therefore, they only depend on the difference rr . In order to investigate a single isolated defect, it is more informative to study local observables instead of averaged quantities as in the presence of a macroscopic concentration of impurities. In particular, in this thesis we shall repeatedly consider the local density of states (LDOS) which can be probed by STM measurements. We recall that by definition, the poles of G(ω) are in one-to-one correspondence with the eigenvalues of the Hamiltonian; therefore, the single-particle Green's function is related to the LDOS as

ρ(r, ω) = - 1 π Im Tr G R (r, r, ω), (1.42)
where G R (ω) = lim η→0 + G(ω + iη) is the single-particle retarded Green's function and Tr should be understood as the trace over the internal degrees of freedom. The modification of the LDOS induced by the impurity is encoded in the second term of Eq. (1.41). In the particular case of s-wave superconductors where the substrate is gapped, the existence of the sub-gap bound states is solely determined by the poles of the T-matrix |ω| < |∆|. Further, we remark that the assumption of a point-like scatter implies that all the spatial features of the sub-gap bound states depend on the bare propagator, i.e., the properties of the substrate. This observation will be crucial in Chapter 3 where we develop a theory to account for the properties of Yu-Shiba-Rusinov states in real space.

Model Hamiltonian

Yu-Shiba-Rusinov states are the response to magnetic impurities embedded in a superconductor. Generally speaking, exogenous atoms can affect the electrons in the host in two different manners [START_REF] Balatsky | Impurity-induced states in conventional and unconventional superconductors[END_REF]. On the one hand, conduction electrons experience a Coulomb interaction because the electronic structure of an impurity is typically different from that of the atoms in the lattice. Usually, one can assume that the Coulomb potential is effectively screened at length scales on the order of the inter-atomic distance, and hence, it is reasonable to model the impurity as a Dirac-delta potential on the impurity site,

V non-mag imp = K σ drδ(r -r 0 )c † r,σ c r,σ , (1.43)
with K the strength of the non-magnetic scattering and r 0 the position of the impurity. Note that this term does not break time-reversal symmetry and, thus, acts on both spin species equivalently. On the other hand, the impurity atom can also have a magnetic moment, which scatters conduction electrons through an effective exchange interaction of amplitude J that can also be assumed local,

V mag imp = - J 2 σ,σ drδ(r -r 0 )c † r,σ S • σc r,σ , (1.44)
where S is the spin on the impurity. In the original analysis of YSR states, the magnetic impurity is described as a classical object. This assumption is formally implemented by taking the limit S → ∞ while keeping the product JS constant, so that the spin operator describing the impurity is replaced by an effective fully local Zeeman splitting of the conduction electrons that, in a slight abuse of notation, we also denote J. We shall discuss the validity of this approximation in detail in Sec. 2.5 of Chapter 2, in the context of YSR states in Van Hove superconductors.

To avoid repetition, here we just mention that the classical treatment provides good qualitative agreement with experimental observations in many situations; however, one should not forget that the quantum nature of the spin on the impurity can induce complex correlations in the solid that originate remarkable effects (most notably, the Kondo resonance). The focus of Chapter 5 will be a set of experimental observations that demand a description of the impurity beyond the classical YSR model.

As we are describing the problem at the single-particle level, the BdG representation of the mean-field BCS Hamiltonian is best suited for developing the T-matrix equations. In the second Nambu basis [Eq. (1.36)], the Hamiltonian reads,

Ĥ0 = ε k τ z σ 0 + ∆τ x σ 0 , (1.45) Vimp = Kτ z σ 0 -Jτ 0 σ z . (1.46)
Exploiting the spin-rotation symmetry of the substrate, we fixed the spin quantization axis of the impurity along ẑ without loss of generality. In this set-up, the BdG Hamiltonian is diagonal in spin-space, and we can work in the restricted particle-hole subspace

[Ψ ↑ k = c † k,↑ , c -k,↓ ].

General properties of YSR states

In the remainder of this section, we consider a two-dimensional, isotropic substrate, with ε k describing a free electron gas. This example will serve to illustrate the use of the mathematical tools introduced above and at the same to discuss the key features of YSR states in the most simple scenario.

For completeness, we explicitly write the bare Green's function in momentum representation, Ĝ0 (k; ω) = 1

ω 2 -ε 2 k -∆ 2 ω + ε k ∆ ∆ ω -ε k , (1.47)
and evaluated at the origin r = 0,

Ĝ0 (0; ω) = - ρ 0 π √ ∆ 2 -ω 2 ω ∆ ∆ ω . (1.48)
In order to compute the previous result, we made the usual assumption that the normal density of states of the substrate is approximately constant on the energy scale of the superconducting gap and set ρ(ε) = ρ 0 with an infinite bandwidth limit.

The poles of T-matrix satisfy the equation det 1 -Vimp Ĝ0 (0, ω) = 0. In our model, we find one sub-gap solution |ω| < |∆| which yields the energy of the YSR bound state as a function 1.2. Introduction to Yu-Shiba-Rusinov states of the system's parameters,

E S = ∆ 1 -J 2 + K 2 (1 -J 2 + K 2 ) 2 + 4 J 2 , (1.49)
where we introduced the dimensionless quantities J = πρ 0 J > 0 and K = πρ 0 K. To interpret the meaning of this result, we recall that the poles of the single-particle Green's function correspond to the eigenenergies of the associated Hamiltonian. It should not be forgotten, however, that here the Green's function is built upon the BdG Hamiltonian whose spectrum represents the single-particle excitations over the BCS ground state; therefore, the in-gap pole corresponds to one of the said excitations. The impurity-mediated scattering of electrons competes with the formation of Cooper pairs: localizing a state and thereby preventing it from entering the coherent wavefunction entails an energy trade-off due to the exchange interaction; therefore, the energy of the lowest excitation is decreased so that it becomes a sub-gap state. Indeed, we note that in the absence of magnetic scattering (J = 0), the in-gap excitation merges in the continuum in consistency with the Anderson theorem, and as shown in Inspired from [START_REF] Farinacci | Tuning the Coupling of an Individual Magnetic Impurity to a Superconductor: Quantum Phase Transition and Transport[END_REF].

At J c = 1 + K 2 , the energy of the in-gap state vanishes, signaling a (first-order) quantum phase transition [START_REF] Sakurai | Comments on Superconductors with Magnetic Impurities[END_REF][START_REF] Salkola | Spectral properties of quasiparticle excitations induced by magnetic moments in superconductors[END_REF]. At weak coupling ( J < J c ), the ground state of the system is a BCSlike state, where all electronic states are paired 9 , and the first excitation corresponds to a state 1. Introduction in which one of the pairs does not enter the coherent wavefunction,

|Ψ GS J< Jc = n u n + v n c † n c † -n |0 , (1.50
)

|Ψ exc J< Jc = γ † -1 |Ψ GS J< Jc = c † -1 n>0 u n + v n c † n c † -n |0 . (1.51)
Here, (n, -n) label time-reversed states as crystal momentum is not a good quantum number in the presence of the scatterer. At the critical point, the strength of the exchange coupling is sufficiently large so that the state with a bound quasiparticle is more energetically favorable than the fully paired BCS state, and hence, the ground state of the system and the first excitation are exchanged,

|Ψ GS J> Jc = |Ψ exc J< Jc , (1.52
)

|Ψ exc J> Jc = |Ψ GS J< Jc . (1.53)
Initially, the spin of the impurity is mostly unscreened, but after the transition, the unpaired localized electron bound to the impurity partially screens it. Therefore, the level crossing entails a change in the total fermion parity and the system's total spin, δS = -1/2. Throughout this discussion, we implicitly assumed an antiferromagnetic exchange coupling, but we note that the YSR model yields analogous results if it is ferromagnetic [the energy of the bound state is even in J, see Eq. (1.49)], in which case we would have δS = 1/2. This is not true for a quantum spin, where a bound state forms for an antiferromagnetic coupling only. In addition, we mention that the classical description that we presented here can be extended to account for multi-orbital impurities, relevant for transition atoms with partially filled d or f shells [START_REF] Arrachea | Yu-Shiba-Rusinov multiplets and clusters of multiorbital adatoms in superconducting substrates: Subgap Green's function approach[END_REF]. In this case, owing to the crystal-field splitting induced by the lattice, scattering occurs in various orbital channels with different hybridization parameters, and multiple in-gap states are expected [START_REF] Ruby | Orbital Picture of Yu-Shiba-Rusinov Multiplets[END_REF].

As argued above, the in-gap state represents a Bogoliuvob quasiparticle, defined as a linear combination of an electron-like and a hole-like operator. The spectral signature of a YSR bound state in an STM experiment is then a double resonance sitting symmetrically with respect to bias energy. Elucidating the tunneling process involved in an STM experiment requires doing a detailed transport calculation where the STM tip is included; nevertheless, the Tmatrix formalism still provides certain insight into the spectral signature of YSR states. For instance, as we show next, it allows us to clarify the origin of the characteristic asymmetry in the spectral weight on the impurity site at the energy of the YSR state that is routinely reported in experiments [START_REF] Ruby | Tunneling Processes into Localized Subgap States in Superconductors[END_REF]. We start by evaluating Eq. (1.41) at the impurity location, Ĝ(0, 0

; ω) = πρ 0 2ω J -(1 -J 2 + K 2 ) √ ∆ 2 -ω 2 ω + ( J + K) √ ∆ 2 -ω 2 ∆ ∆ ω + ( J -K) √ ∆ 2 -ω 2 .
(1.54) The in-gap poles of the total Green's function are naturally given by the expression of E S , 1.2. Introduction to Yu-Shiba-Rusinov states Eq. (1.49). Taking the principal part of the Laurent series expansion at ω = E S we find, Ĝ(0, 0

; ω) ≈ 1 ω -E S u 2 uv uv v 2 , (1.55)
where

u 2 , v 2 = 2 Jπρ 0 ∆ 1 + J ± K 2 (1 -J 2 + K 2 ) 2 + 4 J 2 3/2 (1.56)
Upon identifying the electron and hole components of the Green's function with the differential conductance measured at positive (+E S ) and negative (-E S ) bias energy respectively10 , we can write the in-gap LDOS on the impurity site as

ρ(0, ω) = u 2 δ(ω -E S ) + v 2 δ(ω + E S ). (1.57)
If the scatterer has a non-magnetic component, K = 0, there is an asymmetry in the spectral weight (u 2 = v 2 ). A positive non-magnetic scattering amplitude favors the "electron character" of the excitation and vice-versa if K < 0. It is often said that the non-magnetic scattering potential "locally breaks particle-hole symmetry", but this only means that K acts as a shift of the chemical potential on the impurity site. The BdG Hamiltonian is always particle-hole symmetric by construction, therefore, the poles of the T-matrix come in pairs.

Routine STM measurements allow resolving YSR states in real space. As discussed earlier, under the assumption of a point-like defect, the bare propagator controls the spatial structure of the LDOS. For an isotropic two-dimensional substrate with Fermi momentum k F , in the asymptotic limit far away from the impurity (rk F 1), we find, Ĝ0 (r;

E S ) ∼ ρ 0 2π k F r e -r/ξ g ee g eh g eh g hh , (1.58)
where g ee,hh = sin(γ)

E S √ ∆ 2 -E 2 S ± cos(γ) and g eh = sin(γ) ∆ √ ∆ 2 -E 2 S , with γ = k F r + π 4 .
The in-gap bound state is localized on the impurity, and it decays according to a product of an exponential and a power law. The characteristic length of the propagator is given by

ξ = v F ∆ 2 -E 2 S , (1.59)
which is reminiscent of the superconductor's coherent length. The exponential decay contrasts with the algebraic behavior of charge-screening in normal metals. It is, in fact, a consequence of the gapped nature of the substrate; note, for instance, that when the bound-state merges in the continuum (E S = ∆, in the absence of magnetic scattering), the decay length diverges. Further, the power-law decay is controlled by the dimensionality of the substrate. In the case of bulk impurities in three-dimensional superconductors, the propagator decays as 1 k F r , resulting in shorter-range YSR states [START_REF] Rusinov | Superconductivity near a paramagnetic impurity[END_REF]. This remarkable difference was first evidenced not long ago in YSR states on quasi-two-dimensional NbSe 2 [START_REF] Gerbold | Coherent long-range magnetic bound states in a superconductor[END_REF], which significantly fostered the interest in the field; nevertheless, this property is not inherent to YSR states, but it is a general attribute of the local response to defects. The energy-resolved LDOS modulation due to a single scatterer in a normal metal also exhibits the same power-law decay as the YSR states11 [START_REF] Knorr | Long-range adsorbate interactions mediated by a two-dimensional electron gas[END_REF]. We finally note that Eq. (1.58) contains an oscillatory term with characteristic length λ = 1 k F , reminiscent of the Friedel oscillations in a normal metal. While we do not show it here explicitly, the oscillations of the in-gap LDOS in the far field limit exhibit a phase-difference ϕ = arccos(E S /∆) between the electron (positive bias) and hole (negative bias) components.

To conclude this section, we discuss one assumption we implicitly made during this presentation, namely treating ∆ as a degree of freedom instead of self-consistently. The magnetic impurity has a pair-breaking effect, and indeed, previous studies show that the gap magnitude reduces with increasing strength of the exchange coupling amplitude J, effectively realizing a S-S' junction, which, interestingly, becomes π-type after the quantum phase transition [START_REF] Salkola | Spectral properties of quasiparticle excitations induced by magnetic moments in superconductors[END_REF][START_REF] Meng | Superconducting gap renormalization around two magnetic impurities: From Shiba to Andreev bound states[END_REF]. However, these considerations do not alter the structure of the in-gap spectrum qualitatively. Further, the spatial effects are localized on a length scale on the order of k -1 F , which is much smaller than the range of validity of the asymptotic approximations that we use to calculate the LDOS analytically [START_REF] Flatté | Local electronic structure of defects in superconductors[END_REF]. In the remainder of the manuscript, we assume then that the non-self-consistent treatment of the superconducting order parameter is justified.

Chapter 2

Effect of Van Hove singularities on YSR states in s-wave superconductors

Introduction

The density of states (DOS) is a central concept in solid-state physics. In the context of the electronic structure, it essentially represents the number of existing wave vectors at a given energy per unit interval of energy and volume. A large DOS enhances electronic correlations, which in turn can trigger quantum emergent phenomena such as superconductivity [START_REF] Dzialoshinskii | The superconducting transitions due to van hove singularities in the electron spectrum[END_REF] or chargedensity-wave instability [START_REF] Rice | New Mechanism for a Charge-Density-Wave instability[END_REF]. The specific energy values where the DOS exhibits a divergence are then of particular interest. The energy dispersion determines the DOS, and its singular points coincide with critical points in the Brillouin zone, i.e., the k-points where the electronic group velocity vanishes. In particular, it was first studied by L. Van Hove in the context of phonons that saddle points in two-dimensional systems yield a logarithmic divergence in the DOS, called Van Hove singularity (VHS) [START_REF] Van Hove | The Occurrence of Singularities in the Elastic Frequency Distribution of a Crystal[END_REF]. The notion of VHS was recently extended to higher-order Van Hove singularities (HOVHS), where the expansion of the energy dispersion in the vicinity of the critical point vanishes to (at least) second order [START_REF] Noah | Magic of high-order van Hove singularity[END_REF][START_REF] Noah | Classification of critical points in energy bands based on topology, scaling, and symmetry[END_REF]. Then, the divergence in the DOS is amplified from logarithmic to power-law. HOVHS have been experimentally observed in a kagome superconductor [START_REF] Hu | Rich nature of Van Hove singularities in Kagome superconductor CsV3Sb5[END_REF], and were predicted to be realizable by tuning a single parameter in moiré superlattices [START_REF] Noah | Magic of high-order van Hove singularity[END_REF][START_REF] Noah | Classification of critical points in energy bands based on topology, scaling, and symmetry[END_REF]. In addition, recent experimental progress showcasing the ability to bring a VHS arbitrarily close to the Fermi level in twisted bilayer graphene (TBG) [START_REF] Li | Observation of Van Hove singularities in twisted graphene layers[END_REF][START_REF] Xu | Tunable van Hove singularities and correlated states in twisted monolayerbilayer graphene[END_REF] and in heterostructures [START_REF] Mori | Controlling a Van Hove singularity and Fermi surface topology at a complex oxide heterostructure interface[END_REF] has reawakened the interest in the field.

In this chapter, motivated by the recent discovery of superconductivity in TBG [START_REF] Cao | Unconventional superconductivity in magic-angle graphene superlattices[END_REF] and in Bernal bilayer graphene doped to a VHS [START_REF] Zhou | Isospin magnetism and spin-polarized superconductivity in Bernal bilayer graphene[END_REF], the quest to find YSR states in graphene [START_REF] Cortés-Del Río | Observation of Yu-Shiba-Rusinov states in superconducting graphene[END_REF][START_REF] Lado | Unconventional YuShibaRusinov states in hydrogenated graphene[END_REF][START_REF] Lopez-Bezanilla | Defect-induced magnetism and Yu-Shiba-Rusinov states in twisted bilayer graphene[END_REF], and the promising pathway to VHS tunability in these materials, we explore the influence of conventional and higher-order VHS on YSR states. We find that tuning the Fermi level to a Van Hove singularity is a powerful way to enhance the coupling of YSR states to the substrate, which is crucial to employ them for engineering complex structures, while at the same 2.2. Model Hamiltonian time, it provides a possibility to extract information from the DOS singularity itself. This chapter is organized as follows. In Sec. 4.2, we present the model Hamiltonian for the YSR impurity coupled to a superconductor with a Fermi level tuned near a VHS or HOVHS. Sec. 4.3 contains our analytical and numerical results for both types of Van Hove singularities. In Sec. 2.4, we analyze the effect of a small perturbation in the chemical potential away from the singularity, and finally, in Sec. 2.5, we provide a short conclusion and discussion of the results. Some technical details are presented in App. 2.A.

Model Hamiltonian

We consider a point-like, isotropic, magnetic impurity on a two-dimensional, s-wave superconductor that we describe with the YSR model introduced in Chapter 1. For completeness, we recall the Hamiltonian,

H = k,σ ε k c † k,σ c k,σ + k ∆c † k,↑ c † -k,↓ + h.c. + σ (K -σJ) c † r 0 ,σ c r 0 ,σ , (2.1)
where c † r 0 ,σ creates an electron with spin σ at the impurity site, ε k is the energy dispersion of the electrons in the superconducting substrate measured from the Fermi level, ∆ is the superconducting gap (assumed to be real), K is the amplitude of the non-magnetic scattering potential and J ≡ JS/2 > 0 denotes the coupling strength between the electrons and the magnetic impurity with classical spin S. In the present study, we disregard any quantum effects associated with the magnetic impurity (e.g. Kondo screening), and comment on the model's validity at the end of the chapter.

In this work, we focus on three expressions for the energy dispersion ε k defined in the continuum. Since the information relevant to the YSR states comes from the vicinity of the Fermi momentum, the three energy dispersions are representative of three broad classes of systems: ones with constant DOS at the Fermi energy, ones with a conventional VHS, and ones belonging to a family of HOVHS labeled by a parameter ν, as we detail now. First, owing to the toroidal topology of the Brillouin zone, the energy dispersion of two-dimensional systems is always endowed with at least two saddle points, defined as

∇ k ε k = 0, det D < 0, (2.2)
where D the Hessian matrix of ε k . As it was mentioned in the Introduction, it is well known that such saddle points yield a logarithmic singularity in the DOS at the corresponding energy, henceforth denoted conventional Van Hove singularity:

ρ(ε) = 1 2E c log E c ε , (2.3)
where ε is the energy measured from the Fermi level. Here the prefactor stems from imposing the normalization condition 1 = Ec -Ec ρ(ε)dε, and E c is an energy cutoff which is introduced to delimit the energy range over which Eq. (2.3) is a good description of the system's DOS. In our calculations, we always assume that E c is the largest energy scale, of the order of the bandwidth. Next, for systems with constant density of states at the Fermi energy, we use the constant

ρ(ε) ≡ ρ 0 = 1 2E c , (2.4)
which is conveniently normalized with the use of the same bandwidth. Finally, we also consider higher-order Van Hove singularities, where degenerate saddle points, i.e. fulfilling det D = 0, lead to a power-law divergent DOS,

ρ(ε) = ν + 1 η + 1 1 E ν+1 c |ε| ν ×    η if ε < 0, 1 if ε > 0. (2.5)
Here -1 < ν < 0, ν ∈ Q, is the parameter labeling the HOVHS, E c has the same meaning as above, and η ≡ ρ (-|ε|) ρ(|ε|) is the particle-hole asymmetry ratio. Note that we include the case of particle-hole symmetric HOVHS at η = 1, and that the previous two dispersions are also symmetric. The exponent ν can take an infinite number of rational values in its range, each value having a corresponding η value; a set of possibilities is tabulated in Ref. [START_REF] Noah | Classification of critical points in energy bands based on topology, scaling, and symmetry[END_REF]. Having defined the models, we now calculate the energy of the YSR states by looking for the poles of the T-matrix (see Sec. 1.2.1, Ch. 1 for a description of the method).

Main results

Conventional Van Hove singularity

A logarithmic divergence of the DOS at the Fermi level as in Eq. (2.3), yields the following self-consistent expression for the YSR energy (see Appendix 2.A for details),

E S = ±∆ 1 -J 2 + K 2 4 J 2 + 1 -J 2 + K 2 2 , (2.6)
where the existence of two symmetric eigenvalues stems from the particle-hole symmetry constraint of the BdG Hamiltonian, and J = f vh (∆, E S )J and K = f vh (∆, E S )K are effective coupling energies with

f vh (∆, E S ) = π 2E c log   E c ∆ 2 -E 2 S   , (2.7)
a renormalizing factor proportional to the density of states in Eq. (2.3). An expression for E S of the conventional VHS in the strong-coupling limit E S ∆, and for vanishing K was first obtained in Ref. [START_REF] Tifrea | Classical spins in van Hove superconductors[END_REF]. Remarkably, our expression for the YSR energy in Eq. (2.6) has the same functional form as the one for a constant DOS, namely, taking the dispersion in Eq. (2.4) one obtains the standard result (see Sec. 1.2.3, Ch. 1), i.e., Eq. (2.6) with An important consequence is that a Van Hove singularity in the DOS at the Fermi level enhances the effective coupling energies in the YSR problem, since f const < f vh . As it was discussed in the introductory chapter, in the presence of a magnetic impurity, localizing an electron entails an energy trade-off due to the exchange interaction, therefore the energy of the in-gap excitation E S decreases as J increases (see Fig. 2.2). This competition triggers the YSR quantum phase transition at E S = 0, where the in-gap state becomes occupied and it turns into the new superconducting ground state. The critical magnetic coupling J c prompting the phase transition reads

f const = π 2E c . (2.8)
J j c = K 2 + 1 f j (∆, 0) 2 , (2.9)
where j ∈ {const, vh} labels the chosen dispersion. In all our calculations, we assume K E c , therefore the relation f const (∆, 0) < f vh (∆, 0) implies that a Van Hove singularity in the DOS at the Fermi level reduces J c with respect to its value with a constant DOS. This remains true even when the Fermi level is not exactly tuned to the Van Hove singularity but within an interval on the order of the superconducting gap instead, although the expression for J vh c becomes slightly more complicated. We discuss this scenario in Sec. 2.4.

The DOS enhancement at the VHS enables access to a new regime of the impurity system. Namely, the theory is always applicable in a "small K" regime defined by K 1 f j (∆,0) , which in the case of the constant DOS reiterates the theoretical condition K E c . This regime in the VHS case leads to the critical magnetic coupling J c reduction due to the larger f j (∆, 0) [Fig. 2.2 (a)]. However, for a VHS system, we can also consider a "large K" regime, where K 1 f vh (∆,0) without contradicting any assumption of the model, and then the J j c ∼ K would become independent of the DOS [see Fig. The VHS introduces a dependence of the J c on the superconducting energy gap ∆, in contrast to the standard YSR states where the normal DOS of the substrate is assumed to be constant. Specifically, this implies that lowering the superconducting gap enhances the impurity coupling, which may motivate a search for systems with an optimal size of the superconducting gap given the experimental energy resolution.

The general form of the critical magnetic coupling J j c reveals a quantity independent of the impurity parameters J, K, and instead, it characterizes the DOS itself. Namely, the slope of the E S (J) curve at the quantum phase transition only depends on the DOS and ∆,

Slp c ≡ dE S dJ J=Jc = -∆f j (∆, 0). (2.10)
This is also the case when the chemical potential µ lies on a [-∆, ∆] interval around the singularity. As shown in Sec. 2.4, a small displacement of the chemical potential yields a µ-dependent slope, but its relative change as µ varies along said interval is on the order of 1%. In Fig. 2.3, we illustrate the unchanging nature of the slope, which could be measured in experiments by varying the coupling constant across the transition as it was recently reported [START_REF] Farinacci | Tuning the Coupling of an Individual Magnetic Impurity to a Superconductor: Quantum Phase Transition and Transport[END_REF][START_REF] Hatter | Magnetic anisotropy in Shiba bound states across a quantum phase transition[END_REF][START_REF] Sujoy Karan | Tracking a Spin-Polarized Superconducting Bound State across a Quantum Phase Transition[END_REF][START_REF] Malavolti | Tunable SpinSuperconductor Coupling of Spin 1/2 Vanadyl Phthalocyanine Molecules[END_REF]. It could be possible to obtain an estimate of the renormalization parameter, and subsequently to extract information about the nature of the DOS divergence.

Higher order Van Hove singularity

In the case of a power-law divergent DOS at the Fermi level as in Eq. (2.5), the self-consistent expression for the YSR energy reads,

E S = ±∆ 1 -2f 2,ν K + (f 2 1,ν + f 2 2,ν )(K 2 -J 2 ) 4f 2 1,ν J 2 + (1 -2f 2,ν K + (f 2 1,ν + f 2 2,ν )(K 2 -J 2 )) 2 , ( 2.11)
where the renormalization parameters

f 1,ν (∆, E S ) = π 2E c 1 + ν cos π 2 ν   ∆ 2 + E 2 S E c   ν , (2.12a
)

f 2,ν (∆, E S ) = π 2E c 1 -η 1 + η 1 + ν sin π 2 ν   ∆ 2 + E 2 S E c   ν , (2.12b)
also inherit the structure of the DOS. The seemingly more complicated appearance of the YSRenergy equation stems from the particle-hole asymmetry of the DOS. Indeed, if the DOS is symmetric around the Fermi level, i.e., η = 1, the factor f 2,ν vanishes, and the expression for the YSR-energy simplifies to Eq. (2.6) where now

J = f 1,ν (∆, E S )J and K = f 1,ν (∆, E S )K.
Since -1 < ν < 0, for a sufficiently large energy cutoff E c we have f 1,ν > f vh > f const and we obtain the general result: The effect of a particle-hole symmetric singularity (either a VHS or a HOVHS) in the DOS at the Fermi level is to enhance the effective coupling between the impurity and the electrons in the host.

The critical magnetic coupling for the YSR transition in the case of a HOVHS takes a slightly more complicated form,

J hovh,ν c = K 2 + 1 -2f 2,ν (∆, 0)K f 1,ν (∆, 0) 2 + f 2,ν (∆, 0) 2 .
(2.13)

For the particle-hole symmetric case, η = 1, this expression has the same form as for the VHS, Eq. (2.9). Therefore, by regarding the effect of the non-magnetic scattering K for the particlehole symmetric cases of HOVHS we conclude the same about two possible regimes (small vs. large K) as discussed for the VHS after Eq. (2.9). For the particle-hole asymmetric case, η = 1, we note that for reasonable values of the non-magnetic scattering potential, i.e., K E c , the divergence in the DOS always reduces the critical coupling J c with respect to its value for a constant DOS. As shown in Fig. 2.2 (a), in the small K regime, the stronger the divergence of the DOS, the smaller J c . We note that the f 2,ν (∆, 0)K term stemming from the DOS asymmetry would only become relevant if |K|

1 f 2,ν (∆,0) ∼ E 1+ν c
, i.e., in the large K regime, but then the K 2 term prevails, therefore J c is governed by the non-magnetic scattering amplitude K, and it does not depend on the details of the underlying DOS [Fig. 2.2 (b)]. We remark that when the Fermi level is tuned within an interval of the order of ∆ around the Van Hove singularity, particle-hole symmetric DOSs (η = 1) also lead to a YSR energy and critical J expressions of the form (2.11) and (2.13) respectively, where the f 2,ν factor stems from the asymmetry induced (2.11), as a function of the magnetic coupling strength J zoomed around the YSR transition for logarithmically-divergent (red), power-law divergent with ν = -1/3, η = 2 (orange), and power-law divergent with ν = -1/2, η = 1 (blue) DOS. The tangent, dashed lines indicate the slope at the crossing point, which remains constant for increasing values of K denoted in the color bars and becomes steeper for a stronger divergence. Note that if the DOS is asymmetric, the minimum J c is not at K = 0 [cf. blue and orange curves, and Eq. (2.13) with and without vanishing f 2,ν (∆, 0) factor]. by the displacement of the chemical potential. Nevertheless, the main features presented here remain the same.

Interestingly, we again find a quantity that is independent of the impurity parameters, namely the slope of the E S (J) curve at the YSR transition point:

Slp c ≡ -∆ f 1,ν (∆, 0) 2 + f 2,ν (∆, 0) 2 f 1,ν (∆, 0) , (2.14)
also implying that the particle-hole asymmetry of the HOVHS divergence would be reflected in YSR states (see Fig. 2.3).

Perturbation in the chemical potential away from the Van

Hove singularity

In this section, we assess the robustness of the previous results to a deviation in the chemical potential away from the Van Hove singularity of the order of the superconducting gap ∆. To that purpose, we assume that the expressions (2.3) and (2.5) introduced in the main text remain a good description of the system's DOS and replace the energy dispersion ε k in the Hamiltonian (2.1) by ε k -µ, where µ indicates a shift in the chemical potential from the Van Hove singularity. To be consistent with the previous assumptions, E c must stay the largest energy scale of the problem, therefore, we can only address the situations where the chemical potential shift lies on a comparably small range, i.e., µ ∈ [-∆, ∆].

We find that our main conclusions remain valid even when the Fermi level is not precisely tuned to the Van Hove singularity. The enhancement of the effective coupling constants with respect to the constant DOS scenario continues to exist, we can still access a "large K" regime where the non-magnetic scattering potential controls the critical J, and the slope of the E S (J) curves at the YSR transition remains independent of the impurity parameters. Further, we note that the relative change of the meaningful observables, namely J c and the slope at the transition point, are of the order of a few percent for our range of µ.

In the following, we present the results for the two broad classes of DOS singularities.

Conventional Van Hove singularity

The YSR energy now fulfills a self-consistent equation analogous to that discussed in Section 2.3.2 in the context of asymmetric HOVHS:

E S = ±∆ 1 -2f 2,vh K + (f 2 1,vh + f 2 2,vh )(K 2 -J 2 ) 4f 2 1,vh J 2 + (1 -2f 2,vh K + (f 2 1,vh + f 2 2,vh )(K 2 -J 2 )) 2 , (2.15)
where the f 2,vh factor stems from the asymmetry induced by the perturbation in the chemical potential. We recall that when the DOS is symmetric around the Fermi level f 2 = 0 for any class of Van Hove singularities and the YSR energy equation simplifies to the standard form [Eq. (2.6)]. The renormalizing factors f 1,vh and f 2,vh are obtained by evaluating integrals (2.31a) and (2.31b) in App. 2.A, and while it is possible to obtain a closed form in terms of the dilogarithm, the full expressions are too cumbersome to provide any useful insight so we omit them here. Instead, we present the numerical solution of the self-consistent YSR equation (2.15) in Fig. 2.4, which clearly shows the smallness of the effect. Nevertheless, in the limit µ Ω 1, i.e. for a small perturbation in the chemical potential in the strong-coupling limit where

Ω 2 = ∆ 2 -E 2 S ∼ ∆ 2
, the renormalizing factors have the following approximate compact form (see App. 2.A for details):

f 1,vh (∆, E S , µ) ≈ π 2E c log E c µ 2 + Ω 2 , (2.16a) f 2,vh (∆, E S , µ) ≈ π 2E c µ Ω . (2.16b)
These expressions justify the insignificant variation of the E S (J) curves: the factor f 1,vh inherits the logarithmic form from the DOS as in the µ = 0 case while the correction characterizing the newly introduced asymmetry is such that The approximate factors become most accurate at the YSR transition where E S strictly vanishes, therefore by expanding them to the lowest order in µ ∆ it is possible to capture the behavior of the critical magnetic coupling:

f 2,vh /f 1,vh ∼ µ ∆ 1 log(Ec/∆) 1.
J c J c (µ = 0) = 1 -sgn(K)|α vh | µ ∆ + |β vh | µ ∆ 2 , (2.17)
where

|α vh | ≡ |α vh (K, ∆, E c )| < 1 | log(∆/Ec)| and |β vh | ≡ |β vh (K, ∆, E c )|, with |β(K = 0)| ∼ 1 | log(∆/Ec)| .
Thus, it becomes clear that the relative corrections to J c are small assuming a realistic bandwidth, i.e., ∆ E c . Interestingly, in the absence of a non-magnetic potential scattering (K = 0) the system is particle-hole symmetric1 , and therefore electron or hole doping has the same effect on J c [inset (a) in Fig. 2.4]. In a likewise manner, the slope of the E S (J) curve at the transition point is independent of the impurity parameters, and hence an even function of µ as well [inset (b) in Fig. 2.4]:

Slp c = -∆ f 1,vh (∆, 0, µ) 2 + f 2,vh (∆, 0, µ) 2 f 1,vh (∆, 0, µ) , Slp c Slp c (µ = 0) ≈ 1 -|γ vh | µ ∆ 2 , (2.18)
where

|γ vh | ≡ |γ vh (∆, E c )| ∼ 1 | log(∆/Ec)| .

Higher order Van Hove singularity

Analogously, if the Fermi level is tuned to the vicinity of a HOVHS, the self-consistent YSR energy equation also takes the form of Eq. (2.11), where the renormalization parameters now read

f 1,ν (∆, E S , µ) = π 2E c 1 + ν cos π 2 ν µ 2 + Ω 2 E c ν sin[(π -ϕ)ν] + η sin(ϕν) (1 + η) sin π 2 ν , (2.19a) f 2,ν (∆, E S , µ) = π 2E c 1 + ν sin π 2 ν µ 2 + Ω 2 E c ν µ 2 + Ω 2 Ω sin[(π -ϕ)(ν + 1)] -η sin[ϕ(ν + 1)] (1 + η) cos π 2 ν + µ Ω f 1,ν (∆, E S , µ), (2.19b)
with ϕ = arg(µ + iΩ). We find that the renormalizing parameters also inherit the power-law dependence from the DOS [cf. µ = 0 case, Eq. (2.12)], with µ entering as a correction to Ω as it was the case in the conventional VHS. Therefore, the energy curves when the Fermi level is tuned to the vicinity of a HOVHS maintain the same structure that their µ = 0 counterparts [see Fig. 2.5]. At the YSR transition, Ω = ∆ by definition, therefore a series expansion in powers of µ ∆ captures the behavior of the critical J at small doping. We find

J c J c (µ = 0) = 1 -sgn(K)|α hovh | µ ∆ + β hovh µ ∆ 2 , (2.20)
where

|α hovh | ≡ |α hovh (K, ∆, ν, η, E c )| and β hovh ≡ β vh (K, ∆, ν, η, E c ), with β(K = 0) = -ν 2 .
As discussed in the previous section, in the absence of non-magnetic scattering potential, we obtain an even function of µ, which interestingly only depends on the power-law exponent to the lowest order. Further, for a finite K the linear term does not vanish, and its sign depends on K in the same manner as in the conventional VHS.

The slope of the E S (J) curve at the YSR transition remains independent of the impurity parameters [compare ±K curves at the same µ in Fig. 2.5 inset in (b)], and it takes the form of Eq. (2.14) with the newly-introduced renormalization parameters [Eq. (2.19)]. At low doping,

Slp c Slp c (µ = 0) ≈ 1 + γ hovh µ ∆ + δ hovh µ ∆ 2 , (2.21)
where γ hovh ≡ γ hovh (ν, η) = η-1 η+1 ν cos(νπ/2) sin(νπ/2) and δ hovh ≡ δ hovh (ν, η, ∆, E c ), with γ hovh (ν, η = 1, ∆, E c ) = ν(1 + ν). The slope at the transition is only sensitive to the bulk parameters, therefore, if the DOS is symmetric around the singularity (η = 1), we obtain an even function

Conclusions and outlook

In summary, we found that tuning the Fermi level to a conventional VHS or HOVHS enhances the coupling between the superconducting substrate and the magnetic impurity, thereby offering a pathway to improve YSR states engineering. These results become particularly significant in the advent of twisted transition metal dichalcogenides and graphene heterostructures, which allow tuning the system over large regions of the parameter space. From a practical standpoint, one might inquire about the validity of the results in the vicinity of the Van Hove singularity. We showed that our main conclusions remain valid even when the Fermi level lies on an interval on the order of the smallest energy scale in the problem, i.e., µ ∈ [-∆, ∆]. If the chemical potential deviates from the Van Hove singularity by an energy on the order of the superconducting gap, essentially the same electronic states are mixed when the system enters the superconducting phase; hence similar behavior of the YSR states is expected. The robustness against perturbations in the chemical potential strengthens the experimental relevance of these results.

Additionally, we showed that it is possible to extract an impurity-independent quantity from the E S -J curves, namely the slope of the curve at the YSR transition. State-of-the-art experiments have shown that it is possible to continuously tune the exchange coupling constant between magnetic molecules [START_REF] Farinacci | Tuning the Coupling of an Individual Magnetic Impurity to a Superconductor: Quantum Phase Transition and Transport[END_REF][START_REF] Malavolti | Tunable SpinSuperconductor Coupling of Spin 1/2 Vanadyl Phthalocyanine Molecules[END_REF] or magnetic adatoms [START_REF] Chatzopoulos | Spatially dispersing Yu-Shiba-Rusinov states in the unconventional superconductor FeTe0.55Se0.45[END_REF] to the substrate by varying the distance between the impurity and the STM tip, thereby unambiguously identifying the transition point. We propose that this technique could be employed to compare the strength of the divergence in the DOS of different compounds regardless of the nature of the impurities, and even provide an estimate of the divergence law.

It is fair to remark that here we only considered an s-wave order parameter, while electronic instabilities due to a divergent DOS may yield more exotic pairings; indeed, many theory works have advocated that VHS can provide the pairing mechanism for high-T c superconductivity in cuprates [START_REF] Markiewicz | A survey of the Van Hove scenario for high-tc superconductivity with special emphasis on pseudogaps and striped phases[END_REF]. The situation is complex, and it seems to be far from universal: for instance, doping to Van Hove singularities reportedly destroys superconductivity in twisted trilayer graphene [START_REF] Jeong | Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene[END_REF][START_REF] Hao | Electric fieldtunable superconductivity in alternating-twist magic-angle trilayer graphene[END_REF]. However, it is also a plausible scenario that the divergence in the DOS simply enhances the electron-phonon mechanism, so that, even if certain properties of the superconductor such as the critical temperature are modified, the order parameter remains s-wave. Additionally, one could also envisage inducing an s-wave gap by proximity effect in a system with the Fermi level tuned to a Van Hove singularity.

Before moving on to the next chapter, we must comment on the other crucial approximation we did in this work: treating the spin impurity as a classical degree of freedom. The author in [START_REF] Gogolin | On the theory of the Kondo effect in two-dimensional metals[END_REF] studied the Kondo model in a normal metal near a Van Hove singularity and found nontrivial differences with respect to the standard Kondo-impurity problem in the constant-DOS regime. In particular, he showed that there is an enhancement of the Kondo temperature (T K ), which is the scale below which the quantum effects of the impurity (the Kondo effect) are fully developed:

T const K ∼ e -Ec/J , T vH K ∼ e - √ Ec/J . (2.22)
This result calls our modeling into question. In the standard case of a constant DOS, it is well-established that treating the impurity spin as a classical variable is a successful approach allowing to explain numerous experimental observations [START_REF] Heinrich | Single magnetic adsorbates on s-wave superconductors[END_REF]. To recapitulate the theoretical standpoint, the YSR model follows from the Kondo Hamiltonian (H K = JS • s) by taking the impurity to be classical. This is formally achieved by the semi-classical approximation of taking the spin of the impurity to be large, i.e., the limit S → ∞, while keeping constant the product JS. Paradoxically, it has been shown using numerical renormalization group (NRG) techniques that the actual values of S at which the YSR energy in the Kondo model converges to its classical value are considerably larger than the values of S relevant to experiments [START_REF] Rok Itko | Quantum impurity models for magnetic adsorbates on superconductor surfaces[END_REF]; therefore, the YSR model might seem unreasonably successful. However, in the same work, the author showed that if a strong on-site anisotropy is added to the Kondo Hamiltonian,

H K = JS • s + DS 2 z + E(S 2
x -S 2 y ), a similar sub-gap level structure as in the classical spin limit is recovered. These isotropy-breaking terms are typically generated for adsorbed impurities as inversion symmetry is strongly broken on surfaces. Finally, another supporting point of the YSR model is that the Kondo temperature T K might be sufficiently low so that the measurement temperature is higher than T K -yet, still lower than the superconducting transition temperature T c -, so that the quantum effects of impurity are not pronounced.

Having in mind this well-established wisdom for the case of a constant DOS, we can comment on the effect of the diverging DOS under consideration. We believe that the YSR model in this case is theoretically supported by the semi-classical limit as much as the case of constant DOS. On the other hand, since the T K scale is enhanced as described above, we might expect a reduction in the range of validity of the YSR model with diverging DOS. To go beyond these theoretical speculations, one would need to extend the work in [START_REF] Rok Itko | Quantum impurity models for magnetic adsorbates on superconductor surfaces[END_REF] near a Van Hove singularity which is a highly non-trivial task. Another interesting open challenge is considering the effect of (HO)VHS in a more general Anderson impurity model [START_REF] Huang | Quantum phase transitions and the role of impurity-substrate hybridization in Yu-Shiba-Rusinov states[END_REF].

Appendix 2.A Calculation details 2.A.1 Fermi level precisely tuned to the Van Hove singularity

To obtain the energy of the in-gap YSR state E S , we calculate the poles of the T-matrix as it was shown in the introductory chapter. Owing to the spin-rotation symmetry along the z-axis of the model, we can work in the restricted spin-up subspace of the Hamiltonian in the Nambu representation. In sum, we have to solve the equation

det 1 - dk (2π) 2 E S + ε k τ z + ∆τ x E 2 S -ε 2 k -∆ 2 (Kτ z -J) = 0, (2.23)
where τ α represent the Pauli matrices in particle-hole space, and the identity matrix is omitted. We solve Eq. (2.23) by performing the substitution dk (2π) 2 = ρ(ε)dε and evaluating

I 1 (0) = - Ec -Ec dε ρ(ε) 1 ε 2 + Ω 2 , (2.24a)
I 2 (0) = - Ec -Ec dε ρ(ε) ε ε 2 + Ω 2 , (2.24b)
where Ω 2 = ∆ 2 -E 2 S .

2.A.1.1 Logarithmic DOS (conventional Van Hove singularity)

Here

ρ(ε) = 1 2E c log E c ε . (2.25)
In the infinite band-width limit Ω Ec → 0,

I 1 (0) = 1 E 2 c 1 0 dx log(x) x 2 + (Ω/E c ) 2 ∼ 1 E 2 c ∞ 0 dx log(x) x 2 + (Ω/E c ) 2 = - 1 2E c π Ω log E c Ω , (2.26)
I 2 (0) = 1 2E c 1 -1 dx log(|x|) x x 2 + (Ω/E c ) 2 = 0.
(2.27)

2.A.1.2 Power-law DOS (higher-order Van Hove singularity)

Here

ρ(ε) = 1 1 + η ν + 1 E 1+ν c |ε| ν    η if ε < 0, 1 if ε > 0.
(2.28) 2. Effect of Van Hove singularities on YSR states in s-wave superconductors with -1 < ν < 0, ν ∈ Q . To calculate the integrals we use result 3.251.10 from Ref. [START_REF] Gradshten | Table of integrals, series, and products[END_REF]:

1 0 dx x p-1 (1 -x q ) -p q = π q 1 sin( p q π)
, q > p > 0.

(2.29)

Taking an infinite energy cutoff, E c → ∞, we have

I 1 (0) = - ν + 1 E 1+ν c Ω ν-1 ∞ 0 dx x ν x 2 + 1 = - ν + 1 2E 1+ν c Ω ν-1 1 0 dt t -1+ν 2 (1 -t) ν-1 2 = -π ν + 1 2E 1+ν c cos π 2 ν Ω ν-1 , (2.30a
)

I 2 (0) = - ν + 1 E 1+ν c 1 -η 1 + η Ω ν ∞ 0 dx x ν+1 x 2 + 1 = ν + 1 2E 1+ν c η -1 1 + η Ω ν 1 0 dt t -1-ν 2 (1 -t) ν 2 = π 1 -η 1 + η ν + 1 2E 1+ν c sin π 2 ν Ω ν , (2.30b)
where we performed the changes of variables x = ε Ω and t = 1 x 2 +1 , and we identified q = 1 and p = 1-ν 2 (2.30a) and p = -ν 2 (2.30b).

2.A.2 Fermi level tuned to an interval [-∆, ∆] around the Van Hove singularity

We now allow for a finite chemical potential, and the relevant integrals read

I µ 1 (0) = - Ec -Ec dε ρ(ε) 1 (ε -µ) 2 + Ω 2 ,
(2.31a)

I µ 2 (0) = - Ec -Ec dε ρ(ε) ε -µ (ε -µ) 2 + Ω 2 , (2.31b)
where

Ω 2 = ∆ 2 -E 2 S .

2.A.2.1 Logarithmic DOS (conventional Van Hove singularity)

Substituting Eq. (2.25) and in the limit Ω Ec → 0,

I µ 1 (0) ∼ 1 2E 2 c ∞ 0 dx log(x) [x -(µ/E c )] 2 + (Ω/E c ) 2 + log(x) [x + (µ/E c )] 2 + (Ω/E c ) 2 = -π 2E c Ω log E c µ 2 + Ω 2 ,
(2.32)

I µ 2 (0) ∼ - 1 2E c ∞ 0 dx x log(x) 1 [x + (µ/E c )] 2 + (Ω/E c ) 2 - 1 [x + (µ/E c )] 2 + (Ω/E c ) 2 -µ I µ 1 (0) ∼ π 2E c µ Ω , (2.33)
where in integral (2.33) we further took the limit µ Ω and approximated the term in brackets by

-µ Ec 4x (x 2 +(Ω/Ec) 2 ) 2 .

2.A.2.2 Power-law DOS (higher-order Van Hove singularity)

Substituting Eq. (2.28), the integrals (2.31) are readily solved using the standard keyhole contour of infinite radius with a branch cut on the positive real axis:

I µ 1 (0) ∼ -π(1 + ν) R ν E 1+ν c Ω sin[(π -ϕ)ν] + η sin(ϕν) (1 + η) sin(πν) , (2.34)
I µ 2 (0) ∼ -π(1 + ν) R 1+ν E 1+ν c Ω sin[(π -ϕ)(ν + 1)] -η sin[ϕ(ν + 1)] (1 + η) sin[π(ν + 1)] -µ I µ 1 (0), (2.35)
where Re iϕ = µ + iΩ.

Chapter 3

Quasiparticle focusing of bound states in s-wave superconductors

Introduction

Studying the system's response to an impurity, and more generally, to any kind of defect, constitutes a powerful tool to probe the substrate. Much progress in this direction has been achieved by studying the Fourier transform of scanning tunneling microscopy (STM) maps of disordered samples, which reveals the most significant scattering processes and allows inferring the details of the band structure. This approach will be the subject of Chapter 4. Here, we shall consider an alternative idea brought forward by Weissmann et al. [START_REF] Weismann | Seeing the Fermi Surface in Real Space by Nanoscale Electron Focusing[END_REF], who imaged the Fermi surface (FS) of the host by analyzing scanning tunneling microscopy (STM) topographies around an impurity in real space directly. As we shall show, the local density of states (LDOS) is focused along perpendicular directions to flat sections of the FS, thereby establishing a direct relationship between the anisotropy of the FS and the system's response in what is known as the quasiparticle focusing effect.

In superconductors, the most straightforward response to pair-breaking local defects appears in the form of in-gap YSR states. Since they were first observed, YSR states have been observed in a multitude of substrates. Most notably, YSR states have been realized on NbSe 2 [START_REF] Gerbold | Coherent long-range magnetic bound states in a superconductor[END_REF], which on the one hand, enhances their spatial extent due to its two-dimensional character, and on the other hand, imprints a distinctive six-fold symmetry on the LDOS. Subsequent experiments on similar substrates found analogous responses [START_REF] Kim | Long-range focusing of magnetic bound states in superconducting lanthanum[END_REF][START_REF] Thupakula | Coherent and Incoherent Tunneling into Yu-Shiba-Rusinov States Revealed by Atomic Scale Shot-Noise Spectroscopy[END_REF], and the accumulation of the LDOS along preferential directions has been ascribed to the quasiparticle focusing effect discussed in the first paragraph. However, unlike the charge-density response in a normal metal that decays algebraically and whose anisotropy can only be encoded in an overall prefactor, YSR states are also endowed with an exponential decay length reflecting the anisotropy of FS as well. To understand precisely the link between the band structure and the real-space properties of YSR states, a formal treatment of the quasiparticle focusing effect in superconductors is necessary.

Aside from its fundamental interest, being able to explain the spatial features of the ingap LDOS is a crucial requirement to design complex structures of YSR states in real space, 3.2. Quasiparticle focusing in two-dimensional normal metals potentially hosting topological phases of matter. Indeed, by the time of producing this work, this question was addressed by alternative approaches. In Ref. [START_REF] Ortuzar | Yu-Shiba-Rusinov states in 2D superconductors with arbitrary Fermi contours[END_REF], the authors obtained a general integral expression of the Green's function of the substrate by approximating the Fermi contours by regular polygons. However, an explicit description of the quasiparticle focusing effect in superconductors, i.e., the link between simple geometrical features of the exact energy dispersion and the LDOS at the energy of a YSR state, is lacking. This is precisely the purpose of the work presented in this chapter.

To reach that goal, we perform a saddle-point approximation valid at large distances from the impurity, inspired by the treatment of normal metals in Ref. [START_REF] Lounis | Theory of real space imaging of Fermi surface parts[END_REF]. We unveil a simple analytical relationship between, on the one hand, the real-space anisotropy of decay, oscillations, and amplitude of YSR states and on the other hand, the momentum-space anisotropy of the Fermi surface, Fermi velocity, and pairing function of the substrate. Further, we reveal the underlying scattering mechanisms leading to the formation of YSR states. Our analytical calculations are qualitatively consistent with experimental STM measurements on NbSe 2 , and remarkably, they provide a quantitatively accurate description of tight-binding calculations on the same compound. Hence we provide a complete description of the quasiparticle focusing effect in s-wave superconductors, thereby bringing forth an analytical tool to predict the shape and orientation of YSR states and ultimately aiding the design of collective impurity states.

To conclude this introduction, we mention that in our modeling of the YSR states we assume a local and isotropic impurity. This is a rough assumption, and in fact, the magnetic ion's orbital structure and their adsorption site's peculiarities inflects the YSR states with some degree of anisotropy [START_REF] Ruby | Orbital Picture of Yu-Shiba-Rusinov Multiplets[END_REF]. Nonetheless, the anisotropy of the LDOS that we seek to characterize occurs on length scales of several nanometers, that is, much larger than the typical orbital size. Therefore, addressing this problem within the T-matrix approximation for s-wave scatterers is justified.

The rest of the chapter is organized as follows. We start by presenting the key ideas of the quasiparticle focusing effect in two-dimensional normal metals in Sec. 3.2. The bulk of our work is structured into three parts: first, we discuss the implications derived from the saddle-point approximation in superconductors (Sec. 3.3.2); second, we apply the analytical approximation to two examples of exact tight-binding energy dispersions, and we support these results with numerical calculations (Sec. 3.3.3); and third, we apply the analytical approximation to characterize actual STM data measured on NbSe 2 (Sec. 3.3.4). Additionally, we extend the formalism to k-dependent, gapped pairing functions (Sec. 3.3.5). Finally, in Sec. 3.4, we summarize the conclusions and perspectives of our work. In the appendices, we detail the calculations and the generalization of the saddle-point approximation to the complex plane.

Quasiparticle focusing in two-dimensional normal metals

Part of the analysis and interpretation of the quasiparticle focusing effect in superconductors later in this chapter is performed by studying the ∆/ε F → 0 limit; as such, it is convenient to start by discussing the quasiparticle focusing effect in normal metals. Here we extend the result derived in [START_REF] Lounis | Theory of real space imaging of Fermi surface parts[END_REF] to a two-dimensional substrate with a two-fold goal: first, it is helpful to present the mechanism underneath the quasiparticle focusing effect in a simpler setting, and second, it will clarify the role of the dimensionality. Nevertheless, to avoid repetition, we do not discuss in full detail the implications of the analytical approximation here; this is done later in the chapter in the context of superconductors.

We consider the simplest normal-metal Hamiltonian,

H 0 = kσ ε k c † kσ c kσ , (3.1)
where ε k is the energy dispersion measured from the Fermi level. Further, we consider a nonmagnetic, isotropic, point-like scatterer at the origin r = 0,

H imp = V σ c † r,σ c r,σ . (3.2)
The energy-resolved change in charge density due to the charge impurity embedded in the substrate is given by δn

(r; ω) ∼ Im[G 0 (r, 0; ω)T (ω)G 0 (0, r; ω)], (3.3)
where G 0 (r a , r b , ω) represents the real-space bare propagator from r a to r b at energy ω. In this simple example, both the bare and the impurity Hamiltonians are diagonal in spin space, and naturally, we dispense with the Nambu space; therefore, contrary to the rest of the manuscript, here the terms in the T-matrix equation are scalars. Importantly, under the assumption of a point-like impurity, all the spatial dependence of the response is encoded in the bare propagator, therefore, our task is to calculate the integral

G 0 (r a , r b ; ω) = dk (2π) 2 e ik•(ra-r b ) ω + i0 + -ε k , (3.4)
for a generic energy dispersion ε k . The seminal idea of Lounis and coworkers [START_REF] Lounis | Theory of real space imaging of Fermi surface parts[END_REF] was to assume the limit where the propagator is evaluated far away from the impurity (formally, this implies that rk F,min 1, where k F,min is the minimum Fermi wave vector of an arbitrary Fermi surface) and perform a saddle-point approximation. In the following, we present the key points of the result and leave the mathematical details of the calculation for a two-dimensional substrate to Appendix 3.A.1.

The spirit of the saddle-point approximation is to replace the integral in momentum space by a sum of the integrand evaluated at the critical points k j (θ r ) giving the largest contribution to the integral (3.4), G 0 (r, 0; ω)

∼ j e ik j (θr)•r G 0 [k j (θ r ); ω]. (3.5)
The set of critical points {k j (θ r )} depends on the point in space at which we evaluate G 0 (r, 0, ω), which is, in turn, encoded in the polar angle θ r . In particular, they satisfy the following equations 3.2. Normal metals

ε k j (θr) = ω, (3.6a
)

∇ε k j (θr) = |∇ε k j (θr) |r. (3.6b)
The counter propagator G 0 (0, r; ω) yields an analogous set of equations up to a minus sign in (3.6b).

Equations (3.6) capture the essence of the quasiparticle focusing effect. They provide a rule to determine which are the most relevant quasiparticle states engendering the impurity-induced change in the LDOS in a particular direction r in space. Namely, Eq. (3.6a) expresses that these states belong to iso-energy contours at the LDOS energy, while Eq. (3.6b) and the analogous equation for the counter propagator dictate that scattering predominantly occurs from states whose group velocity ∇ε k is parallel to r into states whose group velocity is anti-parallel to r. Naturally, the relevant iso-energy contour appearing in the expression of observables, such as the expectation value of the electric charge, is the Fermi contour. Then, k j (θ r ) and the corresponding group velocity |∇ε k j (θr) | should be simply interpreted as an angular dependent Fermi momentum and Fermi velocity. Finally, we note that solutions to Eqs. (3.6) need not be unique, and we denote the possible multiple critical points with the subscript j. In particular, there are two types of Fermi contours that give rise to this situation: disconnected Fermi contours, and Fermi contours with negative curvature. In the first case, and if the curvature of the Fermi contours is strictly positive, which we assume in the rest of the chapter, j = 1, . . . , N where N is the number of non-equivalent Fermi pockets in the First Brillouin Zone (FBZ). We note that in the second case, when k j (θ r ) falls on the vanishing-curvature point, the calculation presented in Appendix 3.A.1 needs to be extended to the next order. This pathological situation was discussed in [START_REF] Lounis | Theory of real space imaging of Fermi surface parts[END_REF].

The full expression for the approximate bare propagator reads

G 0 (r, 0; ω) ∼ 1 √ r j Γ j (θ r )e i[k j (θr)•r-π 4 ] , (3.7)
where Γ j (θ r ) =

1 |∇ε k j (θr ) | √ κ k j (θr )
. In this expression |∇ε k j (θr) | and κ k j (θr) denote the norm of the gradient of the energy dispersion and the curvature of the Fermi contour evaluated at the critical point k j (θ r ). The summation in Eq. (3.7) accounts for multiple critical points stemming from a multi-pocket energy dispersion.

The approximate expression of the bare propagator in conjunction with Eqs. (3.6) lays bare the relationship between the band structure and the anisotropy of the real-space response to local defects: a given direction in real space with respect to the scatterer is associated with a certain set of Fermi momenta {k j (θ r )}, which, in turn, determine the characteristic oscillation length through the exponential factor; more importantly, the angular dependent prefactor Γ j (θ r ) enhances the LDOS in directions perpendicular to the flattest sections of the Fermi contour.

We conclude this section by noting two differences with respect to the result for threedimensional systems originally derived in [START_REF] Lounis | Theory of real space imaging of Fermi surface parts[END_REF]. First, the power-law decay in real space of the propagator is slower, as it reflects the dimensionality of the substrate, d. In general1 , the (energyresolved) propagator goes as 1 r d-1/2 . Second, the prefactor in three-dimensional systems does not depend on the angle-dependent Fermi velocity, Γ 3D j (θ r ) =

1 √ κ k j (θr )
. The interplay between the curvature of the Fermi contour and the Fermi velocity will be discussed in the next section in the context of superconductors.

3.3 Quasiparticle focusing in two-dimensional superconductors

Modelling the problem

In this work, we employ the classical YSR model for magnetic impurities presented in the introduction. Let us briefly recall its main features. We describe the two-dimensional superconducting substrate at mean-field level by the standard BCS Hamiltonian for s-wave superconductors,

H 0 = kσ ε kσ c † kσ c kσ + k ∆ k c † k↑ c † -k↓ + h.c. (3.8)
For simplicity, we assume that spin-orbit coupling in the substrate is negligible and therefore that spin is a good quantum number. Nevertheless, if that were not the case Eq. (3.8) would be formally equivalent in a pseudo-spin basis. Further, we will consider a substrate with timereversal symmetry (TRS), and assume that the energy dispersion of the normal electrons ε kσ is spin-independent and even in k. Finally, we choose a gauge such that the superconducting parameter is real, and assume it to be k-independent, ∆ = ∆ * . Since the superconducting substrate has TRS, a non-magnetic potential only does not suffice to induce in-gap states. We will consider a point-like, isotropic, magnetic impurity at r = 0, described by the Hamiltonian

H imp = -J d r c † r↑ c r↑ -c † r↓ c r↓ δ(r), (3.9)
where J is the amplitude of the exchange coupling between the substrate's electrons and the impurity's spin of the impurity. We recall that a complete description of adsorbed atoms and magnetic molecules typically requires adding a non-magnetic scattering potential to the Hamiltonian. The strength of this potential affects the energy of the YSR state and yields some degree of asymmetry between the in-gap DOS at positive and negative bias; however, it does not alter the fundamental properties of the spatial distribution of the quasiparticle excitations, and therefore we will omit it in this chapter to simplify matters. Furthermore, we neglect the spin dynamics of the magnetic impurity and any spatial renormalization of the superconducting gap around the impurity. The Bogoliubov-de Gennes Hamiltonian (BdG) of the system in the Nambu basis Ψ = (ψ

↑ , ψ ↓ , ψ † ↓ , -ψ † ↑ ) T reads Ĥ0 = ε k τ z + ∆τ x -Jσ z . (3.10)
where k and r designate the electron's momentum and position, and Pauli matrices τ i and σ i act on particle-hole and spin space, respectively.

The in-gap contribution to the LDOS due to the impurity is given by

δρ(r, ω) ∼ Tr{Im[ Ĝ0 (r, 0; ω) T (ω) Ĝ0 (0, r; ω)]}, (3.11)
where Ĝ0 (r a ,

r b ; ω) = dk (2π) 2 e ik•(ra-r b ) ω 2 -∆ 2 -ε 2 k ω + ε k ∆ ∆ ω -ε k (3.12)
denotes the real-space bare propagator from r a to r b at energy E in particle-hole space, and T (ω) corresponds to the transfer matrix. Since we assumed that the impurity scattering was fully isotropic, the transfer matrix is momentum-independent, and therefore the spatial structure of the LDOS is encoded in the bare propagator (3.12). This observation further justifies treating the impurity as a classical spin [Eq. (3.9)]. We note that it is possible to express the energy of the YSR state E S in terms of the system's parameters under certain assumptions about the DOS as we saw it in Chapter 2, however, it can be calculated numerically for an arbitrary energy dispersion, or measured in an STM experiment. Therefore, we will treat the energy of the propagator ω as an independent parameter in the calculation and only assume that it is smaller than the superconducting gap. In the following sections, we obtain an approximate expression of the integral in Eq. (3.12) far away from the impurity for an arbitrary anisotropic energy dispersion, and we discuss its implications.

Results

To calculate Ĝ0 (r, 0; ω) and Ĝ0 (0, r; ω) in the large r regime we start from the idea of the saddle-point approximation technique and generalize it to the complex plane (see Appendix 3.A.2 for details). As in the normal-metal calculation, we assume that rk F, min 1 where k F, min is the minimum Fermi wave vector of an arbitrary Fermi surface. For an isotropic Fermi surface, it boils down to the usual condition rk F 1. We do not treat the superconducting gap self-consistently because the gap should be modified only on a short length-scale rk F ∼ 1 [START_REF] Flatté | Local electronic structure of defects in superconductors[END_REF], much below the length-scale on which we apply the saddle-point approximation. Here, the essence of the approximation is also to replace the integral in momentum space (3.12) by a sum of the integrand evaluated at the critical points k j (θ r ) giving the largest contribution to the integral [recall Eq. (3.5)]. This set of critical points depends on the observation direction (θ r ) and in the case of a superconducting substrate, they satisfy the following conditions,

ε k j,± (θr) = ±iΩ, (3.13a
)

∇ε k j,± (θr) = ±|∇ε k j,± (θr) |r, (3.13b)
where Ω 2 = ∆ 2ω 2 . To understand the nature of the critical points, it is insightful to compare Eqs. (3.13) with their analog for a charge impurity embedded in a normal metal discussed in Section 3.2. Similarly to the normal metal scenario, the gradient of the energy dispersion evaluated at the critical points k j,+ (θ r ) is also parallel to the observation direction in the superconductor scenario [compare eqs. (3.6b) and (3.13b)]. Therefore, in both situations, disconnected Fermi contours give rise to multiple critical points which here we denote with the subscript j too. However, there are two crucial differences:

First, we note that for a given observation direction θ r and a given Fermi pocket j there are two critical points in momentum space, namely the gradient being parallel [k j,+ (θ r )] and anti-parallel [k j,-(θ r )] to the observation direction, which yield a significant contribution to both the propagator Ĝ0 (r, 0; ω) and the counter-propagator Ĝ0 (0, r; ω) [see Fig. 3.1 (a)]. In the normal-metal case, only k j,+ (θ r ) contributes to the propagator and only k j,-(θ r ) contributes to the counter-propagator. As we discuss below in the analysis of the LDOS, this duality of critical points increases the number and richness of scattering processes.

Second, the critical points in a normal metal are strictly real and sit on the Fermi contour. However, in a superconductor, it follows from the in-gap constraint on the propagator's energy (i.e., ω < ∆ ⇒ Ω 2 > 0) and Eq. (3.13a) that the critical points k j,± (θ r ) are complex numbers. One can observe in Eq. (3.5) that the real part of the critical points yields the oscillatory behavior of the LDOS, whereas the imaginary part will lead to an exponential decay. Thus, we can define the oscillatory and decay characteristic lengths of the propagator, specifically,

λ j,± (θ r ) = 1 Re[k j,± (θ r )] • r , (3.14a)
ξ j (θ r ) = 1 Im[k j (θ r )] • r . (3.14b)
The former is reminiscent of the Friedel oscillations in a normal metal, while the latter is the natural consequence of evaluating the bare propagator at sub-gap energies. We note that the critical points of the counter-propagator Ĝ0 (0, r; ω) are the complex-conjugate of the critical points of the propagator Ĝ0 (r, 0; ω). Further, we note that owing to the even parity of the energy dispersion we can relate the real and imaginary parts of same-pocket critical points, namely, Re[k j,+ (θ r )] = -Re[k j,-(θ r )] and Im[k j,+ (θ r )] = Im[k j,-(θ r )]. Therefore, we conclude that each pocket contributes to the propagator two terms with the same decay length.

The approximate expression for the bare propagator reads Ĝ0 (r, 0

; ω) ∼ 1 ω √ r j, =± Γ j, (θ r )e -r ξ j (θr ) +i[ r λ j, (θr ) -π 4 ] ω + iΩ ∆ ∆ ω -iΩ , (3.15)
where Γ j,

(θ r ) = 1 |∇ε k j, (θr) | √ κ k j, (θr) . (3.16)
In these expressions |∇ε k j, (θr) | and κ k j, (θr) denote the norm of ∇ε k ≡ ∂ kx ε k , ∂ ky ε k and the curvature of ε k = 0 evaluated at k j, (θ r ) -they are therefore complex numbers. The summation in Eq. (3.15) accounts for multiple critical points.

We emphasize that the observation direction determines the set of critical points k j, (θ r ) through the gradient equation (3.13b). Therefore, the anisotropy of the LDOS at the YSRstate energy is encoded in the exponential decay and the oscillation period, as well as in an overall prefactor which depends inversely on the curvature and the norm gradient of the energy dispersion.

The approximate bare propagator in a superconductor differs from its normal-metal counterpart in its additional exponential decay, and in that also the k-points where the gradient points anti-parallel to the observation direction r contribute to the summation in Eq. (3.15). However, here too, under the assumption of a non-vanishing curvature, we obtain that the power-law decay of the LDOS of the YSR state is isotropic, and it goes as 1/r. We conclude that in generic situations solely the substrate dimensionality determines the power law, while exceptional behavior can occur if the observation direction is perpendicular to a strictly linear segment of the Fermi surface (then the segment forms a continuum of critical points, with vanishing curvature for each), or if the observation direction has critical points lying on zero-curvature points of the Fermi surface (arguably this leads to a slower algebraic decay [START_REF] Lounis | Theory of real space imaging of Fermi surface parts[END_REF]). Furthermore, the dependence of the prefactor Γ j, (θ r ) on the band structure's features is the same in both the normal and superconductor propagators [compare Eqs. (3.7) and (3.16)]. Incidentally, we note that owing to the even parity of the energy dispersion we have the relation Γ j,+ (θ r ) = Γ * j,-(θ r ), thus both critical points (±) belonging to a given pocket j contribute a term with equal amplitude and exponential decay to the propagator in Eq. (3.15). As in the normal metal result, the LDOS inherits its anisotropic features from the Fermi contour, therefore, our approximate expression for the bare propagator (3.15) together with the knowledge of an arbitrary energy dispersion ε k allows to predict the orientation and shape of the YSR state. We leave this discussion to Section 3.3.2.2 where we provide a physical interpretation of the real and imaginary parts of the critical points in terms of the energy dispersion in the limit of a small superconducting gap and we provide a few examples. Next, we continue discussing the scattering processes involved in the LDOS.

Underlying scattering mechanisms

In order to interpret the significance of the critical points k j,± (θ r ) it is insightful to write explicitly the product δ Ĝ(r, r; ω) ∼ Ĝ0 (r, 0; ω) T (ω) Ĝ0 (0, r; ω) up to linear order in the impurity potential. For concreteness, we present the electron-electron component, which corresponds to the LDOS measured at positive bias; the hole-hole entry is analogous up to a phase factor. The full expression can be found at the end of Appendix 3.A.2. The relevant term contributed by Fermi pockets j and j is

δG j,j ee ∼ 1 r , =± Γ j, (θ r )Γ j , (θ r ) • e - r ξ j,j (θr ) +i r λ , j,j (θr ) G 0e,α G 0α,e , (3.17)
where

ξ j,j (θ r ) = 1 ξ j (θ r ) + 1 ξ j (θ r ) -1 , (3.18a)
λ , j,j (θ r ) = 1 λ j, (θ r ) - 1 λ j , (θ r ) -1 . (3.18b)
The summation in α runs over particle-hole space. The products of the matrix entries read G 0e,e G 0e,e = (ω + iΩ) 2 , (3.19a)

G 0 e,h G 0 h,e = ∆ 2 , (3.19b)
G 0e,e G 0e,e = ∆ 2 .

(3.19c)

A term shown in Eq. (3.17) represents one of the possible electron-electron scattering processes up to linear order in the impurity potential. Let us start by considering the case of a single-pocket Fermi contour and thereby dropping the summation in j, j .

Case of a single-pocket Fermi contour, j = 1. As we discussed in the context of the bare propagator, the pair of critical points belonging to the same pocket yields states with the same decay length ξ 1 . Therefore, in this case, all scattering processes have the same decay length ξ 1,1 . Further, scattering processes that reverse the momentum of the excitation, i.e., from k 1, (θ r ) to k 1, (θ r ) exhibit an oscillatory character controlled by λ , 1,1 (θ r ). Within this class, we can distinguish a conventional scattering process [Eq. (3.19a), Fig. 3.1 (c) top] and a condensatemediated scattering process [Eq. (3.19b)]. By taking the ∆ → 0 limit while keeping the energy of the propagator finite, it can be observed that the former is reminiscent of the normal-metal scattering. In contrast, the latter arises due to the superconducting nature of the substrate. On the other hand, scattering processes that conserve the momentum of the excitation, i.e., from k 1, (θ r ) to k 1, (θ r ), do not exhibit an oscillatory character [λ , 1,1 (θ r ) -1 = 0]. The superconducting condensate mediates all processes belonging to this class and therefore their amplitude scales with ∆ 2 [Eq. (3.19b), Fig. 3.1 (c) bottom, and Eq. (3.19c)]. This is consistent with our previous discussion on the critical points, where we pointed out that in the normal-metal scenario only k + (θ r ) and k -(θ r ) contribute to the propagator and counter-propagator respectively. Therefore we conclude that momentum-conserving scattering processes are a distinctive feature of the superconducting medium.

Case of a multi-pocket Fermi contour, j > 1. If there is more than one pocket in the Fermi contour, the discussion of the previous paragraph applies to all intra-pocket processes. Each pocket contributes to the propagator eight terms which decay with ξ j,j (θ r ). Now there also exist inter -pocket scattering processes which decay with ξ j,j (θ r ) and oscillate with λ , j,j (θ r ) [see Fig. 3.1 (b)]. Note that since the LDOS decay length is the harmonic mean of the propagator decay lengths from the existing pockets [Eq. (3.18a)], the largest ξ j,j (θ r ) always belongs to an intra-pocket process, i.e., j = j . Furthermore, in general, all inter-pocket processes have an oscillating character even if = . As predicted for the normal metal, here the existence of several characteristic frequencies also gives rise to a beating pattern. However, the fact that the largest decay length corresponds to an intra-pocket process that has one characteristic frequency only implies that the beating pattern will be suppressed in the very large |r| limit. Finally, we remark that the classification of the scattering processes into normal-metal-like and condensatemediated previously discussed applies to inter-pocket processes as well.

Small-gap limit

To elucidate the meaning of complex critical momenta, it is useful to reconcile the normalmetal solution with the superconductor counterpart. As we discussed at the beginning of Sec. 3.3.2, if ∆ is strictly zero, the critical points are real and sit on the Fermi surface. In Appendix 3.B, we show that in the limit ∆ → 0,

Re[k j,± (θ r )] ∼ ± k j (θ r ), (3.20) 1 Im[k j (θ r )] • r ≡ ξ j (θ r ) ∼ |∇ε k j (θ r )| Ω , (3.21)
where k j (θ r ) ∈ R 2 is the normal-metal critical point, i.e., a point lying on the Fermi contour where the gradient of the energy dispersion lies parallel to r.

The exponential decay of each pocket is hence given by its anisotropic Fermi velocity. This result provides a transparent generalization of the well-known result discussed in the introduction chapter, namely, that if we assume an isotropic energy dispersion, the YSR LDOS decays with the superconducting coherence length, ξ iso ∼ v F ∆ [START_REF] Rusinov | Superconductivity near a paramagnetic impurity[END_REF][START_REF] Gerbold | Coherent long-range magnetic bound states in a superconductor[END_REF]. The second source of anisotropy in the propagator is the prefactor Γ j, (θ r ) which itself depends on two quantities [see Eq. (3.16)]: it goes inversely with the norm of the gradient of the energy dispersion, and inversely with the curvature, both evaluated at the critical point. The inverse curvature causes a phenomenon discussed in the context of charge impurities in threedimensional metals [START_REF] Weismann | Seeing the Fermi Surface in Real Space by Nanoscale Electron Focusing[END_REF][START_REF] Lounis | Theory of real space imaging of Fermi surface parts[END_REF]: quasiparticle focusing. Namely, the inverse curvature is highest on the flattest parts of the Fermi surface, and the prefactor Γ will be enhanced for observation directions perpendicular to such segments. The saddle point approach makes this connection explicit: if the observation direction is perpendicular to such a flatter segment, and hence it is aligned with the energy gradient there, the critical point will indeed be on the segment [see Eq. (3.13b)] and its inverse curvature will be high. The quasiparticle focusing in our theory for superconductors hence justifies why previous experimental works show an enhancement of the LDOS along directions perpendicular to the flattest segments of the Fermi surface [START_REF] Gerbold | Coherent long-range magnetic bound states in a superconductor[END_REF][START_REF] Kim | Long-range focusing of magnetic bound states in superconducting lanthanum[END_REF][START_REF] Thupakula | Coherent and Incoherent Tunneling into Yu-Shiba-Rusinov States Revealed by Atomic Scale Shot-Noise Spectroscopy[END_REF].

The norm of the gradient of the energy dispersion plays a crucial role in the anisotropy of the YSR LDOS. Firstly, through the decay length which is enhanced where the gradient is the highest, as we discussed in the beginning of this subsection. Previous studies of superconductors failed to point out this dependence, which constitutes a fundamental difference with respect to the normal-metal scenario where the impurity response lacks any exponential decay length. Secondly, as a quantity entering inversely in the prefactor Γ j, (θ r ), which stems from the reduced dimensionality of the substrate. Hence the norm of the gradient reduces the prefactor Γ j, (θ r ) for observation directions for which it enhances the decay length, and naively one would expect a competition. Nevertheless, we observed in all studied examples that overall the prefactor Γ j, (θ r ) and the characteristic length ξ j (θ r ) grow and shrink in phase as the observation direction θ r varies. This behavior is possible because the reduction of Γ due to the inverse norm gradient can be more than compensated by the inverse curvature. Next, we provide a scaling argument to justify that indeed overall Γ j, (θ r ) varies as the inverse curvature (e.g., it is highest on the flattest segments of the Fermi surface).

We recall that

ξ(θ r ) ∼ |∇ε k (θ r )|, (3.22a) Γ(θ r ) = 1 |∇ε k (θ r )| κ(θ r ) , (3.22b)
where it is understood that the right-hand side of the equations is evaluated at the critical point on the Fermi contours (in the small-gap approximation), and we dropped the j, labels to lighten the notation. The curvature of the energy contour is given by

κ = r⊥ • D(ε k ) • r⊥ |∇ε k | , (3.23)
where r⊥ is a unit vector perpendicular to the observation direction r (hence perpendicular to the gradient of ε k at the critical points) and D(ε k ) is the Hessian matrix of the normal energy dispersion.

Let us consider the coordinate transformation

x = γx, (3.24a)
y = y γ , (3.24b)
whereby we create an anisotropy of a previously isotropic dispersion while preserving the total area of the Fermi surface. This operation decreases the curvature and increases the norm of the gradient of the energy dispersion along the x-direction, and vice versa along the y-direction (see Fig. 3.2).

For concreteness, let us consider a critical point on the x axis such that θ r = 0. It follows that

|∇ε k (θ r = 0)| = γ|∇ε k (θ r = 0)|, (3.25a) κ (θ r = 0) = 1 γ 3 κ(θ r = 0). (3.25b)
Putting all together, we find

ξ (θ r = 0) = γ ξ(θ r = 0), (3.26a) Γ (θ r = 0) = √ γ Γ(θ r = 0). (3.26b)
For the chosen coordinate transformation and observation direction, both the prefactor and the decay length are enhanced. Obviously, in the perpendicular observation direction, one has γ → γ -1 so both are diminished. Therefore, as observed in Fig. 3.4, the prefactor and the decay length are in phase as we vary the observation direction θ r . The underlying reason is that even though the prefactor inversely depends on the norm of the gradient, this is compensated by an opposite behavior of the curvature. The opposite occurs at k x = 0.

Beyond the small-gap limit

To better understand the meaning of complex critical points, it is illustrative to consider a toy model for which Eqs. (3.13) can be solved analytically, namely an ellipsoidal energy dispersion:

ε k = τ (αk 2 x + k 2 y ) -µ, (3.27)
where α ∈ (0, 1] controls the anisotropy (for α = 1 the Fermi surface is a circle and in the limit α → 0 the Fermi surface becomes a line), µ is the chemical potential which effectively controls the "size" of the Fermi surface, and τ =

[k x j ], Re[k y j ] = R cos ϕ cos 2 θ r + α sin 2 θ r 1 √ α cos θ r , √ α sin θ r , (3.29
)

Im[k x j ], Im[k y j ] = R sin ϕ cos 2 θ r + α sin 2 θ r 1 √ α cos θ r , √ α sin θ r , (3.30)
where

R = µ 2 +Ω 2 τ 2 1/4
, and ϕ = 1 2 arctan Ω µ . The negative solution of Eq. (3.13) is analogous up to an overall minus sign for the real part, in agreement with the relationship discussed after Eqs. (3.14) for even energy dispersions.

a Re = R cos ϕ √ α , b Re = R cos ϕ, (3.31)
a Im = R sin ϕ √ α , b Im = R sin ϕ. (3.32)
To first order in Ω we have (a Re , b Re ) ≈ (a FS , b FS ) and (a Im , b Im ) ≈ Ω 2µ (a FS , b FS ). As discussed in Section 3.3.2.2, in the small-gap limit the real part of the critical points collapses to the Fermi surface, while the imaginary counterpart is linear in the superconducting parameter. The relevant quantities characterizing the anisotropy of the bare propagator follow:

ξ decay (θ r ) ≡ 1 Im[k(θ r )] • r = 1 R sin ϕ α cos 2 θ r + α sin 2 θ r , (3.33)
λ oscill. (θ r ) ≡ 1 Re[k(θ r )] • r = 1 R cos ϕ α cos 2 θ r + α sin 2 θ r , (3.34)
Γ(θ r ) ≡ 1 |∇ε k(θr) | √ κ k(θr) = 1 2 √ R(cos 2 θ r + α sin 2 θ r ) 1/4 e -i ϕ 2 . (3.35)
As shown in Fig. 3.3 (b) the propagator is enhanced along directions perpendicular to flatter sections of the Fermi surface, and the decay length ξ decay is in phase.

Applying the theory: two examples

In this section, we illustrate the key ideas of the theory by applying it to two specific tight-binding energy dispersions. We start by considering a simple model where we can easily interpolate from an almost isotropic to a highly anisotropic Fermi contour, and later, we analyze a model for NbSe 2 motivated by recent experiments.

To test the validity of the theory, we shall perform two calculations. We start from the exact tight-binding dispersion, and we calculate the LDOS at the YSR energy by solving the T-matrix equations numerically [START_REF] Gerbold | Coherent long-range magnetic bound states in a superconductor[END_REF]. We extract the exponential decay length by fitting radial cuts at a given θ r to the envelop function of the LDOS, namely, a r e -r/ξ LDOS . Then, we compare this quantity against the analytical prediction from the theory: for each θ r we find the set of relevant critical points, i.e. lying on the Fermi surface and fulfilling ∇ε k j (θ k ) r, and subsequently we evaluate expression (3.21) at said points. As we shall see, the fitted ξ LDOS and the analytical model are in excellent quantitative agreement.

Single-pocket model

Let us consider a nearest-neighbors tight-binding energy dispersion on a square lattice,

ε k = µ -2t(cos k x + cos k y ), (3.36)
where µ is the chemical potential and t is the hopping amplitude. As we tune the chemical potential away from the mid-band point, the Fermi contours become more isotropic [see Fig. For a given doping, the LDOS prefactor is most prominent along directions perpendicular to the flattest sections of the Fermi contour, namely θ r = ± π 4 [Fig. 3.4 (b)]. Nevertheless, we recall that the prefactor does not only depend on the curvature of the Fermi contour but also on the inverse of the angle-dependent Fermi velocity. Compare, for instance, the pale-orange (µ/t = 1/80) and brown (µ/t = 7/8) curves in Fig. 3.4 (a) and (b) at θ r = 0. Although the curvature of the µ/t = 1/80 contour is larger for that direction, the Fermi velocity is substantially smaller, leading to a larger prefactor.

On the other hand, the exponential decay length [color line in Fig. 3.4 (c-f)] is wholly governed by the angle-dependent Fermi velocity and as we argued at the end of Section 3.3.2.2, it is in phase with the prefactor. Finally, we remark the excellent agreement between the decay length calculated with the small-gap analytical approximation and the decay length extracted from the tight-binding calculation [solid color line and markers in Fig. 3.4 (c-f) respectively]. In the next subsection, we perform the same analysis of a realistic tight-binding model, thereby showing all the power of the analytical approximation.

Multi-pocket model

Next, we apply our theory to 2H-NbSe 2 . The motivation to choose this particular compound is two-fold: first, readily available STM data shows that YSR states on this material exhibit a marked anisotropy, hence it is a relevant platform to test the theory. Second, the Fermi surface of NbSe 2 has multiple disconnected Fermi pockets, which serves to illustrate how the theory can be pushed beyond the toy model discussed above. Niobium diselenide is a transition dichalcogenide constituted by alternating planes of Nb and Se atoms arranged in a triangular lattice (see Fig. 3.5). These layers are coupled by weak van der Waals forces, and the band structure hardly disperses along the c-axis, therefore it can be considered a two-dimensional superconductor. Ab initio simulations show that the states in the vicinity of the Fermi surface are mostly contributed by two Nb 4d orbitals, therefore, in practice, the band structure of NbSe 2 is often described with an effective two-band, fifth-nearest neighbors tight-binding energy dispersion on a triangular lattice,

ε k = µ + 5 i=1 t i f i (k), (3.37)
where t i is the hopping amplitude to the i-th nearest neighbour and All parameters in meV, for a = 3.444 Å. Extracted from Ref. [START_REF] Rahn | Gaps and kinks in the electronic structure of the superconductor 2H-NbSe 2 from angleresolved photoemission at 1 K[END_REF] The model captures the shape of the Fermi surface, which is crucial in our theory; however, it has certain limitations. Most notably, it does not incorporate spin-orbit coupling, and the two bands cannot be identified with different spin projections. Here, we follow previous works and assume that the adatom primarily couples to one of the bands only [START_REF] Gerbold | Coherent long-range magnetic bound states in a superconductor[END_REF][START_REF] Liebhaber | YuShiba-Rusinov states in the Charge-Density Modulated superconductor NbSe 2[END_REF]. Specifically, we take Band 2, which we treat as a spin-degenerate energy dispersion. In addition, we note that some recent works claim that 2H-NbSe 2 might be a two-gap superconductor [START_REF] Noat | Quasiparticle spectra of 2 H NbSe 2 : Two-band superconductivity and the role of tunneling selectivity[END_REF]; nevertheless, here we adhere to a simpler scenario, namely an isotropic s-wave gap.

f 1 (k) = 2 cos ζ cos η + cos 2ζ, (3.38a) f 2 (k) = 2 cos 3ζ cos η + cos 2η, (3.38b) f 3 (k) =
The energy dispersion evaluated at the relevant hopping parameters yields a disconnected Fermi surface that has three non-equivalent Fermi pockets, specifically at Γ, K, and K points. Therefore, for a given observation direction θ r , we have three pairs of critical points [see Fig. As predicted by our theory, the LDOS at the YSR-state energy is enhanced along directions perpendicular to flatter sections of the Fermi contours [Fig. 3.6 (b)]. Remarkably the analytical approximation for the exponential decay length and the numerical fits are also in excellent agreement in this case, despite the complexity of the Fermi surface. As we discussed in Sec. 3.3.2, an N j = 3 Fermi contour yields six different decay lengths. The color line in Fig. 3.6 (b) represents the largest ξ j,j ; nevertheless, we note that for the present energy dispersion, the difference between the various decay lengths is of a few lattice sites only, and therefore negligible. This example showcases the ability of this method to predict the shape and orientation of a YSR state on an arbitrary substrate.

Analysis of STM data

In this section, we analyze the anisotropy of YSR states on 2H-NbSe 2 observed in an STM measurement. We consider two data sets provided by F. Massee from the NS2 group of the LPS. The crystals contain a few tens of ppm of magnetic atoms (Fe, Cr, Mn) (for details concerning the fabrication and measurement conditions, see [START_REF] Gerbold | Coherent long-range magnetic bound states in a superconductor[END_REF][START_REF] Thupakula | Coherent and Incoherent Tunneling into Yu-Shiba-Rusinov States Revealed by Atomic Scale Shot-Noise Spectroscopy[END_REF]). In Fig. 3.7, we plot the raw dI/dV spectra at the impurity location for each data set acquired with a metallic tip. Data set 1 and 2 exhibit respectively an in-gap peak at E S = 0.1 meV and E S = 0.25 meV, evidencing the presence of YSR states. In both samples, the amplitude of the in-gap DOS has a strong particle-hole asymmetry, stemming from the non-magnetic component of the impurity scattering potential. In Fig. 3.8, we present the spatial data around the impurity location. The topography [panels (a) and (d)] clearly shows the triangular geometry of the Se lattice. Further, we can observe a pattern of brighter spots inconmensurate with the underlying lattice, signaling the charge-density-wave (CDW) order that coexists with superconductivity [START_REF] Liebhaber | YuShiba-Rusinov states in the Charge-Density Modulated superconductor NbSe 2[END_REF][START_REF] Zheng | First-principles study of charge and magnetic ordering in monolayer NbSe 2[END_REF]. The topography does not show any imperfection implying that the defect giving rise to the YSR state is most likely an atom substitution in the Nb layer. We can assume then that the impurity couples primarily to one of the trilayers, which justifies the simplified model without inter-layer coupling presented in the previous section. In panels (b,c,d,e), we plot the current measured at ±0.6 mV, just before the onset of the superconducting coherent peak. Measuring the current amounts to integrating the DOS from zero to the measuring voltage; since the sole contribution within this range comes from the YSR state, we obtain an analogous spatial pattern to the dI/dV at the YSR energy (or the LDOS), but with higher signal-to-noise ratio [98]. Panels (c) and (e) show a strong maximum on one lattice site at the center of the YSR state. The position of this peak is located between three maxima in the topography [see blue circle in panels (a) and (d)], further supporting the hypothesis that defects are atom substitutions in the Nb atomic layer (recall Fig. 3.5). Finally, as it was first appreciated in [START_REF] Gerbold | Coherent long-range magnetic bound states in a superconductor[END_REF], we also note that due to the two-dimensional character of the substrate, the YSR states extend over a much larger region in space (on the order of tens of nanometers) than in fully three-dimensional substrates [START_REF] Yazdani | Probing the local effects of magnetic impurities on superconductivity[END_REF]. The LDOS in both data sets exhibits a striking anisotropy, constituting an excellent example of the quasiparticle focusing effect. As discussed in this chapter, the anisotropy of the LDOS is encoded in its exponential decay length, oscillations, and prefactor. Unfortunately, due to the small superconducting gap (a few millielectron-volts), the coherence length ξ = v F ∆ is about 50 nm. This distance exceeds the field of view and typical impurity separation, therefore, we cannot reliably extract the exponential decay length from the data set. Additionally, we note that the oscillation characteristic length of the signal is on the order of k F ∼ 5 nm -1 in consistence with our theory; however, the signal-to-noise ratio is not enough to resolve the angular anisotropy in the oscillations. For these reasons, in order to quantitatively capture the anisotropy of the LDOS, we resort to the prefactor's angular dependence.

We recall that within our analytical approximation the LDOS can be expressed as

ρ(r, θ) = A(θ)f osc r λ(θ) e -r/ξ(θ) r , (3.39)
where the prefactor is related to the Fermi velocity and curvature of the Fermi surface of the substrate

A(θ) = max ξ j,j (θr)   1 |∇ε k j, (θr) | κ k j, (θr) • 1 |∇ε k j , (θr) | κ k j , (θr)   , (3.40) f osc r λ(θ)
is a bounded periodic function with angle-dependent characteristic length λ(θ), and ξ(θ) is the angle-dependent exponential decay length. In Eq. (3.40), the Fermi velocity ∇ε k and curvature of the Fermi surface κ k are calculated from the tight-binding energy dispersion for NbSe 2 presented in Sec. 3.3.3. Further, the term inside [. . . ] is evaluated at the pair of critical points k j, (θ r ), k j , (θ r ) yielding the maximum decay length ξ j,j (θ r ).

We shall consider the following quantity

ρ R 0 ,δ int (θ) = 1 δ R 0 +δ R 0 drrρ(r, θ) ≡ 1 N n∈M (θ) I n |R n |, (3.41)
where M (θ) is a set of N points belonging to an angular sector around the impurity of radius R 0 , radial width δ and infinitesimal angular width (in practice, we set ∆θ = 2π/450). Further, I n is the measured current at point n, and |R n | is the distance from the impurity to point n.

The analysis of the experimental data is based on three important assumptions: R 0 is large enough to make the asymptotic approximation reasonable. In practice this

means R 0 1/k min F ∼ 0.2 nm.
The integration width δ is sufficiently large so that the oscillatory function is washed-out.

In practice, this requires δ > λ max ∼ 0.4 nm, where λ max is associated with an intra-pocket process (see discussion at the end of Section 3.3.2.1).

The exponential decay length is sufficiently large so that e -r/ξ(θ) ∼ 1 over the integration range. This demands R 0 + δ ξ min ∼ 50 nm.

Noting that the r factor in the integrand compensates the power-law decay of the LDOS, ρ R 0 ,δ int (θ) becomes approximately δ-independent under these assumptions, and further, we have

ρ R 0 ,δ int (θ) ∼ A(θ). (3.42)
In Fig. 3.9 we present the result of our analysis in comparison with the model curve

A model (θ) = aA(θ) + a 0 . (3.43)
Here a is a free parameter in the theory, and a 0 is constant shift on the order of the background signal. We estimate the latter by taking the signal average far away from the impurity times the typical integration length

R n ∼ R 0 + δ min +δmax 2
, because in a clean superconductor the integrated current up to a sub-gap energy should vanish in theory. Experimental data points ρ R 0 ,δ int (θ) above the background signal collapse onto a universal curve, thereby supporting the assumptions stated earlier. Incidentally, below the background signal ρ R 0 ,δ int (θ) curves do not coincide [see for instance Fig. 3.9 (c)]. Indeed, the background signal does not decay as a power law a priori, and therefore the quantity defined in Eq. (3.41) is not δ-independent.

Moreover, we note that the relative angular position of the model curve with respect to the data points is not a degree of freedom, but it is determined by the orientation of the lattice vectors (see legend insets), the latter being extracted from the topography. We emphasize the agreement between the positions of the maxima and minima of the model and the experimental curves. While the six-fold nature of the YSR state may follow trivially from the symmetry of the band structure, its orientation with respect to the lattice does not2 , and it constitutes a significant success of the theory.

Finally, we remark that the periodicity of the data points exhibits certain higher harmonics, namely, the signal becomes thinner at the crests and wider at the valleys. This non-trivial feature is also present in the model curve [see Fig. 3.9 (d) in particular, where all data points are above the background signal].

To conclude this section, we mention that the legs of the YSR signal exhibit a remarkable petal shape [see specially Fig. 3.8 (b)], which our theory fails to capture. Nevertheless, we emphasize that this is not a shortcoming of the analytical approximation but of the model itself. Indeed, the analytical approximation is in quantitative accord with the LDOS calculated numerically from the exact tight-binding energy dispersion, but already the latter does not display a petal shape [see Fig. 3.6 (b)]. Interestingly, this particular silhouette is also present in YSR states on an Sn/Si [START_REF] Manchon | New perspectives for Rashba spinorbit coupling[END_REF] surface (see Fig. 2 (d) in [START_REF] Ming | Evidence for chiral superconductivity on a silicon surface[END_REF]), suggesting that it might be a universal feature.

Generalization to extended s-wave pairing

Up to this point, we restricted our considerations to conventional s-wave superconductors. Nevertheless, the formalism developed in the previous sections allows to treat more involved situations where the superconducting gap function is momentum dependent. In order to conserve the structure of the Green functions (3.12), we will stick to singlet pairing and simply incorporate a k in ∆, but we emphasize that in principle, the technique could be employed in arbitrarily gapped superconductors. Note that ∆ k must be an even function as required by the fermionic anticommutation rules. It is useful to introduce the BdG energy dispersion, E k = ε 2 k + ∆ 2 k , which allows to express the critical-point conditions in a compact form:

E k j,± (θr) = ±ω, (3.44a) ∇E k j,± (θr) = ±|∇E k j,± |r(θ r ). (3.44b)
Naturally, Eqs. (3.44) reduce to Eqs. (3.13) if ∆ is independent of k. The setting discussed earlier can be formally understood as a particular case of the present situation; however, formulating the solution for k-independent pairing in terms of the normal electron energy dispersion and the actual Fermi surface was more illuminating and permitted a clearer physical interpretation.

The structure of the bare propagator is analogous to the solution discussed in the preceding sections. The power-law goes as 1/ √ r as dictated by the dimensionality of the substrate, and the anisotropy is encoded in the argument of the exponential function and in the prefactor Γ j, (θ r ) = (θr ) . Now the curvature and the norm of the gradient refer to the BdG dispersion E k .

1 |∇E k j, (θr ) | κ k j,
To further understand the effect of a nodeless, anisotropic superconducting gap on the spatial structure of the YSR state, it is insightful to treat the k dependent part as a perturbation of a constant background,

∆ k = ∆ + ∆ f ∆ (k), (3.45)
with ∆ ∆ and f ∆ (k) an even function of k, and further, to consider the small-gap limit discussed in Section 3.3.2.2. We find that the exponential decay length is corrected as follows

ξ j, (θ r ) ∼ |∇ε k j (θ r )| Ω 1 + ∆∆ Ω 2 f ∆ [ k j, (θ r )] -1 , (3.46)
where k j, (θ r ) ∈ R 2 is the critical point in the normal metal limit.

In order to exemplify this result, we benchmark the analytical approximation for the decay length against a tight-binding calculation of the LDOS at the YSR-state energy. We choose the energy dispersion introduced in Eq. (3.36) describing a nearest-neighbors tight-binding model on a square lattice. Further, we take a nearest-neighbors superconducting coupling such that

f ∆ (k) = cos k x + α cos k y , (3.47)
where α ∈ [0, 1] is a parameter to control the anisotropy. In the limit α = 1, this pairing function is known under the name of extended s-wave or unconventional s ± -wave, and it has been proposed as a candidate to describe iron-based superconductors [100,[START_REF] Mashkoori | Impact of topology on the impurity effects in extended s -wave superconductors with spin-orbit coupling[END_REF]. However, we choose α = 0 in our calculations to maximize the variation of the gap along the Fermi contour, thereby enhancing the effect of the pairing function anisotropy on the LDOS of the YSR state.

Results are presented in Fig. 3.10. In the absence of gap-anisotropy [∆ = 0, panel (b)] the LDOS naturally exhibits the four-fold rotational symmetry of the underlying lattice model. As discussed in Sec. 3.3.3.1, the decay length is the most prominent along the θ r = ± π 4 directions for which the gradient of the energy dispersion is the largest. When we switch on an anisotropic texture on the superconducting gap, the symmetry of the LDOS is reduced accordingly [see panel (c) in Fig. 3.10]. We recall that within the working approximation, the critical points sit on the Fermi contour; therefore, if we set the observation direction along θ r = 0 for instance, we have that f ∆ [ k(0)] < 0 [see panel (a) in Fig. 3.10]. This leads to an enhancement of the decay length as predicted in Eq. (3.46). We note that the approximation captures very well the intricacies of the spatial structure of the LDOS despite the seemingly large value of ∆ /∆ employed in the numerical calculations.

Conclusions and outlook

In this chapter, we provided a precise explanation of the role of the Fermi surface in the spatial anisotropy of YSR states in two-dimensional s-wave superconductors. To summarize, the anisotropy of the LDOS is encoded in an overall prefactor, the exponential decay length and oscillations. The prefactor, which also arises in the charge-density response in normal metals, depends inversely on the angle-dependent Fermi velocity and the curvature of the Fermi contour, meaning that YSR states show prominent features along directions perpendicular to flatter sections of the Fermi contours. The decay length is proportional to the angle-dependent Fermi velocity constituting an elegant generalization of the superconducting coherence length, which governs the exponential decay of YSR states on isotropic substrates. Through a simple scaling argument, we showed that the prefactor and the decay length are always in phase; therefore the knowledge of the energy dispersion allows predicting the shape and orientation of YSR states, even for STM measurements whose field of view is too small to encompass the exponential decay. Understanding how the Fermi surface shapes the spatial structure of YSR states eases the path toward the optimal design of collective impurity states on superconductors.

We found that the most likely scattering processes involve excitations whose momenta lies on the Fermi contour where the Fermi velocity is parallel and anti-parallel to the observation direction. This result implies that each pocket of the Fermi surface contributes twice as many meaningful scattering momenta than in the normal-metal scenario. The emerging scattering processes are indeed mediated by the condensate and constitute a distinctive feature of the superconducting nature of the substrate. Unfortunately, all the contributions to the bare propagator stemming from the same pocket decay similarly, therefore it does not seem plausible to arbitrarily enhance the Andreev-like processes; nevertheless, our analysis via the saddle-point technique deepens the current understanding of the underlying scattering mechanisms in the YSR problem.

We emphasize that contrary to previous works [START_REF] Ortuzar | Yu-Shiba-Rusinov states in 2D superconductors with arbitrary Fermi contours[END_REF] we did not make any approximations regarding the Fermi surface, but instead, we applied our analytical expression to arbitrarily complex energy dispersions. We achieved an accurate quantitative comparison with a tightbinding calculation using a realistic energy dispersion for NbSe 2 which showcases the power of our analytical approximation. Aside from reproducing the observed symmetry of the YSR states on TMDs, we showed that our approximation for the LDOS' prefactor is consistent with STM measurements on NbSe 2 . Further, we found that under the assumption of a Fermi surface with strictly positive curvature, the power-law decay of the LDOS at the YSR-state energy goes as 1/r and depends on the substrate dimensionality only. Earlier works [START_REF] Kim | Long-range focusing of magnetic bound states in superconducting lanthanum[END_REF][START_REF] Ortuzar | Yu-Shiba-Rusinov states in 2D superconductors with arbitrary Fermi contours[END_REF] had suggested that quasiparticle focusing could lead to a slower decay. It remains an open question to investigate the next leading terms in the saddle-point expansion, which will arguably have a slower algebraic decay. These terms will become relevant for observation directions for which the critical points lie on straight segments of the Fermi surface or with nearly vanishing curvature. Incidentally, the method presented in Ref. [START_REF] Ortuzar | Yu-Shiba-Rusinov states in 2D superconductors with arbitrary Fermi contours[END_REF] becomes more accurate in this limit, therefore, we conclude that the two theoretical descriptions complement each other.

Finally, we demonstrated that the analytical approximation also offers quantitatively correct results in superconductors with a gapped, momentum-dependent pairing function. This generalization opens the door to studying more complex situations, for instance, d-wave pairing functions relevant for cuprates and other high-Tc superconductors. Here, the DOS of the clean system does not vanish below the gap, and strong non-magnetic scatters engender virtual bound states that decay algebraically along the nodal directions and exponentially otherwise [START_REF] Balatsky | Impurity-induced virtual bound states in d -wave superconductors[END_REF]. Performing our approximation scheme in this scenario is possible, however, special care must be taken to analyze the critical-point equations at the gap nodes.

To conclude this chapter, we propose investigating the manifestation of the quasiparticle focusing effect in other phenomenology in superconductors. In particular, we mention Type-II superconductors under a magnetic field, where scattering of confined quasiparticles in the vortex core results in another type of in-gap bound states, known as Caroli-de Gennes-Matricon (CdGM) states [START_REF] Caroli | Bound Fermion states on a vortex line in a type II superconductor[END_REF]. These excitations lie at energy E µ = ±µ ∆ 2 ε F , with µ a half-integer; therefore, they are difficult to resolve in typical superconductors where ∆/ε F 1, although there exist some notable exceptions such as FeTe 0.55 Se 0.45 where their observation has been recently reported [START_REF] Chen | Discrete energy levels of Caroli-de Gennes-Matricon states in quantum limit in FeTe0.55Se0.45[END_REF]. In STM experiments on superconductors far from the quantum limit, the spectral hallmark of CdGM states is a DOS peak at zero bias localized at the vortex core. Remarkably, measurements on NbSe 2 [START_REF] Hess | Vortex-core structure observed with a scanning tunneling microscope[END_REF] and similar compounds [START_REF] Kim | Anisotropic non-split zero-energy vortex bound states in a conventional superconductor[END_REF] show that the LDOS at the vortex core has a distinctive six-fold symmetry which inevitably reminds of the shape of YSR states in the same materials. Authors in Ref. [START_REF] Kim | Anisotropic non-split zero-energy vortex bound states in a conventional superconductor[END_REF] showed by means of a numerical simulation that the electronic structure of the substrate controls the LDOS anisotropy; nevertheless, it would be insightful to obtain an explicit link between tight-binding band structures and the spatial features of vortex cores. We note that CdGM states were originally studied by solving the Bogoliubov-de Gennes equation in real space [START_REF] Caroli | Bound Fermion states on a vortex line in a type II superconductor[END_REF], and that there exist alternative methods to describe the LDOS at low energies in vortex cores that make use of a quasiclassical Green function [START_REF] Nagai | Analytical Formulation of the Local Density of States around a Vortex Core in Unconventional Superconductors[END_REF][START_REF] Hayashi | Star-Shaped Local Density of States around Vortices in a Type-II Superconductor[END_REF]. However, unlike in the T-matrix approximation, the scattering mechanism underlying the formation of the in-gap states is not explicit in any of these approaches: clarifying how to use the machinery developed in this chapter -namely, a complex saddle-point approximation of an object describing the bare substrate-to address this problem is an open challenge. and its counterpart G 0 (0, r; ω) for an arbitrary energy dispersion. We introduce an auxiliary variable t and write

G 0 (r, 0; ω) = -i|r| dk (2π) 2 ∞ 0 dte i|r|φ(k,t) , (3.49)
where

φ(k, t) = k • r + t(ω + i0 + -ε k ). ( 3.50)
The critical points (k j , t j ) giving the largest contribution to the the integral (3.49) fulfill

(∇, ∂ t )φ(k, t) = 0, (3.51)
and naturally, they depend on the observation direction (θ r ). This yields the following conditions,

ε k j (θr) = ω, (3.52a)
∇ε k j (θr) = |∇ε k j (θr) |r, (3.52b)
t j (θ r ) = 1 |∇ε k j (θr) | . (3.52c)
Note that Eq. (3.52a) follows from taking the limit ∆ → 0 while keeping ω finite in the superconductor critical equation (3.13a). Contrary to the superconductor scenario, the "anti-parallel" solution does not contribute to the integral in the normal metal case [compare (3.13b) and (3.52b)]. Further, we remark that Eq. (3.52a) yields an iso-energy contour at the propagator energy, the relevant contour corresponding to ω = 0, i.e., the Fermi contour. Finally, we note that the counter propagator G 0 (0, r; ω) yields an analogous set of equations up to a minus sign in (3.52b), thereby spawning the shaded entries in Fig. 3.1, panel (b).

From now on, it is understood that critical points depend on the observation direction, therefore we drop (θ r ) to lighten the notation. Next, we expand the phase (3.50) up to second order around the critical points. The zeroth order contribution follows trivially,

G 0 (r, 0; ω) ∼ -i|r|e ik j •r . (3.53)
To second order, the integral in t reads

∞ 0 d∆t j exp -i|r| 1 2 α ∂ε k ∂k α k j ∆k α j ∆t j ∼ 4π |r||∇ε k j | δ(∆k rj ), (3.54)
where ∆• j are the integration variables that denote a small interval around the critical point • j , δ represents the Dirac delta distribution, and k rj is the projection of k j along r. To approximate (3.54) we extended the integration bounds from [0, ∞) to (-∞, ∞) and we used the fact that

δ(∇ε k j • ∆k j ) = δ(∆k rj )
|∇ε k j | which follows from the critical-point equation (3.52b). Finally, by rotating the integration axes so that they lie tangent and normal to an iso-energy contour (3.52a), the integral in k reads

∞ -∞ d∆k rj d∆k r⊥ j δ(∆k rj ) exp -i|r| t j 2 αβ ∂ 2 ε k ∂k α ∂k β k j • • ∆k α j ∆k β j ∼   2 π|r||∇ε k j | 2 t j ∂ 2 k 2 r⊥ ε k j   1/2 e iϕ j , (3.55)
where

ϕ j = -π 4 sign(∂ 2 k 2 r⊥ ε k j ) and ∂ 2 k 2
r⊥ ε k j is the second directional derivative tangent to the iso-energy contour evaluated at the critical point. Putting everything together

G 0 (r, 0; ω) ∼ 1 √ r j Γ j (θ r )e i[k j (θr)•r+ϕ j ] , (3.56)
where Γ j (θ r ) =

1 |∇ε k j (θr ) | √ κ k j (θr )
. In this expression |∇ε k j (θr) | and κ k j (θr) denote the norm of the gradient of the energy dispersion and the curvature of the Fermi contour evaluated at the critical point k j (θ r ). The summation in Eq. (3.56) accounts for multiple critical points stemming from a multi-pocket energy dispersion.

3.A.2 Superconductors

In this appendix, we detail the calculation of the Fourier transforms of the superconducting bare propagator. Since critical points in the saddle-point approximation lie somewhere in the complex plane outside the original integration range, the procedure differs slightly from that presented above. To perform the approximation, we introduce a new integration variable t and re-express the bare propagator as

G αβ 0 (r, 0; ω) = -|r| dk (2π 2 ) f αβ (k) ∞ 0 dt e -|r|φ r (k,t) , (3.57)
where f αβ (k) comprises the matrix entries in Eq. (3.12), and

φ r(k, t) = -ik • r + (ε 2 k + Ω 2 )t, (3.58)
with θ r the polar angle determined by r. The critical points (k j , t j ) giving the largest contribution to the integral (3.57) satisfy (∇, ∂ t )φ r(k, t) = 0. For a given radial direction θ r this equation yields the set of conditions (3.13) presented in the main text, which we rewrite here for completeness along with an extra condition for the t variable:

ε k j,± (θr) = ±iΩ, (3.59a
)

∇ε k j,± (θr) = ±|∇ε k j,± (θr) |r, (3.59b)
t j,± = 1 2Ω|∇ε k j,± (θr) | . (3.59c)
Next, we recall the following compact expression to perform the saddle-point approximation on a multivariate complex integral. We refer the reader to [START_REF] Bleistein | Saddle point contribution for an n-fold complex-valued integral[END_REF] for details on its derivation. Let I n (λ) be an integral over n complex variables defined as

I n (λ) = dzf (z)e -λφ(z) , z = (z 1 , z 2 , . . . , z n ).
(3.60)

In the "large" λ limit we have

I n (λ) ∼ 2π λ n/2 f (z s ) det{H n [φ(z s )]} e -λφ(zs) , (3.61)
where z s is defined through the equation ∇ z φ(z)

zs = 0 and H n [φ(z s )] = ∂ 2 φ(z) ∂zα∂z β zs , α, β = 1, . . . , n, as long as det{H k [φ(z s )]} = 0 for 1 ≤ k ≤ n.
In our problem the large parameter is |r|, namely the distance from the impurity, n = 3 with z ≡ (k x , k y , t), and f (z) and φ(z) are defined in Eqs. (3.57) and (3.58).

The approximate expression for the bare propagator reads

G αβ 0 (r, 0; ω) ∼ 1 ω √ r j, =± Γ j, (θ r )f αβ [k j, (θ r )]e i[k j, (θr)-π 4 ] , (3.62)
where Γ j, (θ r ) is the anisotropic prefactor introduced in Sec. 4.3.

The critical conditions for the counter-propagator G αβ 0 (0, r; ω) read as follows,

ε k j,± (θr) = ∓iΩ, (3.63a)
∇ε k j,± (θr) = ±|∇ε k j,± (θr) |r, (3.63b)
t j,± = 1 2Ω|∇ε k j,± (θr) | . (3.63c)
By comparing the sets of equations (3.59) and (3.63) it becomes evident that the critical points of the counter-propagator G αβ 0 (0, r; ω) are, in fact, the complex-conjugate of the critical points of the propagator G αβ 0 (r, 0; ω). It follows,

G αβ 0 (0, r; ω) ∼ 1 Ω √ r j, =± Γ j, (θ r ) * f αβ [k j, (θ r )] * e -i[k j, (θr) * -π 4 ] , (3.64)
where we used the fact that Γ j, (θ r ) and f αβ [k j, (θ r )] are real functions evaluated at complex values.

The full expression for the electron-electron component for the LDOS in terms of the parameters introduced in the main text reads

δG ee ∼ 1 r j,j e - r ξ j,j (θr ) , Γ j, (θ r )Γ j , (θ r ) exp i r λ , j,j (θ r ) -( -) π 4 α G 0e,α G 0α,e . (3.65)
Appendix 3.B Small-gap approximation

In this appendix, we provide an interpretation of the exponential part of the bare propagator e ik j, (θr)•r by relating the real and imaginary parts of critical points to the normal energy dispersion. For concreteness, we consider the positive solution = + and the pocket j = 1, and we drop those labels. Generalizing to other solutions is straightforward.

We extend the domain of the normal energy dispersion to the complex plane, and we separate it into its real and imaginary parts,

ε k = u ε (k Re x , k Re y , k Im x , k Im y ) + iv ε (k Re x , k Re y , k Im x , k Im y ), (3.66)
where k = (k x , k y ) and k ν = k Re ν + ik Im ν for ν = x, y. Further, we assume that ε k is holomorphic and satisfies the Cauchy-Riemann relations. Let us introduce

P = {k Re x (θ r ), k Re y (θ r ), 0, 0}, (3.67)
a point living in C 2 ∼ = R 4 whose projection on the real plane {k Re x (θ r ), k Re y (θ r )} is a critical point of the normal-metal bare propagator. From the normal-metal solution we know that the critical points lie on the Fermi contours, therefore P satisfies the following equations,

u ε (P ) = 0, (3.68a)
v ε (P ) = 0, (3.68b) ∇ε k |∇ε k | P = r. (3.68c)
We consider the ansatz that the critical points of the superconducting propagator are a perturbation from P , namely

P = {k Re x (θ r ) + δk Re x (θ r ), k Re y (θ r ) + δk Re y (θ r ), δk Im x (θ r ), δk Im y (θ r )}. (3.69)
This assumption will remain valid in the small-gap limit, i.e., as long as ∆ ε F . Naturally, P fulfills Eqs. (3.13a) and (3.13b), therefore we have

u ε (P ) = 0, (3.70a) v ε (P ) = Ω, (3.70b)
g x (P ) sin θ r = g y (P ) cos θ r , (3.70c)
where

g ν = ∂ k Re ν u ε + i∂ k Re ν v ε for ν = x, y.
Next, we develop to first order the Eqs. (3.70). To zeroth order, the energy equations (3.70a) and (3.70b) are trivial, while the gradient equation (3.70c) yields the following relations,

∂ k Re x u ε P sin θ r = ∂ k Re y u ε P cos θ r , (3.71a)
∂ k Re x v ε P sin θ r = ∂ k Re y v ε P cos θ r . (3.71b)
To first order, we obtain the following condition,

A       δk Re x (θ r ) δk Re y (θ r ) δk Im x (θ r ) δk Im y (θ r )       =       0 Ω 0 0       , (3.72)
where A is a four-by-four matrix whose non-zero entries read

a 11 = a 23 = ∂ k Re x u ε P , (3.73a)
a 12 = a 24 = ∂ k Re y u ε P , (3.73b)
a 31 = a 43 = ∂ 2 k Re x k Re x u ε P sin θ r -∂ 2 k Re y k Re x u ε P cos θ r , (3.73c)
a 32 = a 44 = ∂ 2 k Re x k Re y u ε P sin θ r -∂ 2 k Re y k Re y u ε P cos θ r , (3.73d)
where the equalities between different matrix entries follow from applying the Cauchy-Riemann relations. By expressing ε k as a power series, it becomes clear that the first and second-order derivatives of u ε with respect to k Im x and k Im y evaluated at P vanish, therefore all the other matrix entries are zero.

To this order of approximation, we find

{δk Re x (θ r ), δk Re y (θ r )} = {0, 0}, (3.74)
that is, the real part of the critical points sits on the Fermi contours mimicking the normal-metal critical points. For the imaginary part, which yields the exponential decay length, we have

δk Im • r = Ω ∂ k Re x u ε P cos θ r + ∂ k Re y u ε P sin θ r = Ω |∇ε k (θ r )| . (3.75)
where the last equality follows from noting that the gradient of the energy dispersion is parallel to r at the critical point.

Chapter 4

Quasiparticle interference in non-centrosymmetric superconductors

Introduction

The lack of an inversion center has fundamental consequences for the physical properties of crystalline solids. This symmetry reduction is associated with the emergence of internal electric fields which lock the spin of the electron to its orbital motion -a relativistic effect known as spinorbit coupling (SOC)-, and eventually results in spin-split bands even in non-magnetic materials [START_REF] Rashba | Properties of semiconductors with an extremum loop. i. cyclotron and combinational resonance in a magnetic field perpendicular to the plane of the loop[END_REF][START_REF] Manchon | New perspectives for Rashba spinorbit coupling[END_REF]. In superconductors, inversion symmetry breaking precludes the classification of the pairing function into even and odd under parity transformation, and hence it allows the existence of a mixed spin-singlet and spin-triplet order parameter [START_REF] Gor | Superconducting 2D System with Lifted Spin Degeneracy: Mixed Singlet-Triplet State[END_REF][START_REF] Smidman | Superconductivity and spinorbit coupling in non-centrosymmetric materials: a review[END_REF][START_REF] Bauer | Non-Centrosymmetric Superconductors[END_REF].

The quest for non-centrosymmetric superconductors started in the early years of this century after the discovery of CePt 3 Si, whose unusually high paramagnetic limiting field propelled the idea that this class of materials may harbor unconventional superconductivity [START_REF] Bauer | Heavy fermion superconductivity and magnetic order in noncentrosymmetric cept 3 Si[END_REF][START_REF] Frigeri | Superconductivity without inversion symmetry: MnSi versus CePt 3 Si[END_REF]. To this day, superconductivity has been observed in a plethora of different non-centrosymmetric materials (e.g., other heavy-fermion compounds, metal carbides, oxide insulator interfaces, etc. See Ref. [START_REF] Smidman | Superconductivity and spinorbit coupling in non-centrosymmetric materials: a review[END_REF] for a comprehensive review); however, many of these non-centrosymmetric superconductors exhibit properties which are consistent with conventional s-wave superconductivity [START_REF] Tay | s-wave superconductivity in the noncentrosymmetric w3al2c superconductor: an nmr study[END_REF][START_REF] Mayoh | Superconducting and normal-state properties of the noncentrosymmetric superconductor re 6 Zr[END_REF][START_REF] Singh | Superconducting properties of the noncentrosymmetric superconductor re 6 Hf[END_REF]. Indeed, it should be emphasized that the presence of spin-orbit coupling alone does not imply the existence of a triplet component in the order parameter. As shown in Ref. [START_REF] Samokhin | Gap structure in noncentrosymmetric superconductors[END_REF], a standard attractive BCS interaction in a non-centrosymmetric crystal yields a vanishingly small pairing strength in the triplet channel. Stabilizing singlet-triplet mixing in the order parameter requires the combination of broken inversion symmetry and the appropriate pairing mechanism. In addition to this ambiguity, experimentally determining the degree of spin-triplet mixing is a challenging task because a potentially dominant singlet component in the order parameter may lead to a gapped density of states (DOS), hindering the interpretation of routine spectroscopic measurements [START_REF] Haim | Signatures of triplet correlations in density of states of Ising superconductors[END_REF]. This chapter addresses this question in the context of transition metal dichalcogenides (TMDs). Materials belonging to this class have a layered structure, where layers are coupled by weak van der Waals forces. Each layer comprises three stacked planes in a trigonal prismatic structure: a triangular lattice of a 4d transition metal atom sandwiched between two identical triangular lattices of chalcogen atoms. Even though bulk crystals are centrosymmetric, a monolayer of TMD breaks inversion symmetry, which, in combination with the heavy 4d atoms, results in a spin-orbit coupling splitting on the order of hundreds of meV (recall Fig. 3.5 in Ch. 3). Due to the basal mirror symmetry with respect to the transition-metal plane, the crystal electric field is restricted to the monolayer plane; therefore, spins are momentum-locked in the out-of-plane direction, realizing the so-called Ising spin-orbit coupling [START_REF] Manzeli | 2d transition metal dichalcogenides[END_REF]. Superconductivity has been observed in monolayer NbSe 2 [START_REF] Xi | Ising pairing in superconducting NbSe2 atomic layers[END_REF] and TaS 2 [START_REF] Barrera | Tuning ising superconductivity with layer and spin-orbit coupling in two-dimensional transition-metal dichalcogenides[END_REF], as well as gated monolayer MoS 2 [START_REF] Lu | Evidence for two-dimensional ising superconductivity in gated mos2[END_REF] and WS 2 [START_REF] Lu | Full superconducting dome of strong ising protection in gated monolayer ws2[END_REF], where the large in-plane paramagnetic limiting field suggests that the Ising SOC plays a vital role in the formation of Cooper pairs [START_REF] Wickramaratne | Ising superconductivity and magnetism in NbSe 2[END_REF]. These considerations have triggered a flurry of activity advocating for a mixed singlet-triplet order parameter in these compounds [START_REF] Möckli | Robust parity-mixed superconductivity in disordered monolayer transition metal dichalcogenides[END_REF][START_REF] Möckli | Ising superconductors: Interplay of magnetic field, triplet channels, and disorder[END_REF][START_REF] Wickramaratne | Ising superconductivity and magnetism in NbSe 2[END_REF], and there have been many theoretical works proposing that TMDs may be a suitable platform to realize topological superconductivity upon doping [START_REF] Hsu | Topological superconductivity in monolayer transition metal dichalcogenides[END_REF][START_REF] Noah | Possible Topological Superconducting Phases of MoS 2[END_REF] or the application of an external magnetic field [START_REF] He | Magnetic field driven nodal topological superconductivity in monolayer transition metal dichalcogenides[END_REF][START_REF] Shaffer | Crystalline nodal topological superconductivity and Bogolyubov Fermi surfaces in monolayer NbSe 2[END_REF][START_REF] Wang | Platform for nodal topological superconductors in monolayer molybdenum dichalcogenides[END_REF]. On the experimental side, the picture remains less clear: a recent analysis of collective modes in monolayer NbSe 2 suggests that the order parameter might be a mixture of singlet s-wave and triplet f -wave [START_REF] Wan | Observation of Superconducting Collective Modes from Competing Pairing Instabilities in SingleLayer NbSe 2[END_REF], while the two-fold anisotropy in the magnetic response [START_REF] Hamill | Two-fold symmetric superconductivity in few-layer NbSe2[END_REF][START_REF] Cho | Nodal and Nematic Superconducting Phases in NbSe 2 Monolayers from Competing Superconducting Channels[END_REF] and other transport measurements [START_REF] Kuzmanovi | Tunneling spectroscopy of few-monolayer NbSe 2 in high magnetic fields: Triplet superconductivity and Ising protection[END_REF] has been interpreted as a signature of a more exotic nematic phase. To summarize, the degree of singlet-triplet mixing in TMDs -let alone the actual structure of the pairing functionis far from settled.

In view of this uncertainty, and motivated by the recent scanning tunneling microscopy (STM) studies in monolayer TMDs and similar compounds [START_REF] Raphaël T Leriche | Misfit layer compounds: a platform for heavily doped 2d transition metal dichalcogenides[END_REF], we explore how the local response to defects can be used to determine the existence of subdominant triplet correlations in non-centrosymmetric superconductors. In particular, we study the quasiparticle interference (QPI) spectrum, which is readily obtained by Fourier-transforming a spatial STM measurement. This probe measures the modulations in the local density of states (LDOS) induced by point-like impurities in the substrate. These modulations result from the interference of Bloch states mixed by the scatterers; therefore, they contain information about the pristine system. In practice, the wavelengths of the standing waves resulting from the scattering process show as peaks at specific wave vectors q in the Fourier transform of the LDOS at a given energy bias [START_REF] Capriotti | Wave-vector power spectrum of the local tunneling density of states: Ripples in a d -wave sea[END_REF]. A quantitative interpretation of the QPI spectrum is unfeasible, as it depends on the strength of the scattering potentials and the tunneling matrix elements which are unknown in general [START_REF] Antonio Sulangi | Revisiting quasiparticle scattering interference in high-temperature superconductors: The problem of narrow peaks[END_REF]; nevertheless, qualitative analyses of the q-pattern have proven useful to infer details of the band structure and the momentum dependence of the pairing function in superconductors [START_REF] Hoffman | Imaging quasiparticle interference in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF][START_REF] Wang | Quasiparticle scattering interference in hightemperature superconductors[END_REF][START_REF] Sharma | Momentum-resolved superconducting energy gaps of Sr 2 RuO 4 from quasiparticle interference imaging[END_REF][START_REF] Ming | Evidence for chiral superconductivity on a silicon surface[END_REF]. In addition, in recent years there have been more sophisticated proposals to extract information on the gap structure: for instance, authors in [START_REF] Hirschfeld | Robust determination of the superconducting gap sign structure via quasiparticle interference[END_REF][START_REF] Böker | Phase-sensitive determination of nodal d -wave order parameter in single-band and multiband superconductors[END_REF][START_REF] Sharma | Multi-atom quasiparticle scattering interference for superconductor energy-gap symmetry determination[END_REF] showed that antisymmetrizing the QPI spectrum with respect to the bias energy could serve to identify sign-changes in the superconducting gap, and it has been suggested that bias-asymmetry might be a signature of odd-frequency superconducting pair correlations [START_REF] Chakraborty | Quasiparticle Interference as a Direct Experimental Probe of Bulk Odd-Frequency Superconducting Pairing[END_REF].

In this chapter, we consider a conventional scenario of singlet-triplet admixture in a TMD, namely an s-wave order parameter with a subdominant f -wave component, and we assess if the subdominant triplet component can be pinpointed in the QPI spectrum of both conventional and spin-polarized LDOS. We find that this methodology is insufficient for that purpose; nevertheless, we show that the substrate's inversion symmetry-breaking manifests in the spin-polarized QPI spectrum, and we provide a detailed analysis of the wave vector q-pattern in systems with lifted spin-degeneracy, opening the way for subsequent studies on QPI in TMDs.

The rest of the chapter is structured as follows: in Sec. 4.2, we introduce the model Hamiltonian; in Sec. 4.3, we present our results; and in Sec. 4.4, we discuss their implications and limitations.

Model

We introduce a simple toy model of a two-dimensional superconductor with D3h point-group crystal symmetry, representing monolayer transition metal dichalcogenides. In the Nambu basis

Ψ k = (c k,↑ , c k,↓ , c † -k,↓ , -c † -k,↑ ) T the bare Hamiltonian reads Ĥ0 (k) = ε(k)τ z + γ(k)τ z σ z + ∆ s τ x + d z (k)τ x σ z , (4.1)
where τ α and σ α are the Pauli matrices acting in particle-hole and spin space, respectively. Here, ε(k) represents the spin-degenerate energy dispersion, and γ(k) an antisymmetric, Isinglike spin-orbit coupling (SOC) field. We treat superconductivity at the mean-field level, and we choose the superconducting order parameter to transform according to the trivial representation of D3h, i.e., to be a mixture of s-wave spin-singlet (∆ s ) and f -wave S z = 0 spin-triplet [d z (k)] which does not break any lattice symmetries. We further assume that the superconducting state does not break time-reversal symmetry; in other words, the order parameter is a linear combination of the s-wave and f -wave components without any phase difference, and we choose an overall gauge such that ∆ s is real.

Importantly, we note that by modeling the system with Hamiltonian (4.1) we treat broken inversion symmetry as a symmetry-breaking field [γ(k)], and consequently, we have to introduce singlet-triplet mixing of the superconducting order parameter by hand. When doing so, we must enforce the well-established fact that the spin-orbit coupling field is parallel to the pseudo-vector characterizing the triplet pairing [START_REF] Frigeri | Superconductivity without inversion symmetry: MnSi versus CePt 3 Si[END_REF], γ(k) d(k), with γ(k) = γ(k)ẑ and d(k) = d z (k)ẑ in our system. This condition follows immediately by phrasing the problem in the band basis and introducing an intra-band pairing interaction between the eigenstates of the normal Hamiltonian [START_REF] Samokhin | Gap structure in noncentrosymmetric superconductors[END_REF]. Nonetheless, our approach is valid insofar as we are not interested in the origin of the triplet pairing, nor do we intend to treat superconductivity self-consistently. We restate that our goal is to find a signature of subdominant triplet correlations in the QPI spectrum, regardless of their microscopic origin, and addressing this question in the spin basis is more natural. Incidentally, for this particular choice of SOC, spin remains a good quantum number, and it is 4.2. Model straightforward to define a band gap,

∆ ± (k) = ∆ s ± d z (k) for band ε ± (k) = ε(k) ± γ(k), (4.2)
In subsequent sections, we complement our analytical results with numerical calculations on a lattice. To perform the latter, we choose a toy model with the following tight-binding dispersions:

ε(k) = t[2 cos(ξ) cos(ν) + cos(2ξ)] + µ, (4.3a
)

γ(k) = 2α sin(ξ)[cos(ξ) -cos(ν)], (4.3b)
d z (k) = 2∆ t sin(ξ)[cos(ξ) -cos(ν)], (4.3c)
where ξ = kxa 2 and ν = √ 3kya

2 , with a = 1 the lattice spacing. Note that the energy dispersion (4.3a) stems from (symmetric) nearest-neighbors hopping on a triangular lattice with amplitude t, and the SOC (4.3b) and the f -wave gap function (4.3c) stem from (antisymmetric) nearest-neighbors coupling with amplitudes α and ∆ t respectively. As we seek to investigate subdominant triplet correlations, we shall restrict the analysis to the ∆ s > ∆ t regime. Further, we fix the chemical potential µ so that the Fermi contour (FC) is essentially one pocket centered at the Γ point [see Fig. 4.1 (a)]. While it may be unnatural for the gap function to have maxima at the K, K points in the absence of Fermi-energy electronic states in these regions, it is insightful to test general claims on a simple toy model. We emphasize that our goal is not to reproduce the QPI spectrum of a particular material but to find a general smoking gun of subdominant triplet correlations. As shown in Fig. 4.1 (b), the superconducting gap varies continuously over the FCs due to the triplet component. This variation represents a small percentage of the singlet component, and therefore, the DOS of the bare system is gapped [Fig. 4.1 (c)]; however, it induces some remanent states below the DOS maximum, which contain information about the triplet correlations and hence will be the target of the QPI analysis. Finally, we introduce a local and isotropic scatterer at the origin in order to study the QPI spectrum within the T-matrix formalism. We consider both a non-magnetic (V 0 ) and a magnetic potential (V β ) of strength J. They are described by the Hamiltonian V0 (r) = J 0 δ(r)τ z σ 0 , (4.4a)

Vβ (r) = J β δ(r)τ 0 σ α , (4.4b)
where β = x, y, z denotes the orientation of the magnetic impurity.

Results

As we discussed earlier, the modulations of the LDOS due to a point-like scatterer at r = 0 at energy ω are given by

δρ α,β (r, ω) = - 1 π Im Tr Mα Ĝ0 (r, 0, ω) Tβ (ω) Ĝ0 (0, r, ω), (4.5)
where we trace over spin and particle-hole degrees of freedom. Here, in contrast to the previous chapter, we include the STM-tip potential, M α = τ 0 +τz 2 σ α , which selects the electron component, and crucially, distinguishes between conventional (α = 0) and spin-polarized (α = x, y, z) measurements. For completeness, we recall that Ĝ0

(r a , r b , ω) = dk (2π) 2 e ik•(ra-r b ) Ĝ0 (k, ω), (4.6)
with Ĝ0 (k, ω) = ω + i0 + -Ĥ0 (k)

-1

the usual retarded Green function of the clean system, and

Tβ (ω) = 1 -Vβ dk (2π) 2 Ĝ0 (k, ω) -1 Vβ , (4.7)
the T-matrix. The assumption of a fully local and isotropic impurity leads to a k-independent T-matrix.

The QPI spectrum is given by the Fourier transform of the LDOS,

δρ α,β (q, ω) = dre -iq•r δρ α,β (r, ω). (4.8)
After some manipulation, Eq. (4.8) reads

δρ α,β (q, ω) = - 1 2πi Λ α,β (-q, ω) -Λ * α,β (q, ω) = - 1 π Im Λ even α,β (q, ω) + i Re Λ odd α,β (q, ω) , (4.9)
with * denoting complex conjugation, Λ α,β (q, ω) = dk (2π) 2 Tr Mα Ĝ0 (k, ω) Tβ (ω) Ĝ0 (k + q, ω) , (4.10)

Results

and "even" and "odd" referred to q. Importantly, the form δρ α,β (q, ω) = -1 π Im Λ α,β (q, ω) that is sometimes found in the literature is only valid if the substrate and the scattering potential are inversion-symmetric. Further, we note that, in principle, the LDOS modulations can be directly probed in an experiment, and as such, δρ α,β (r, ω) is a real quantity. On the contrary, the QPI spectrum is not directly observable, and therefore, δρ α,β (q, ω) can take complex values.

In the following, we shall restrict our analysis to the Born approximation (i.e., the assumption of a weak scattering potential, J β BW, the typical bandwidth energy scale), where we have that Tβ (ω) ≈ Vβ . We note that impurities usually scatter in both the non-magnetic (β = 0) and magnetic (β = x, y, z) channels with arbitrary strengths and further, the orientation of the spin-polarized tip with respect to the substrate is generally unknown. However, the weak-scatterer approximation allows us to express the full QPI spectrum as a linear combination ρ tot (q, ω) = α,β δρ α,β (q; ω) of all the possible tip-impurity combinations. In the following subsections, we analyze these components separately, pinpointing the effect of a subdominant triplet order parameter.

Singlet-triplet admixture with vanishing spin-orbit coupling

We start by setting a negligible spin-orbit coupling, i.e., γ(k) = 0 in Eq. (4.1). We emphasize that a singlet-triplet mixed order parameter that spontaneously breaks inversion symmetry in an otherwise centrosymmetric superconductor is only possible in a fine-tuned scenario that has not been detected to date to the best of our knowledge. Nevertheless, the results in this section provide valuable insight into the role of a triplet component.

Tracing over the spin and particle-hole degrees of freedom in expression (4.10) yields the following results: Non-magnetic x-magnetic ŷ-magnetic ẑ-magnetic Non-SP Λ 0,0 (q, ω) 0 0 0 SP-X 0 Λ x,x (q, ω) Λ x,y (q, ω) 0 SP-Y 0 Λ y,x (q, ω) Λ y,y (q, ω) 0 SP-Z 0 0 0 Λ z,z (q, ω) Table 4.1: Components of the QPI spectrum for a singlet-d z -triplet admixed order parameter in the absence of spin-orbit coupling in the Born approximation.

The entries in Table 4.1 read Λ 0,0 (q, ω)

= σ=± dk (2π) 2 [G σ (k, ω)G σ (k + q, ω) -F σ (k, ω)F σ (k + q, ω)], (4.11)
Λ j,j (q, ω)

= σ=± dk (2π) 2 [G σ (k, ω)G σ (k + q, ω) + F σ (k, ω)F σ (k + q, ω)],
for j = x, y, z, (4.12)

Λ y,x (q, ω) = i σ=± σ dk (2π) 2 [G σ (k, ω)G σ (k + q, ω) + F σ (k, ω)F σ (k + q, ω)], (4.13)
and Λ y,x (q, ω) = -Λ x,y (q, ω), with

G ± (k, ω) = ω + ε(k) ω 2 -ε(k) 2 -∆ ± (k) 2 + i sign(ω)0 + , (4.14)
F ± (k, ω) = ∆ ± (k) ω 2 -ε(k) 2 -∆ ± (k) 2 + i sign(ω)0 + , (4.15)
where ∆ ± (k) was defined in Eq. (4.2). First, we note that within the Born approximation, magnetic impurities do not couple to a standard STM tip, and conversely, spin-polarized tips are oblivious of non-magnetic impurities [START_REF] Hirschfeld | Robust determination of the superconducting gap sign structure via quasiparticle interference[END_REF]. Second, we notice that for a vanishing triplet component,

G + (k) = G -(k) and F + (k) = F -(k)
, therefore, the off-diagonal entries of Table 4.1 are zero [Eq. (4.13)]. Third, we remark that

G + (-k) = G -(k) and F + (-k) = F -(k) owing
to the antisymmetry of the triplet order parameter under spatial inversion, and therefore, the non-zero, off-diagonal entries of Table 4.1 are odd in q, whereas the diagonal entries are even in q. Then, it follows from Eq. (4.10) that δρ α,α (q, ω) is purely real and δρ y,x (q, ω) is purely imaginary.

This qualitative difference regarding the parity of the QPI signal is exemplified in Fig. 4.2, where we plot Re[δρ x,x (q; ω)] [panel (a)] and Im[δρ y,x (q; ω)] [panel (b)] for a non-zero ∆ t /∆ s ratio in the toy model presented in Section 4.2. In a STM experiment, it would be challenging to measure the LDOS for three perpendicular spin polarizations and, further, the actual orientation of the magnetic impurity with respect to the triplet pseudo-vector d(k) is unknown, however, we conclude that a non-zero signal in the imaginary channel of the FT-LDOS is a clear signature of a subdominant triplet component under the assumption of a vanishing spin-orbit coupling. Further, we note that the FT-LDOS is a scalar, and within the Born approximation, it is linear in d(k) |d(k)| , V mag and M SP , the last two being two (pseudo)vectors representing the magnetic impurity and the spin-polarized tip. As it must be invariant under rotations 1 , we can express the relevant result of this section as follows:

Im[δρ SP (q, ω)] ∼ [M SP × V mag ] • d(k) |d(k)| . (4.16)

Singlet-triplet admixture with nonvanishing spin-orbit coupling

Next, we turn to the more realistic scenario where a non-zero antisymmetric spin-orbit coupling is present. For the toy model presented in Eq. (4.1), the QPI spectrum has the same structure as in the absence of spin-orbit coupling [cf. Table 4.1 and Eqs. (4.11)-(4.13)], where now we take into account the spin dependence in the energy dispersion

G ± (k, ω) = ω + ε ± (k) ω 2 -ε ± (k) 2 -∆ ± (k) 2 + i sign(ω)0 + , (4.17)
F ± (k, ω) = ∆ s ± ∆ t (k) ω 2 -ε ± (k) 2 -∆ ± (k) 2 + i sign(ω)0 + . ( 4

Analysis of the q-vector pattern

Next, we show how to extract further insight into the nature of the subdominant triplet order parameter in spite of the presence of the spin-orbit coupling term by analyzing the specific q-pattern in the spin-polarized FT-LDOS.

In order to understand the q-pattern, we recall that under the assumption of elastic scattering, an impurity mixes Bogoliubov quasiparticles with different k wave-vector but the same energy. The LDOS contains terms accounting for the interference of the incoming (k 1 ) and scattered (k 2 ) eigenstates, which emerge as a non-zero signal at the wave-vector q = k 1k 2 in the FT-LDOS.

In particular, scattering predominantly happens between Bogoliubov quasiparticles satisfy-

4.3. Results ing E l,σ 1 (k 1 ) = E l,σ 2 (k 2 ) = ω, (4.19)
where ω is the bias energy. The points in k-space fulfilling Eq. (4.19) define the so-called BdG contours, in analogy with the Fermi contour. For the simple toy model discussed so far, the BdG energy dispersion reads

E l,σ (k) = l ε σ (k) 2 + ∆ σ (k) 2 , (4.20)
where l, σ = {±} denote the four BdG bands. Within the framework of the Born approximation, the mixing between different spin bands (σ 1 , σ 2 ) depends on the joint action of the impurity potential and spin polarization of the tip. One can heuristically reason that the QPI signal is maximal where the joint density of states (JDOS) is. It follows from

ρ(k, E) ∼ 1 |∇ k E l,σ (k)| , (4.21)
that the highest signal in the FT-LDOS is at q-points joining k-vectors where the gradient of the BdG energy dispersion is minimal, or equivalently where its curvature is maximal. 2 This heuristic analysis of the QPI pattern was pioneered by J. Hoffman et al. [START_REF] Hoffman | Imaging quasiparticle interference in Bi 2 Sr 2 CaCu 2 O 8+δ[END_REF] in the study of BSCOO. Indeed, in d-wave superconductors, the nodal character of the gap leads to the emergence of "banana-shaped" BdG contours, with a highly peaked DOS at the tips of the contours.

Here we extend their analysis to gapped superconductors with a subdominant triplet order parameter by noting that "banana-shaped" BdG contours also emerge in this scenario, provided that the bias energy lies within the variation of the superconducting gap with respect to the constant s-wave background over the Fermi surface [recall Fig It follows from Eq. (4.20) that for a pure singlet s-wave order parameter, BdG contours emerge at ω = ∆ s and coincide with the normal Fermi contour, defined by

FC ± ≡ ε ± (k) = 0. ( 4.22)
Setting ω ≥ ∆ s leads to a pair of BdG contours surrounding the normal FC, which do not intersect each other [Fig. 4.4 (a)]. If we turn on the triplet component, and we set the bias energy to satisfy ∆ s + min[d z (k)] FC < ω < ∆ s + max[d z (k)] FC , the pair of BdG contours surrounding the normal FC intersect at the normal FC precisely, giving rise to a BdG-contour tip. In other words, in our toy model, the tips of the BdG contours are at k-points satisfying

ε ± (k) = 0, (4.23)
∆ ± (k) = ω, (4.24)
simultaneously. In the limit ω = ∆ s the contours' tips sit at the intersection of the triplet orderparameter nodes and the normal FC, and as we decrease the bias energy, the BdG-contour's tips slide over the normal FC until they reach the point where d z (k) is minimal. Conversely, if we increase the bias energy, the BdG-contour's tips move toward max[d z (k)] on the FCs. These two cases are exemplified in Fig. 4.4 (b) and (c), respectively.

As shown in Fig. 4.4, the toy model under consideration yields twelve contour tips, which, in turn, yield 72 distinct q-vectors. These can be generated from a set of nine non-equivalent q-vectors via six-fold rotations and mirror operations. We distinguish two types of scattering processes among this collection of q-vectors: inter-band scattering, where the q-vector joins two contour tips belonging to bands with opposite spin number (σ) [e.g. green arrow in Fig. 4

Results

This distinction is crucial to interpret the QPI patterns. Within the Born approximation, standard FT-LDOS (relevant for non-magnetic impurities) filters the inter-band scattering out. In contrast, spin-polarized FT-LDOS (relevant for magnetic impurities) precludes the intra-band processes [Fig. 4.5]. One can readily understand this effect by noting that the expression describing the former [Eq. (4.11)] contains products of propagators belonging to the same band only, namely G σ (k)G σ (k + q) and F σ (k)F σ (k + q), while it is the opposite for the latter [Eqs. (4.12) and (4.13)], i.e., they contain the products G σ (k)G σ (k + q) and F σ (k)F σ (k + q) only. Furthermore, it should be noted that the processes involving σ → σ and σ → σ exhibit the same sign when contributing to the even component of the spin-polarized FT-LDOS, as indicated in Eq. (4.12). Conversely, these processes contribute with opposite signs to the odd component, as specified in Eq. (4.13). By observing the relationship between the position of the BdG contours' tips and the bias energy, as depicted in Fig. 4.4 (b) and (c), we can account for the overall difference in sign between Im[δρ y,x (q, ω)] for ω < ∆ s and ω > ∆ s shown in Fig. 4.5 (e) and (f) at the relevant q-vectors. for a non-magnetic potential. Second (third) row: spin polarization along x ( ŷ) for a magnetic impurity along x. The signal strength is much weaker for a non-magnetic scatterer than for a magnetic impuritiy (see color-scale); hence the q-pattern is less clear. Parameters (in units of t): µ = -2.0, α = 0.4, ∆ s = 0.1, ∆ t = 0.05, J x = J 0 = 0.2, ω = 0.092 (a, c, e), ω = 0.11 (b, d, f), η = 1.5 • 10 -4 , k-space discretization: 2500 x 2500 grid.

Results

We emphasize that the emergence of "banana-shaped" BdG contours cannot stem from the presence of spin-orbit coupling alone; it requires a variation of the superconducting gap over the normal FC, and the bias energy to be tuned within the bounds of such variation. It is precisely the lack of contour tips that leads to a diffuse QPI spectrum [recall panels (a) and (b) in Fig. 4.3]. Nevertheless, it should be noted that the emergence of contour tips is not inherent to the triplet correlations in the order parameter. It solely requires a variation of the superconducting gap over the Fermi contours; therefore, an extended s-wave OP with spin-orbit coupling might also give rise to "banana-shaped" contours and lead to a peaked signal in the antisymmetric channel Im[δρ y,x (q, ω)] signal.

To exemplify this crux, we introduce an additional extended s-wave pairing ∆ s + ∆ ext-s (k), and we choose ∆ ext-s (k) = 2∆ s cos(2ξ) cos(2ν) cos(4ξ), (

with ξ and ν as defined above. This gap function originates from third-nearest-neighbors coupling, which is highly artificial; however, it also yields a continuous variation of the superconducting gap over the FCs on the same order of magnitude as in Fig We conclude this section by mentioning a subtle difference between the BdG contours in an (s+f )-wave and an extended s-wave superconductor. In the first case, it follows from Eqs. This intimate relation between the quantum number of the BdG band and the position of the contours' tip with respect to the sign of d z (k) is a fundamental consequence of the triplet nature of the subdominant order parameter. Indeed, in the second case, the BdG energy dispersion reads E ext-s l,σ

(k) = l ε σ (k) 2 + [∆ s + ∆ ext-s (k)] 2 ,
and therefore, the tips of the BdG contours will sit on regions where ∆ s (k) < 0 if ω < ∆ s and vice-versa if ω > ∆ s , for both bands [see Fig. 4.6 (a), (b)]. Alas, this is a minimal difference that does not have a striking consequence in the expressions of the QPI spectrum deduced at the beginning of the section; in accordance, we do not find any universal differences between the two gap functions considered here in the spin-polarized FT-LDOS at varying energy bias.

Discussion

In this chapter, we attempted to develop a methodology to detect a subdominant triplet component in an admixed s+f -wave superconductor. The clear-cut conclusion of this work is that inversion-symmetry breaking fields in the normal Hamiltonian manifest as an imaginary, qodd component in the spin-polarized FT-LDOS. Therefore, simply checking whether this channel of the QPI spectrum is non-zero does not suffice to certify that the antisymmetric spin-orbit coupling induces triplet correlations in the superconducting order parameter. Nevertheless, this approach would be fruitful to detect the subdominant triplet component if an accidental degeneracy, i.e., singlet-triplet mixing in a centrosymmetric crystal, were to occur. This result is consistent with Ref. [START_REF] Hofmann | Theory of quasiparticle scattering interference on the surface of topological superconductors[END_REF] where the spin-polarized FT-LDOS of topological superconductors was analyzed. In view of this conclusion, it is a natural question to ask if additional universal laws relating the symmetries of the normal state Hamiltonian to the properties of QPI spectrum exist. This type of analysis could be advantageous to detect more exotic gap functions that spontaneously break lattice symmetries, contrary to the f -wave case considered in this work. Performing a reliable symmetry-based analysis of the QPI spectrum would require considering the effect of symmetry-breaking scatterers [START_REF] Böker | Quasiparticle interference and symmetry of superconducting order parameter in strongly electron-doped iron-based superconductors[END_REF].

Moreover, we provided an interpretation of the QPI pattern in terms of the BdG contours in a system where spin-degeneracy is lifted. By carefully analyzing these patterns and having a precise knowledge of the band structure of the substrate, it would be possible to infer the symmetry of the order parameter. However, this scheme requires high-precision measurements in the bias energy and in q-space. The smaller the triplet component, the more fine-tuned in energy one needs to be in order to probe "banana-shaped" contours (instead of BdG rings); therefore a JDOS-inspired analysis seems unfeasible for TMD superconductors, where the superconducting gap spans over a few milli-electron volts only. Nevertheless, it has worked in the past for BSCCO and other cuprate superconductors where the gap is on the order of tens of meV.

If tracking the evolution of the q-vectors with varying bias energy in TMDs may not be doable from a practical standpoint, it could be possible to measure a single QPI snapshot featuring a q-peaked pattern. While that would be a promising observation, we showed that there exist pathological scenarios -for instance, the combination of antisymmetric spin-orbit coupling and an extended s-wave gap function-that can result in a peaked q-pattern in the imaginary, qodd channel of the spin-polarized FT-LDOS, even in the absence of triplet correlations. However, we showed that there exists a fundamental difference between the structure of the BdG contours in (k-dependent) singlet and singlet-triplet superconductors. Namely, in the former, the contour tips always lie on regions where the subdominant gap function has the same sign, while in the latter, the tips of spin-up and spin-down BdG contours lie on regions where the subdominant triplet component has the opposite sign. This distinction is subtle but it offers a path for further investigation: an analysis in the spirit of the HAEM technique for gapless superconductors [START_REF] Böker | Phase-sensitive determination of nodal d -wave order parameter in single-band and multiband superconductors[END_REF] may prove helpful in this situation.

T (n) β (ω) Ĝ0 (k + q, ω) , (4.26)
where

T (n) β (ω) = Vβ dk (2π) 2 Ĝ0 (k, ω) n Vβ . (4.27)
For a standard tip and a magnetic impurity, we recall that the zeroth order contribution to Λ 0,β (q, ω) vanishes (see Table 4.1). The next leading term reads

Λ (1) 0,β (q, ω) = σ=± dk (2π) 2 {AG σ (k, ω)G σ (k + q, ω) (4.28a) +BF σ (k, ω)F σ (k + q, ω) (4.28b) +C[G σ (k, ω)F σ (k + q, ω) + F σ (k, ω)G σ (k + q, ω)]}, (4.28c)
where G σ (k, ω) and F σ (k, ω) were defined in Eqs. (4.17) and (4.18) in the main text and

A = (J β ) 2 1 2 σ=± dk (2π) 2 G σ (k, ω), (4.29a) B = -(J β ) 2 1 2 σ=± dk (2π) 2 G * σ (k, -ω), (4.29b)
C = (J β ) 2 1 2 σ=± dk (2π) 2 F σ (k, ω). (4.29c)
Chapter 5

Tunable multiple in-gap states: beyond the classical impurity model

Introduction

Scanning tunneling microscopy (STM) measurements on the surroundings of a magnetic impurity on a superconductor yield a frozen snapshot of the spectroscopic and spatial properties of the impurity-bound state, which result from the local coupling to the environment, among other factors. This parameter depends, in turn, on the nature of the contaminant and the adsorption site; therefore, performing additional scans on other impurities on the sample provides a discrete set of couplings, albeit unknown, and ultimately, it leads to a more complete description of the bound states. But the STM probe itself can also serve as a tuning knob of the impurity-substrate coupling. This scheme has been achieved, for instance, in molecular complexes with a magnetic center which are loosely attached to the substrate and whose coupling is therefore sensitive to the mechanical force exerted by approaching the STM tip to the sample [START_REF] Farinacci | Tuning the Coupling of an Individual Magnetic Impurity to a Superconductor: Quantum Phase Transition and Transport[END_REF][START_REF] Malavolti | Tunable SpinSuperconductor Coupling of Spin 1/2 Vanadyl Phthalocyanine Molecules[END_REF]. This operation allows varying the energy of the in-gap bound states continuously, thereby offering the possibility to investigate the quantum phase transition that occurs when an in-gap excitation crosses zero energy. This chapter analyzes a set of STM observations showing tunable multiple in-gap states on magnetic impurities in FeTe 0.55 Se 0.45 . This material belongs to the broader class of high-T c iron-based superconductors and it has sparked a great deal of interest from the community owing to its potential to host topological superconductivity and Majorana zero modes on vortex cores [START_REF] Zhang | Observation of topological superconductivity on the surface of an iron-based superconductor[END_REF][START_REF] Peng Fan | Observation of magnetic adatom-induced Majorana vortex and its hybridization with field-induced Majorana vortex in an iron-based superconductor[END_REF]. In addition, this compound has a low carrier density; therefore, local defects can be substantially affected by the electrostatic field generated by the STM tip, and indeed, it has proven to be a convenient platform to tune the energy of in-gap impurity-bound states [START_REF] Chatzopoulos | Spatially dispersing Yu-Shiba-Rusinov states in the unconventional superconductor FeTe0.55Se0.45[END_REF].

As we shall see shortly, the experimental data that motivated this study has the particularity of exhibiting several in-gap resonances whose energy can be continuously altered. This multiplicity can stem from various origins: distinct angular-momentum scattering channels [START_REF] Ji | High-Resolution Scanning Tunneling Spectroscopy of Magnetic Impurity Induced Bound States in the Superconducting Gap of Pb Thin Films[END_REF], multi-band superconductivity [START_REF] Li | Magnetic impurities in the two-band s ± -wave superconductors[END_REF], higher-spin impurities and magnetic anisotropy [START_REF] Hatter | Magnetic anisotropy in Shiba bound states across a quantum phase transition[END_REF], or crystal-field split d levels coupled to different symmetry channels of the Bloch states in the sub-strate [START_REF] Ruby | Orbital Picture of Yu-Shiba-Rusinov Multiplets[END_REF][START_REF] Choi | Mapping the orbital structure of impurity bound states in a superconductor[END_REF]. The last two are especially relevant for impurity transition atoms (as is the case in this study), where the reduced symmetry of the adsorption site lifts the degeneracy among the d orbitals, and Hund's occupation rule enforces a higher-spin configuration.

The classical model for magnetic impurities that we have employed throughout this thesis can be extended to characterize multiple in-gap YSR states by considering independent impurity potentials that scatter conduction electrons within different channels [START_REF] Arrachea | Yu-Shiba-Rusinov multiplets and clusters of multiorbital adatoms in superconducting substrates: Subgap Green's function approach[END_REF][START_REF] Moca | Spin-resolved spectra of Shiba multiplets from Mn impurities in MgB 2[END_REF]. However, this description precludes the appearance of correlated behavior of the in-gap resonances with a varying parameter, that, as we shall see, is the hallmark of the experimental observations at hand. Some studies have incorporated the quantum nature of the impurities by considering an anisotropic higher-spin Kondo Hamiltonian. This model is usually solved by means of highthroughput numerical renormalization group (NRG) calculations [START_REF] Itko | Effects of magnetic anisotropy on the subgap excitations induced by quantum impurities in a superconducting host[END_REF], and only very recently it was thoroughly examined with analytical tools [START_REF] Felix Von Oppen | Yu-Shiba-Rusinov states in real metals[END_REF].

However, the observation of negative differential conductance in the STM data suggests that Coulomb interactions play a decisive role in the impurity system [START_REF] Thielmann | Super-Poissonian noise, negative differential conductance, and relaxation effects in transport through molecules, quantum dots, and nanotubes[END_REF]; therefore, we consider instead a multi-orbital Anderson Hamiltonian, which can be understood as a parent model of the higher-spin Kondo Hamiltonian, with the added advantage that interaction effects are explicit, and that it encompasses both the local-moment and the valence-fluctuation regimes [START_REF] Huang | Quantum phase transitions and the role of impurity-substrate hybridization in Yu-Shiba-Rusinov states[END_REF][START_REF] Rubio-Verdú | Coupled Yu-Shiba-Rusinov States Induced by a Many-Body Molecular Spin on a Superconductor[END_REF]. As we shall see, this model qualitatively captures the evolution of the in-gap states observed in the experiment, which, we propose, reflects a new type of quantum phase transition characterized by a change of the impurity's mean occupation and total spin and driven by Hund's coupling.

The rest of the chapter is structured as follows. In Sec. 5.2, we present the analysis of the STM data, and we continue with the theoretical modeling in Sec. 5.3. Finally, in Sec. 5.4, we summarize the conclusions and perspectives of our work.

Experimental motivation 1

Excess Fe impurities can be recognized as bright protrusions among the mixture of Te and Se atoms (bright and dark respectively) in Fig. 5.1 (a), which shows a typical constant current image of cleaved FeTe 0.55 Se 0.45 .

The energy of the in-gap peaks varies for different impurities, highlighting the variation in coupling parameters with the local environment. Differential conductance spectra taken at the center of two different Fe impurities are shown in Fig. 5.1 (b,c). Interestingly, not only peaks are observed, but a large fraction of excess Fe also displays negative differential conductance (NDC) at one or more energies. Previous studies on YSR states have reported NDC; however, in those cases, NDC results from a convolution effect with the superconducting DOS of the STM tip [START_REF] Liebhaber | YuShiba-Rusinov states in the Charge-Density Modulated superconductor NbSe 2[END_REF]. Here, measurements are performed with a normal tip whose density of states can be assumed to be constant on an energy range of the order of the superconducting gap. Lastly, all excitation energies can be shifted by changing the tip-sample distance, i.e., modifying the coupling parameters [ Having identified multiple tunable in-gap states, we can next use the tip-impurity distance to shift low-lying states through zero-bias energy and track their evolution. To find the location where the coupling parameters can be more dramatically tuned, we first take a line cut across the excess Fe atom in Fig. 5.2 (a) at different junction resistances. As Figs. 5.2(b-d) show, at large tip-sample distance (high-junction resistance), the impurity system is little affected by the tip, and therefore, the in-gap states hardly shift as a function of position. Upon nearing the tip to the surface, however, the states move towards zero energy, and the first one eventually crosses when the tip is directly above the Fe impurity and sufficiently close (at a low-junction resistance) [Fig. 5.2(d)]. To analyze the crossing in more detail, we perform a more detailed junction resistance dependence on top of the Fe impurity, see Fig. 5.2 (e). For large tip-sample distance, the in-gap state closest to zero energy is strongest in intensity at negative bias, and is followed by additional peaks and a relatively prominent negative differential conductance dip [similar to Fig. 5.1 (b)]. Importantly, the intensity of all peaks is stronger at negative bias than at positive bias. As we reduce the tip-sample distance, all features shift closer to zero bias. Then, as soon as the first peak crosses zero-bias energy, the intensity of all other peaks switches polarity and the NDC disappears. Additionally, whereas the peak that crossed continues to shift to higher energy, the others never cross, but instead an ever-increasing full gap is formed below the crossing point. For independent in-gap states this would not have been possible: higher-energy states would simply have continued shifting until they also would have crossed zero.

Finally, we present in Fig. 5.3 a detailed analysis of the zero-bias energy crossing of the lowest-lying in-gap state, where we show for two different impurities that the slope at the transition of the spectral weight curve with varying parameter is discontinuous.

Theoretical model

To understand the observed behavior, we construct a phenomenological model focused on explaining two key observations: (1) The simultaneous switch of nature of in-gap modes between hole-like (dominant intensity at E < 0) and electron-like (dominant intensity at energy E > 0), and (2) the appearance of negative differential conductance. The latter phenomenon occurs even though the experiment is well within the tunneling regime, implying that the tunneling current cannot be described by the single-particle density of states. The impurity must involve at least two states with variable electron occupation, which allows the blocking of tunneling current (thereby yielding negative dI/dV ) through one state due to interactions with electrons present in the other state [START_REF] Thielmann | Super-Poissonian noise, negative differential conductance, and relaxation effects in transport through molecules, quantum dots, and nanotubes[END_REF]. Therefore, the impurity cannot be modeled by a (classical or quantum) spin, and we consider the more general Anderson impurity model.

The minimal model capturing the observed phenomenology needs at least two orbitals interacting through an effective Hund's coupling that favors higher-spin configurations. However, in an attempt to make this presentation more pedagogical, we start by discussing the one-orbital case. We present the full model in Sec. 5.3.2.

Preliminaries: single-orbital Anderson model

The Anderson impurity model describes an atomic site coupled to a fermionic bath, in this case, the superconductor at the mean-field level. Since we focus exclusively on in-gap states, we simplify matters by treating the substrate as an s-wave superconductor with zero bandwidth, where the BCS density of states is replaced by two quasiparticle states at energy ∆. This approximation is known to perform very well for in-gap bound states of spin impurities [START_REF] Felix Von Oppen | Yu-Shiba-Rusinov states in real metals[END_REF]; however, outside-the-gap excitations cannot be faithfully described. The impurity site is characterized by the orbital energy level ε and the Coulomb interaction U > 0, and it is coupled to the bath via a tunneling rate Γ = πt 2 , with t the hopping amplitude. The full Hamiltonian reads,

H = H 0 + H imp + H T , (5.1)
with

H 0 = ∆c † ↑ c † ↓ + h.c. , (5.2)
H imp = σ ε d ndσ + U nd↑ nd↓ , (5.3)
H T = σ tc † σ d σ + h.c. . (5.4)
Here, c † σ and d † σ are the spin-σ electron-creator operator on the superconducting site and the impurity site respectively, and nσ = d † σ d σ the particle-number operator on the impurity. This Hamiltonian conserves fermion parity and has full spin-rotation symmetry. Therefore, eigen-states can be classified according to the magnitude and z-component of the total spin,

S tot = σ,σ c † σ s σ,σ c σ + d † σ s σ,σ d σ , (5.5)
with s = 1 2 σ the vector of Pauli matrices, and total parity,

P = (-1) n, (5.6)
with n = σ c † σ c σ + d † σ d σ . Finally, we introduce the local density of states (LDOS) on the impurity site that we use to characterize the system,

ρ imp (ω) = - 1 π Im σ G R σ;σ (ω). ( 5.7)
In this case where the impurity is an interacting quantum object, we cannot employ the Tmatrix formalism as in the previous chapters. However, owing to the reduced size of the Hilbert space, it is possible to diagonalize the many-body Hamiltonian and obtain the retarded Green function directly from the Lehmann representation,

G R σ;σ (ω) = 1 Z n,n (e -βEn + e -βE n ) n|d † σ |n n |d σ |n ω + E n -E n + i0 + , (5.8)
where the sum goes over the eigenstates of H, and Z = n e -βEn is the partition function. In the zero-bandwidth limit, the absence of a "continuum" precludes the appearance of a selfenergy; therefore the broadening of the LDOS is controlled by the phenomenological parameter 0 + . In an STM experiment, one can expect that k B T ∆, the smallest energy scale in our problem. In this case, the contributions to the LDOS from transitions between excited states are exponentially small, and they would only induce a slight broadening at the level crossings, hence, we assume T = 0. In this limit,

G R,T =0 σ;σ (ω) = 1 d GS GS n | GS|d † σ |n | 2 ω + E n -E GS + i0 + + | n|d † σ |GS | 2 ω + E GS -E n + i0 + , , (5.9)
where d GS denotes the degeneracy of the ground state.

Since the impurity site has one orbital only, we can exploit the reduced size of the Hilbert space and gain some insight by considering the eigenstates of the substrate and the impurity in the absence of coupling. Owing to the Pauli principle, the superconducting site and the impurity can host two electrons each at most; therefore, the Hilbert space is spanned by {|0 , |↑ , |↓ , |2 } sc ⊗ {|0 , |↑ , |↓ , |2 } imp where the basis states are labeled according to their occupations. The impurity Hamiltonian H imp conserves the number of particles, therefore, it is diagonal in the Fock basis. The eigenenergies are

E |0 = 0, E |↑ = E |↓ = ε d and E |2 = 2ε d + U .
It is precisely in the parameter regime where the doublet {|↑ , |↓ } becomes the ground state that the atomic site develops a local moment, and thus the model effectively describes a magnetic impurity. On the contrary, the eigenstates of H 0 do not have a defined number of particles.

The eigenenergies are E

|0 -|2 √ 2 = -∆, E |↑ = E |↓ = 0 and E |0 +|2 √ 2
= ∆. We mention while passing how the zero-bandwidth model captures the essence of the mean-field description of superconductors: the eigenstates are connected by single-particle operators that are a linear combination of an electron and a hole, and the ground state can be constructed by removing these quasiparticles from the vacuum |0 .

The effect of the impurity-substrate coupling is to mix the eigenstates of the free Hamiltonians. In addition, there is a competition between the Coulomb repulsion and the hybridization, so that, roughly speaking, the local moment only develops when the former outweighs the latter. In Fig. 5.4, we show the typical evolution of the system as a function of the impurity energy level ε for a set of parameters allowing the local moment formation. The ground-state mean occupation of the impurity reflects the eigenspectrum of H imp : it decreases from fully occupied (favored at very negative ε) to empty (favored at positive ε). The emergence of a local momentevidenced by the single occupation of the impurity-prompts a quantum phase transition: there is a change in the system's ground state, signaled by the total spin and parity change. An analogous level crossing occurs when the local moment fades into an empty impurity site. In particular, the relevant lowest-lying eigenstates are a singlet state, |s tot = 0, m = 0, P = + , and doubly degenerate spin-1/2 state, |s tot = 1/2, m = ±1/2, P = -, that we shall denote "doublet". These eigenstates are represented in Fig. 5.4 (b) in the Fock basis at various points in the vicinity of the quantum phase transition. This representation allows drawing an analogy with the classical YSR model (see Chapter 1, Sec. 1.2.3): the singlet state can be understood as the situation where a BCS quasiparticle is effectively screening the impurity and the doublet state as the unscreened regime, where the ground state is approximately the tensor product of the unperturbed BCS ground state and the impurity spin. However, this analogy should not be pushed too far. For instance, the classical YSR model does not capture the degeneracy of the unscreened state. Finally, in Fig. 5.4 (c), we plot the LDOS on the impurity site. The in-gap spectral weight corresponds to the single-particle excitation between the ground state and the first excited state; naturally, at the quantum phase transition, these two states become degenerate, and accordingly, the spectral peak crosses zero-bias energy. The spectral weight exhibits a marked asymmetry between the positive and the negative bias energy. This feature reflects the occupation of the impurity: hole-like excitations are favored when the impurity is mostly occupied, while electron-like excitations are more likely in the empty regime. As we shall see in the next section, this property of the Anderson impurity model will be crucial to explaining the experimental observations. Further, the LDOS is particle-hole symmetric2 with respect to ε = -U 2 . For completeness, we show an example where the hybridization parameter is much larger than before, up to the point of impeding the appearance of the local moment [see Fig. 5.5]. As a consequence, the quantum phase transition cannot be triggered, and the in-gap excitation does not cross zero-bias with varying ε. This scenario is generally irrelevant for transition adatoms in solids, where we expect the Coulomb interaction to be dominant. However, we mention a recent study on a delicate nanostructure of silver atoms on superconducting Nb that reportedly realizes an effective non-interacting Anderson model [START_REF] Schneider | Proximity superconductivity in atom-by-atom crafted quantum dots[END_REF].

Eq. (5.1), where now This Hamiltonian conserves fermion-parity per channel and has full spin rotation symmetry too.

H 0 = i=A,B ∆c † i,↑ c † i,↓ + h.c. , (5.10
The key interaction that couples the orbitals is represented as the Hund's coupling, J H > 0, which favors higher-spin configurations typical of transition-element impurities. Here, S α=a,b = σ,σ d † α,σ s σ,σ d α,σ is spin operator for each orbital. In the absence of Hund's coupling, the system consists of two copies of the single-orbital model discussed in the previous section. As we increase the average impurity energy level ε, both orbitals evolve from double to single occupation and then become empty. The nonzero Hund's coupling favors a spin-triplet configuration when both orbitals are in the singleoccupation regime. In addition, we set J H |δε| ≡ |ε bε a |, the energy difference between the orbital energy levels, in order to enforce a sharp transition into the spin-triplet regime. The multi-channel Hamiltonian also has a particle-hole symmetry point at ε = -U 2 , therefore, as it was noted earlier, the transitions from double to single occupation and from single occupation to an empty orbital with varying ε are formally equivalent. Here we shall only focus on the former. The numerical relationship among the parameters of the model are within the physically reasonable limits, and such that the quantum phase transition is triggered. To summarize, we set J H U ∼ |ε| > |δε| Γ i,α > ∆, (5.13) with the superconducting gap the smallest energy scale.

The experimental topography indicates that the excess Fe atoms we study are on top of the Se/Te surface layer, and therefore, it is safe to assume that any movement of the tip exclusively affects the energy levels of the impurity orbitals. We further assume for simplicity that δε remains constant, while the average impurity energy level ε varies linearly with the tip-sample distance -thereby controlling the energy of the sub-gap excitations and representing the key experimentally tunable parameter.

with the generic spin-impurity model quantum phase transition (e.g. associated to the YSR model, or the single-orbital model discussed above), but is actually not the universal signature of our MCQPT. Namely, we find that by adding a third orbital to our model, the same complete phenomenology can be recovered even though the parity flip of all three channels makes the total parity change sign across the transition. The essential property of our MCQPT is the large change of the average orbital occupation from nearly 4 (both orbitals fully occupied) to nearly 2 (both singly occupied), see Fig. 5.6 (c). The cause is the strong Hund's coupling which drives (as ε changes) the transition from nearly fully occupied orbitals with the system in a total spin-singlet state (s tot = 0) into a state of orbitals each having nearly a single electron and forming a higher-spin configuration so the total spin of system becomes triplet (s tot = 1). This mechanism is allowed by the reasonable assumption that the impurity is in a mixed-valence state, i.e., the energy cost of the change of orbital occupation can be compensated by the gain of Hund's energy, leading to the requirement stated in Eq. (5.13). The distinctive switch of all in-gap excitations from hole-like to electron-like is due to the depletion of the average electron occupation of the impurity orbitals.

To address the STM observations, we calculate the total single-particle LDOS of the impurity orbitals. Expressions for the LDOS are analogous to those defined in Sec. 5.3.1, where now, there is an additional sum over the orbital degree of freedom. Only excitations from the many-body ground state to states with opposite total parity can have a non-zero spectral weight. To demonstrate the phenomenology of the MCQPT, we focus on the two lowest in-gap excitations which result from the four lowest many-body eigenstates described above [see arrows in Another key feature of the MCQPT model is that the single-particle excitations appearing in the in-gap LDOS result from different ground states before and after the MCQPT; therefore, in general there is a discontinuity of slope of the in-gap state energy at the point it crosses zero-bias with changing ε [see Fig . 5.6 (d)]. Indeed, this property is a common feature for the experimentally measured Fe impurities where a MCQPT occurs (recall Fig. 5.3), further supporting our interpretation. Note that the discontinuity of the slope at the MCQPT is a priori independent of the precise functional dependence of ε on the electric field (assumed linear).

We introduce next a systematic study to confirm that the phenomenology is not fine-tuned, but is present for a range of values of phenomenological parameters as long as we stay in the regime given by Eq. (5.13). We start by noting one consequence which derives from the nature of this new multi-channel QPT: since the excitations involve transitions between different states before and after the QPT (and not a simple crossing as in the classical YSR model or in the oneorbital Anderson model), the lowest-lying excitation does not need to cross zero-bias energy at the transition. In the situation presented in Fig. 5.6, the energy of the first excitation satisfies that E |1/2,-,+ -E GS 0 at the MCQPT, therefore, there is a zero-bias crossing in the ingap LDOS, although it is not protected. Nevertheless, we show in Fig. 5.7 that there exists a wide region in the parameter space within the model assumptions for which the lowest-lyingexcitation gap is below the experimental resolution. In Fig. 5.8 (a,c) we give an example of a MCQPT happening without zero-energy crossing. We emphasize that the essential features of our new MCQPT are robust. Specifically, the large change of the average orbital occupation at the transition, which yields a simultaneous change in the polarity of all in-gap states (from hole-like to electron-like). Finally, we note that there exists an additional mechanism yielding a discontinuous in-gap LDOS. Namely, the system can undergo two consecutive QPTs, from |0, +, + into |1/2, -, + , and subsequently, from |1/2, -, + into |1, -, -. Far away from the transition, the level ordering is analogous to that of the scenario discussed so far, and crucially, the impurity also experiences a large change of the occupation [see Fig. 5.8 (d-f)]. Nonetheless, there exists again a wide region in the parameter space in which the QPTs are sufficiently close in ε, so that they result in an in-gap LDOS indistinguishable from the scenario discussed above.

Spectral function

Energy spectrum

Impurity mean occupation

Spectral function

Energy spectrum Impurity mean occupation

Toy-model illustration of negative differential conductance in a multilevel system

In this section, we present a minimal transport calculation to account for the negative differential conductance (NDC) observed in the STM measurements. As noted in [START_REF] Thielmann | Super-Poissonian noise, negative differential conductance, and relaxation effects in transport through molecules, quantum dots, and nanotubes[END_REF], NDC in multi-channel quantum-dot-like systems originates from a combination of the Coulomb interaction and an asymmetrically-suppressed coupling to the reservoir. We claim that some of the in-gap bound states are comparably weakly coupled to the STM tip, therefore, their spectral footprint in the multi-channel impurity model (MCIM) becomes a NDC peak in the transport calculation.

To illustrate this behavior, we represent tunneling into the in-gap bound states emerging from a complex impurity-superconductor system as tunneling into simple energy levels. Specifically, we map the four lowest-lying many-body eigenstates of the MCIM presented in the main with nl the particle-number operator. The energy levels in this effective toy model should be interpreted as the in-gap bound states stemming from the magnetic impurity in the superconducting substrate; they should not be confused with the orbital energy levels in the MCIM employed to explain the emergence and behaviour of the in-gap bound states. This is a oneto-one mapping, implying that the re-ordering of the many-body states as we tune the system through the multi-channel QPT is reflected in a re-ordering of the energy levels in the effective model [see Fig. 5.9]. In addition, K should be interpreted as an effective repulsive interaction which penalizes double occupation of the bound states.

Further, we couple the two-level system to two reservoirs which describe the substrate and the STM tip respectively. Crucially, we assume that the tunneling matrix element between the tip and one of the bound states is strongly suppressed; this inefficient coupling could arise, for example, if the wavefunction of the tip and the corresponding bound state had an orthogonal symmetry (impurity-bound states have been shown to inherit the symmetry of the adatom's orbitals [START_REF] Ruby | Orbital Picture of Yu-Shiba-Rusinov Multiplets[END_REF][START_REF] Choi | Mapping the orbital structure of impurity bound states in a superconductor[END_REF]). Therefore, we assume (T 1,S = T 2,S and T 1,tip T 2,tip ) in the following. For the sake of simplicity, we set T 1,S = T 1,tip ; however, results do not depend on this condition. In Fig. 5.10 we present the differential conductance calculated numerically for this effective toy model with a method based on the Lindblad equation. Before the transition [Fig. 5.10 (a)], the two lowest-lying peaks in the calculated dI/dV for the effective toy model correspond to single-particle transitions from an empty state to singleoccupied levels 1 and 2 respectively. If the tunneling rate from the left reservoir (STM tip) to level 2 (an in-gap bound state) is much smaller than the rest, electrons cannot flow from level 2. Since the large K penalizes the double occupation of level 1, electrons get stuck in level 2. Therefore, the current collapses and the corresponding excitation appears as a NDC peak at negative bias. After the transition [Fig. 5.10 (b)], the two lowest-lying excitations for the effective toy model correspond to single-particle transitions from a doubly-occupied level to single-occupied levels 1 and 2 respectively. The differential conductance is not negative now, because these transitions involve both levels, therefore, electrons can always escape through level 1.

Before the MCQPT After the MCQPT

Bias voltage Bias voltage

Finally, we note that the Lindblad equation method we employed to calculate dI/dV relies on the assumption that all the coupling strengths are much smaller than the temperature, T l,r k B T , with l = 1, 2 and r = S, tip, which is a reasonable assumption in our experiment. We emphasize again that the couplings T l,r are not directly related to the couplings between the impurity and the superconducting site in the MCIM Γ i,α , which are typically larger than ∆ but instead to the intrinsic broadening of the bound states; the latter is typically much smaller than the temperature (in Ref. [START_REF] Thupakula | Coherent and Incoherent Tunneling into Yu-Shiba-Rusinov States Revealed by Atomic Scale Shot-Noise Spectroscopy[END_REF], the relevant energy-scale was experimentally estimated to be Λ ∼ 1µeV).

Conclusions and outlook

In this chapter, we analyzed a set of STM data corresponding to in-gap bound states on Fe impurities on superconducting FeTe 0.55 Se 0.45 . In summary, each impurity adatom yields multiple in-gap resonances whose energy can be continuously tuned by approaching the STM tip. There are two crucial aspects that distinguish these experimental observations from previous results of impurity-bound states in superconductors : (1) as soon as the lowest-lying in-gap state crosses zero-bias energy, the spectral weight of all in-gap states switches from negative to positive bias, and (2) the differential conductance spectra exhibit dips taking negative values. Effect (2) had been previously studied in quantum dots and ascribed to the combination of Coulomb interaction and an asymmetrically-suppressed coupling to the reservoir; therefore, we chose to model the impurity system with a two-orbital Anderson model where interaction effects are explicit, and we enforce the higher-spin configuration typical of transition atoms by including a Hund's inter-orbital coupling. Further, owing to the low carrier density of FeTe 0.55 Se 0.45 , the STM tip's electric field penetrates the top-most layers, therefore, we assume that the orbital energy level of the impurity is shifted by varying the tip-sample distance.

Our model captures the concurrent switch of the spectral weight of all in-gap states: we showed that it is the manifestation of a multi-channel quantum phase transition, characterized by the depletion of the average electron occupation of all the impurity orbitals, which in turn results from a large inter-orbital Hund's coupling. We note that a realistic description of a Fe impurity demands a larger number of orbitals, which, subsequently, would yield a larger number of in-gap states; however, the two-orbital model discussed in this chapter contains the minimal ingredients to capture the distinctive feature of the MCQPT. Further, multiple in-gap states appear already in other impurity models, e.g. due to a higher impurity spin (classical [START_REF] Ruby | Orbital Picture of Yu-Shiba-Rusinov Multiplets[END_REF][START_REF] Arrachea | Yu-Shiba-Rusinov multiplets and clusters of multiorbital adatoms in superconducting substrates: Subgap Green's function approach[END_REF] or quantum [START_REF] Felix Von Oppen | Yu-Shiba-Rusinov states in real metals[END_REF]) and splitting by anisotropy [START_REF] Hatter | Magnetic anisotropy in Shiba bound states across a quantum phase transition[END_REF][START_REF] Itko | Effects of magnetic anisotropy on the subgap excitations induced by quantum impurities in a superconducting host[END_REF], and notably in iron-based compounds due to multiple superconducting gaps [START_REF] Li | Magnetic impurities in the two-band s ± -wave superconductors[END_REF]. In the classical YSR model, there is no reason to have a change in spectral weight of higher in-gap excitations when the lowest excitation crosses zero energy, since the multiple scattering channels are independent from one another. In a model of a quantum higher spin on the impurity, there may be total-parity-conserving QPT in which two channels undergo a screening transition simultaneously, either due to anisotropy, or due to symmetry-protected degeneracy of some impurity-substrate couplings [START_REF] Felix Von Oppen | Yu-Shiba-Rusinov states in real metals[END_REF]. In such a transition, the spectral weights of multiple in-gap states may shift between hole-and electronlike too. However, as noted above, such a model could never explain a negative conductance. We emphasize that in our model the transition is characterized by a change of occupation and total spin driven by Hund's interaction, rather than a change in screening of impurity by the substrate.

In addition, our model with its single-site single-gap s-wave superconductor obviously cannot address the spatial features of impurity-bound states, but the fact that the key features are found at various distances from the impurity and in both horizontal and vertical movement indicates that we are observing universal features driven by the field effect, and not fine-tuned features due to spatial-or momentum-dependent properties of the impurity and substrate. Nevertheless, it is an interesting open challenge to corroborate the conclusions drawn in this work by doing a numerical renormalization group calculation, which could eventually take into account the multi-band nature of FeTe 0.55 Se 0.45 . Finally, we note that while the observation of NDC justified the modeling choice, we did not account for both experimental observations (1) and (2) in a unified manner. Our calculation in Sec. 5.3.3 is a mere illustration of how an asymmetric coupling between the tip and the multiple in-gap bound states would yield negative differential conductance. There remains, thus, the (not inconsiderable) challenge of performing a transport calculation of the tunneling current that encompasses a superconducting substrate and a multi-orbital interacting impurity.

Figure 1 . 1 :

 11 Figure 1.1: (a) Radial cut of the excitation energy dispersion E k . The normal state is assumed to be a 2D free electron gas. (b) Excitation density of states. The divergence coincides with the flatness of the spectrum, and the DOS converges to the normal-state value at high energies.

1. 2 .

 2 Introduction to Yu-Shiba-Rusinov states tude of probability of propagating directly, being scattered once by the impurity, twice, thrice, etc.

Figure 1 . 2 :

 12 Figure 1.2: Evolution of the positive branch of the YSR energy in Eq. (1.49) for an arbitrary fixed K. Blue represents the BCS-like state [Eq. (1.50)] and green the broken-pair state [Eq. (1.51)]. The vertical dashed line indicates the quantum phase transition, where E S = 0.Inspired from[START_REF] Farinacci | Tuning the Coupling of an Individual Magnetic Impurity to a Superconductor: Quantum Phase Transition and Transport[END_REF].

Figure 2 . 1 :

 21 Figure 2.1: Various examples of Van Hove singularities in the density of states. The red curve represents a conventional VHS (logarithmic divergence), while the blue and the orange curves are higher-order VHS (power-law divergence) with the parameters specified in the legend. The vertical axis is normalized by the constant DOS [see Eq. (2.4)].

Figure 2 . 2 :

 22 Figure 2.2: Positive branch of the YSR energy, Eqs. (2.6) and(2.11), as a function of the magnetic coupling strength J for a constant (green), logarithmically-divergent (red), power-law divergent with ν = -1/3, η = 2 (orange) and power-law divergent with ν = -1/2, η = 1 (blue) DOS. The black dot indicates the value of K/E c . In the small K regime (a), J c depends strongly on the underlying DOS, while for large K regime (b), J c ∼ K. The solid lines indicate the numerical solution of the self-consistent equations (2.6) and (2.11) while the dashed lines represent the zeroth-order approximation in E S .

  2.2 (b)].

Figure 2 . 3 :

 23 Figure 2.3: Positive branch of the YSR energy, Eqs. (2.6) and(2.11), as a function of the magnetic coupling strength J zoomed around the YSR transition for logarithmically-divergent (red), power-law divergent with ν = -1/3, η = 2 (orange), and power-law divergent with ν = -1/2, η = 1 (blue) DOS. The tangent, dashed lines indicate the slope at the crossing point, which remains constant for increasing values of K denoted in the color bars and becomes steeper for a stronger divergence. Note that if the DOS is asymmetric, the minimum J c is not at K = 0 [cf. blue and orange curves, and Eq. (2.13) with and without vanishing f 2,ν (∆, 0) factor].

Figure 2 . 4 :

 24 Figure 2.4: Variation of the positive branch of the YSR energy, Eq. (2.15) as a function of the magnetic coupling strength when the Fermi level tuned to a [-∆, ∆] interval around a conventional VHS. The red (green) area corresponds to a vanishing (large) non-magnetic scattering potential K. The solid line indicates the µ = 0 curve while the dashed-dotted lines correspond to µ = ±∆. Inset (a) shows the normalized critical J as a function of the chemical potential for different values of K. The markers indicate the corresponding curves in the main plot. Inset (b) depicts the normalized slope of the E S (J) curve at the transition point as a function of the chemical potential. In both insets, the solid line represents the exact numerical solution, and the dashed line the series expansion to the lowest order in µ/∆ [Eq. (2.17)]. Plot parameters: E c = 1000∆.

Figure 2 . 5 :

 25 Figure 2.5: Variation of the positive branch of the YSR energy as a function of the magnetic coupling strength when the Fermi level is tuned to a [-∆, ∆] interval around different HOVHS as indicated in the legend. The solid lines indicate the µ = 0 curves, and the dashed-dotted lines correspond to µ = ±∆ (omitted in the bottom panel to improve readability). Top panel: Zero non-magnetic scattering potential. Bottom panel: Large non-magnetic scattering potential. Note that an asymmetric DOS (η = 1, in orange) leads to different curves for ±K but whose slope at E S = 0 is the same. Insets show the normalized critical J as a function of the chemical potential for different HOVHS. The relative change in J c at K = 0 [inset in (a)] is one order magnitude larger than in the other considered scenarios because the actual values of J c are close to 0. The markers indicate the corresponding curves in the main plot. The dashed lines correspond to the series expansion to the lowest order [Eq. (2.20)], which was omitted in the inset in panel (b) to improve readability. Note that the ±K curves in the inset are symmetric upon electron or hole-doping only if the DOS is symmetric around the singularity (η = 1). Further, the relative change of the absolute value of J c is symmetric around µ = 0 only if the DOS is symmetric around the singularity. Plot parameters: E c = 1000∆.

Figure 2 . 6 :

 26 Figure 2.6: Normalized slope of the E S (J) curve at the transition point as a function of the chemical potential for the HOVHS presented in Fig. 2.5. The solid line represents the exact solution using the renormalization parameters in Eq. (2.19) and dashed curve the series expansion to lowest order in µ/∆ [Eq. (2.21)]. Electron or hole doping has the same effect on the slope only if the DOS is symmetric around the singularity. Plot parameters: E c = 1000∆.

Figure 3 . 1 :

 31 Figure 3.1: Scattering processes for a Fermi contour with two pockets, j = 2. (a) The black dashed lines represent the Fermi contour. The gray arrows indicate the normalized gradient of the energy dispersion at the Fermi contour. The black arrow marks an arbitrary vector in real space from the scatterer to the point where the propagator is evaluated. In the text, this is sometimes referred to as "observation direction", and it is determined by the angle θ r . The color markers indicate the real part of the critical points on the Fermi contours (see Sec. 3.3.2.2 for details) for the observation direction θ r and each color corresponds to a different Fermi pocket. The color arrows represent the normalized gradient of the energy dispersion evaluated at the critical points, which is parallel and anti-parallel to the observation direction. (b) Summary of the oscillation and decay lengths in the relevant scattering. Colored frames correspond to intra-pocket processes. Shaded entries indicate the processes present in a normal metal. We dropped the redundant labels in λ , j,j to lighten the notation. (c) Examples of normal-metal-like and condensate-mediated scattering processes [cf. Eqs. (3.19a) and (3.19b) respectively]. Color markers indicate the corresponding entry in panel (b).

Figure 3 . 2 :

 32 Figure 3.2: Area-preserving coordinate transformation. The dashed red line represents the Fermi contour. The arrow represents the magnitude of the Fermi velocity at that point on the contour. At k y = 0, the transformation increases the Fermi velocity and decreases the curvature, but both the prefactor and decay length of the LDOS are enhanced [see Eqs. (3.26a), (3.26b)]. The opposite occurs at k x = 0.

2 2m*

 2 is a trivial prefactor to adjust the units. The Fermi surface is the ellipse b FS = µ τ . The positive solution of Eq.(3.13) reads Re

Figure 3 . 3 :

 33 Figure 3.3: Saddle-point approximation beyond the small-gap limit for the ellipsoidal toy model. (a) Real and imaginary parts of the set of critical points as a function of the observation direction θ r which is color-coded. The dashed line indicates the Fermi surface. (b) Polar plot of the dimensionless quantities encoding the anisotropy of the bare propagator. The system's parameters are µ = Ω = τ = 1 a.u. and α = 0.25

3 . 4 (

 34 a)], and so does the LDOS at the YSR-state energy [see color maps in Fig. 3.4 (c-f)].

Figure 3 . 4 :

 34 Figure 3.4: (a) Fermi contours of the energy dispersion in Eq. (3.36) at several doping values. The long, black arrow indicates an arbitrary observation direction θ r , and the small, colored arrows the normalized gradient of the energy dispersion at the corresponding critical points. For a circular Fermi contour, the critical point would sit at θ r exactly. (b) Polar plot of the LDOS prefactor in log-scale for the energy dispersions represented in (a). (c-f) Electron part of the LDOS at the YSR-state energy calculated numerically from the energy dispersions in (a). The field of view is 401 by 401 lattice sites around the impurity. The color bar is in arbitrary units, log-scale. The color curve is a polar plot of the decay length. The solid line indicates the analytical approximation, and the circular markers are extracted from fitting cuts of the numerical LDOS to the exponential-times-power-law enveloppe. The scale of the polar curves differs from the scale of the underlying color maps; the circumscribing circle of the color curve in (c) corresponds to 65 lattice sites. Numerical parameters: t = 200, ∆ = 5, J = 285 (in meV).

Figure 3 . 5 :

 35 Figure 3.5: Atomic structure NbSe 2 . (a) Lateral view of the bulk bilayer. Note that the unit cell (indicated by the black rectangle) has an inversion center. (b) Monolayer system. The Nb atoms (green) are disposed on a triangular lattice that is the basis for the tight-binding description. Figure reproduced with permission from [94].

Figure 3 . 6 :

 36 Figure 3.6: (a) Fermi surface for fifth-nearest neighbors tight-binding model on a triangular lattice describing NbSe 2 , parameters from band 2 in [95]. The black hexagon marks the first Brillouin zone. The long, black arrow indicates an arbitrary observation direction θ r , and the small, orange arrows mark the gradient of the energy dispersion evaluated at the critical points on the Fermi surface for the choice of θ r . (b) Same as panels (c-f) in Fig. 3.4 for the NbSe 2 energy dispersion, with a field of view of 500 by 500 lattice sites. The circumscribing circle of the orange curve corresponds to 59 lattice sites.

Figure 3 . 7 :

 37 Figure 3.7: Raw dI/dV spectrum measured at the impurity location. Panel (a), (b) correspond to data set 1, 2, respectively.

Figure 3 . 8 :

 38 Figure 3.8: Raw spatial data. (a) Topography for data set 1. The blue circle indicates the impurity position as inferred from the current measurements. (b) and (c) Current measured at ±0.6 mV respectively. Lower row: same for data set 2. The blue segments in the bottom-left corner indicate the crystallographic axes of the Nb layer. The field of view is 26 nm and 15 nm, respectively.

Figure 3 . 9 :

 39 Figure 3.9: Angular dependence of the LDOS prefactor. Top (bottom) row corresponds to data set 1 (2). Left (right) column corresponds to positive (negative) bias. Purple-to-yellow curves represent experimental data points ρ R 0 ,δ int (θ) for δ values specified in the legend. Legend inset shows the real-space I measurement with a lattice vector aligned with the horizontal axis. We excluded a range of data points in data set 1 due to interference with another YSR state (partially visible on the bottom right of the image).

Figure 3 . 10 :

 310 Figure 3.10: (a) The color map represents f ∆ (k) on the FBZ. The dashed line represents the Fermi contour of the tight-binding energy dispersion (3.36) with µ/t = 0.25. (b) and (c) Same as panels (c-f) in Fig. 3.4 for the tight-binding model discussed in the present section. In panel (b) ∆ = 0 while in panel (c) ∆ /∆ = 0.8. The field of view of the LDOS plots is 401 x 401 sites while the largest ξ LDOS in the red curves is ∼ 83 sites. Numerical parameters: t = 200 meV, ∆ = 5 meV, J = 285 meV.

Figure 4 . 1 :

 41 Figure 4.1: Characterization of the bare system [Eq. (4.3)]. (a) The color map represents the triplet f -wave order parameter. The red (blue) curve indicates the spin-up (spin-down) Fermi contour, and the black hexagon indicates the FBZ. (b) Variation of the superconducting gap over the Fermi contours. Red and blue correspond to spin-up and spin-down bands, respectively. (c) DOS of the bare system. The vertical magenta lines delimit the relevant region in bias energy for the QPI analysis. Parameters: α = 0.4t, ∆ s = 0.1t, ∆ t = 0.05t, µ = -2t.

Figure 4 . 2 :

 42 Figure 4.2: Numerical calculation of the spin-polarized FT-LDOS for a magnetic impurity along x. The black hexagon delimits the first Brillouin Zone. (a) Spin polarization along x. (b) Spin polarization along ŷ. Parameters (in units of t): µ = -2.0, α = 0, ∆ s = 0.1, ∆ t = 0.05, J x = 0.2, ω = 0.11, η = 1.5 • 10 -4 , k-space discretization: 2500 x 2500 grid.

. 18 )Figure 4 . 3 :

 1843 Figure 4.3: Imaginary part of the ŷ spin-polarized FT-LDOS for a magnetic impurity along x calculated numerically, with vanishing (a) and non-zero (b) triplet component. The black hexagon delimits the first Brillouin Zone. Compare panels (a) and (b) with Fig. 4.4 (b) and (c) respectively. Parameters (in units of t): µ = -2.0, α = 0.4, ∆ s = 0.1, ∆ t = 0.05, J x = 0.2, ω = 0.11, η = 1.5 • 10 -4 , k-space discretization: 2500 x 2500 grid.

  . 4.1 (b)].

Figure 4 . 4 :

 44 Figure 4.4: Constant energy contours for the energy dispersions in Eq. (4.3). Solid (dashed) lines indicate the BdG (Fermi) contours. Red and blue correspond to the spin-up (σ = +1) and spin-down (σ = -1) bands respectively. (a) Constant gap function. (b) Non-zero ∆ t and ω < ∆ s . (c) Non-zero ∆ t and ω > ∆ s . The green (brown) arrows represent one of the meaningful q-vectors corresponding to an inter-band (intra-band) process. The color map in the background represents the subdominant triplet component. Parameters (in units of t): µ = -2.0, α = 0.4, ∆ s = 0.1, ∆ t = 0.05 (b, c), ω = 0.11 (a, c) and ω = 0.092 (b).

  .4 (b), (c)], and intra-band scattering, where the contour tips belong to bands with the same spin number [e.g. brown arrow in Fig. 4.4 (b), (c)].

Figure 4 . 5 :

 45 Figure 4.5: Numerical calculation of the FT-LDOS for a local impurity. The black hexagon delimits the first Brillouin Zone. The green (brown) circles indicate the q-vectors representing the dominant inter(intra)-band scattering processes, relevant for the spin-polarized (non-polarized) FT-LDOS. Left column: ω < ∆ s . Right column: ω > ∆ s . First row: non-polarized FT-LDOSfor a non-magnetic potential. Second (third) row: spin polarization along x ( ŷ) for a magnetic impurity along x. The signal strength is much weaker for a non-magnetic scatterer than for a magnetic impuritiy (see color-scale); hence the q-pattern is less clear. Parameters (in units of t): µ = -2.0, α = 0.4, ∆ s = 0.1, ∆ t = 0.05, J x = J 0 = 0.2, ω = 0.092 (a, c, e), ω = 0.11 (b, d, f), η = 1.5 • 10 -4 , k-space discretization: 2500 x 2500 grid.

  . 4.1 (with the same parameters and ∆ s = ∆ t ), which makes it suitable for comparison purposes. When setting min[∆ ext-s (k)] FC < ω < max[∆ ext-s (k)] FC , we obtain a set of BdG contours akin to that observed in (s+f )-wave superconductor [Fig. 4.6 (a), (b)]. As vaticinated before, we find a peaked signal in the antisymmetric channel due to the combination of spin-orbit coupling and k-dependence of the gap function, even though the order parameter is purely singlet [Fig. 4.6 (c), (d)].

  (4.23), (4.24) that if ω < ∆ s , the tips of the BdG contours σ = + and σ =sit on regions where d z (k) < 0 and d z (k) > 0 respectively [Fig. 4.4 (b)] and vice-versa if ω > ∆ s [Fig. 4.4 (c)].

Figure 4 . 6 :

 46 Figure 4.6: Extended s-wave gap function. Left column: ω < ∆ s . Right column: ω > ∆ s . Top row: Constant energy contours. Solid (dashed) lines indicate the BdG (Fermi) contours. Red and blue correspond to the spin-up (σ = +1) and spin-down (σ = -1) bands respectively. The black, solid line indicates the node of the gap function. Bottom row: Imaginary part of the ŷ spinpolarized FT-LDOS for a magnetic impurity along x calculated numerically. The black hexagon delimits the first Brillouin Zone. The green (brown) circles indicate the q-vectors representing inter(intra)-band scattering processes. Parameters (in units of t): µ = -2.0, α = 0.6, ∆ s = 0.1, ∆ s = 0.05, J x = 0.2, ω = 0.09 (a, c), ω = 0.108 (b, d), η = 1.5 • 10 -4 , k-space discretization: 2500 x 2500 grid.

Figure 4 . 7 :

 47 Figure 4.7: Real part of the non-polarized FT-LDOS for a magnetic impurity along x calculated numerically beyond the Born approximation. Left column: ω < ∆ s . Right column: ω > ∆ s . The black hexagon delimits the first Brillouin Zone. The green (brown) circles indicate the q-vectors representing inter(intra)-band scattering processes. The latter shows a higher signal. Parameters same as in the main text (in units of t): µ = -2.0, α = 0.4, ∆ s = 0.1, ∆ t = 0.05, J x = 0.2, ω = 0.092 (a), ω = 0.11 (b), η = 1.5 • 10 -4 , k-space discretization: 2500 x 2500 grid.

  Fig. 5.1(d)].

Figure 5 . 1 :

 51 Figure 5.1: (a) Constant current image of FeTe 0.55 Se 0.45 with several interstitial excess Fe atoms (bright protrusions). V set = 5 mV, I set = 100 pA. (b, c) Differential conductance spectra taken at the excess Fe impurities marked 1 and 2 in panel a, respectively (setup resistance = 100 MΩ). Both show a multitude of peaks highlighting the multi-orbital nature of the Fe sites. The appearance of negative differential conductance in both spectra reflects interactions between sub-gap states. (d) Height dependence of differential conductance on the Fe impurity marked 3 in panel a. Upon lowering the setup resistance, the peaks shift closer to zero.

Figure 5 . 2 :

 52 Figure 5.2: (a) Constant current image of FeTe 0.55 Se 0.45 with an excess Fe atom in the middle. (b-d) Differential conductance taken along the line in panel a for 150 MΩ, 45 MΩ and 25 MΩ, respectively. Directly on top of the Fe impurity, the in-gap states shift most dramatically and cross zero. (e) Junction resistance dependence of the normalized differential conductance directly above the Fe impurity ranging from 150 MΩ (top) to 5 MΩ (bottom). The step size in resistance uses a logarithmic scale to enhance the visibility of the crossing. Note that the electric field is linearly proportional to the tip-sample distance, and the current (and thus resistance) depends exponentially on the distance.

Figure 5 . 3 :

 53 Figure 5.3: Slope change upon crossing E = 0. (a) Normalized differential conductance spectra taken at different junction resistances on the same excess Fe atom as Fig. 5.2. Crosses are fitted locations of the most prominent peaks. The junction resistances run from 5-100 MΩ on a logarithmic scale. (b) Fitted peak voltages for the in-gap state that crosses E = 0. The line indicates the slope of the in-gap energy versus junction resistance after crossing E = 0: upon crossing, the slope changes. (c, d) Same as (a, b) for another excess Fe impurity. Again, upon crossing E = 0, the slope changes.

Figure 5 . 4 :

 54 Figure 5.4: Quantum phase transition in the single-orbital Anderson model. (a) Evolution of the ground-state observables with the impurity energy level ε. For the spin we represent s, such that s(s + 1) = S 2 (b) Modulus square of the three lowest-energy eigenstates. Rows correspond to different values of ε indicated by dashed vertical lines in panels (a) and (c). Columns correspond to different energies in ascending order (from left to right). Each square represents a component in the tensor Fock basis (see the legend on the bottom-left panel). The color scale is normalized between 0 (faint) and 1 (bright). Green (orange) represents the singlet (doublet) state. The m = ∓ 1 2 partner of the doublet is represented on the left (right), but we recall that they are degenerate. (c) In-gap LDOS on the impurity site as a function of ε. The colored horizontal line at zero-bias energy represents the current ground state. The continuous vertical lines in (a) and (c) mark the particle-hole symmetry point ε = -U 2 . Calculation parameters: U = 15∆, Γ = 5∆ 2 .

)U

  α nα,↑ nα,↓ -J H S a • S b , (5.11)H T = i,α,σ t i,α c † i,σ d α,σ + h.c. . (5.12)

Fig. 5 .

 5 6 (b), (d)]. To be precise, before the MCQPT the in-gap excitations correspond to excitations from |0, +, + to |1/2, -, + and to |1/2, +, -; there is a + →parity flip in channels (A, a) or (B, b), respectively. After the MCQPT, they correspond to excitations from |1, -,to |1/2, -, + and |1/2, +, -. Panel Fig. 5.6 (d) shows clear agreement with the STM results [Fig. 5.2 (e)], where the MCQPT is signaled by the concurrent change of spectral weight in all the in-gap modes. To confirm our interpretations and the significance of multi-orbital interactions, in Fig. 5.6 (e) we present the LDOS result of the transition in our model with the Hund's coupling switched off (J H = 0), where obviously the experimental phenomenology is not captured. In this case, the LDOS is essentially the linear superposition of the LDOS calculated in the previous section, for two different values of ε [recall Fig. 5.4 (c)].

Figure 5 . 7 :

 57 Figure 5.7: Magnitude of the deviation from zero-bias crossing of the lowest-lying excitation at the MCQPT (a) δε/∆ = -6 (b) δε/∆ = -3. The rest of the parameters are the same in both plots (U = 15, J = 30, in units of ∆), but note that the axis span a slightly different range. The red contours correspond to δE/∆ = 0.13, i.e. the experimental resolution for ∆ = 1.5 meV and electron temperature T = 0.4 K. The markers in (a) correspond to various choices of the tunneling rates presented in the text: Fig. 5.6 (circle) and Fig. 5.8 (diamond and cross).

Figure 5 . 8 :

 58 Figure 5.8: Robustness of the MCQPT. Top row: (a) Simulated in-gap LDOS at the impurity site as a function of ε/∆. (b) Evolution of the energy of the four lowest-lying many-body states.(c) Evolution of the average total electron occupation on the impurity for all four many-body state. The color-coding and the arrows represent the same than in Fig.5.6 in the main text. The parameters correspond to the diamond marker in Fig.5.7. Bottom row: Same as top row but with tunneling rates corresponding to the cross marker in Fig.5.7. In this case, the deviation from zero-bias crossing δE/∆ is defined as the separation of the two QPT (black dashed lines), which yields a discontinuity in the in-gap excitations.

Figure 5 . 9 : 2 E

 592 Figure 5.9: Effective two-level model to account for the NDC. (a) Schematic mapping between the energy levels in the effective model and the lowest-lying many-body eigenstates in the multi-channel impurity model. The arrow crossing indicates the MCQPT. (b) Schematic of the transport model. The coupling between the left reservoir and level 2 is strongly suppressed.

Figure 5 . 10 :

 510 Figure 5.10: Simulated differential conductance in the effective two-level model. (a) Level ordering analogous to right-hand side of Fig. 5.9. (b) Level ordering analogous to left-hand side of Fig.5.9. Solid (dashed) lines represent assymetric (equivalent) coupling strengths. Only in the former case there can be NDC. The numerical calculation was performed using the python package QmeQ 1.0 [161]. Parameters (in meV): (a) E 1 = 0.5, E 2 = 1.0, K = 3, T 1,tip = 0.005 (adim.), k B T = 0.05. (b) E 1 = -4.0, E 2 = -3.5, the rest same as (a).

  .[START_REF] Kaladzhyan | Characterizing p-wave superconductivity using the spin structure of Shiba states[END_REF] which implies that for every eigenstate |φ(k) of the BdG Hamiltonian with energy E k , there exists another eigenstate P|φ(-k) with energy -E -k . Therefore, the spectrum of the BdG Hamiltonian takes the form {E k,+ , E k,-, -E -k,+ , -E -k,-}, where the apparent doubling of the spectrum is an artifact of the formalism. Indeed, if we go back to the second-quantization representation by setting Υ k = U k Ψ k , with U k a unitary transformation that renders the BdG Hamiltonian diagonal, ĤD (k), we find

Table 3 .

 3 1: Fitted tight-binding parameters for the two lowest-lying Nb 4d bands of 2H-NbSe 2 .

		2 cos 2ζ cos 2η + cos 4ζ,			(3.38c)
	f 4 (k) = cos ζ cos 3η + cos 5ζ cos η, + cos 4ζ cos 2η,	(3.38d)
	f 5 (k) = 2 cos 3η cos 3ζ + cos 6ζ,			(3.38e)
	with ζ = 1 2 k x a and ζ =	√ 2 k y a, and a is the lattice constant. The hopping amplitudes are 3
	extracted from a fit to angle-resolved photoemission spectroscopy (ARPES) data:
		µ	t 1	t 2	t 3	t 4	t 5
	Band 1 10.9 86.8 139.9 29.6	3.5	3.3
	Band 2 203.0 46.0 257.5 4.4 -15.0 6.0

These were, maybe not so surprisingly, the same people who performed the STM studies on cooper mentioned above...

Ce sont, peut-être pas étonamment, les mêmes chercheurs qui réalisèrent les études STM sur Cooper mentionnées ci-dessus...

This is a pure collective effect which results from the interplay of the electron's pair with the filled Fermi sea through the Pauli exclusion principle. The binding of two bodies interacting via an inverse-square law F = -g/r

(expected in three dimensions) is not attained unless the potential energy of each body in the system is larger than its kinetic energy.

We adopt the usual convention to express the tensor product of two one-particle spin wavefunctions in a matrix form. Here | ↑ and | ↓ are the eigenstates of Sz.

This implies that ε k F = 0, with kF the Fermi momentum. We note that in the literature the energy dispersion is usually denoted ξ k when it incorporates the chemical potential. In this thesis, we do not follow this convention to avoid confusion with the characteristic length of impurity-bound states ξ.

While this is standard practice, we note that gauge transformations in superconductors are a thorny issue. The order parameter defined as the anomalous average ∼ cc is not invariant under local gauge transformations either, suggesting that it is not a good choice for the theory. See[START_REF] Beekman | An introduction to spontaneous symmetry breaking[END_REF] for a comprehensive review.

There exist exact solutions of the interacting BCS Hamiltonian, e.g. the work of Richardson and Gaudin that followed the publication of the BCS theory[START_REF] Combescot | BCS ansatz for superconductivity in the light of the Bogoliubov approach and the RichardsonGaudin exact wave function[END_REF]. However, they require a piece of exceedingly complicated mathematical machinery compared to the mean-field approximation.

The fermionic nature of the electrons imposes this restriction, and it ensures that V σ 1 ,σ 2 ;σ 3 ,σ 4 k,k satisfies the

We picked a gauge such that the pairing term in Eq. 1.25 has an overall positive sign.

By "all" it is meant "all the states that participate in the coherent wavefunction in the absence of impurity", i.e., states in the vicinity of the Fermi surface.

Note that we would obtain the same result by expanding Ĝ(0, 0; ω) around the negative energy pole ω = -ES.

This should not be confused with the standard result for the Friedel oscillations, where the integration in energy to calculate the expectation value of the electric charge gives a final 1/r d power-law decay, with d the dimensionality of the substrate.

See discussion in Sec.1.2.3

The lattice symmetries can originate exceptions to this rule: for instance, the energy-resolved Friedel oscillations in graphene decay as 1/

r 2 despite its two-dimensional character[START_REF] Vadim | Friedel Oscillations, Impurity Scattering, and Temperature Dependence of Resistivity in Graphene[END_REF][START_REF] Bena | Friedel oscillations: Decoding the hidden physics[END_REF].

One could generally expect YSR states to be oriented according to high-symmetry lines; however, it is a priori equally reasonable for the "legs" to be along the lattice vectors or in-between. The theory shows how this is, in fact, determined by the geometry of the Fermi surface and Fermi velocity.

Strictly speaking, only rotations belonging to the symmetry group of the Hamiltonian.

Recall the relationship between gradient and curvature of the Fermi surface discussed in the quasiparticle focusing effect (Ch.

3).

Measurements and data processing presented in this section were performed by F. Massee.

Naturally, this property can be traced back to the Hamiltonian that satisfies the relationship [H, Ξ] = 0 upon setting U + 2ε = 0, with Ξ a unitary operator defined as Ξd † σ Ξ † = dσ and Ξc † σ Ξ † = -cσ.

Acknowledgements

I thank my mentors Andrej and Pascal, Andrej for his enthusiasm and optimism and encouragement, and sharing his passion for science, and the many hours discussing broken gauge symmetry that I hope one day I will understand, and Pascal for welcoming me to this house already quite some time ago, and his pragmatism, and his will for me to progress.

Acknowledgements i

General overview v

Organization of this thesis xii

Contents Appendix 3.A Saddle-point approximation 3.A.1 Normal metal

In this appendix, we present the details of the saddle-point approximation for a twodimensional normal metal. We recall that the crux of the problem lies in calculating the integral

Appendix 4.A Beyond the Born approximation

The analytical results in this chapter were derived by treating the T-matrix within the Born approximation. To ensure that our claims are sufficiently general, we verified that the QPI spectra obtained by numerical simulations did not differ if we resummed all terms in the Tmatrix calculation in opposition to cutting to zeroth order, as long as J β BW. Interestingly, we found one exception to this rule, namely, that magnetic impurities couple to a non-polarized tip beyond the Born approximation as exemplified in Fig. 4.7, and further, only intra-band processes are allowed. We justify this observation in this appendix by calculating the relevant QPI spectrum to the next leading order in the impurity potential.

We recall that G

0,β (q, ω) is even in q, and as per Eq.(4.9), it is real. Importantly, Eq. (4.28) contains propagators belonging to the same band σ only; hence, as illustrated in Fig. 4.7, intra-band processes dominate in this channel.

This result shows that the next-leading terms in the Born approximation, which are often overlooked in the literature, can have non-trivial effects. Further, we conclude that by switching between non-polarized and spin-polarized STM measurements, we could distinguish between intra-band and inter-band q-vectors, even in the likely scenario that the impurity scatters in both non-magnetic and magnetic channels. Note, however, that the relative strength of the signal would not be comparable, as the intra-band and inter-band q-vectors involve different channels of the scattering potential, which have different magnitudes in general. We emphasize that within the single-orbital Anderson model, the energy of the in-gap state (and hence, triggering the quantum phase transition), can be done by either varying the impurity energy level ε (at fixed U and Γ) or the hybridization parameter Γ (at fixed U and ε). This model has been previously employed to explain the observed change in energy of an impurity in-gap state in FeTe 0.55 Se 0.45 with varying STM tip-impurity distance [START_REF] Chatzopoulos | Spatially dispersing Yu-Shiba-Rusinov states in the unconventional superconductor FeTe0.55Se0.45[END_REF]. In that work, it was assumed that the STM tip acts as an effective local gate that influences the impurity energy level so that by varying the tip-impurity distance, ε evolves linearly. Iron-based superconductors are a class of materials with low density of carriers, which leads to poor screening [160]. In [START_REF] Chatzopoulos | Spatially dispersing Yu-Shiba-Rusinov states in the unconventional superconductor FeTe0.55Se0.45[END_REF], it was shown that the Thomas-Fermi screening length is on the order of the interlayer distance; therefore, it is reasonable to expect that the electric field of the tip will penetrate the topmost layers of the sample. In the following section, we shall adopt this view. On the contrary, assuming that varying the STM tip-sample distance affects the substrate-impurity coupling is more likely for adsorbed molecular complexes [START_REF] Huang | Quantum phase transitions and the role of impurity-substrate hybridization in Yu-Shiba-Rusinov states[END_REF][START_REF] Rubio-Verdú | Coupled Yu-Shiba-Rusinov States Induced by a Many-Body Molecular Spin on a Superconductor[END_REF].

LDOS on the impurity

Multi-channel Anderson model

In this section, we minimally extend the single-orbital Anderson model to account for the experimental observations. First, we focus on the simultaneous switch from hole-like to electronlike of all in-gap modes with varying tip-impurity distance. As we shall see shortly, this is the manifestation of a new type of quantum phase transition -denoted multi-channel (MCQPT) in the following. The MCQPT results from the simultaneous depletion of multiple orbitals, which, in turn, is induced by a Hund's interaction among them. Then, in Sec. 5.3.3, we present a qualitative transport model built upon the calculated in-gap modes, and we discuss the origin of the observed negative differential conductance.

To address the necessary interactions, we extend the model discussed in Sec. The essence of the multi-channel quantum phase transition is reflected in the evolution of the four lowest-lying many-body eigenstates, which we can label as |s tot , p A,a , p B,b , due to the conserved total-spin s tot , and conserved electron-number parity p M,m of channels (i, α) = (A, a) or (B, b). The z-component of the total spin m also serves to label the eigenstates, but we drop it here to lighten the notation. The four states are |0, +, + , |1/2, +, -, |1/2, -, + , |1, -, -(zero-, two-, two-and three-fold degenerate, respectively), and the evolution of their energies with ε is shown in Fig. 5.6 (b). The transition occurs as the state |0, +, + exchanges places with the state |1, -, -. Hence, at the transition, both channels are involved, and both undergo a parity flip, so that the total parity does not change. This property is in stark contrast