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Doctoral Thesis

Topological Kondo insulators: materials at the interface oftopology and strong correlations

Author(s): Legner, Markus

Publication Date: 2016

Permanent Link: https://doi.org/10.3929/ethz-a-010779690

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Diss. ETH No. 23831

Topological Kondo insulatorsMaterials at the interface of topology and strong correlations

A thesis submitted to attain the degree of

DOCTOR OF SCIENCES of ETH ZÜRICH(Dr. sc. ETH Zürich)

presented by

Markus Legner

MSc ETH in Physics

born on October 15, 1989

citizen of Austria

accepted on the recommendation of

Prof. Dr. Manfred Sigrist, examinerProf. Dr. Titus Neupert, co-examinerProf. Dr. Ronny Thomale, co-examiner

2016

Nothing in life is to be feared, it is only to be understood. Nowis the time to understand more, so that we may fear less.

(Marie Curie)

Abstract

The concept of topology has been introduced into modern condensed matterphysics during the last decades, with research efforts especially increasing afterthe discovery of topological insulators ten years ago. Since then, these materialshave been studied in much detail, leading to the identification of many novel prop-erties and promising opportunities for application. The remarkable characteristicsof the conducting surface states, inherent to topological insulators, make them idealfor use in the development of low-power electronics or spintronics. In addition, thesurfaces of topological insulators have been proposed as building blocks to realizetopological quantum computers.Recently, topological Kondo insulators have emerged, a new class of topological

materials in which the energy gap is formed due to correlations of localized elec-trons. This interplay of topology and interactions gives rise to unique features anda very rich phase diagram including magnetically ordered phases and collectiveexcitations. Samarium hexaboride (SmB6), a material extensively studied for over50 years, has recently received much attention as the potentially first topologicalKondo insulator. In contrast to most previously known topological insulators, thismaterial is completely insulating in the bulk at low temperatures, which is a crucialproperty for future applications. In addition, SmB6 exhibits other novel properties;for example, it has multiple topologically protected states at the surface instead ofa single one that is found in other topological insulators.

In this thesis, we give an overview over the many intriguing characteristics oftopological Kondo insulators and their interrelations. We start by introducing asimple model for this class of materials in the form of a periodic Anderson model,

vi | Abstract

where the particular form of the hybridization between conduction electrons andlocalized states is the crucial difference to previously used models for (topologi-cally trivial) Kondo insulators. After identifying different important topologicalinvariants and relating them to points with band inversion, we discuss the relatedsurface states in further detail.

The position of the topologically protected surface states in momentum spaceallows for a novel type of spin texture in addition to the helical spin texture foundin other topological insulators. We explain the characterization of the spin textureby means of a winding number and analyze its relationship with the mirror Chernnumbers of the system. These mirror Chern numbers are topological invariantsthat can be defined in the presence of mirror symmetry and are themselves closelyconnected to fundamental properties of the system. In particular, they depend di-rectly on the type of orbitals at the Fermi level and the pattern of hybridization. Ourresults therefore provide the possibility to extract information about microscopicmaterial parameters from the measurement of the spin texture at the surface, e.g.,by spin- and angle-resolved photoemission spectroscopy.

We proceed by discussing different consequences of the interactions between thestrongly localized electrons in Kondo insulators. The first effect we consider is therenormalization of model parameters, leading to an energy shift of the localizedstates and, in turn, to interaction-driven topological phase transitions. It has alsobeen shown that SmB6 and other Kondo insulators exhibit instabilities towardsmagnetic ordering for certain external parameters. We adopt multiple approachesfor detectingmagnetic phases in these materials and discuss the impact of differentchoices of models. In addition, collective excitations, so-called spin excitons, havebeen detected in Kondo insulators; we present both experimental and theoreticalinsights into this subject, with particular focus on their size and behavior close tothe surface of the material.

Finally, we discuss the relationship of topological invariants, mainly in two di-mensions, to seemingly unrelated concepts of entanglement, quantum geometry,and statistical physics.

Zusammenfassung

Das Konzept der Topologie wurde während der letzten Jahrzehnte in die moder-ne Festkörperphysik integriert, wobei das Forschungsinteresse besonders nach derEntdeckung von topologischen Isolatoren vor zehn Jahren zugenommen hat. Seit-dem wurden die Eigenschaften dieser speziellen Materialien sehr detailliert unter-sucht, was zu zahlreichen vielversprechenden Anwendungsmöglichkeiten geführthat.Wegender aussergewöhnlichenCharakteristiken der leitendenOberflächenzu-stände, die in diesen Materialien natürlicherweise auftreten, sind diese besondersfür die Entwicklung von elektronischen Bauteilen mit sehr geringer Leistungs-aufnahme oder sogenannten „Spintronic“-Komponenten geeignet. Zudem wurdevorgeschlagen, die Oberflächen von topologischen Isolatoren für die Konstruktionvon topologischen Quantencomputern zu verwenden.Vor wenigen Jahren wurden topologische Kondo-Isolatoren entdeckt, eine neue

Klasse von topologischenMaterialien, in denen die Energielücke zwischen Valenz-und Leitungsband durch Wechselwirkungen zwischen den lokalisierten Elektro-nen entsteht. Dieses Zusammenspiel von Topologie und elektronischen Wechsel-wirkungen führt zu neuartigen Eigenschaften und einem sehr vielseitigen Phasen-diagramm,welches unter anderemmagnetischeOrdnung und kollektiveAnregun-gen beinhaltet. Samariumhexaborid (SmB6), ein Material, das bereits seit mehr als50 Jahren untersucht wird, erhält in letzter Zeit als möglicherweise erster topolo-gischer Kondo-Isolator besondere Aufmerksamkeit. Im Gegensatz zu den meistenbisher bekannten topologischen Isolatoren ist dieses Material bei tiefen Tempera-turen im Inneren absolut isolierend, eine Eigenschaft, die besonders wichtig fürzukünftige Anwendungen ist. Ausserdem weist SmB6 weitere Besonderheiten auf,

viii | Zusammenfassung

wie z. B. mehrere topologisch geschützte Oberflächenzustände anstatt – wie beianderen topologischen Isolatoren – einem einzelnen.

In dieser Arbeit geben wir einen Überblick über die Vielzahl faszinierender Ei-genschaften von topologischen Kondo-Isolatoren und deren wechselseitige Bezie-hungen. Zu Beginn führen wir ein einfaches Modell für diese Materialklasse ein,welches auf das bekannte periodische Anderson-Modell aufgebaut ist. Es unter-scheidet sich von bisher untersuchten Modellen für (topologisch triviale) Kondo-Isolatoren vor allem durch die spezielle Form der Hybridisierung von Leitungs-elektronen und lokalisierten Zuständen. Nachdemwir verschiedenewichtige topo-logische Invarianten identifiziert und deren Zusammenhang mit Bandinversionenan denHochsymmetriepunkten diskutiert haben, wendenwir uns einer genauerenBetrachtung der damit verbundenen Oberflächenzustände zu.

Die besondere Position dieser topologisch geschützten Oberflächenzustände imImpulsraum ermöglicht zusätzlich zur von anderen topologischen Isolatoren be-kannten helikalen Spintextur eine bisher unbekannte andere Textur. Wir erklären,wie die Spintextur durch eine Windungszahl beschrieben werden kann, und stel-len einen Zusammenhang zu sogenannten Spiegel-Chern-Zahlen her. Diese sindtopologische Invarianten, die bei bestehender Spiegelsymmetrie definiert werdenkönnen und direktmit fundamentalen Eigenschaften desMaterials zusammenhän-gen; insbesondere die Art der Orbitale in der Nähe der Fermi-Energie sowie dieStruktur der Hybridisierung sind von grosser Bedeutung. Unsere Resultate ermög-lichen also Rückschlüsse über mikroskopische Materialeigenschaften, ausgehendvon experimentellen Messungen der Oberflächenzustände – z. B. durch Spin- undWinkel-aufgelöste Photoemissionsspektroskopie.

Weiterhin untersuchen wir die Auswirkungen von elektronischen Wechselwir-kungen in Kondo-Isolatoren. Der erste Effekt, den wir betrachten, ist eine Re-normalisierung von Modellparametern, welche zu einer Energieverschiebung derlokalisierten Elektronen und dadurch zu wechselwirkungsgetriebenen topologi-schen Phasenübergängen führt. In der Vergangenheit wurde zudem gezeigt, dassSmB6 und andere Kondo-Isolatoren unter gewissen Umständen instabil gegenübermagnetischer Ordnung sind. In dieser Arbeit stellen wir mehrere verschiedeneMethoden vor, mit denen magnetische Phasen untersucht werden können, und

Zusammenfassung | ix

diskutieren die Resultate von unterschiedlichen Modellen. Ausserdem wurden inKondo-Isolatoren kollektive Anregungen, sogenannte Spin-Exzitonen, nachgewie-sen; wir präsentieren diesbezüglich sowohl theoretische als auch experimentelleErkenntnisse, insbesondere in Bezug auf die Grösse dieser Zustände und derenVerhalten in der Nähe der Oberfläche des Materials.Schliesslich diskutieren wir den Zusammenhang von topologischen Invarianten

mit anderen zentralen physikalischen Konzepten, z. B. Verschränkung, Quanten-geometrie und statistischer Physik.

Table of contents

Abstract v

Zusammenfassung vii

1 Motivation 11.1 A selective history of condensed matter physics . . . . . . . . . . . 11.2 The interface of strong correlations and topology . . . . . . . . . . . 31.3 Impacts of condensed matter physics . . . . . . . . . . . . . . . . . . 4

2 Introduction to topological phases of matter 72.1 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 The quantum Hall effect and Chern insulators . . . . . . . . 72.1.2 The quantum spin Hall effect . . . . . . . . . . . . . . . . . . 82.1.3 Three-dimensional topological insulators . . . . . . . . . . . 92.1.4 Other topological systems . . . . . . . . . . . . . . . . . . . . 9

2.2 Properties of noninteracting topological systems . . . . . . . . . . . 112.2.1 Equivalence of topological phases . . . . . . . . . . . . . . . 112.2.2 Symmetry classification . . . . . . . . . . . . . . . . . . . . . 122.2.3 Surface states and bulk–boundary correspondence . . . . . . 14

2.3 The Chern number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Topological insulators with time-reversal symmetry . . . . . . . . . 17

2.4.1 Topological invariant for the 2D TI . . . . . . . . . . . . . . . 17

xii | Table of contents

2.4.2 Topological invariants in 3D TIs . . . . . . . . . . . . . . . . 192.4.3 Surface states in 3D TIs . . . . . . . . . . . . . . . . . . . . . 20

2.5 Topological crystalline insulators . . . . . . . . . . . . . . . . . . . . 212.5.1 Calculation of mirror Chern numbers . . . . . . . . . . . . . 212.5.2 Surface states in TCIs . . . . . . . . . . . . . . . . . . . . . . . 22

3 Introduction to Kondo insulators 253.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 The Kondo effect . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.2 Heavy-fermion metals and Kondo insulators . . . . . . . . . 263.1.3 Diverse properties of heavy-fermion materials . . . . . . . . 273.1.4 Samarium hexaboride . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Models for Kondo insulators . . . . . . . . . . . . . . . . . . . . . . . 303.2.1 Periodic Anderson model . . . . . . . . . . . . . . . . . . . . 303.2.2 Kondo lattice model . . . . . . . . . . . . . . . . . . . . . . . 313.2.3 Schrieffer-Wolff transformation . . . . . . . . . . . . . . . . . 32

3.3 Topological Kondo insulators . . . . . . . . . . . . . . . . . . . . . . 32

4 Outline 35

I Topology in Kondo insulators 39

5 Topological phases in Kondo insulators 415.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.1 Cubic topological Kondo insulator . . . . . . . . . . . . . . . 425.1.2 Noninteracting band structure . . . . . . . . . . . . . . . . . 445.1.3 Relation to SmB6 . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2 Topological classification . . . . . . . . . . . . . . . . . . . . . . . . . 485.2.1 Topological phases from band inversions . . . . . . . . . . . 485.2.2 Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 505.2.3 Topological invariants . . . . . . . . . . . . . . . . . . . . . . 53

5.3 Surface states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Table of contents | xiii

6 The spin texture of topological surface states 616.1 Mirror Chern numbers define pseudospin texture . . . . . . . . . . 626.2 Hybridization matrix defines mirror Chern numbers . . . . . . . . 656.3 Relation between physical spin and pseudospin . . . . . . . . . . . 686.4 Model calculations for SmB6 . . . . . . . . . . . . . . . . . . . . . . . 69

6.4.1 Calculations for the Γ8 model . . . . . . . . . . . . . . . . . . 706.4.2 Calculations for the Γ7 model . . . . . . . . . . . . . . . . . . 736.4.3 Calculations for the full model . . . . . . . . . . . . . . . . . 756.4.4 Simple model with NNN hybridization . . . . . . . . . . . . 76

II Interaction effects in Kondo insulators 79

7 Interaction-driven topological phase transitions 817.1 Mean-field treatment of Kotliar–Ruckenstein slave bosons . . . . . . 827.2 Renormalization of band parameters . . . . . . . . . . . . . . . . . . 857.3 Phase transitions and phase diagram . . . . . . . . . . . . . . . . . . 86

8 Response functions from fluctuation calculations 898.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.2 Spin-rotational invariant Kotliar–Ruckenstein slave bosons . . . . . 908.3 Mean-field solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.4 Gaussian fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.5 Spin susceptibility and spin structure factor . . . . . . . . . . . . . . 103

9 Kondo lattice model and RKKY interaction 1059.1 Schrieffer–Wolff transformation . . . . . . . . . . . . . . . . . . . . . 105

9.1.1 Derivation of the Kondo model . . . . . . . . . . . . . . . . . 1069.1.2 Real-space representation . . . . . . . . . . . . . . . . . . . . 1089.1.3 Spin–spin interaction . . . . . . . . . . . . . . . . . . . . . . . 110

9.2 RKKY interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.2.1 The Kondo lattice model . . . . . . . . . . . . . . . . . . . . . 1119.2.2 Calculation of the RKKY interaction . . . . . . . . . . . . . . 1129.2.3 RKKY interaction in real space . . . . . . . . . . . . . . . . . 116

xiv | Table of contents

9.3 Model calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169.3.1 Comparison of different hybridizations . . . . . . . . . . . . 1169.3.2 Multiple conduction bands . . . . . . . . . . . . . . . . . . . 118

10 Spin excitons in SmB6 12110.1 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12210.2 Size and magnetic moments of excitonic state . . . . . . . . . . . . . 125

III Properties of topological invariants 127

11 Entanglement spectra 12911.1 The entanglement spectrum . . . . . . . . . . . . . . . . . . . . . . . 13011.2 The sublattice entanglement spectrum . . . . . . . . . . . . . . . . . 134

11.2.1 The SLES for Chern insulators . . . . . . . . . . . . . . . . . 13611.2.2 The SLES for Z2 topological insulators . . . . . . . . . . . . . 139

11.3 Entanglement spectrum and quantum geometry . . . . . . . . . . . 14011.4 Model calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

11.4.1 The π-flux model . . . . . . . . . . . . . . . . . . . . . . . . . 14411.4.2 The Kane-Mele model . . . . . . . . . . . . . . . . . . . . . . 146

12 Scaling theory of topological phase transitions 15112.1 Scaling laws associated with Chern numbers . . . . . . . . . . . . . 15212.2 Scaling laws and RG flow in a TKI . . . . . . . . . . . . . . . . . . . 154

13 Conclusions and outlook 15913.1 The significance of the hybridization . . . . . . . . . . . . . . . . . . 15913.2 Interplay of topology and interactions . . . . . . . . . . . . . . . . . 16113.3 Unconventional spin texture of surface states . . . . . . . . . . . . . 16213.4 Magnetic phases and collective excitations . . . . . . . . . . . . . . . 16313.5 Properties of topological invariants . . . . . . . . . . . . . . . . . . . 16413.6 Real materials and applications . . . . . . . . . . . . . . . . . . . . . 165

Table of contents | xv

IV Appendices 167

A Topological invariants and surface states 169A.1 Calculation of surface states . . . . . . . . . . . . . . . . . . . . . . . 169A.2 The k·p theory on the (110) surface . . . . . . . . . . . . . . . . . . . 171A.3 Mirror operators and sign choice of the mirror Chern numbers . . . 173A.4 Spin operator for f orbitals . . . . . . . . . . . . . . . . . . . . . . . . 175

B Details of slave-boson calculation 177B.1 Constraints in the spin-rotational-invariant slave-boson representation 177B.2 Correct mean-field results in the noninteracting limit . . . . . . . . 185B.3 Maximization with respect to µ0 and β0 . . . . . . . . . . . . . . . . 185B.4 Slave-boson number operator . . . . . . . . . . . . . . . . . . . . . . 186B.5 Calculation of derivatives in momentum space . . . . . . . . . . . . 187B.6 Derivation of Equation (8.35) . . . . . . . . . . . . . . . . . . . . . . 188B.7 Details of Matsubara summations . . . . . . . . . . . . . . . . . . . . 190

C Dispersion of excitonic states 193C.1 Ansatz for the excitonic state . . . . . . . . . . . . . . . . . . . . . . . 194C.2 Matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195C.3 Energy and size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197C.4 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Bibliography 199

List of acronyms 215

List of materials 217

List of publications 219

Curriculum vitæ 221

Acknowledgments 223

Chapter1

Chapter 1

Motivation

1.1 A selective history of condensed matter physics

Already at the beginning of the 19th century, first attempts to understand thestructure and properties of materials were made by Humphry Davy, which mightbe considered the first studies of condensed matter physics [1]. A first theoreticalmodel for the movements of electrons through a solid was developed by PaulK. L. Drude in 1900 [2]. Shortly thereafter, Heike Kamerlingh Onnes, using hisnewly developed method to liquefy helium, discovered superconductivitywhen theresistivity in mercury dropped to zero below the critical temperature of 4.2 K [3].At that time, the inner structure of materials was still largely unknown. In 1912,Walter Friedrich, Paul Knipping, and Max von Laue discovered the effect of x-raydiffraction [4], a tool that was subsequently used byWilliam H. Bragg andWilliamL. Bragg to study materials in detail and confirm their crystal structure [5].However, only after the development of quantum mechanics, modern many-

body theory led to a satisfactory theoretical understanding of many propertiesof condensed matter. In 1929, Felix Bloch laid the foundation for the immenselysuccessful band theory of solids [6]: Electronic states form energy bands, whichare then filled successively with electrons while respecting the Pauli exclusionprinciple [7]. The structure and filling of these energy bands determines manyproperties of the respective materials. For example, if there is a finite energy gapbetween the highest occupied and the lowest unoccupied state, the system is aninsulator or semiconductor, while in the opposite case we speak of a metal. Many

2 | 1 Motivation

questions of condensed matter physics could already be solved in these early yearsof quantum mechanics [8].

Band theory is inherently a single-particle theory that neglects the (Coulomb)interaction among electrons. A justification for applying this strongly simplifiedapproach to metals has been given much later by Lev D. Landau through hisFermi-liquid theory, which he proposed in 1956 [9]. In the presence of (not neces-sarily small) interactions, electrons form noninteracting quasiparticles, which havewell-defined momentum and spin but renormalized properties. Consequently, thesingle-particle approach is legitimate as long as the Fermi-liquid theory is applica-ble.

For many interesting phenomena that are found in materials, however, interac-tions play an essential role and therefore band theory alone cannot provide a fulldescription. A first understanding of magnetic phases at a microscopic level wasobtained by Ernst Ising for ferromagnetism [10] and Louis E. F. Néel for antiferromag-netism [11] in the first half of the 20th century. On a phenomenological level, phasetransitions have been described by Lev D. Landau using the concept of spontaneoussymmetry breaking [12]. In this theory, an underlying symmetry of a system can bespontaneously broken and the strength of this symmetry-breaking is described by alocal order parameter. This theory has been extended to describe superconductivityby Vitaly L. Ginzburg and Lev D. Landau [13].

Later, almost 50 years after the discovery of superconductivity, a first completemicroscopic description of this phenomenonwas developed by John Bardeen, LeonN. Cooper, and John Robert Schrieffer [14]. In their theory, electrons form Cooperpairs via an attraction produced by lattice deformations, which can then condenseandmove through thematerial without scattering. TheKondo effect, experimentallyfirst observed in 1934 [15], remained for three decades until it was studied theoreti-cally by Jun Kondo in 1964 [16]. In 1972, PhilipW. Anderson explained the relationof many-body physics and single-particle physics in his article on emergence, enti-tled “More is different” [17]. Even today, there remain numerous unsolved problemsin strongly-correlated systems, e.g., a full understanding of high-temperature su-perconductivity, which was first observed in 1986 by Johannes G. Bednorz and KarlA. Müller [18].

Chapter1

1.2 The interface of strong correlations and topology | 3

Superconductors also provided one of the first appearance of topology in con-densed matter physics, when Alexei A. Abrikosov studied vortices in supercon-ductors in 1957 [19]. These vortices are topological defects carrying a quantizedmagnetic flux and had been discussed earlier by Lars Onsager and Richard P. Feyn-man in the context of superfluids [20, 21]. As discovered at the beginning of the1970s by Vadim L. Berezinskii, John M. Kosterlitz, and David J. Thouless, vor-tices also appear in the Kosterlitz–Thouless transition of spin systems [22, 23]. JohnM. Kosterlitz and David J. Thouless use the term “topological order” to describe thisnew type of phases.In 1980, the field of topological phases of matter was extended with the obser-

vation of the quantum Hall effect by Klaus von Klitzing [24, 25]. Already two yearslater, the precise quantization of the Hall conductivity was related to a topologicalinvariant, the Thouless–Kohmoto–Nightingale–den Nĳs invariant [26], which was lateridentified as the Chern number [27–29]. At the same time, the fractional quantumHall effect, an interaction effect with topological order, was first observed by DanielC. Tsui, Horst L. Strömer, and Arthur C. Gossard [30] and subsequently explainedby Robert B. Laughlin [31].The field of noninteracting topological phases advanced through the theoreti-

cal models for topological systems without external magnetic field [32] and withtime-reversal symmetry [33, 34]. In subsequent years, this field received increasingattention, with ever more and new topological phases of matter to be discovered.These discoveries revealed that symmetries and local order parameters are notsufficient to define different phases of matter, as topological phases are indistin-guishable using these criteria. In addition, topological insulators create a bridgebetween the distinct classes of insulators and metals, as they are insulating in thebulk but possess conducting states on the surface [25].

1.2 The interface of strong correlations and topology

Today, many of the actively studied problems in condensed matter physics arerelated to either strongly-correlated physics or topology. Of particular interestare systems in which both aspects play a significant role, such that correlationeffects have been studied for many different topological phases [35]. Important

4 | 1 Motivation

examples include fractional topological insulators [36], which are generalizations ofthe fractional quantum Hall effect, and topological superconductors [25].

In 2010, a new class of materials emerged, when Maxim O. Dzero, Kai Sun,VictorM. Galitski, and Piers Coleman suggested that certain Kondo insulatorsmightactually be topological insulators [37]. These topological Kondo insulators constitutean ideal platform to study both correlations and topology in the samematerial, andhave attracted much attention from both theoretical and experimental physicists inrecent years [38].

One material in particular, SmB6, has been studied extensively both theoreticallyand experimentally. Since the first studies 50 years ago [39, 40], a rich set ofunique and unexpected features has been discovered in this material, includingintermediate valence [41], spin excitons [42], andmagnetism [43]. Using the theoryof topological Kondo insulators, some previously mysterious properties could beexplained satisfactorily, such as the resistivity plateau at low temperatures [44].

This set of diverse properties make the material interesting for fundamentalresearch as well as for possible applications, where its truly insulating bulk is ofparticular relevance. However, in order to develop its full potential, it is crucial tofully understand the effects and interplay of strong correlations and topology inthis material.

1.3 Impacts of condensed matter physics

Over the last century, condensed matter physics has had a tremendous impacton both natural sciences and society through numerous inventions and technolo-gies; many researchers involved in these discoveries were later awarded the Nobelprize [45]. The transistor, invented in 1947 by John Bardeen, Walter H. Brattain, andWilliam B. Shockley, is one of the core components of all modern electronics andarguably constitutes one of the most important inventions in the past century [45].The storage of data, in particular on hard disc drives, relies heavily on the under-standing of magnetic materials. Superconductors are now used to produce strongmagnetic fields for applications in, e.g., medicine (magnetic resonance imaging),transportation (maglev trains), or research (particle accelerators, fusion reactors,nuclear magnetic resonance) [45]. Lasers, first hypothesized by Albert Einstein

Chapter1

1.3 Impacts of condensed matter physics | 5

in 1917 [47], are now ubiquitous throughout the world and integrated in variousmodern devices, as are light-emitting diodes [45, 48]. These are but a few examplesof the long list of applications enabled by research in condensed matter physicsthat have fundamentally affected technology and society during the last century.Considering the large delay between fundamental discoveries and their appli-

cations, it is very difficult to predict future impact of current fields of research incondensedmatter physics. However, some topics have been identified as promisingcandidates for future technological developments: Due to the suppression of scat-tering at the boundaries of topological insulators, these materials could be used toconstruct electronic devices with very low power consumption. Another possibleapplication related to topological insulators is the field of spintronics [49]; these aredevices that make use of the electronic spin in addition to its charge are expectedto eventually replace current electronics in the future. Topological insulators arepromising candidate materials due to the spin-polarization of their protected sur-face states [25, 50, 51]. In this context, topological Kondo insulators may be ofparticular relevance due to the possibly unusual spin texture that may occur inthese materials and is studied in this thesis.In 2008 Liang Fu and Charles L. Kane proposed a setup of a topological insulator

in contact with a superconductor in order to create non-Abelian anyons, whichmight be used to create topologically protected quantum bits and enable topologicalquantum computation [52–54]. Other suggested applications of topological insu-lators include the construction of terahertz emitters/detectors [55] and catalysis inchemistry [56].

In this chapter, we have presented multiple different motivations for studyingtopological Kondo insulators such as SmB6. They are of profound interest fromthe point of view of fundamental research due to their interplay of two importantphysical concepts, namely topology and strong electronic correlations. In addition,these fundamental properties may be of immense significance in the context ofpossible future applications where topological insulators and SmB6 in particularmay play a central role.

Chapter2

Chapter 2

Introduction to topological phases of matter

2.1 Historical overview

Topological phases are a relatively new field of condensed matter physics (CMP)and have only been studied since the 1980s. In the following we will review howthe discovery of the quantumHall effect (QHE) and Chern insulators eventually led tothree-dimensional (3D) topological insulators (TIs) andmanymore topological phaseswith unique properties.

2.1.1 The quantum Hall effect and Chern insulators

In 1980, while performing measurements on a silicon metal-oxide-semiconductorfield-effect transistor at low temperature and in strong magnetic fields, Klaus vonKlitzing made an astounding discovery: The Hall conductance did not increasecontinuously but showed plateaus at very precise integer multiples of e2/h, wheree is the elementary charge and h the Planck constant [24]. While these plateaushad already been predicted by Tsuneya Ando, Yukio Matsumoto, and YasutadaUemura in 1975 [57], the unexpected precision of the values and their insensitivitytowards the sample geometry were remarkable and could not be explained at first.Ayear later, Robert B. Laughlin proposed an argument based ongauge invariance

to explain this exact quantization [58]. Shortly thereafter, a first connection totopology was made by David J. Thouless, Mahito Kohmoto, M. Peter Nightingale,and Marcel den Nĳs through their Thouless–Kohmoto–Nightingale–den Nĳs (TKNN)invariant [26]. This invariant was written as an integral over the crystal momentumk in the Brillouin zone (BZ) mapped to the Bloch states of the system. It was

8 | 2 Introduction to topological phases of matter

shown that this integer invariant coincides with the number of chiral edge states,see Figure 2.1 (a), and could only change when the energy gap closes, therebyexplaining the exact quantization of the plateaus of the QHE. Later, it becameapparent that the TKNN invariant is the first Chern number that had been studiedpreviously by Shiing-Shen Chern in the context of complex vector bundles [27–29].

A significant progress was made in 1988, when F. Duncan M. Haldane proposeda model on the honeycomb lattice with nontrivial Chern numbers, which did notrequire any external magnetic field [32]. This effect became known as the quantumanomalous Hall effect (QAHE). For 25 years, Chern insulators, the class of materialsexhibiting the QAHE, were only a theoretically considered possibility. Only in 2013,Cui-Zu Chang, Jinsong Zhang, Xiao Feng, Jie Shen et al. finally reported the firstexperimental observation of the QAHE in a magnetic TI.

2.1.2 The quantum spin Hall effect and two-dimensional topologicalinsulators

While Chern insulators, such as the model proposed by F. Duncan M. Haldane,do not require any external magnetic field, time-reversal symmetry (TRS) has to beintrinsically broken. In 2005, CharlesL.KaneandEugene J.Mele aswell asB.AndreiBernevig and Shou-ChengZhangproposedmodels of two-dimensional (2D) systemwith TRS and strong spin–orbit coupling (SOC) exhibiting the quantum spin Hall effect(QSHE) [34, 59]. This effect can be understood intuitively as “two copies” of theHaldane model, one for spin up and one for spin down, with opposite Chernnumbers. This construction leads to counterpropagating edge states with oppositespin, so-called helical edge states, see Figure 2.1 (b).In a seminal paper, Charles L. Kane and Eugene J. Mele showed that one pair

of helical edge states is protected even when spin conservation is broken, e.g., inthe presence of Rashba-type SOC [33]. Instead of the Chern number, which couldtake any integer value (Z invariant), this phase is characterized by a Z2 topologicalinvariant that can only take two values: ν 0, corresponding to a trivial state, andν 1, corresponding to the topological phase. In order to set them apart from theQSHE that requires spin conservation, these topological systems are usually called2D TIs.

Chapter2

2.1 Historical overview | 9

The theoretical discovery of theQSHE lead to increased research efforts in the fieldof topological phases of matter. In 2006, B. Andrei Bernevig, Taylor L. Hughes, andShou-Cheng Zhang proposed their famous Bernevig–Hughes–Zhang model formercury telluride (HgTe) quantum wells exhibiting the QSHE [60]. A prediction ofedge states in this system was already made in 1987 by Oleg A. Pankratov, SergueiV. Pakhomov, and B. A. Volkov [61]. In 2007, Markus König et al. reported the firstexperimental realization of the QSHE [62].

2.1.3 Three-dimensional topological insulators

All systems showing the QHE, QAHE, or QSHE that were considered so far, are 2D

systems. However, in 2006 multiple groups independently proposed a generaliza-tion of the 2D TI to three dimensions [63, 64]. Soon afterwards, several bismuth andantimony compounds were predicted to be 3D TIs [65, 66]. In 2008, angle-resolvedphotoemission spectroscopy (ARPES) measurements by David Hsieh et al. showed thetopological surface states of bismuth antimony (Bi1−xSbx), confirming thismaterialas the first 3D material with nontrivial topology [67]. In subsequent years, manymore compounds, including bismuth selenide (Bi2Se3), bismuth telluride (Bi2Te3),and antimony telluride (Sb2Te3), have been shown to be TIs [25, 68]. Most of thesematerials are weakly correlated semiconductors with heavy elements that providestrong SOC.

2.1.4 Other topological systems

BesideChern insulators and 2D/3D topological insulatorswith TRS, there existmanymore topological systems that have been studied in recent years. We will give ashort overview over these classes of materials in the following, although they haveless connection to the remainder of this thesis.

Fractional topological insulators and topological order

The fractional quantum Hall effect (FQHE), experimentally discovered in 1982, is afundamentally different system than the integer quantum Hall effect [30, 31]. In apartially filled Landau level, interactions between electrons lead to quasiparticles


with fractional electric charge as well as fractional statistics [69]. This is relatedto plateaus in the Hall conductivity at fractional multiples of e2/h, which wasthe initial experimental observation [30]. After the discovery of noninteractingtopological systems without external magnetic field, states analogous to the FQHE

have also been found there in the form of fractional Chern insulators and fractionaltopological insulators [36].

Collective states such as the FQHE are called “topologically ordered” in contrastto noninteracting topological phases. They exhibit long-range entanglement and arobust degeneracy of the ground state [70, 71]. While these topologically orderedphases are unique condensed-matter systems and may be utilized as buildingblocks for future topological quantumcomputers, theywill not be discussed furtherin this thesis.

Topological superconductors and Majorana fermions

Superconductors can be topologically classified analogously to band insulatorswhen adopting the Bogoliubov–de Gennes formalism [25, 72]. The formalismguarantees that the system has intrinsic particle–hole symmetry (PHS) leading todifferent topological phases with remarkable features. In topological superconductors(TSCs), the bulk–boundary correspondence manifests itself via the appearance ofgaplessMajorana states at the boundary, which may be used to perform topologicalquantum computation as suggested by Alexei J. Kitaev [52, 73, 74]. A naturalexample for TSCs are chiral p-wave superconductors, themost prominent candidatematerial of this class being strontium ruthenate (Sr2RuO4) [75].

Liang Fu and Charles L. Kane suggested the artificial creation of such a TSC byinducing superconductivity via the proximity effect on the surface of a 3D TI [53].The first experimental signature of Majorana fermions was observed in 2012 in aone-dimensional system built of nanowires [76].

Weyl semimetals

Topological effects are not restricted to systems with energy gaps (insulators orsuperconductors). Recently, Weyl semimetals, a new class of topological materialswhere the low-energy excitations are described by the Weyl equation, have been

Chapter2

2.2 Properties of noninteracting topological systems | 11

discovered [77–79]. These materials show unique properties such as Fermi arcs onthe surface, which lead to novel transport properties. In 2014, multiple groupspredicted the material tantalum arsenide (TaAs) to be a Weyl semimetal [80, 81],which was confirmed experimentally in 2015 [82].

Floquet topological insulators

Asmaterialswith intrinsic topological properties are rare, many research efforts aredevoted to engineering artificial systems with the desired properties. One routethat has been considered in recent years are Floquet topological insulators, wheretime-periodic driving of topologically trivial systems is used to create effectiveHamiltonians with nontrivial topology [83–86]. Some experimental evidence forthis approach to creating topological phases has been shown in photonic systems,on the surfaces of TIs, and in optical lattices of cold-atoms [87–89].

2.2 Properties of noninteracting topological systems

After Lev D. Landau had introduced his general theory of phase transitions [12],it was believed that different phases of matter could always be distinguished bytheir different symmetry and a locally defined order parameter. This view had to bechanged radically after the emergence of topological phases of matter. Differenttopological phases cannot be distinguished by any local observable, similar to thefact that with a local measurement it is impossible to tell the difference betweena (topological) sphere or torus. Instead, the topological invariants differentiatingbetween phases are calculated from the global properties of the ground state of thesystem.

2.2.1 Equivalence of topological phases

As discussed in Section 2.1.4, topological phases come in many forms. For thefollowing discussion and the rest of the thesis, we will only consider systems withan energy gap between different electronic bands, i.e., band insulators (BIs). For thesesystems, two phases are said to be topologically equivalent if they can be connected byadiabatically changing the parameters of the underlying Hamiltonian, i.e., without


closing the energy gap.1 Otherwise, they belong to different topological classesand are characterized by different values of the topological invariants.

The allowed adiabatic transformations can be restricted to complywith an under-lying symmetryof the system, e.g., TRS. In this case, otherwise equivalentphases canbe separated into different topological classes and we speak of symmetry-protectedtopological (SPT) phases [90]. This definition of topological phases can be summa-rized as the following statement: Topological phases are characterized by topolog-ical invariants belonging to a discrete set of numbers that can only change when(i) the gap separating the bulk energy bands closes or (ii) the protecting symmetryof the SPT phase is broken. Therefore, assuming any protecting symmetry remainsunbroken, topological phase transitions are only possible by closing the bulk energygap.

2.2.2 Symmetry classification

While, on the one hand, symmetries can protect topological phases, they can alsoimpose certain constraints on the electronic states that can cause topological invari-ants to vanish; for example, in the presence of TRS, the Chern number of a systemnecessarily vanishes, as we will discuss below. Also, many topological invariantscan only be defined in certain dimensions. This means that the symmetries ofthe Hamiltonian as well as the dimensionality of the system determine whethertopological phases are possible or not.

There exist two global antiunitary symmetries, time-reversal symmetry (TRS) andparticle–hole symmetry (PHS), represented by the operators T and P, respectively.According to the presence or absence of these, as well as chiral symmetry C TP,noninteracting fermionic systems can be categorized in ten different symmetryclasses.2 It has been shown by Alexander Altland and Martin R. Zirnbauer in 1997that these classes correspond to the classes of symmetric spaces in differential ge-ometry introduced by Élie J. Cartan [91]. In 2008, Andreas P. Schnyder, Shinsei Ryu,Akira Furusaki, and Andreas W. W. Ludwig used this Altland–Zirnbauer (AZ) clas-sification to identify all symmetry classes with possible topological phases in three1The transformation also needs to be local in the sense that it is generated by a local Hermitian operator.2Note that a Hamiltonian can have chiral symmetry even in the absence of both TRS and PHS. This is classAIII in the classification discussed here.

Chapter2

2.2 Properties of noninteracting topological systems | 13

Table 2.1: Periodic table of topological phases according to References [93] and [95]. The firstcolumn is the name of the symmetry class according to the AZ classification. The nextthree columns describe the absence () or presence (±1) of TRS (T ), PHS (P), and chiralsymmetry (C). When the symmetry is present, the value ±1 denotes the square of thesymmetry operator. The last eight columns show, in which dimension topological phaseswith integer (Z) and Z2 invariants are possible. The most important symmetry classes forthis thesis are printed in bold (A and AII).

symmetry dimension

AZ T 2 P2 C2 1 2 3 4 5 6 7 8

A – ZZZ – Z – Z – Z

AIII 1 Z – Z – Z – Z –

AII −1 – Z2Z2Z2 Z2Z2Z2 Z – – – Z

DIII −1 1 1 Z2 Z2 Z – – – Z –D 1 Z2 Z – – – Z – Z2

BDI 1 1 1 Z – – – Z – Z2 Z2

AI 1 – – – Z – Z2 Z2 Z

CI 1 −1 1 – – Z – Z2 Z2 Z –C −1 – Z – Z2 Z2 Z – –CII −1 −1 1 Z – Z2 Z2 Z – – –

dimensions [92]. Shortly afterwards, the same group as well as Alexei J. Kitaevrealized that, when considering arbitrary dimensions, the topological classifica-tion leads to a repeating pattern that has been called periodic table of topologicalphases [93–95]; see Table 2.1 for full details.Classes with PHS (P2 ±1) are generally superconductors, where this symmetry

appears naturally and whose topological features we discussed briefly above. Thefour remaining classes A, AI, AII, and AIII are TIs. Themost important ones for thisthesis are classes A and AII with topologically nontrivial phases in two dimensionsand two or three dimensions, respectively. The 2D/3D TIs discussed above belong toclass AII according to the AZ classification.3 Due to their Z2 topological invariant,they are usually referred to as “Z2 topological insulators”, whichwewill also use inthis thesis. Systems exhibiting the QAHE belong to class A without any underlying

3Note that the QSHE requires spin conservation in addition to TRS.


symmetries (besides global particle number conservation) and are called “Cherninsulators”. An example of a topological system belonging to symmetry class AIIIis the Su–Schrieffer–Heeger model. In this thesis, we will use the term “topologicalinsulator (TI)” in a general sense to include all insulators with nontrivial topologyof symmetry classes A, AI, AII, and AIII (in particular including Chern insulators).

2.2.3 Surface states and bulk–boundary correspondence

One of the most central properties of TIs are their surfaces states4. While tradi-tionally materials could be divided into insulators and conductors by the presenceor absence of a gap in the energy spectrum, this notion is more complicated fortopological systems. In a TI, the bulk of the material is insulating due to the energygap separating the valence and conduction bands. However, on the surface of thematerials there exist topologically protected surface states (TSSs) connecting the valenceand conduction band, which close the energy gap and make it conducting.

The appearance of gapless surface states has a very intuitive explanation whenconsidering the definition of topological invariants given above: By definition, thevacuum is topologically trivial, whereas there is a nontrivial invariant in the bulk ofa TI. As stated above, topological invariants can only change when the energy gapcloses or the protecting symmetry is broken. Therefore, at any (spatial) interfacewith different topological invariants that does not break the protecting symmetry,the energy gap must close, which creates the TSSs.

This relationship between topological invariants, which are properties of thebulk, and the TSSs, which exist on the surface, is known as bulk–edge correspondenceor bulk–boundary correspondence and has been considered in different forms since the1980s [25, 96]. While, in the most general form, it does not predict any propertiesof the TSSs beyond their existence, it is a central property of all noninteractingtopological systems. A comparison of edge states of the QHE and the QSHE ispresented in Figure 2.1.

4Here, we use “surface” in the general sense of “boundary”, meaning the end points in one, the edges intwo, and the surface in three dimensions.

Chapter2

2.3 The Chern number | 15

E

k

(a) The number of edge states is given by the Chern number for the Chern insulator,here C 2.

↑↓↓

↑↓

↑

E

k

(b) Helical edge states in the QSHE. In a 2D TI with Rashba-type SOC, i.e., withoutspin conservation, a single pair of helical edge states remains protected in thetopological state, ν 1, but in general with a more complicated spin structure.

Figure 2.1: Comparison of a Chern insulator (a) and the QSHE (b) showing chiral and helicaledge states, respectively, in real (left) and momentum space (right).

2.3 The Chern number

The Chern number of a 2D system is the first topological invariant to be discussedin the context of CMP as the TKNN integer [26]. It can most easily be understood asan analogon to the genus of a compact 2D Riemannian manifold without bound-ary [97]: According to the Gauss-Bonnet theorem, the Gaussian curvature FG(x) of themanifold M (a geometric quantity) can be related to its genus g (an integer valueand topological invariant) via

2(1 − g) 12π

∫M

d2x FG(x) . (2.1)

Here, the Gaussian curvature is calculated by considering parallel transport oftangential vectors and describes the change of the tangential space of the manifold.In CMP, the analogon to the genus g is the Chern number C, the manifold M

is replaced by the BZ, and the Gaussian curvature FG(x) by the Berry curvature


F (k) [98]:C

12π

∫BZ

d2k F (k) . (2.2)

The Berry curvature itself is related to the fiber bundle of Bloch states as a functionof the crystal momentum in a similar way as the Gaussian curvature is related tothe tangent bundle of the manifold. It can be calculated as the sum of the Berrycurvatures of the a ≤ N occupied energy bands,

F (k) N∑

a1Fa(k) , (2.3a)

Fa(k) ∂Aa ,2(k)∂k1

− ∂Aa ,1(k)∂k2

, (2.3b)

where the Berry connection A is defined as

Aa(k) i 〈ua(k) | ∇k | ua(k)〉 . (2.4)

Here, |ua(k)〉 is the Bloch state at crystal momentum k in band a, using bra–ket notation [99]. A numerical method for computing Chern numbers using adiscretized BZ has been developed [100]. For a two-band model of the form

H(k) d(k) · σ , (2.5)

where σ (σ1 , σ2 , σ3), the Berry curvature can be calculated as

F (k) 12 d ·

(∂d∂k1× ∂d∂k2

), (2.6)

with d d/||d ||.Different conventions for the phase of A as well as the sign of F and C are

possible; for example, the convention used in Reference [100] differs from the onepresented in this thesis by a global minus sign. For the Berry curvature (andconsequently the Chern number) to be well-defined, the occupied and unoccupiedenergy bands must be separated by a finite energy gap. Note that, while the Berrycurvature is a gauge-invariant quantity, the Berry connection depends on the phase

Chapter2

2.4 Topological insulators with time-reversal symmetry | 17

choice for the Bloch states. Also, note that in the calculation presented here, we usetranslational invariance only in the interest of convenience; it is not a condition forthe definition of the Chern number.

2.4 Topological insulators with time-reversal symmetry

According to the periodic table, 2D or 3D systems with TRS belonging to class AIIin the AZ notation can host topological phases characterized by Z2 invariants. Inthe following, we will discuss the construction and calculation of these topologicalinvariants as well as their implications for the existence of surface states.

2.4.1 Topological invariant for the 2D TI

We have shown above, how the Chern number, the topological invariant for 2D

systems belonging to class A in the AZ notation, is defined. In the presence of TRS,the Hamiltonian fulfills

T h(k)T −1 h(−k) , (2.7)

where the operator T is antiunitary and fulfills T 2 −1 (which also leads toT † T −T −1). This implies that the Berry curvature is an odd function of thecrystal momentum,

F (k) −F (−k) , (2.8)

which, due to the integration over thewhole BZ, leads to a vanishingChern number,C 0.For the QSHE it is possible to calculate Chern numbers separately for spin up

and spin down due to spin conservation; these invariants are called spin Chernnumbers and are similar to the mirror Chern numbers discussed below. As theChern numbers for opposite spins have different signs, their sum vanishes and theconstraint imposed by TRS is fulfilled.This procedure breaks down as soon as spin-nonconserving processes are in-

troduced, e.g., Rashba-type SOC. However, TRS allows for the construction of adifferent topological invariant by considering overlaps of Bloch states with their


time-reversal partners,

ma ,b(k) : 〈ua(k)| T |ub(k)〉 , (2.9)

with a and b labeling occupied bands. This is an antisymmetric matrix, whichmotivated Charles L. Kane and Eugene J. Mele to analyze its Pfaffian [33]

p(k) : Pf [m(k)] . (2.10)

At the four time-reversal-invariant momenta (TRIM) k Γi ∈ 0, π2 , i 1, . . . , 4,where −Γi Γi + G for a reciprocal lattice vector G and therefore

T h(Γi)T −1 h(Γi) (2.11)

in the presence of TRS, there is a two-fold degeneracy of all bands, called Kramer’sdegeneracy. The subspace spanned by occupied Bloch states is the same as that oftheir time-reversal partners, which implies

p(Γi) 1.

It is possible that there exist zeros p(±k∗) 0 of the Pfaffian at time-reversal pairs±k∗. Considering continuous transformations of the Hamiltonian, these zeros canonly vanish if two of these points meets. However, if there is only a single pair ofzeros, they would have to meet at one of the TRIM where

p(Γi) 1. Therefore, the

parity of pairs of zeroes of the Pfaffian

ν n02 mod 2 (2.12)

is a Z2 topological invariant that distinguishes between the trivial and topologicalphase of the 2D TI [33].5

A simpler formula for ν can be obtained by considering instead

wa ,b(k) : 〈ua(k)| T |ub(−k)〉 , (2.13)

5If the set of zeros of p(k) is not discrete, ν can be expressed as the winding number of the phase of p(k)along a time-reversal invariant path γ enclosing half the BZ [33].

Chapter2

2.4 Topological insulators with time-reversal symmetry | 19

which coincides with the matrix ma ,b defined in Equation (2.9) at the four TRIM.There, it is antisymmetric and we can define

δi Pf(w(Γi))√det(w(Γi))

±1 . (2.14)

It can be shown, that the Z2 invariant of the 2D TI can then be calculated as [101]

(−1)ν

4∏i1

δi . (2.15)

The calculation can be further simplified in the presence of inversion symmetryin addition to TRS, as shown by Liang Fu andCharles L. Kane [65]. In that case, eachKramer’s pair (labeled by a) at the four TRIM has a well-defined parity ξi ,a ±1and the Z2 invariant is given by

(−1)ν

4∏i1

∏aξi ,a , (2.16)

where the index a runs over all occupied Kramer’s pairs.

According to the bulk–boundary correspondence, the topological invariant mustbe accompanied by gapless edge states. In the QSHE and 2D TIs, these differ fromthose in the QHE in that there are pairs of counterpropagating, so-called helical edgestates, see Figure 2.1. Note that, while in the QHE and QSHE an arbitrary numberof edge states can occur, at most one pair of edge states is protected by TRS in the2D TI.

2.4.2 Topological invariants in 3D TIs

We have seen above that there should also exist nontrivial topological phases in3D systems of class AII. Indeed, it is possible to generalize the results and thetopological invariant of the 2D TI to three dimensions [63, 64]. In contrast to twodimensions with a single topological invariant ν, a 3D TI is characterized by fourZ2 invariants (ν0; ν1 , ν2 , ν3). The so-called weak topological invariants νi (i 1, 2, 3)correspond to stackingof 2DTI layers and require translational invariance. However,


the strong topological invariant ν0 is a true 3D invariant and is only protected by TRS,similar to the invariant ν in two dimensions.

In the presence of inversion symmetry, Equation (2.16) can be generalized tothree dimensions, where there are eight TRIM Γi ∈ 0, π3. The strong topologicalinvariant is then given by

(−1)ν0

8∏i1

∏aξi ,a , (2.17a)

where the second product runs over all occupied Kramer’s pairs and ξi ,a againis the inversion eigenvalue of the Kramer’s pair a at k Γi . Similarly, the weakindices νi are defined by a product of parities on the planes ki π:

(−1)νi

∏j;kiπ

∏aξ j,a . (2.17b)

If the system is not inversion symmetric, there exist other ways to calculate thetopological invariants, which are more complicated [25].

2.4.3 Surface states in 3D TIs

According to the bulk–boundary correspondence, the Z2 invariants (ν0; ν1 , ν2 , ν3)imply gapless boundary states on all or only certain high-symmetry surfaces. In astrong topological insulator (STI), where ν0 1, an odd number of TSSs is protected inthe surface Brillouin zone (SBZ). Unlike 2D systems such as graphene, these surfacestates are Dirac cones which are spin-polarized, an effect called spin–momentumlocking. This is unique to TIs and only possible due to the fact that the surface is theboundary of a 3D system. The fermion-doubling theorem, proven byHolger B. NielsenandMasaoNinomiya in 1981, states that, in two dimensions, chiral (spin-polarized)Dirac cones can only appear in pairs [78, 102]. This still holds for 3D TIs; however,there these cones appear at two opposite surfaces such that they are separated bya macroscopic distance in real space [25].

Strictly speaking, in the presence of disorder, only the strong topological invari-ant ν0 iswell-defined, since its definition only requires TRS and charge conservation.However, it has been shown that both the weak Z2 invariants as well as other topo-

Chapter2

2.5 Topological crystalline insulators | 21

logical invariants protected by crystal symmetries (see below) retain their meaningif the disorder is sufficiently small and preserves the protecting symmetries onaverage [103, 104]. In this case, also the weak topological invariants are meaningfuland a Dirac cone is present at all (high-symmetry) points in the SBZ onto which anodd number of bulk inverted high-symmetry points (HSPs) is projected [65, 105].

2.5 Topological crystalline insulators

In addition to the discrete symmetries discussed in Section 2.2.2, it has been shownthat alsopoint-group symmetries such as rotational andmirror symmetries can leadto SPT phases called topological crystalline insulators (TCIs) [106, 107]. If the point-group symmetry containsmirror operations, it is possible to define new topologicalinvariants called mirror Chern numbers (MCNs), which protect gapless surface statessimilarly to Chern numbers and the Z2 invariants [106]. Their definition andconsequences for a 3D system with TRS are discussed in the following.

2.5.1 Calculation of mirror Chern numbers

There is one plane in momentum space that is invariant under the mirror operationfor each mirror plane in real space.6 Wewill write the mirror operator as M IC2,where I is the inversion operator and C2 describes a rotation by π about an axisperpendicular to the mirror plane. Because M2 −1 for spin-1/2 particles, theBloch states in a mirror plane can be chosen to have eigenvalues ±i under themirror operation,

Mu±a (k)⟩ ±i

u±a (k)⟩ , (2.18)

where the momentum k lies in this mirror plane. There, we can define the Berryconnection of the band with mirror eigenvalue ±i,

A±a (k) i

⟨u±a (k)

∇k

u±a (k)⟩, (2.19)

6Note that one mirror operation can have multiple mirror planes in real space leading to multiple mirror-invariant planes in momentum space.


and the corresponding Berry curvature

F ±a (k) ∂A±a ,2(k)∂k1

−∂A±a ,1(k)∂k2

, (2.20)

where ∇k denotes the gradient in the mirror plane and we again follow the con-ventions of Reference [100].

Then, we can define theMCN associated with a particular mirror operation as theChern number C+ of the occupied states with eigenvalues +i:

C+

12π

∫BZ

d2k F +(k) . (2.21)

Here, BZ is the mirror-invariant plane (MIP) in the BZ and F + is the sum of the Berrycurvatures of all occupied bands with mirror eigenvalue +i. Note that the totalChern number C C+ + C− necessarily vanishes due to the constraints of TRS (seeSection 2.4) such that C− −C+. Therefore, we could have also chosen C− to definethe MCN, leading to an additional global sign.

2.5.2 Surface states in TCIs

According to the bulk–boundary correspondence discussed above, topological in-variants defined in the bulk are related to gapless states on the boundary of thesystem. A nonzero MCN implies that at least |C+ | Dirac cones exist along thehigh-symmetry line (HSL) in the SBZ, which is invariant under the respective mirroroperation [108]. The sign of the MCN is called the mirror chirality [106, 108]. It de-termines the mirror eigenvalue (±i) of the surface bands crossing from the valenceto the conduction band in the positive direction of the respective HSL. If, e.g., theMCN is C+ +2, two bands with eigenvalue +i cross from valence to conductionband, see Figure 2.2.

The positive direction is defined by the orientation of the mirror-invariant planein the calculation of the MCN and the chosen surface as follows. First, in orderto calculate the MCN, a right-handed coordinate system with a unit vector nmpperpendicular to the mirror-invariant plane needs to be defined. Second, for thesurface, we define nsf as the outward pointing normal vector. Then, the positive

Chapter2

2.5 Topological crystalline insulators | 23

+i

−i+i

−i +i

−i+i

−i

E

k

(a) C+ 2.

+i

−i+i

−i

E

k

(b) C+ −1.

Figure 2.2:Gapless surface states (blue) connecting bulk valence and conduction bands alongthe projection of a MIP in the SBZ for different MCNs.

direction of the HSLs (for our sign conventions) is defined by the vector-productnpos nsf × nmp.As discussed above, the surface states protected by mirror symmetries are still

present in real systems with disorder as long as the symmetries are fulfilled onaverage. This robustness against weak disorder is also consistent with the recentexperimental observations of an even number of TSSs7 in tin telluride (SnTe), leadtin selenide (Pb1−xSnxSe), and lead tin telluride (Pb1−xSnxTe), where disorder iscertainly present [109–111].

7An even number of TSSs indicates a TCI, as an odd number of surface states occurs in a STI.

Chapter3

Chapter 3

Introduction to Kondo insulators

3.1 Overview

The physical properties of materials known as Kondo insulators (KIs) are closelyrelated to the Kondo effect. In the following, we will give an overview of both topicsand their relationship, following References [112] and [113] for the Kondo effect aswell as References [114] and [115] for KIs. We will focus on samarium hexaboride(SmB6) in particular, as an example of a KI, and also give an overview of otherrelated material classes.

3.1.1 The Kondo effect

In 1934, Wander J. de Haas, Jan de Boer, and G. J. van den Berg performed mea-surements on gold and discovered a minimum in the resistivity as a functionof temperature [15, 113]. This result was unexpected, as the resistivity in met-als generally increases monotonically with temperature due to electron–phononand electron–electron scattering processes, while impurity scattering leads to atemperature-independent contribution to the resistivity. It was unclear how thiseffect, which was later discovered in other materials as well, could be reconciledwith Matthiessen’s rule and known physical principles [112, 113].

For 30 years, this remained an unsolved problem until in 1964 Jun Kondo per-formed a perturbative calculation for a model of magnetic impurities, later knownas theKondomodel. He showed thatmagnetic impurities give rise to a contribution tothe resistivity scaling as − log(T), thereby explaining the resistivity minimum [16].

26 | 3 Introduction to Kondo insulators

However, Jun Kondo’s result diverges when approaching zero temperature and istherefore not reliable at low temperatures.

The Kondo problem, i.e., how to obtain a valid theory of scattering processesinvolving magnetic impurities at low temperatures, occupied theoretical physicistsduring the following years. Extending the perturbative approachwas unsuccessful,as it led to a divergence even at a finite temperature, which became known as theKondo temperature TK [112]. In 1970, Philip W. Anderson, who had earlier proposedthe Anderson impurity model to describe magnetic impurities [116], proposed the“Poor man’s scaling” to describe the scattering below the Kondo temperature [117].This approach provided an intuitive understanding to the problem: An increasingeffective coupling between conduction electrons and the impurity, when loweringthe temperature, leads to a screening of the magnetic moment of the impurityand therefore a constant contribution to the resistivity at very low temperatures.However, it was still perturbative such that it was not applicable at T TK. TheKondo problem was finally solved in 1974, when Kenneth G. Wilson applied thenumerical renormalization group, a non-perturbative technique he had developedhimself [118]. He was able to show that magnetic impurities form singlets withconduction electrons, so-called Kondo singlets, and indeed behave as non-magneticimpurities in the limit T → 0.

3.1.2 Heavy-fermion metals and Kondo insulators

In 1969, Anton Menth, Ernest Buehler, and Theodore H. Geballe discovered achange from metallic to semiconducting behavior in the rare-earth compoundSmB6, a material that had already been studied previously by É. E. Vainshtein,S. M. Blokhin, and Y. B. Paderno [39, 40]. Shortly thereafter, it was observed thatthe samarium atoms are in a state of mixed valence, i.e., their f orbitals are frac-tionally occupied [41]. A similar effect was also reported for other compoundssuch as cerium aluminium (CeAl3) [119] and samarium sulfide (SmS) in the high-pressure phase [120]. In 1975, Klaus Andres, John E. Graebner, and Hans RudolfOtt observed very large specific-heat coefficients in CeAl3, corresponding to effec-tive masses of the order of m∗ ∼ 1000 me, where me is the bare electron mass [121].

Chapter3

3.1 Overview | 27

It is this property that later led to these materials to be referred to as heavy-fermionmaterials or heavy-electron materials [122].In 1977, Sebastian Doniach proposed the Kondo lattice model (KLM) to describe

mixed-valence compounds [123]; this model had been introduced 20 years earlierby Tadao Kasuya in the context of the Ruderman–Kittel–Kasuya–Yosida (RKKY) inter-action, based on a previous model by Clarence M. Zener [124, 125]. He suggestedthat every local moment, in the form of an f electron, interacts with the conductionelectrons via the Kondo effect. The result of this effect is a hybridization of conduc-tion and localized electrons with the opening of a gap of the order of 10 meV at lowtemperatures. Depending on the position of the chemical potential, the materialis either a heavy-fermion metal or a heavy-fermion semiconductor, see Figure 3.1 [115].Gabriel Äppli and Zachary Fisk later proposed the term “Kondo insulators” forheavy-fermion semiconductors due to the effect that produces the hybridizationgap [126].Since the discovery of SmB6 as the first KI, a large number of materials has been

considered as KIs, including ytterbiumhexaboride (YbB6), ytterbiumdodecaboride(YbB12), plutonium hexaboride (PuB6), as well as many uranium- and cerium-based compounds [114]. As correlation effects are a crucial ingredient for KIs, thesematerials usually include a heavy element from the lanthanoid and actinoid serieswith strongly localized f electrons. With exceptions of cerium nickel tin (CeNiSn),cerium rhodium antimony (CeRhSb), and cerium rhodium arsenide (CeRhAs),which were later discovered to be semimetals, all candidate materials for the classof KIs have a cubic crystal structure [114].

3.1.3 Diverse properties of heavy-fermion materials

The field of heavy-fermion materials is very large and includes a long list of mate-rials with very different properties. In addition to metallic and insulating behaviorthat we discussed above, heavy-fermion materials also show superconductivity,antiferromagnetic ordering as well as quantum criticality and non-Fermi-liquidbehavior [115].After the first discovery of heavy-fermion superconductivity in cerium copper

silicon (CeCu2Si2) by Frank Steglich et al. in 1979 [122], unconventional supercon-


E

k

(a) Band structure without hybridization,showing the conduction band (red) andthe flat band representing the localizedstates (blue).

E

k

(b) The hybridization opens a small energygap. If the chemical potential lies in thegap (continuous line), the system is a KI;if it lies in one of the hybridized bands(dashed line) it is a heavy-fermionmetal.

Figure 3.1: In heavy-fermion materials, the hybridization of conduction with localized elec-trons opens a small gap in the band structure. Depending on the position of the chemicalpotential, the material is a heavy-fermion metal or Kondo insulator.

ductivity was found in many more heavy-fermion materials [115, 127]. Recently,the material cerium cobalt indium 5 (CeCoIn5) has been considered as an systemin which the Fulde–Ferrel–Larkin–Ovchinnikov state occurs [128]. This is an exoticstate in superconductors, where Cooper pairs with finite center-of-mass momen-tum are formed in a strong magnetic field [129, 130].

Antiferromagnetic ordering has been observed in multiple heavy-fermion mate-rials. Some of them, including CeAl3 and uranium platinum (UPt3), exhibit veryweak magnetic order with magnetic moments of the order of 10−2µB, where µB isthe Bohr magneton [131, 132].

Several studies also suggest that many heavy-fermion materials can be tuned toa quantum-critical point, where a phase transition occurs between a paramagneticand a magnetically ordered phase at zero temperature [115]. Hilbert von Löh-neysen et al. were the first to report non-Fermi-liquid behavior for a heavy-fermionmaterial [133]. The tendency towards the development of superconductivity in thevicinity of quantum-critical points [134] led to increasing interest in this field andsimilar results were obtained later for many more heavy-fermion materials [115].

Chapter3

3.1 Overview | 29

Sm

B

Figure 3.2: Crystal structure of SmB6 showing the samarium atoms (blue) at the corners andthe boron octaehedron (green) at the center of the unit cell.

3.1.4 Samarium hexaboride

One of the best studied KIs is SmB6, in part, because it was the first KI to be discov-ered, but also due to its many remarkable properties. It has a cubic crystal structureof the cesium-chloride type with a lattice constant of aSmB6 ≈ 4.13Å, where thedifferent lattice sites are occupied by samarium atoms and boron octahedra, seeFigure 3.2.At high temperature, the material behaves like a metal, but when reducing the

temperature below ∼ 50 K, it exhibits insulating behavior [40]. The hybridizationgap has been measured at 15−20 meV [135–137]. This constitutes typical behaviorfor a KI; however, SmB6 shows some unusual properties in addition.

Already in the first paper on SmB6, Anton Menth, Ernest Buehler, and TheodoreH. Geballe observed a plateau-like feature of the resistivity below 3 K [40]. JamesW. Allen, Bertram Batlogg, and Peter Wachter later found the same behavior atvery low temperatures [44]. This residual conductivity was attributed to impuritystates [40, 138] but did not seem to be affected by sample quality [38]. It remainedmysterious formany years and only after the discovery of topological Kondo insulators(TKIs), the origin of the resistivity plateau could be explained by the existence ofTSSs, see Section 3.3.


Multiple studies also indicate remarkable magnetic properties of SmB6. Ininelastic-neutron-scattering experiments, Pavel A. Alekseev et al. found a mag-netic excitation with an energy of ∼ 14 meV in the band gap [139, 140], which haslater been interpreted as a spin exciton [114, 141–143]. In recent years, additionalneutron-scattering [144, 145] and muon spin relaxation (µSR) studies [146] consis-tently produced results consistent with spin-exciton states in SmB6 [147, 148].

How close SmB6 is to magnetic instabilities is also shown by its behavior underhigh pressure: Alessandro Barla et al. showed that, at a pressure of ∼ 6 GPa, aphase transition from insulating tometallic state occurs,where long-rangemagneticorder appears [43]. Similar results have been obtained by Julien Derr et al. whosedata suggests antiferromagnetic ordering under high pressure [149]. Very recently,ferromagnetic orderingwas observed on the surface of SmB6 below∼600 mK [150].

3.2 Models for Kondo insulators

The two fundamental microscopic models for magnetic impurities in metals arethe Anderson model [116] and the Kondo model [16]. For heavy-fermion materialswith localized states in the form of f orbitals in every unit cell, these models canbe modified to obtain the periodic Anderson model (PAM) and the Kondo lattice model(KLM) [125, 151]. In the following, we will introduce the two models and discusstheir relationship via the Schrieffer-Wolff transformation [112, 152].

3.2.1 Periodic Anderson model

The PAM can be formulated in a simple way in second quantization language. Inthis model, there are two different species of electrons: conduction electrons andlocalized electrons, often originating from d and f orbitals, respectively, and repre-sented by creation (annihilation) operators c†σ (cσ) and f †σ ( fσ), respectively. Here,the index σ ↑, ↓ represents the spin or pseudospin of the electrons, where the latteris relevant for the localized electrons with generally large SOC. The Hamiltonianconsists of three parts, the bare hopping of the individual orbitals, H0, the hy-bridization between d and f orbitals, Hhyb, and an onsite repulsion for f electrons,

Chapter3

3.2 Models for Kondo insulators | 31

Hint:HPAM H0 + Hhyb + Hint . (3.1a)

Using spinors c (c↑, c↓)t and f ( f↑, f↓)t, the different parts can be written as

H0

∑k

[c†k hc(k)ck + f †k h f (k) fk

], (3.1b)

Hhyb

∑k

c†kΦ(k) fk + h.c. , (3.1c)

Hint ∑

i

U f †i↑ fi↑ f †i↓ fi↓ , (3.1d)

where hc , f (k) are Bloch matrices,Φ(k) is the hybridization matrix, and U is the on-site repulsion of f electrons. The sums runover all values for the crystalmomentum(k) or all lattice sites (i).As this model treats the localized electrons on equal footing with the conduction

electrons, this model is ideal for the description of mixed-valence compounds andis frequently applied to KIs. Before the discovery of TKIs, this Hamiltonian wasusually simplified by using Φ(k) V1, h f (k) ε f 1, and hc(k) εd(k)1, where1 is the 2×2 identity matrix. This leads to band structures similar to those shownin Figure 3.1. The interaction term can be studied nonperturbatively by variousmethods, including dynamical mean-field theory, Gutzwiller-projected variationalwave-functions, or slave-particle representations [112, 115]. The original Andersonimpurity model is obtained by removing all f states except one.

3.2.2 Kondo lattice model

In the KLM, the f electrons are represented by local spin operators Si . In compar-ison to the PAM, the dynamics of conduction electrons is unchanged, but all othercontributions are replaced by a spin-exchange term:

HKLM H0 + Hspin , (3.2a)

H0

∑k

c†k hc(k)ck , (3.2b)


Hspin

∑k ,q

J(k , q)Sq ·(c†k ,σσσσ′ ck−q ,σ′

). (3.2c)

Here, σ (σ1 , σ2 , σ3) is the vector of Pauli matrices, J(k , q) is the spin-exchangeparameter, and we use the Einstein summation convention for the sums over σ andσ′ [153].A frequently applied simplification is a momentum-independent spin exchange,

J(k , q) J, which corresponds to an onsite spin–spin interaction and is similar tothe simplifications in the PAM discussed above. By applying second-order pertur-bation theory, an effective interaction between the local spins, the RKKY interaction,can be derived [125, 154, 155]. The Kondomodel for a single impurity, also referredto as the s–d model, can be obtained by considering a single local spin, i.e., byreplacing Sq → S. It should be noted that the KLM is not well suited to studymixed-valence effects, as each localized orbital is assumed to be occupied by asingle spin.

3.2.3 Schrieffer-Wolff transformation

The Anderson and Kondo models are closely related to each other, which hasbeen shown by John R. Schrieffer and Peter A. Wolff in 1966 [152]. Starting fromthe PAM with a half-filled f band in the ground state, the KLM can be derived byconsidering virtual excitations of the localized states and projecting out empty anddoubly occupied states. This results in a spin-exchange term of the form givenin Equation (3.2c) and an additional potential-scattering term [112]. Assuming adiagonal hybridization Φ(k) Vk1 as well as h f (k) ε f 1 and hc(k) εd(k)1 inthe PAM, the transformation leads to a spin-exchange parameter [112]

J(k , q) V∗kVk−q

(1

U + ε f − εd(k − q) +1

εd(k) − ε f

). (3.3)

3.3 Topological Kondo insulators

In 2010, MaximO. Dzero, Kai Sun, VictorM. Galitski, and Piers Coleman suggestedthat KIs could actually be TIs and proposed the name topological Kondo insulators(TKIs) for this novel type of materials [37]. They argued that, as all known KIs have

Chapter3

3.3 Topological Kondo insulators | 33

inversion symmetry, the Z2 invariants can be calculated from knowledge of theparities of occupied states at the HSPs, as discussed in Section 2.4.2. Due to thefact that conduction (d) and localized ( f ) electrons have opposite parities, a bandinversion at one of theHSPs should lead to a STI. The essential topological propertiescanbe studiedusing amodel basedon the PAM.However, it is necessary to introducea spin-dependent hybridization betweennearest neighbors (NNs) instead of the onsitehybridization that was commonly used before [37, 156]. We will discuss a minimalmodel for TKIs in detail in Chapter 5.

Although other materials such as PuB6 [157], YbB6, and YbB12 [158–160] werealso discussed in the context of TKIs, both experimental and theoretical researchmainly focused on SmB6 in recent years. Shortly after the theoretical character-ization of TKIs, several theoretical studies predicted SmB6to be a TKI with threetopologically protected Dirac cones at the surface [161–166]. This finally provideda satisfactory interpretation of the low-temperature resistivity plateau that hadbeen a long-standing puzzle.Early-on, multiple transport experiments showed that the residual conductiv-

ity at low temperatures, which we discussed in Section 3.1.4, indeed takes placesolely at the surface [167–170]. At the same time, ARPES [105, 171–175], quantum-oscillation [176], and scanning-tunneling-microscopy measurements [177] con-firmed the existence of gapless surface states. Nevertheless, some controversiespersisted, corresponding to different interpretations of quantum-oscillation mea-surements [176, 178, 179]. Furthermore, different values have been reported forthe effective mass from quantum oscillations [176, 178], ARPES [105, 172–174] andmagnetothermoelectric transport [180]. Due to the small bulk gap and strong elec-tronic correlations, a detailed characterization of the nature of the surface states isdifficult and may require the consideration of additional concepts such as atomicreconstruction [181], Kondo breakdown [179, 182], or excitonic scattering [183].Some groups even challenged the scenario of a TKI, suggesting other mechanismsfor the surface conductance [137, 184].To date, the most conclusive evidence for the topological nature of the sur-

face states in SmB6 is provided by spin-resolved ARPES measurements of the (001)


surface showing that the surface states around the X point of the SBZ are spin-polarized [136].

The ideal TI for real-world applications features a truly insulating bulk combinedwith surface conduction due to the TSSs. However, many “conventional” TIs haveshown a very large residual conductivity in the bulk due to band overlaps ordoping [25, 185]. In contrast, due to the robust energy gap created by the Kondomechanism, SmB6 has a truly insulating bulk. Another remarkable feature of SmB6is that it has three surface Dirac cones instead of a single one, as in all other knownTIs. This fact opens up new possibilities for the spin texture of the surface statesand interesting novel properties.

Chapter4

Chapter 4

Outline

This thesis strives to explore the field of topological Kondo insulators and theconsequences of its various remarkable characteristics. In particular, we want toadvance the understanding of topological properties of Kondo insulators, correla-tion effects, and their interplay. Most results are applicable to topological Kondoinsulators in general or even to “conventional” topological insulators, i.e., materi-als whose band gap is not created by the Kondo mechanism. However, in manyplaces we will adjust our models and calculations to fit the features of SmB6, theonly experimentally verified example of a topological Kondo insulator to date. Themain content of this thesis is divided into three parts as follows.

In Part I, various topological properties of topological Kondo insulators will bediscussed. We will start by introducing a simplified model for a topological Kondoinsulator in Chapter 5 and explain the essential theoretical considerations. Whilethis model does not take into account material-specific details for, e.g., SmB6, it isable to capture many of the essential features from the point of view of topologicalinsulators. Focusing on this model, we will classify topological phases by calcu-lating the Z2 invariants as well as mirror Chern numbers, which were discussedin Sections 2.4 and 2.5, respectively. We move on to studying the relationship ofthese topological invariants with the gapless surface states that arise in the differenttopological phases.These surface states, in particular their spin texture, are the core subject of Chap-

ter 6. Focusing on the example of SmB6, we will characterize the spin texture of thetopologically protected surface states by a winding number and study its relation-

36 | 4 Outline

shipwith the differentmirror Chern numbers in the system andmicroscopicmodelparameters. Our theoretical findings, which constitute a core result of this thesis,are complemented by calculations of different models fitted to the band structureof SmB6.

Part II addresses the diverse physical effects of the interactions among thestrongly localized electrons in Kondo insulators. One important consequence ofelectronic correlations, assuming a “local Fermi liquid”, are renormalizations ofthe band structure. This effect is studied in Chapter 7 by performing mean-fieldcalculations for the Kotliar–Ruckenstein slave-boson scheme. An important findingof this chapter is the evidence of topological phase transitions as a function of theinteraction strength of the localized electrons, mainly due to a renormalization ofthe onsite energy.

In Chapter 8, we will modify the slave-boson calculation to be spin-rotationallyinvariant and extend it by taking into account Gaussian fluctuations around thesaddle point. After presenting a very general calculation, we will discuss how theeffective action obtained from this approach can be related to response functionssuch as the (dynamic) spin susceptibility and the spin structure factor. In turn,these can be used extend the phase diagram by magnetically ordered phases anddetect collective excitations.

Adifferent approach towardsmagnetic phases is presented inChapter 9. Startingfrom the simplified model for topological Kondo insulators in the form of theperiodic Anderson model, we will perform the Schrieffer–Wolff transformation inorder to obtain the related Kondo lattice model. Integrating out the conductionelectrons in thismodel leads to an effective exchange coupling of the localized spins,the Ruderman–Kittel–Kasuya–Yosida interaction. We consider differentmodels forthe hybridization, including onsite and nearest neighbor hybridization, and discusstheir effects on the spin–spin interaction.

In Chapter 10, we will discuss recent muon spin relaxation measurements withvarying penetration depth on SmB6 in the context of spin excitons. These arecollective excitations due to the electronic interactions that have been observedby different experiments in SmB6, see Section 3.1.4. The data allow us to extractinformation on the size of the excitonic states as well as their magnetic moments.

Chapter4

| 37

Furthermore, the behavior of magnetic fields as a function of penetration depth canbe related to magnetic ordering at the surface.

In Part III, we will widen our perspective and study general properties of topo-logical invariants and topological phase transitions. Topological invariants can belinked to the concept of entanglement via the entanglement spectrum. We discussthis relationship in detail in Chapter 11, where we will review the basic propertiesof the entanglement spectrum and introduce a novel variant called the sublatticeentanglement spectrum. Furthermore, we will derive a connection of the entangle-ment spectrum with quantum geometry and complement our analytical findingswith model calculation for two-dimensional topological systems.Lastly, we will draw a connection from topological phase transitions to statistical

physics in Chapter 12. In particular, we will discuss critical exponents and scalinglaws related to Chern numbers and consider a renormalization-group procedurefor topological phases. Wewill relate our results to the rest of the thesis by applyingour methods to the mirror Chern numbers in a topological Kondo insulator.

A short summary of the contents is inserted at the beginning of each part andevery chapter ends with concluding remarks. In Chapter 13, we will concludethe thesis by summarizing our findings and discuss their relevance in the contextof current research in condensed matter physics. Finally, we will review openquestions and give an outlook on future research related to the results of thisthesis.

ITopology in Kondo insulators

In this part, we will study topological propertiesof Kondo insulators. After discussing a simpli-fied model for topological Kondo insulators andits topological phases, we will focus on the surfacestates. The relationship of their spin texture withmirror Chern numbers and microscopic model pa-rameters is a central result of this thesis.The contents of this part of the thesis are partiallyincluded in References [186] and [187].

Chapter5

Chapter 5

Topological phases in Kondo insulators

From a theoretical point of view, the crucial ingredient to realize a topologicalKondo insulator (TKI) (with inversion symmetry) is that the hybridization matrixΦ(k) between the localized and itinerant electrons is an odd function of k [37]. Thismeans that the hybridization is necessarily non-onsite, in contrast to early modelsfor Kondo insulators (KIs), which used an onsite hybridization, see Section 3.2. Thisreflects the fact that for many mixed-valance compounds the localized degrees offreedom are typically derived from the atomic f orbitals, which are odd underparity, and the itinerant degrees of freedom from the atomic d orbitals, which haveeven parity. Consequently, if the system has inverted bands at an odd number oftime-reversal-invariant momenta (TRIM), a topological insulator (TI) is realized, seeSection 2.4.2.In this chapter, we construct and analyze a simple tight-binding model for a KI

on a cubic lattice in the form of a periodic Anderson model (PAM). It consists of twospin-degenerate orbitals, which couple via an odd-parity hybridization, modeling,e.g., the localized f and itinerant d electrons of Sm in SmB6 [156, 161]. As a result ofthe cubic symmetry, we find that eight different gapped band insulators (with theirrespective charge-conjugate partners) are possible at half filling, realizing varioustrivial and topological phases. These different phases are all distinguished by theirinversion eigenvalues at the eight TRIM and also differ in the nature of their surfacestates (if present). We show that a complementary classification of these phasesis possible using two mirror Chern numbers (MCNs), which we have discussed inSection 2.5. TheseMCNs also uniquely determine (a) the strongZ2 invariant ν0 (pro-tected by time-reversal symmetry (TRS) and charge conservation) and (b) the three

42 | 5 Topological phases in Kondo insulators

weak Z2 invariants (ν1 , ν2 , ν3) (which require additional translation symmetry) inthis simple model.

The effect of of electron–electron interactions of the f orbitals can be consideredwithin the quasiparticle approximation assuming a “local Fermi liquid” [188–192].The quasiparticle excitations of the interacting system are then accurately describedby a renormalized quadratic Hamiltonian which can be classified topologically. Inthis part, we assume our model parameters to be renormalized due to correlationeffects, the details of which will be studied in Chapter 7.

The remainder of this chapter is organized as follows. In Section 5.1, we introducethe model and discuss its noninteracting band structure, assuming renormalizedparameters due to electron–electron interactions. In Section 5.2, we show how dif-ferent topological phases emerge from band inversions at high-symmetry points(HSPs) and characterize them by different symmetry-protected topological invari-ants. Furthermore, in Section 5.3 we show the gapless surface modes in differentphases and discuss how they are protected by the different symmetries.

5.1 Model

5.1.1 Cubic topological Kondo insulator

Let us start by defining a minimal model for a (topological) Kondo insulator ona simple cubic lattice with one spin-degenerate orbital per lattice site each for dand f electrons [156]. The general Hamiltonian has the form of the PAM defined inEquation (3.1a) and is given by

H H0 + Hhyb + Hint , (5.1)

where H0, Hhyb, and Hint describe the tight-binding energy of d and f electrons,the hybridization between d and f electrons, and the interactions, respectively.We include up to third-neighbors hopping in H0, which we model by an s-typehopping, and assume an imaginary and spin-dependent hybridization betweennearest-neighboring d and f electrons, reflecting their different parity and thestrong spin–orbit coupling (SOC) of the f orbitals. Furthermore, we assume thatthe f electrons locally interact via a Hubbard-U repulsion while the d electrons are

Chapter5

5.1 Model | 43

noninteracting. Therefore, we can write the individual parts of the total Hamilto-nian (5.1) as

H0

∑i

ε f f †i fi −∑〈i , j〉

(td c†i c j + t f f †i f j + h.c.

)−

∑〈〈i , j〉〉

(t′d c†i c j + t′f f †i f j + h.c.

)−

∑〈〈〈i , j〉〉〉

(t′′d c†i c j + t′′f f †i f j + h.c.

),

(5.2a)

Hhyb

∑αx ,y ,z

∑〈i , j〉α

[iVc†i σα f j+iV f †i σαc j+h.c.

], (5.2b)

Hint ∑

i

U f †i↑ fi↑ f †i↓ fi↓ , (5.2c)

where i , j ∈ 1, . . . ,N label the lattice sites and 〈i , j〉, 〈〈i , j〉〉, and 〈〈〈i , j〉〉〉 denotepairs of nearest neighbors (NNs), next-to-nearest neighbors (NNNs), and next-to-next-to-nearest neighbors (NNNNs). The notation 〈i , j〉α stands for a NN bondin the α-direction and σα are the Pauli matrices in spin space. The annihilation(creation) operators ci , fi (c

†i , f †i ) for the conduction d electrons and f electrons,

respectively, are spinors ci (ci↑, ci↓)t and fi ( fi↑, fi↓)t. The particular form ofthe hybridization has been chosen in order to be odd under parity and to obey cubicsymmetry as well as TRS. The Brillouin zone (BZ) for systems with cubic symmetryis shown in Figure 5.1 with the positions of the four different HSPs Γ, X, M, and R.We note that the model (5.2) ignores the complicated multiplet structure of the

d and f orbitals usually encountered in real KIs such as SmB6. But importantly,most topological properties of cubic KIs do not depend on the particular shape ofthe orbitals or the precise form of the hopping and hybridization matrix elements.Instead, they follow directly from the points with band inversion, as we discuss inthe course of this chapter. However, we note that the details of the hybridizationcan be important for theMCNs and therefore the the spin texture of the topologicallyprotected surface states (TSSs), as we will discuss in Chapter 6. Also, the multipletstructure may be important for identifying possible topological phases of specificmaterials as the orbital degeneracy can prevent the exchange of parity eigenvaluesbetween valence and conduction bands at certain HSPs. For example, the cubic


Γ X

MX

X M

RM

ky

kz

kx

Figure 5.1: BZ for systems with cubic symmetry showing the four different HSPs Γ, X, M, andR in blue.

symmetry enforces a twofold degeneracy of the relevant Γ8 and Eg orbitals at theΓ and R points in SmB6, such that the product of parity eigenvalues is alwayspositive at those points [164]. In this chapter, we do not consider such material-specific questions but instead we focus on the universal topological properties thatare in principle possible in the presence of cubic symmetry.

The bandwidth of the f electrons is much smaller than the bandwidth of theconduction electrons andwe therefore assume that |t f | |td | and similar relationshold for second- and third-neighbor hopping amplitudes. The hybridization ischaracterized by the parameterV forwhichwe typically use |V | . |td |. Throughoutthe whole chapter, we choose td to be the unit of energy, td 1, and assume halffilling.

5.1.2 Noninteracting band structure

During the remainder of this chapter, we will discuss the noninteracting model,assuming Hint 0, and therefore analyze the Hamiltonian

Hni H0 + Hhyb . (5.3)

Chapter5

5.1 Model | 45

Here, we assume that allmodel parameters are renormalizeddue to the correlationsof f electrons as discussed later in Chapter 7. In order to simplify the followingexpressions, we used the definitions

c1(k) : cx + cy + cz , (5.4a)

c2(k) : cx y + cyz + czx , (5.4b)

c3(k) : cx cy cz , (5.4c)

where cα cos(kα), cαβ cαcβ , and sα sin(kα) for α, β x , y , z.For periodic boundary conditions, we can perform a Fourier transform which

leads toHni

∑k

Ψ†(k) h(k)Ψ(k) . (5.5)

Here, we defined the 4-spinor Ψ(k) (c↑, c↓, f↑, f↓

) t and the 4 × 4 Bloch matrixh(k):

h(k) hd(k) 1 + τz2 + h f (k) 1 − τz

2 +Φ(k) τx ©«

hd(k) Φ(k)Φ(k) h f (k)

ª®¬ . (5.6)

The Pauli matrices in orbital space are denoted by τi and 1 is the 2×2 identitymatrix. The dispersion of the d and f electrons is described by hd(k) and h f (k),respectively, and the hybridization by the matrix Φ(k) as follows:

hd(k) [−2td c1(k) − 4t′d c2(k) − 8t′′d c3(k)

]1 , (5.7a)

h f (k) [ε f − 2t f c1(k) − 4t′f c2(k) − 8t′′f c3(k)

]1 , (5.7b)

Φ(k) −2V(sxσx + syσy + szσz

). (5.7c)

Note that the hybridization matrix (5.7c) is an odd function of k, Φ(k) −Φ(−k),but in order to preserve TRS, it also couples to the physical spin of the electron.These properties are crucial for realizing a time-reversal-invariant TKI [37], as wewill discuss in more details in the following.


Diagonalizing the Bloch matrix (5.6) yields the energy eigenvalues

E± Ed + E f

2 ±√(Ed − E f

2

)2+ E2

hyb , (5.8)

where we suppressed the k-label in the interest of better readability. Each band istwofold degenerate because of the combination of time-reversal and inversion sym-metry. Ed , E f , and Ehyb are the eigenvalues of hd(k), h f (k), and Φ(k), respectively,and are given by

Ed(k) −2td c1(k) − 4t′d c2(k) − 8t′′d c3(k) , (5.9a)

E f (k) ε f − 2t f c1(k) − 4t′f c2(k) − 8t′′f c3(k) , (5.9b)

Ehyb(k) −2V√

s2x + s2

y + s2z . (5.9c)

For future use, we also define the weight of the d orbitals for a state vector u as

w(u) : u† 1 + τz2 u . (5.10)

We say that u has d character ( f character) if w(u) 1 [w(u) 0]. We will also usethe shorthand notation

wa(k) : w[ua(k)] , (5.11)

where ua(k) is the state of band a at momentum k.Figure 5.2 illustrates two exemplary band structures forwhich the narrow f band

lies within the conduction band. For these examples, we included onlyNN hoppingand in both cases the nonzero hybridization opens a direct gap. However, only ifthe sign of t f is opposite to the sign of td , also an indirect gap opens at half filling(and a weak TI is found for the parameters of Figure 5.2). Instead, if the signs arethe same, the bands overlap and we obtain a heavy-fermion metal. If additionalfurther-neighbor hoppings are considered, insulating phases are possible if theratios between first, second and third-neighbor hoppings for f electrons are similarto the corresponding ratios for d electrons, t′d/td ≈ t′f /t f and t′′d /td ≈ t′′f /t f . Wewill therefore assume t′f t f (t′d/td) and t′′f t f (t′′d /td) in the following.

Chapter5

5.1 Model | 47

(a) (b)

Figure 5.2: Energy spectrum of the model (5.3) for td 1, |t f | 0.1, t′ t′′ 0, V 0.5, andε f 0. The difference between the metallic (a) and insulating phase (b) is the sign of t f :It is positive for the metallic and negative for the insulating phase. The shown insulatingphase isWTI(ΓX) (see Table 5.1). The color shows w(k), where red and blue denote d and fcharacter, respectively. The thin dashed lines show the bare energies of d and f bands forthe same parameters but with vanishing hybridization, V 0. The relatively large valuefor V is chosen for a better visibility of the hybridization gap.

5.1.3 Relation to SmB6

Ab initio calculations indicate that in SmB6 a band inversion of d and f bandsoccurs at the X HSPs [163, 193–195]. While our simplified model is not able tocapture details of the band structure, the band inversion at the X points can berealized when assuming non-vanishing NNN hopping. This phase, which we callSTI(X), is shown in Figure 5.3 (a) and its properties are of particular interest withregard to experiments. For comparison, we show the band structure of the phaseTCI(ΓM) in Figure 5.3 (b), which has band inversions at the Γ and the three Mpoints. All the Z2 invariants are trivial in this phase but it has nontrivial MCNs andis therefore an example of a topological crystalline insulator (TCI), which we havediscussed in Section 2.5.The labeling of the different phases follows the convention discussed in Sec-

tion 5.2.1 and Table 5.1. Note that band energies, such as shown in Figures 5.2 and5.3, are given in accordance with Equation (5.3). Thus, the Fermi level at half fillingis located at different energies for different choices of band parameters.


(a) Phase STI(X) for t′ −0.4t, t′′ 0, andε f −2.

(b) Phase TCI(ΓM) for t′ 0, t′′ 0.3t,and ε f 0.

Figure 5.3: Energy spectrum of the model (5.3) for different topological phases. The phaseSTI(X) (a), which is relevant for SmB6, requires NNN hoppings, and the phase TCI(ΓM) (b)can only be realized when t′′ > 0.25t. For both plots, we chose the parameters td 1,t f −0.1, and V 0.5. The color coding and the different lines follow the same conventionas in Figure 5.2.

5.2 Topological classification

In the following sections, we provide a complete topological classification of thetime-reversal-invariant gapped phases obtained in the noninteracting model (5.3).For notational simplicity, the band parameters are denoted with their bare values,but they can equally well be understood as the renormalized values in the interact-ing model, see Chapter 7 for details. Depending on the relative magnitude of NN,NNN, and NNNN hopping for d and f electrons, as well as the onsite potential andthe interaction between f electrons, a metallic, trivial insulating or one of severaldifferent topological phases is realized. An overview of the different insulatingphases is given in Table 5.1. In Section 5.2.3, we characterize these different phasesusing different symmetry-protected topological invariants.

5.2.1 Topological phases from band inversions

Because our model respects the inversion symmetry of the cubic lattice, we firstdiscuss how the different phases are distinct by the inversion eigenvalues atHSPs inthe BZ. The odd-parity property of the hybridization function (5.7c) implies that thetwo orbitals ( f and d electrons) have opposite parity. Hence, the inversion operator

Chapter5

5.2 Topological classification | 49

(a) (b) (c)

Figure 5.4: Topological phase transitions occur when the gap closes at on of the HSPs. This isonly possible for a non-onsite hybridization, which vanishes at the HSPs.

is represented as I τz and the Bloch Hamiltonian (5.6) satisfies

I−1h(−k)I h(k) . (5.12)

Because of cubic symmetry, only four out of eight TRIM Γi ∈ 0, π3 are indepen-dent. Following standard notation, we denote them as Γ (0, 0, 0), X ∈ (π, 0, 0),(0, π, 0), (0, 0, π), M ∈ (π, π, 0), (0, π, π), (π, 0, π), and R (π, π, π), see Fig-ure 5.1. The three X points are equivalent for symmetry reasons, as are the three Mpoints. At these HSPs, we have a vanishing hybridization, Φ(Γi) 0, and thereforeeach Kramers pair has pure d or f character. Also, a vanishing hybridization is anecessary condition for the energy gap to close, thereby allowing for topologicalphase transitions, see Figure 5.4.We define the points with band inversion as thoseHSPswhere the occupied states

have d character (instead of f character) and label the corresponding phase withthe HSP(s), at which the band inversion occurs (see Table 5.1). If there are morethan two of those points, we instead list the points where the occupied states havef character and denote the respective phase with a bar. With this convention, eachphase is labeled with at most two HSPs with a band inversion. In total, there are16 different phases with different occupations at the HSPs, but always two phasesare related to each other by inverting the occupations at all the HSPs, which can beachieved by flipping the sign of all the hopping amplitudes. In Table 5.1, we list theremaining eight independent phases, which can be grouped into (trivial) BI, TCI,WTI, and STI.


Table 5.1: Topological invariants and MCNs for different insulating phases grouped into BI,TCI, WTI, and STI. The different phases are labeled by the HSPs with an occupied d band.For an inversion of f electrons at the respective points, theMCNs must be multiplied by −1,and we will denote the respective phase by a bar, e.g., STI(M). The second column shows,which hopping amplitudes of our model are required to be nonzero in order to create therespective phases. For example, the phase TCI(ΓM) only occurs when including NNNNhopping.

phase requiredhopping (ν0; ν1 , ν2 , ν3) C0 Cπ Cd

BI none (0; 0, 0, 0) 0 0 0

TCI(ΓM) t′′ (0; 0, 0, 0) 2 −2 0

WTI(ΓX) t (0; 1, 1, 1) −1 1 0WTI(ΓR) t′ (0; 1, 1, 1) 1 1 2

STI(Γ) t (1; 0, 0, 0) 1 0 1STI(X) t , t′ (1; 1, 1, 1) −2 1 −1STI(M) t , t′ (1; 0, 0, 0) 1 −2 −1STI(R) t (1; 1, 1, 1) 0 1 1

5.2.2 Phase diagrams

We now discuss how the different trivial and topological phases depend on thetight-binding parameters. First, we consider the case of vanishing NNN and NNNN

hopping, t′d t′f t′′d t′′f 0. Then, the two remaining parameters are t f andε f . The phase diagram for this case is shown in Figure 5.5. We observe that withonly NN hoppings, a single band inversion is possible at the Γ or the R point butnot at the X or M points. At the phase transitions, the energy gap closes, which issketched in Figure 5.4. This must happen at a TRIM, as these are the only pointswhere there is no hybridization gap. Note that this is only possible due to thefact that the hybridization is non-onsite; with an onsite hybridization, which wasfrequently used before the characterization of TKIs, no topological phase transitionsare possible.

From the general expression of the band energies (5.8), the condition for a gapclosing at Γi is obtained as

t f (Γi) td +ε f

2c1(Γi) . (5.13)

Chapter5


Figure 5.5: Phase diagram of the noninteracting model (5.3) for td 1 and t′ t′′ 0. Thevalue of the hybridization (as long as V , 0) does not influence the phase diagram. Thelabels for the different phases correspond to the convention of Table 5.1. In addition, thereis a metallic phase where d and f bands overlap for t f > 0. The thick blue lines show phasetransitions and are labeled according to the HSP at which the energy gap closes. These arecontinued inside the metallic phase as thin, dashed gray lines, where the direct bandgapcloses at the respective points. The thick, dashed, black lines show where the bands startto overlap.

For the four different HSPs, Equation (5.13) reads as

t f (Γ) td +16 ε f , t f (X) td +

12 ε f , (5.14a)

t f (M) td − 12 ε f , t f (R) td − 1

6 ε f . (5.14b)

In Figure 5.5, these are the lines between the different topological regions for t f < 0.The transition from insulating tometallic behavior at t f 0 is not associatedwith aclosing of the direct band gap, but by the closing of the indirect gap, see Figure 5.2.The lines given in Equation (5.14) therefore extend also into the metallic region att f > 0.

Now, we want to analyze how the situation changes when we allow for nonzeroNNN hopping amplitudes. As discussed in Section 5.1.2, we consider arbitraryNNN hopping which obeys the condition t′d/td t′f /t f . We are interested in theinsulating region, so we fix t f < 0 and V , 0. The resulting phase diagram for thischoice is shown in Figure 5.6 (a) in the (t′/t)-ε f plane and in 5.6 (b) in the (t′/t)-n f


(a) (b)

Figure 5.6: Phase diagram of the noninteracting model (5.3) for td 1 and t f −0.1, as afunction of theNNN hopping t′d and the chemical potential ε f (a), and of the filling fractionn f of f orbitals (b). The ratios of NN and NNN hopping are equal for d and f electrons,t′d/td t′f /t f and we assume vanishing NNNN hopping, t′′ 0. Although the value of thehybridization (as long as V , 0) does not influence the t′-ε f phase diagram, it influencesthe relation between ε f and n f ; the (t′/t)-n f diagram was created using V 0.5. Thelabeling conventions are the same as in Figure 5.5.

plane, where n f denotes the occupation of the f orbitals:

n f

∑a

1(2π)3

∫BZ

d3k [1 − wa(k)] . (5.15)

Here, the sum is taken over all occupied bands and we integrate the weight func-tion (5.11) over the BZ. Because 0 ≤ w ≤ 1 and because we always consider twooccupied bands, the f -orbital filling must satisfy 0 ≤ n f ≤ 2.

In the (t′/t)-ε f diagram, we can again analytically obtain the phase transitionlines by considering the energy (5.8) at the HSPs. The general condition for the gapclosing is

(t′/t)(Γi) − c1(Γi)2c2(Γi) −

ε f

4(td − t f )c2(Γi) , (5.16)

Chapter5


which leads to the following four different lines shown in Figure 5.6:

(t′/t)(Γ) −12 −

112(td − t f ) ε f , (t′/t)(X) +

12 +

14(td − t f ) ε f , (5.17a)

(t′/t)(M) −12 +

14(td − t f ) ε f , (t′/t)(R) +

12 −

112(td − t f ) ε f . (5.17b)

Because the relation between ε f and n f is nonlinear, these lines map onto com-plicated curves in the (t′/t)-n f diagram. Note, however, that the topology of thephase diagram is the same in both cases.If the f orbitals are half filled, n f 1, the narrow f band necessarily lies in the

middle of the d band, which for almost all choices of parameters leads to two HSPswith band inversion. This explains why a weak TI phase is favored in this regime,which is consistent with the finding in Reference [37].

Table 5.1 also lists the hopping amplitudes which are required to be nonzerofor the eight different phases. All phases can be realized with only NN and NNN

hopping, except for the TCI(ΓM) phase. This phase, which is characterized byvanishing Z2 invariants but has nonzero MCNs, can only be realized when t′′ >0.25t. The bulk and surface band structure are shown in Figures 5.3 (b) and 5.12,respectively, but the phase is not present in any of the phase diagrams where wealways assumed t′′ 0.

5.2.3 Topological invariants

Wedemonstrate that the band inversions uniquely define a set of topological invari-ants. These are the four Z2 invariants (ν0; ν1 , ν2 , ν3) as well as three MCNs C+

kz0,C+

kzπand C+

kxkyassociated with three independent mirror planes.

Z2 invariants

Our model (5.3) belongs to the class AII in the Altland–Zirnbauer (AZ) classifica-tion and is therefore characterized by a Z2 topological invariant, as discussed inSection 2.2.2. This is the strong topological index ν0 which is protected by TRS andparticle conservation. In addition, because of (discrete) translation symmetry, wecan also define three weak topological Z2 indices (ν1, ν2, ν3), see Section 2.4.2.


As discussed in Section 2.4.2, these four invariants can be calculated bymultiply-ing inversion eigenvalues of occupied bands at the TRIM. Using cubic symmetry,the expressions (2.17) simplify to

(−1)ν0

∏aξΓ,a ξ

3X,a ξ

3M,a ξR,a

∏aξΓ,a ξX,a ξM,a ξR,a , (5.18a)

(−1)νi

∏aξX,a ξ

2M,a ξR,a

∏aξX,a ξR,a . (5.18b)

Note that because of cubic symmetry, the weak indices are all equal, ν1 ν2 ν3.As we always calculate products over an even number of values ±1 for the topolog-ical invariants, the result is not changed by switching the parity of d and f orbitals(I → −I). Moreover, the strong index only depends on the parity of the number ofband inversions (it is nonzero for an odd number of band inversions and vanishesfor an even number) and according to Equation (5.18b), the weak indices dependonly on the inversion eigenvalues at X and R. It is thus apparent that knowledge ofthe Z2 invariants does not uniquely determine the band inversions (see Table 5.1).Nevertheless, in the presence of cubic symmetry, this finer classification can beobtained from the MCNs, which will be discussed next.

Mirror Chern numbers

For cubic symmetry, there are three independent mirror planes in real space, z 0,z 1/2, and x y, leaving invariant the planes kz 0, kz π, and kx ky ,respectively, in momentum space. For our definition of the spinor, the mirroroperators are given by

Mz −i τzσz , (5.19a)

Mx−y −i τzσx − σy√

2, (5.19b)

see Appendix A.3 for more details. As discussed in Section 2.5, for each of thedifferent planes, there exists an associated MCN. We note that for planes with Cn

symmetry, the (mirror) Chern number can be calculated up to a multiple of n bymultiplying rotational eigenvalues at the n-fold rotation-invariant points in the

Chapter5


surface Brillouin zone (SBZ) [166, 196]. Here, we instead compute C+ exactly forour model by using a numerical method for a discretized BZ [100].1 The results areshown in Table 5.1 for all different phases.The (momentum-resolved) Berry curvature of the bands with mirror eigenvalue

+i is shown in Figure 5.7 (a) for the three different planes kz 0, kz π, andkx ky and three different sets of parameters specified in Figure 5.7 (b). When theenergy gap closes and reopens at one of the HSPs, thereby creating an additionalband inversion at that point, the Berry flux changes by ±1 at the respective pointwhich leads to a change of theMCN by ±1. We can therefore view the inverted HSPsas sources of the Berry flux (monopoles).It is then possible to formulate a simple rule, illustrated in Figure 5.8, which

provides the three MCNs for all eight different phases in Table 5.1 for our model.Every MCN can be obtained by summing up all the “charges” ±1 at the invertedHSPs, which lie in the respective mirror plane. The relationship of the MCNs withmodel parameters is discussed in detail in Section 6.2.The picture shown in Figure 5.8 is equivalent to the following formulas in terms

of the band-inversions at the HSPs for the MCNs:

C0 ≡ C+

kz0 w(Γ) − 2w(X) + w(M) , (5.20a)

Cπ ≡ C+

kzπ w(X) − 2w(M) + w(R) , (5.20b)

Cd ≡ C+

kxky w(Γ) − w(X) − w(M) + w(R) . (5.20c)

We observe that the knowledge of the two MCNs for kz 0 and kz π suffices touniquely identify the topological phase and the parities of occupied bands at allHSPs. In particular, also the Z2 invariants [Equation (5.18)] follow:

(−1)ν0 (−1)C0+Cπ , (5.21a)

(−1)ν1,2,3 (−1)Cπ . (5.21b)

Equations (5.20) also directly provide the sum rule Cd C0 + Cπ for the threeMCNs.

1Note that Reference [100] uses a different sign convention for the Chern number.


-30 -20 -10 0 10 20 30

(a) Theparameters for the plots are td 1, t f −0.1, t′/t −0.8, t′′ 0, andV 0.5;the chemical potential ε f varies as shown in (b). Close to the phase transitions,the Berry curvature is strongly peaked at the HSP where the gap closes (columnsA and C). At the phase transition it has a jump of ±1 at the respective HSP. Awayfrom the phase transitions, the Berry curvature is delocalized (column B). Notethat the Berry curvature only has twofold rotational symmetry around the Xpoints (M points) on the kz 0 and kx ky (kz π and kx ky ) planes.

ε fX M ΓBI STI(X) WTI(ΓR)

A B C

(b) Different phases and phase transitions for the chosen parameters and variableε f , as well as the choices for ε f in the different columns, ε f −5 (A), ε f −3(B), and ε f −0.5 (C).

Figure 5.7: Plots of the mirror Berry curvature F +(k) as defined in Equation (2.20).

Chapter5

5.3 Surface states | 57

Γ

X

MX

XM

R

M

+1

−1

Figure 5.8: Illustration of the relations (5.20) for the MCNs: As the Berry flux at the HSPschanges by ±1 when creating a band inversion at the respective point, any MCN can beobtained by summing up all the “charges” ±1 lying in the considered mirror plane forthose HSPs with band inversion. The charges ±1 are shown as red and blue dots. Note thatfor the X and M points, the charges are different for the different mirror planes, reflectingthe smaller rotational symmetry group at those points. While there is a freedom to choosean overall sign for the different charges, their relative sign is fixed by the condition that thesum of all charges in any plane must vanish.

Note that in contrast to the Z2 indices, the MCNs depend on the parity of theoccupied bands; they are multiplied by −1 when switching d and f electrons.

5.3 Surface states

We have discussed the implications of the Z2 invariants (ν0; ν1 , ν2 , ν3) and theMCNs on the TSSs in Sections 2.4 and 2.5, respectively. For example, a Dirac cone ispresent at all (high-symmetry) points in the SBZ onto which an odd number of bulkinverted HSPs is projected [65, 105]. On the other hand, a nonzeroMCN implies thatat least |C+ | Dirac cones exist along the high-symmetry line (HSL) in the SBZ, whichis invariant under the respective mirror operation [108].Figure 5.9 shows the SBZ for the (100) and (110) surface along with the projected

bulk HSPs and projected mirror planes. Consulting Table 5.1, it is now simple todetermine if and where gapless Dirac points are expected in the SBZ. For the (100)surface, the location of the Dirac points predicted by the Z2 invariants agrees withthe values of the three MCNs. Thus, all the Dirac points are located at HSPs of the(100) SBZ. On the other hand, for the (110) surface and if there is a band inversion


ky

kz

k yπ

kz π

k y

0

kz 0

k y

k z

ΓX XM

XM MR

Γ X

X M

(a) (100) surface.

k y

kz

kz π

k y

k x

kz 0ΓM X2

XR M2

Γ X

Y S

(b) (110) surface.

Figure 5.9: SBZs of a simple cubic lattice. The dashed blue lines are orthogonal projections ofmirror planes. The bulk HSPs which are projected to each surface HSP are displayed in red.

at X (M), the Z2 invariants predict one Dirac cone located at Y (Γ). However, due tothe nonzero MCNs |C0 | 2 (|Cπ | 2), there must be two additional Dirac cones onthe kz 0 (kz π) HSL, which are protected by mirror symmetry [166].

We have confirmed these expectations by explicitly calculating the surface statesin thin film geometries, see Appendix A.1 for further details. The results of thecalculations for three different topological phases are shown in Figures 5.10, 5.11,and 5.12.

Figure 5.10 shows the energy spectrum for surfaces of the phase STI(X). Asexpected, there are Dirac cones at Γ and the two X points for the (100) surface. Onthe other hand, for the (110) surface, we observe additional two Dirac cones on theX–Γ–X line protected by the mirror symmetry [166].2

We now turn to the surface spectrum of the phaseWTI(ΓR), shown in Figure 5.11,which has a remarkable feature on its (110) surface. For this phase, theZ2 invariantspredict Dirac cones at the Γ and X points. These Dirac cones are also required bythe MCNs. Nevertheless, we find two additional (and unexpected) band crossingsalong the S–Y–S line. The solution to this puzzle lies in the sign of the MCN, the

2Note that only one Dirac cone is visible because only half of the HSL corresponding to the kz 0 planeis shown.

Chapter5

5.3 Surface states | 59

(a) (100) surface. (b) (110) surface.

Figure 5.10: Energy spectrumof a 100unit cells thick slab in the phase STI(X) (td 1, t′d −0.6,t f −0.1, t′f 0.06, V 0.1, ε f −4). All gapless modes predicted by the topologicalinvariants are present. In the plot for the (110) plane, there is clearly visible the Dirac coneat the Γ–X line, which is protected by mirror symmetry.


Figure 5.11: Energy spectrum of a 100 unit cells thick slab in the phaseWTI(ΓR) (hopping andhybridization as in Figure 5.10, ε f 0). All gapless modes predicted by the topologicalinvariants are present. On the (110) surface there is an additional crossing on the Y–S linethat occurs due to the mirror chirality associated with the kz π plane.

mirror chirality. The MCN for the kz π plane is −1, which means that along theline S–Y–S, one band with mirror eigenvalue +i crosses from the conduction to thevalence band. However, as shown in Appendix A.2 within the k · p theory, thevelocity of this band at the Y point is given by

v

4|V |√−t f td

td − t f, (5.22)

which is positive for td > 0 and t f < 0. Therefore, the two bands with eigenvalues±i crossing at the Y point must cross two additional times along the S–Y–S line



Figure 5.12: Energy spectrum of a 100 unit cells thick slab in the phase TCI(ΓM) (td 1,t f −0.1, t′ 0, t′′/t 0.3, V 0.1, ε f 0). There are four Dirac cones on each of thesurfaces, which are all protected by mirror symmetry only (note that only two are visibleon the (110) surface as only half of each HSL is plotted).

in order to fulfill the constraint from the mirror chirality. We note that a similarobservation wasmade for the surface states of a time-reversal-invariant topologicalsuperconductor [108].

Finally, Figure 5.12 shows the surface states of the topological crystalline insulatorphase TCI(ΓM). In this phase, allZ2 invariants vanish. Therefore, all gapless surfacestates are protected by mirror symmetry. Due to the MCNs C0 2 and Cπ −2,there are two Dirac cones on each of the corresponding HSLs.

Conclusion

In this chapter, we constructed a minimal two-orbital model in the form of the PAM

for a cubic TKI, which for certain parameters is able to model the band structure ofSmB6. We found eight topologically distinctKIphases, depending on the inversionsof the two bands, which can be characterized by Z2 invariants (protected by TRS)andMCNs (protected by different mirror symmetries). These topological invariantshave a direct influence on the existence and position of gapless surface modes,which was demonstrated by numerically diagonalizing the Hamiltonian for a slabof finite thickness.

Chapter6

Chapter 6

The spin texture of topological surface states

Model-based and ab initio calculations predict the material SmB6 to have a bandinversion at the X HSPs [161–164]. As discussed in Chapter 5, the X-invertedphase has a nontrivial strong Z2 index ν0 1 and weak topological indices ν

(1, 1, 1), leading to protected surface Dirac cones as shown in Figure 6.1 (a) for the(001) surface. The experimental work in Reference [136] is consistent with thesepredictions and furthermore suggests that the spin texture of the surface statesis as sketched in Figure 6.2 (a). Interestingly, however, several theoretical studiesreached conflicting conclusions about the nature of the spin texture [194, 197, 198],which is not uniquely determined by the Z2 invariants. In fact, for linear Diraccones, two situations are compatible with the cubic symmetry, see Figures 6.2 (a)and 6.2 (b). They are distinguished by opposite winding numbers wX ±1 of

the planar unit spin (nx , ny) (Sx , Sy)/√

S2x + S2

y around the X point, where thewinding number around the HSP K 1 is defined as

wK 1

2π

∮γK

∇[Im log(nx + iny)

] · ds , (6.1)

with γK a contour encircling K in an anticlockwise fashion. This discrepancybetween different theoretical models and approaches raises the important questionof what determines the spin texture in cubic TKIs.In this chapter, we provide two answers to this question: First, we show that

there is a close connection between the spin texture and the MCNs, see Section 2.5.

1The HSP K has the property −K K + G, where G is a reciprocal lattice vector. On the (001) surface,there are three different HSPs: Γ, X, and M.

62 | 6 The spin texture of topological surface states

Γ X

MX

(a) Sketch of the Dirac cones in the (001)SBZ.

kx

ky

Γ X

X M

(b) Positive directions of mirror invariantlines in the (001) SBZ..

Figure 6.1: Topologically protected surface states (a) and positive directions of mirror-invariant lines in the SBZ for the (001) surface (b). We assume an outward pointing normalvector nsf ez , see Appendix A.3 for further details.

In particular, knowledge of the MCNs allows us to distinguish between the twosituations shown in Figures 6.2 (a) and 6.2 (b). Second, we provide analytical ex-pressions relating the surface-state spin texture to the hybridization parameters ofspecific models. These relations demonstrate that the number and type of includedorbitals in the effective model does not uniquely define the winding number; in-stead, the relative strength of different-range hybridization parameters is equallyimportant. In addition, we show how the system can be tuned across topologicalphase transitions, during which the surface-state spin texture changes while all theZ2 invariants remain unaffected.

We will also apply our approach to different multiorbital models with itinerantEg and localized Γ7 and Γ8 electrons, as in Reference [198], as well as the simplifiedmodel introduced inChapter 5. We remark that similar results, which are consistentwith the study in this chapter, have been obtained independently by Pier PaoloBaruselli and Matthias Vojta [199].

6.1 Mirror Chern numbers define pseudospin texture

To start, we review certain facts about theMCNs in SmB6. TheMCNs are topologicalinvariants, which are protected by mirror symmetries, as discussed in Section 2.5

Chapter6

6.1 Mirror Chern numbers define pseudospin texture | 63

kx

ky

Γ X

X M

(a) wX +1.

kx

ky

Γ X

X M

(b) wX −1.

Figure 6.2: Sketch of the spin (or pseudospin, see page 62) textures in the (001) SBZ. While atΓ the winding number is always wΓ 1, at the X points it can be wX ±1, depending onthe configuration of the mirror Chern numbers.

and Chapter 5. In a cubic system, there are three distinct MCNs: C0 ≡ C+

kα0,Cπ ≡ C+

kαπ, and Cd ≡ C+

kαkβ, with α, β ∈ x , y , z and β , α, where C+

S refersto the Chern number of the Bloch states on the mirror-invariant plane (MIP) Swith eigenvalue +i under the mirror operation, see also Chapter 5. As was shownin Reference [166], the cubic symmetry implies that the MCNs in the X-invertedphase are C0 2 mod 4, Cπ 1 mod 4 and Cd 1 mod 2. These values implytwo additional Dirac nodes along the Γ–X line on the (110) surface, as discussedin Chapter 5 and Reference [166]. In the following, we show that the MCNs alsodetermine the spin texture on the (001) surface.2

The projections of themirror planes onto the (001) surface correspond to theHSLsshown in Figure 6.1 (b). Along these mirror-invariant lines (MILs), we can classifythe surface states according to their mirror-eigenvalues ±i. The bulk–boundarycorrespondence for each MIP then states that the MCN C is equal to the number ofright-moving (C > 0) or left-moving (C < 0) surface modes with mirror-eigenvalue+i, see Figures 2.2 and 6.3. There exists a certain freedom to choose signs in thecalculation of the MCNs. We use a convention (see Appendix A.3), which leads tothe positive directions shown in Figure 6.1 (b).The mirror eigenvalues also define a pseudospin of the surface states µ in the fol-

lowing way: On the ky 0 or ky π MIL, we choose a basis u1 , u2 in which2A related argument for Hg-based topological insulators was presented in Reference [200].


+i

−i+i

−i +i

−i+i

−i

E

X Γ X

(a)

+i

−i+i

−i

E

M X M

(b)

Figure 6.3: Chiral surface states with mirror eigenvalues ±i along the Γ–X (a) and X–M line(b) in positive direction (see Figure 6.2) for C0 −2 and Cπ 1.

the mirror operator takes the form My −iµy , where µα is the α-th Pauli matrix.Furthermore, on the kx 0 and kx π MILs we can choose the mirror operatorMx −iµx . The pseudospin is then given by the spinor u au1 + bu2 ≡ (a , b)t. Itsrelation to the physical spin of the electron is detailed in Section 6.3. It follows that,along theMILs, the pseudospin lies in the surface plane and is always perpendicularto the MIL. In order to make the connection to the pseudospin texture, it is usefulto consider the effective Hamiltonian close to the Dirac node at the HSP K Γ orK X:

HK(q) vxKµy qx − v y

Kµx qy i(vxK My qx − v y

K Mx qy) . (6.2)

Here, we measure the momentum relative to the respective HSP, q k − K. At theΓ point, the cubic symmetry implies that vx

Γ v y

Γand the resulting pseudospin

texture necessarily has a winding number wΓ 1. But at the X points, vxX, v y

Xin general, and the winding number of the pseudospin texture is wX sgn(vx

Xv y

X).

Because theMCNs fix the direction of the pseudospin at the points where the Fermilines cross the MILs, the MCNs also fix the relative sign between vx

Xand v y

Xand

hence the winding number wX. It is then easy to see that the set (C0 , Cπ) (2, 1)implies the pseudospin texture shown in Figure 6.2 (a), while (C0 , Cπ) (−2, 1)implies the pseudospin texture shown in Figure 6.2 (b). For linear Dirac cones atΓ and X, there are no other possibilities, i.e., higher MCNs imply additional Diracnodes along HSLs.3

3Note that in Figure 6.2 we assume a chemical potential above the Dirac nodes.

Chapter6

6.2 Hybridization matrix defines mirror Chern numbers | 65

d

f

f

d

E

X

E

X

E

X

Figure 6.4: The creation of band inversions at theX point bymoving the d and f bands relativeto each other.

6.2 Hybridization matrix defines mirror Chern numbers

We now analyze the connection between microscopic parameters of the electronicHamiltonian and the set of MCNs. From ab initio calculations, it is known thatthe states near the Fermi energy in SmB6 are predominantly formed by the Sm 5delectrons of Eg symmetry and the Sm 4 f electrons in the J 5/2 multiplet [163, 194,

195, 198]. The latter splits further into a Γ8 quartet,Γ(1)8,±

⟩

√56± 5

2⟩+

√16∓ 3

2⟩

andΓ(2)8,±

⟩

± 12⟩, and a Γ7 doublet,

Γ7,±⟩

√16± 5

2⟩ −√

56∓ 3

2⟩, where the index

± is the orbital pseudospin.4 Our strategy is to start in the trivial insulating phasewithout band inversion and consider the effective model, which describes the gapclosing and subsequent band inversion at the X points, see Figure 6.4.

The little co-group at the X point is isomorphic to the tetragonal symmetry groupD4h. Thus, all the irreducible representations are at most two-dimensional and theband inversion occurs between the energetically highest single Kramers pair off electrons fX,± and the energetically lowest single Kramers pair of d electronsdX,↑↓. Near the transition between the trivial and the topological phase, the low-energy electronic structure can be obtained from an effective 4×4BlochHamiltonianaround the X points,

HXeff(q)

©«εd

q Φ†qΦq ε

fq

ª®¬ . (6.3)

4This orbital pseudospin should not be confused with the surface-state pseudospin defined before. Thedetailed relation between the orbital pseudospin and the physical spin of the f electrons is given inAppendix A.4.


Equation (6.3) is given for a spinor ψ (dX,↑, dX,↓, fX,+ , fX,−

) t and q is measuredfrom X, q k − X. The simultaneous presence of inversion symmetry and TRS

allows us to choose the hybridization matrix in the form Φq iφq · σ, with φq

φ∗q −φ−q and σ the Pauli matrices in spin space. In the following, we consider

X (0, 0, π), and expand to lowest order in q: εdq εd1, ε

fq ε f 1 and

Φq i[φ1(σx qx + σy qy) + φ2σz qz

]. (6.4)

In the following, we will show that the relative sign between the two independentparameters φ1 and φ2 of the linearized hybridization matrix (6.4) determines theset of MCNs and hence the surface-state spin texture in the X-inverted phase.

First, we address the MCN C0 and therefore consider the mirror plane kx 0.The mirror operator in the basis of Equation (6.3) is Mx −iτz ⊗ σx . Thus, in thesubspace Mx +i, Equation (6.3) reduces to

H+

eff,kx0(q) ε1 − φ1qyµx + φ2qzµy − ∆ µz , (6.5)

where µα are the Pauli matrices acting on the basis vectors (1,−1, 0, 0)/√2 and(0, 0, 1, 1)/√2 and we have defined ε ≡ 1

2 (εd + ε f ) and ∆ ≡ 12 (ε f − εd). The total

Berry flux contribution of the lower band of a Dirac model h(k) ε(k) d(k) ·µwithd d/||d || is given by

CDirac

14π

∫dk1 dk2 d(k) ·

(∂d∂k1× ∂d∂k2

), (6.6)

see Equation (2.6). For our case with d(k) (−φ1k1 , φ2k2 ,−∆), this leads to

CDirackx0

12 sgn(∆φ1φ2) . (6.7)

Therefore, starting from the trivial phase with ∆ < 0 and creating a band inversionat X (∆ > 0) leads to aMCN of C0 2 sgn(φ1φ2), where the factor 2 comes from thefact that there are two X points in the kx 0 plane.

Chapter6

6.2 Hybridization matrix defines mirror Chern numbers | 67

For the mirror plane kz π, the mirror operator is Mz −iτz ⊗ σz and theeffective Hamiltonian in the Mz +i subspace is

H+

eff,kzπ(q) ε1 + φ1(qyµx + qxµy) − ∆ µz , (6.8)

using the basis vectors (0, 1, 0, 0) and (0, 0, 1, 0). It only depends on the parameterφ1 and the contribution to the total Berry flux amounts to

CDirackzπ

12 sgn(∆φ2

1) 12 sgn(∆) . (6.9)

Using the same argument as for C0 and with the fact that there is only one X pointin the kx π plane, the MCN is therefore always Cπ 1 in the X-inverted phase.Finally, consider the mirror plane kx ky with Mx y −iντz

1√2(σy − σx), where

ν ±1 is the orbital rotation eigenvalue in the considered subspace. There, weobtain the effective Hamiltonian

H+

eff,kxky(q) ε1 + νφ1qx yµx + φ2qzµy − ∆ µz , (6.10)

where we use qx y :√

2qx with qx qy and we chose the basis vectors ((1 +

i)/2, ν/√2, 0, 0) and (0, 0, (1 + i)/2,−ν/√2). The choice nmp 1√2(ey − ex) corre-

sponds to k1 ≡ qz and k2 ≡ qx y . Then, analogous to Equation (6.7), we obtain theBerry flux

CDirackxky

12 ν sgn(∆φ1φ2) , (6.11)

corresponding to a MCN Cd ν sgn(φ1φ2) in the X-inverted phase.In summary, we obtain C0 2 sgn(φ1φ2), Cπ 1, and Cd ν sgn(φ1φ2),

where ν −1 for a band inversion between (x2 − y2) and a linear superpositionof Γ(1)8 and Γ7, and ν 1 for a band inversion between (3z2 − r2) and Γ(2)8 . Hence,if sgn(φ1φ2) 1 (−1), we recover the set of MCNs which imply the pseudospintexture in Figure 6.2 (a) [Figure 6.2 (b)]. In general, we obtain

wX sgn(φ1φ2) . (6.12)


6.3 Relation between physical spin and pseudospin

In Section 6.1, we have derived the close relationship between MCNs and the pseu-dospin texture at the surface. However, it is the texture of the physical spin that weare interested in, which we will discuss in the following. Because the f electronsexperience strong SOC, the orbital pseudospin is not equivalent to the physical spinof the electrons and the mirror and spin operators do not commute. The rela-tion between physical and orbital pseudospin for the J 5/2 multiplet is given inAppendix A.4.

According to the definition of the pseudospin in Section 6.1, a surface pseudospinin positive n direction corresponds to an eigenvalue −i of Mn . In order to find arelation between the physical-spin and pseudospin texture of the surface states, wetherefore consider the effect of the projector Pps

n ≡ 12 (1+ iMn) on the physical-spin

operator Sn , where Ppsn projects onto the subspace Mn −i and n is the normal

vector of the mirror plane. One can show that, for the Eg and J 5/2 multiplets,

Ppsn Sn′P

psn ≡ 0 for n ⊥ n′ , (6.13)

which states that on a MIL, the physical spin is always parallel (or antiparallel)to the surface-state pseudospin. Whether the two are parallel or antiparallel isdetermined by the eigenvalues of the projected spin operator,

Spsn ≡ Pps

n SnPpsn . (6.14)

For the d orbitals we have S σ leading to eigenvalues +1 of Spsn , while for the Γ7,

theΓ8, and the fullmodel, we obtain the (approximate) spectra −0.24, 0.52, 0.14,and 0.71, 0.14,−0.43, respectively, see Appendix A.4. As all eigenvalues are posi-tive for the Γ8 model, the physical spin is indeed always parallel to the surface-statepseudospin and all findings concerning the pseudospin are directly transferable tothe physical spin. This is not the case if we also consider the Γ7 orbital, because theprojected spin operator of f electrons also has negative eigenvalues. In these cases,the relation between pseudospin and physical spin of the surface states depends onthe orbital character of the state. In all cases we have studied, the winding numberof the physical spin sufficiently close to the Dirac node is nevertheless identical to

Chapter6

6.4 Model calculations for SmB6 | 69

the winding number of the pseudospin. However, the direction may be reversedaround some of the Dirac points. Indeed, we find that this may occur for the Γ7model, signaling a dominant (in terms of spin) Γ7 character of the surface states,see Section 6.4.2.

6.4 Model calculations for SmB6

In Chapter 5, we have discussed a simplifiedmodel for a TKI, an extension of whichwe will study in Section 6.4.4. However, for a more realistic description of SmB6,usually the two Eg orbitals and one or several of the spin–orbit coupled f orbitals Γ7and Γ8 are used [161, 194, 197, 198]. The complete Hamiltonians can be constructedfrom the individual intra- and inter-orbital hopping amplitudes. Here, we startfrom the model defined in Reference [198] with a selection of nonzero parametersas a basis for our numerical calculations. In the following definitions, τα and σαwill denote the Pauli matrices in orbital and spin space, respectively.Then, with hopping and hybridization defined by

hd(k) σ0

(− 3

2 (c1+c2)(t(1)d +2t(2)d c3

) √3

2 (c1−c2)(t(1)d −2t(2)d c3

)√

32 (c1−c2)

(t(1)d −2t(2)d c3

)−4t(2)d c1c2−2t(1)d c3− 1

2 (c1+c2)(t(1)d +2t(2)d c3

) ) , (6.15a)

h7(k) σ0[ε7−2t(1)7 (c1+c2+c3)−4t(2)7 (c1c2+c2c3+c3c1)−8t(3)7 c1c2c3

], (6.15b)

h8(k) σ0

(ε8− 3

2 (c1+c2)(t(1)8 +2t(2)8 c3

) √3

2 (c1−c2)(t(1)8 −2t(2)8 c3

)√

32 (c1−c2)

(t(1)8 −2t(2)8 c3

)ε8−4t(2)8 c1c2−2t(1)8 c3− 1

2 (c1+c2)(t(1)8 +2t(2)8 c3

) ) , (6.15c)

h78(k) (

t(1)78 σ0(−2c3+c1+c2)+2t(2)78 ((c1+c2)c3+−2c1c2)σ0+2√

3it(2)78 s3(s1σ2−s2σ1) ···√

3t(1)78 σ0(c1−c2)−t(2)78

(2√

3σ0(c1−c2)c3−4iσ3s1s2+2is3(σ2s1+σ1s2)) )

,

(6.15d)

Φ7(k) − i(

V (1)7 (2s3σ3−(s1σ1+s2σ2))+V (2)7

(2√

3c3(s1σ1+s2σ2)−2√

3(c1+c2)s3σ3

)···

V (1)7

(−√3(s1σ1−s2σ2)

)+V (2)7 (c3(s1σ1−s2σ2)+4(c2s1σ1−c1s2σ2)−2(c1−c2)s3σ3)

),

(6.15e)


Φ8(k) −i(

3/2 V (1)8 (s1σ1+s2σ2)+3V (2)8 [(c1+c2)s3σ3+c3(s1σ1+s2σ2)] ···−√3/2 V (1)8 (s1σ1−s2σ2)+

√3V (2)8 [(c1−c2)s3σ3+c3(s1σ1−s2σ2)] ···

−√3/2 V (1)8 (s1σ1−s2σ2)+√

3V (2)8 [(c1−c2)s3σ3+c3(s1σ1−s2σ2)]V (1)8 [2s3σ3+1/2(s1σ1+s2σ2)]+V (2)8 [(c1+c2)s3σ3+4(c2s1σ1+c1s2σ2)+c3(s1σ1+s2σ2)]

),

(6.15f)

and the spinor

ψ

(dx2−y2 ,↑↓, d3z2−r2 ,↑↓, f

Γ(1)8 ,± , f

Γ(2)8 ,± , f

Γ7 ,±) t, (6.16)

the Hamiltonian of the full model can be written as

Hfull

©«hd Φ8

† Φ7†

Φ8 h8 h78†

Φ7 h78 h7

ª®®®®¬. (6.17)

Compared to Reference [198], our parameters are (for γ 1, 2, 3) t(γ)d tdηdγz ,

t(γ)7 t f ηf γ7 , t(γ)8 t f η

f γz , t(γ)78 t f η

f γx7 , V(γ)7 vηvγ

z7 , V(γ)8 vηvγzz , ε7 ε

f

Γ7,

ε8 εfΓ8, and we set εd 0. We use t(1,2) (V(1,2)) to denote first and second

neighbor hopping (hybridization) parameters, respectively.Reduced models including only the Γ8 or only the Γ7 orbitals can be obtained by

removing one line and column of the Bloch matrix in Equation (6.17) and will bediscussed in Sections 6.4.1 and 6.4.2.

6.4.1 Calculations for the Γ8 model

In the following, we will illustrate our theoretical findings by calculations withan effective lattice model for SmB6, restricting ourselves to the Γ8 quartet for felectrons. The model is similar to those used in References [161] and [198]. TheBloch Hamiltonian is an 8×8 matrix

H(Γ8) ©«

hd Φ8†

Φ8 h8

ª®¬ , (6.18)

Chapter6


where the hopping of d and f electrons and the hybridization are given in Equa-tion (6.15) with the spinor

ψ

(dx2−y2 ,↑↓, d3z2−r2 ,↑↓, f

Γ(1)8 ,± , f

Γ(2)8 ,±

) t. (6.19)

The hopping and hybridization parameters should be considered as renormalizeddue to a strong local Coulomb interaction for the f electrons [188–190], see alsoChapter 7. A typical band structure in theX-inverted phase (without hybridization)is shown in Figure 6.5 (a).For kx ±ky , the offdiagonal elements of both hd and h8 vanish and the d and f

electrons are split into (x2−y2) and (3z2−r2) orbitals, and Γ(1)8 and Γ(2)8 , respectively.Therefore, at the point X (0, 0, π)we obtain

hd(X) σ0 diag[−3

(t(1)d − 2t(2)d

), t(1)d − 2t(2)d

], (6.20)

and similarly for the Γ8 orbitals. Ab initio calculations suggest that t(1)d , t(2)8 > 0 andt(2)d , t(1)8 < 0, such that the band inversion occurs between the (x2 − y2) and the Γ(1)8orbitals [157]. The hybridization matrix for these two orbitals can be expanded tofirst order at the X point:

−iΦq 32 (σx qx + σy qy)

(V(1)8 − 2V(2)8

)−6V(2)8 σz qz . (6.21)

Therefore, according to Equation (6.12), we obtain

wX − sgn[V(2)8

(V(1)8 − 2V(2)8

)], (6.22)

leading to the phase diagram shown in Figure 6.5 (b). As discussed in Section 6.2,ν −1 for (x2 − y2) and Γ(1)8 orbitals, such that we expect (C0 , Cπ , Cd) (2, 1,−1)in phase I, leading to a spin texture with wX 1, while in phase II, we expect(C0 , Cπ , Cd) (−2, 1, 1) and wX −1. At the phase transitions V(2)8 0, thehybridization vanishes along the Γ–X line, for V(2)8

12 V(1)8 it vanishes at both

the X–M and X–R lines. This causes the hybridization gap to close and the MCNs(C0 , Cπ , Cd) to change by (±4, 0,∓2). We remark that due to the fact that kx , ky

along the X–M, there is a finite mixing of the different Eg and Γ8 orbitals, such


(a) Band structure without hybridization.

V(1)8

V(2)8

II

I(c)

(d)

(b) Phase diagram showing parame-ters for (c) and (d).

(c) Spin texture in phase I where(C0 , Cπ , Cd) (2, 1,−1) for para-meters (V (1)8 ,V (2)8 ) (−0.1, 0.1).

(d) Spin texture in phase II where(C0 , Cπ , Cd) (−2, 1, 1) for para-meters (V (1)8 ,V (2)8 ) (0.3, 0.07).

Figure 6.5: Results for the Γ8 model defined in Equation (6.18) with t(1)d 1, t(2)d −0.2,t(1)8 −0.03, t(2)8 0.02, and ε8 −3.

Chapter6


that the energy gap does not close for exactly the same parameters as along theX–R line. Therefore, there exists an additional phase with higher values for theMCNs between phases I and II, which is not shown in Figure 6.5 (b), similar to thosefound in Section 6.4.4 [201]. Figures 6.5 (c) and (d) show the physical-spin texture inphases I and II, respectively. They were calculated for a slab of 500 unit cells andfit the expected spin texture according to Sections 6.1 and 6.3.

6.4.2 Calculations for the Γ7 model

Now, we want to consider only the Γ7 doublet for f electrons and the Eg quartetfor d electrons. Then, the Bloch Hamiltonian is given by

h(Γ7) ©«

hd Φ7†

Φ7 h7

ª®¬ , (6.23)

where the hopping and hybridization parts are defined in Equation (6.15).As in Section 6.4.1, we consider the situation where the (x2 − y2) orbitals are

lower in energy at the point X (0, 0, π), such that the inversion occurs betweenthose and the Γ7 orbitals. The hybridization matrix for these two orbitals can beexpanded to first order at the X point:

−iΦq (σx qx + σy qy)(V(1)7 + 2

√3V(2)7

)+ σz qz

(2V(1)7 − 4

√3V(2)7

). (6.24)

Therefore, we obtain

C0 2 sgn[ (

V(1)7)2 − 12

(V(2)7

)2], (6.25)

leading to the phase diagram shown in Figure 6.6 (b). For dominant nearest-neighbor hybridization (phase I), we therefore expect a spin texture with wX 1,while we expect wX −1 for dominant next-to-nearest neighbor hybridization(phase II), see also Figure 6.2. At the phase transition V(1)7 2

√3V(2)7 , the hy-

bridization vanishes along the Γ–X line, for V(1)7 −2√

3V(2)7 at both the X–M andX–R lines.Our numerical calculations for the Γ7 model confirm both the MCNs andthe expected spin texture, see Figures 6.6 (c) and 6.6 (d).



V(1)7

V(2)7

I

II(d)

(c)

(b) Phase diagram indicating parame-ters for (c) and (d).

(c) Spin texture in phase I where(C0 , Cπ , Cd) (2, 1,−1) for para-meters (V (1)7 ,V (2)7 ) (0.3, 0).

(d) Spin texture in phase II where(C0 , Cπ , Cd) (−2, 1, 1) for para-meters (V (1)7 ,V (2)7 ) (0.1, 0.1).

Figure 6.6: Results for the Γ7 model defined in Equation (6.23) and (6.15) with t(1)d 1,t(2)d −0.2, t(1)7 −0.03, t(2)7 0.02, and ε7 −3. Due to the negative eigenvalue ofthe restricted spin operator for f electrons [Equation (6.14)], the spin direction is reversedaround all HSPs when compared to Figure 6.2. Note that the magnitude of the spinexpectation value around the Γ is relatively small in (d) due to the mixing of f and dorbitals.

Chapter6


6.4.3 Calculations for the full model

Assuming that t(1)d , t(2)8 > 0 and t(2)d , t(1)8 < 0, the MCNs and therefore the spintexture depend on the energy of Γ7 and Γ(1)8 orbitals at the X point, εX

7 and εX8 ,

respectively. If the energy difference ∆εX ≡ εX8 − εX

7 is much larger than thehopping between Γ7 and Γ8 orbitals, ∆εX t78 ≡ max(|t(1)78 |, |t

(2)78 |), the band

inversion essentially occurs between the (x2 − y2) and the Γ(1)8 orbitals. Then,hopping and hybridization of the Γ7 orbitals is irrelevant for the topology and allresults for the Γ8 model in Section 6.4.1 are applicable. Instead, for ∆εX −|t(1)78 |,the Γ(1)8 orbitals are not involved in the band inversion such that we can apply theresults of the Γ7 model.However, at the X point there exists a non-vanishing mixing between those two

orbitals, which we need to take into account for comparable energies of Γ7 andΓ(1)8 orbitals. This leads to a continuous crossover between the two situations. For∆εX 0, we obtain an avoided crossing at the X point with the splitting beingdefined by t(1)78 and t(2)78 . Then, the highest f band is an equal superposition of Γ7

and Γ(1)8 orbitals,

ψX± 1√2

(f XΓ7± f XΓ(1)8

), (6.26)

where the sign ± depends on the parameters t(1)78 and t(2)78 .For this orbital, the hybridization matrix with the (x2 − y2) at the X point to first

order is given by a superposition of the matrices defined in Eqs. (6.21) and (6.24):

−i√

2Φq (σx qx + σy qy)(V(1)7 + 2

√3V(2)7 ± 3

2 V(1)8 ∓ 3V(2)8

)+ σz qz

(2V(1)7 − 4

√3V(2)7 ∓ 6V(2)8

).

(6.27)

Then, the winding number of the spin texture at the X points depends on all fourhybridization parameters,

wX sgn[(

V(1)7 + 2√

3V(2)7 ± 32 V(1)8 ∓ 3V(2)8

) (V(1)7 − 2

√3V(2)7 ∓ 3V(2)8

)]. (6.28)

This is a complicated phase diagram where the two different phases are separatedby two hyper planes. For illustration purposes, we show the phase diagrams for the


V(1)7

V(1)8

III

(a) V (2)7 V (2)8 0.

V(2)7

V(2)8

II

I

(b) V (1)7 V (1)8 0.

Figure 6.7: Phase diagrams for the full model in the case where ψX+ defined in Equation (6.26)

is the highest f orbital. The phases I and II correspond to wX 1 and wX −1, respectively.

positive sign in Equation (6.26), and only nearest and only next-to-nearest neighborhybridizations in Figures 6.7 (a) and 6.7 (b), respectively.

6.4.4 Simple model with NNN hybridization

In Chapter 5, we have defined a simplified two-orbital model to describe SmB6. Inorder to be able to discuss different spin textures of the surface states, we add a

NNN hybridization term to it. Then, using the spinor ψ

(d↑, d↓, f↑, f↓

) t, the Bloch

Hamiltonian is given by

H(s)(k) ©«h(s)d (k) Φ(s)(k)Φ(s)(k) h(s)f (k)

ª®¬ , (6.29a)

where the hopping matrices are the same as in Equation (5.7) and the modifiedhybridization is defined as

Φ(s)(k) −2[σx sx

(V1 + V2(cy + cz)

)+ σy sy

(V1 + V2(cz + cx)

)+σz sz

(V1 + V2(cx + cy)

) ].(6.29b)

Here, we again use the definitions cα ≡ cos(kα), cαβ ≡ cαcβ , and sα ≡ sin(kα). Theband structure of this model without hybridization is shown in Figure 6.8 (a). In

Chapter6


Table 6.1: Values for V1 and V2 and conditions for k such that Φ(k) 0 according to Equa-tion (6.29b) for the simple model (HSPs are not shown). k∗ is a free parameter and permu-tations of the kα are always allowed. Also shown are the HSLs to which the shown valuesfor k correspond.

V1 k HSL

0 (0, π, k∗) X–M−2V2 (0, 0, k∗) Γ–X+2V2 (π, π, k∗) M–R

−V2(1 + cos k∗) (0, k∗ ,±k∗) Γ–M+V2(1 − cos k∗) (π, k∗ ,±k∗) X–R−2V2 cos k∗ k ∈ ±k∗3 Γ–R

this simple model, the rotation operator is Rorbn 1 for all mirror planes. If there is

only a NN hybridization (V2 0), then Φ(k) 0 is only possible at one of the HSPs,k ∈ 0, π3 leading to the MCNs discussed in Chapter 5.However, in general there are multiple other possibilities for Φ(k) 0, if we

allow for nonzero NNN hybridization, see Table 6.1. If there exists a band crossingof the bare bands on one of the HSLs, phase transitions are possible by changing themodel parameters V1 and V2. While the Z2 topological indices are invariant, as noband inversion at the HSPs is changed, the MCNs can change with this procedure.This leads to a richer phase diagram which includes phases with higher values forthe MCNs and a large number of protected gapless surface modes. An examplewith two additional phases for fixed hopping parameters is shown in Figure 6.8 (b).The phase III exists due to the fact that the gaps at the X–M and X–R lines do

not close simultaneously as also indicated for the Γ8 model. The line separating itfrom phase II is obtained from calculating the crossing point of d- and f -electronbands along the X–R line and using the formula from Table 6.1. Similarly, the linesseparating phases I and III are obtained by calculating the two crossings of the dand f bands along the Γ–M lines. The two other lines, V2 − 1

2 V1 and V1 0correspond to gap closings at the Γ–X and X–M lines, respectively, similar to theresults from the Γ8 model in Section 6.4.1. The MCNs for all different phases werecalculated numerically using the same method as above [100].



V1

V2

I

II

IV

III

(b) Phase diagram.

Figure 6.8: Results for the simplified model of a TKI defined in Equation (6.29) with td 1,t′d −0.5, t′′d 0, t f −0.2td , and ε f −2. The band crossing along the Γ–M linecombined with the fact that the hybridization can vanish along all HSLs leads to a richphase diagram. Phases I and II are the same as for the Γ8 model in Section 6.4.1. Forthe two additional phases III and IV, the MCNs (C0 , Cπ , Cd) are (−2,−3, 1) and (−2, 1, 3),respectively. Note that all phases are consistent with the general results of Reference [166].

Conclusion

We have derived a close relationship between the hybridization matrix at the Xhigh-symmetry points, the mirror Chern numbers, and the spin texture of thetopologically protected surface states in TKIs. Although we have motivated ourstudy with SmB6, our theoretical calculations also applies to other topologicalinsulators. Explicit calculations for different models for SmB6 showed that thespin texture of the surface states does not only depend on the orbitals that areincluded in the effective model, but also depend on the magnitude of differenthybridization parameters. This fact needs to be kept in mind when interpretingab initio or effective-model-based calculations for this type of materials. Finally,our results can be used to infer mirror Chern numbers from spin-resolved ARPES

measurements and predict further observables.

IIInteraction effects in Kondo insulators

This part addresses the effects of interactionsamong localized electrons in Kondo insulators. Af-ter discussing the renormalization of parameters ina mean-field calculation, we will study magnetisminKondo insulatorsusing twodifferent approaches.Furthermore, we will analyze measurements of ex-citonic states in SmB6.The contents of this part of the thesis are partiallyincluded in References [186], [202], and [203].

Chapter7

Chapter 7

Interaction-driven topological phase transitions

In Part I, we have analyzed the topological properties of a noninteracting Hamilto-nian modeling a topological Kondo insulator (TKI), while keeping in mind that theband parameters are renormalized by the interactions among the f electrons. Ingeneral, these interactions can have various effects on the system, stabilizing, e.g.,magnetically ordered states, heavy or non-Fermi liquids, excitonic states, as wellas unconventional superconductors. Some of these effects will be discussed in thefollowing chapters.Here, we restrict our analysis to the situation typically found in Kondo insulators

(KIs): Interactions strongly renormalize the band parameters, but the low-energyexcitations are still described bywell-defined Fermi-liquid quasiparticles. Thus, wetreat the interactions in the quasiparticle approximation to the periodic Andersonmodel (PAM) [188–192], assuming a k-independent self-energy for the f electronsof the Fermi-liquid type Σ f (ω) a + bω + O(ω2). The Fermi-liquid quasiparti-cles in such a state are then accurately described by a noninteracting Hamiltonianwith renormalized parameters, which depend on the interaction U and the non-interacting band parameters. Specifically, all f -electron hopping amplitudes arerenormalized by t f → z2t f , the hybridization by V → zV , and the onsite-energyof f electrons by ε f → ε f + λ, where the parameters z and λ are related to thecoefficients in the expansion of the self-energy by z (1− b)−1/2 and λ a/(1− b).As mentioned in Section 3.2, the dependence of z and λ on the band parametersand U can be studied nonperturbatively by different methods.In this chapter, wewill apply themethod of slave particles. In contrast to previous

studies where the U → ∞ approximation was applied [156, 161, 162], we use the

82 | 7 Interaction-driven topological phase transitions

Kotliar–Ruckenstein (KR) slave-boson scheme at finiteU [192] to compute the renor-malization factors in a selfconsistent manner. The topological properties can thenbe analyzed in the same way as those of the noninteracting Hamiltonian discussedin Chapter 5 and we thereby demonstrate that interactions can drive topologicalphase transitions [204–208]. Remarkably, for fixed band parameters, the f -orbitaloccupation number at the topological transitions is essentially independent of theinteraction strength, thus yielding a robust criterion to discriminate between differ-ent phases. In the following, we want to explicitly study how the renormalizationparameters depend on the interaction U and the noninteracting band parameters.For this purpose, we now consider again the full Hamiltonian (5.1), including theinteraction part Hint given in Equation (5.2c). We treat the resulting problem,whichis now quartic in creation and annihilation operators, within the KR slave-bosonscheme [192].

7.1 Mean-field treatment of Kotliar–Ruckenstein slave bosons

We extend the Hilbert space of our system by introducing slave bosons with anni-hilation operators ei , siσ , and di for empty, singly occupied, and doubly occupiedf orbitals at site i, respectively. The fermionic creation and annihilation operatorsf †iσ and fiσ are replaced by

f †iσ → f †iσ z†iσ , fiσ → fiσ ziσ , (7.1a)

where we have defined new pseudofermion operators f and the boson hoppingoperator

ziσ : s†iσ di + e†i siσ . (7.1b)

The physical subspace of this extended Hilbert space is recovered by imposing theconstraints

nei + ns

i↑ + nsi↓ + nd

i 1 , (7.2a)

nsiσ + nd

i f †iσ fiσ , (7.2b)

Chapter7

7.1 Mean-field treatment of Kotliar–Ruckenstein slave bosons | 83

where we use the number operators na a† a for a ∈ e , s , d. The constraint (7.2a)states that any site must be either empty, singly occupied, or doubly occupied. Thesecond constraint (7.2b) connects the presence of an f electron to a singly or doublyoccupied site.As long as the constraints (7.2) are imposed exactly, there exist different choices

for the boson hopping operators which produce the same physical spectrum. Ithas been shown [192] that the definition

ziσ :(1 − nd

i − nsiσ

)−1/2ziσ

(1 − ne

i − nsiσ

)−1/2(7.3)

produces the correct spectrum in the mean-field approximation for the noninter-acting case, see also Appendix B.2.In order to simplify the Hamiltonian, we now assume no spatial dependence of

the boson operators and in addition replace the them by their expectation values

e 〈ei〉〈e†i 〉 , ne e2 〈ne

i 〉 , (7.4a)

s 〈siσ〉〈s†iσ〉 , ns s2 〈ns

iσ〉 , (7.4b)

d 〈di〉〈d†i 〉 , D : nd d2 〈nd

i 〉 . (7.4c)

The originally local constraints (7.2) are now only imposed on average and lead to

ns n f /2 − D , (7.5a)

ne 1 − n f + D . (7.5b)

Note that n f was defined as n f 〈 f †i↑ fi↑〉 + 〈 f †i↓ fi↓〉 and it holds 0 ≤ n f ≤ 2 while0 ≤ na ≤ 1 for a ∈ e , s , d.

Using Equations (7.4) and (7.5), the boson hopping operators (7.3) can now bereplaced by

z

√D

( n f2 − D

)+

√(1 − n f + D)

( n f2 − D

)√

n f2

(1 − n f

2

) , (7.6)


which is the form known from the Gutzwiller approximation [190]. According toEquation (7.1a), the hopping of f electrons is therefore reduced by a factor z2 andthe hybridization by a factor of z. To the resulting mean-field Hamiltonian we haveto add a Lagrange multiplier λ in order to enforce the relation n f 〈 f †i fi〉.

The variables λ, n f , and D can now be determined selfconsistently by the saddle-point equations for the free energy. For T 0, these are equivalent to

0

⟨∂H∂λ

⟩

⟨∂H∂n f

⟩

⟨∂H∂D

⟩, (7.7)

where the expectation value 〈·〉 is calculated for the occupied bands. For ourmodel,the three equations can be written as

0 1 − n f − S1 , (7.8a)

0 −λ + 2zd†zn f S2 + d†zn f S3 , (7.8b)

0 U + 2zd†zD S2 + d†zD S3 , (7.8c)

where S1, S2, and S3 are the expectation values

S1 1N

∑k

∑a

u†a(k)τz2 ua(k) , (7.9a)

S2 1N

∑k

∑a

u†a(k)1 − τz

2 h f (k)ua(k) , (7.9b)

S3 1N

∑k

∑a

u†a(k)τxΦ(k)ua(k) . (7.9c)

Here, the functions h f (k) and Φ(k) are given by Equation (5.7) and the sums∑

a

run over the two occupied bands.As the expectation values (7.9) implicitly also depend on λ, n f , and D, an analytic

treatment of these saddle-point equations is not possible. Instead, we solve themnumerically by an iterative method: We solve Equation (7.8a) for λ as a function ofn f and D, and use this to minimize the total energy as a function. The expectationvalues (7.9) are calculated for a discrete set of k-points.

Chapter7

7.2 Renormalization of band parameters | 85

(a) Plot of occupation numbers and hoppingrenormalization as a function of the inter-action strength.

(b) Comparison of the renormalization of thechemical potential and the Hartree contri-bution.

Figure 7.1: Interaction effects for td 1, t′d −0.4, t f −0.1, t′f 0.04, t′′ 0, ε f −8, andV 0.5.

7.2 Renormalization of band parameters

In summary, the mean-field treatment of KR slave bosons leads to a new (nonin-teracting) mean-field Hamiltonian with the renormalized parameters ˜t f z2t f ,˜t′f z2t′f ,

˜t′′f z2t′′f , V zV , and ε f ε f + λ, and a total energy offset ofN(DU − λn f ):

H → Hni Hni( ˜t f , V , ε f ) + N(DU − λn f ) . (7.10)

The renormalization factor z, given in Equation (7.6), reduces the hopping of felectrons and the hybridization as 0 ≤ z ≤ 1,.In Figure 7.1 (a), we show the selfconsistent values of D, n f , and z as a function

of the interaction U for noninteracting band parameters in the BI phase, see Sec-tion 5.2.1 for details. For a repulsive interaction, U > 0, the double occupancy D isreduced below its noninteracting value

D0 〈ni↑〉〈ni↓〉 n f

2

4 , (7.11)

and hence D/D0 < 1. This suppression of D leads to a reduction of the renormal-ization factor, z < 1. As a consequence, the (weak) dispersion of the f electrons


Figure 7.2: The renormalized chemical potential ε f + λ is shown in red for different values ofthe interaction U. The noninteracting band parameters were chosen as in Figure 7.1 andremain fixed. At U 0 the system is in the trivial phase BI, for different finite values of Uit undergoes topological phase transitions to the phases STI(X) and WTI(ΓX). The labels ofthe different phases follow the convention of Section 5.2.1.

is further suppressed and the hybridization gap is reduced by the interactions. InFigure 7.1 (b), we show the shift λ of the f -orbital level. For small U, this shiftfollows the Hartree contribution λH Un f /2, but clearly deviates for strongerinteractions where it saturates. This dependence of λ on U is also reflected in thedependence of n f on U shown in Figure 7.1 (a): n f is reduced for small interactionsbut then approaches a constant value that corresponds to a half-filled f band.

7.3 Phase transitions and phase diagram

The shift of the f -electron level by λ as a function of U can fundamentally affect thetopological properties of the renormalized quasiparticle Hamiltonian, because thepoints with band inversion may change, see Section 5.2.1. In fact, the interactionmay drive topological phase transitions between various trivial and topologicalphases. Figure 7.2 shows the topological phase transitions for one particular choiceof noninteracting band parameters.

In Figure 7.3, we show the ε f –U phase diagram for a fixed choice of noninter-acting band parameters, illustrating how the lines of phase transitions bend with

Chapter7

7.3 Phase transitions and phase diagram | 87

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

X Γ M R

BI

STI(X)

WTI(ΓX) STI(R) BI

0

2

4

6

8

10

ε f

U

Figure 7.3: Phase transitions of the full Hamiltonian (5.1) in a mean-field treatment of theslave bosons for td 1, t′d −0.4, t f −0.1, t′f 0.04, t′′ 0, and V 0.5. The labeling ofphases and phase transitions follows the same convention as in Figure 5.5. The occupationof f orbitals is virtually constant along the lines of phase transitions as the renormalizedchemical potential for the f electrons remains constant along those lines (see also Table 7.1).

increasing U. Similar interaction-driven phase transitions have also been observedin interacting versions of quantum spin Hall models [204–206] and a 2D model ofa topological Kondo insulator [207, 208]. We note that in this context, the renor-malization of the hopping of f electrons by a factor z2 and the hybridization by afactor of z barely affects the topology of the system.The results in Figure 7.3 show that, for a fixed onsite potential ε f , the topological

phase also depends on the interaction strength U. Therefore, the value of ε f doesnot provide a robust indicator of the topological phase. Remarkably, however,along the lines of phase transitions in the ε f –U diagram, the filling of f orbitalsstays almost constant, see Table 7.1. This is due to the fact that it much morestrongly depends on the renormalization of the chemical potential than on therenormalization of the hopping and hybridization parameters. Therefore, for fixednoninteracting hopping and hybridization parameters, the filling n f is a moreuseful indicator of the topological phase than ε f .

We also remark that when starting from fixed noninteracting band parametersin the mixed-valence regime with n f > 1, one generically ends up in the local-moment regime at n f ≈ 1 for large interactions, see Section 7.2. This is because theelectron–electron interaction among the f electrons suppresses double occupancy


Table 7.1: Comparison of occupation of f orbitals for different values of the interaction forthe four lines of phase transitions shown in Figure 7.3. The occupations change very littlewhen moving along a line of phase transition for varying U. Therefore, the filling n is agood indicator of the phase for fixed hopping and hybridization.

X Γ M R

U=0ε f −3.96 −1.32 0.44 11.88

n f /2 0.865 0.578 0.387 0.016

U=10ε f −12.98 −8.42 −1.24 11.79

n f /2 0.836 0.559 0.375 0.016

but also pushes electrons from the correlated f band into the dispersive d bandby renormalizing the f -orbital level (see Figure 7.1). As discussed in Section 5.2.2and shown in Figure 5.6 (b), the weak topological insulator phases are stronglyfavored if n f ≈ 1. As a consequence, a strong topological insulator is less likely tobe realized in the large-U limit.

Conclusion

We studied the effect of interactions among the localized f electrons by perform-ing a mean-field calculation in the KR slave-boson scheme. We demonstrated howthe renormalization of the chemical potential of f electrons changes the band-inversions and thereby the topology of the system. Depending on the noninter-acting band parameters, the interactions among electrons can destroy or facilitatea topological phase. As the occupation of f orbitals is mainly determined by therenormalized chemical potential and only weakly depends on the hopping and hy-bridization amplitudes, it stays almost constant along the lines of phase transitionsfor variable U. It therefore provides a more robust criterion for the topologicalphase than, e.g., the bare chemical potential.

Chapter8

Chapter 8

Response functions from fluctuation calculations

The approach discussed in the previous chapter did not allow us to study any or-dered phases, such as antiferromagnetism, which is known to occur in SmB6 underhigh pressure [43, 149]. Following Sebastian Doniach’s picture of the competitionbetween Kondo screening and Ruderman–Kittel–Kasuya–Yosida (RKKY) interac-tion, a quantum phase transition between the Kondo insulating and a magneticallyordered phase can occur [123]. TKIs may show novel intriguing behavior at thistransition.In this chapter, wewill adjust our approach, in order to obtain response functions

such as the spin susceptibility and the spin structure factor and study collectiveexcitations as well as ordered phases. To this end, we use an SU(2)-invariant gen-eralization of the KR slave bosons [209–211] and perform a fluctuation calculationin order to obtain dynamic response functions. A similar calculation has been per-formed previously for the Hubbardmodel [212] as well as the (standard) PAM [213].Our derivations will closely follow References [214], [212], and [213].Our first goal is to use the dynamic spin susceptibility in order to detect spin

excitons in KIs, which will also be discussed in Chapter 10. In addition, the staticspin susceptibility as well as the spin structure factor can be utilized for detectingmagnetic ordering in the system. Wewill keep our calculation as general as possibleuntil the end, such that it can be applied to a variety ofmodels even beyond the fieldof TKIs. We will present some model calculations, for which we use the simplifiedmodel for a TKI that we introduced in Chapter 5.

90 | 8 Response functions from fluctuation calculations

8.1 Problem description

We want to consider a very general model on an arbitrary lattice in an arbitrarydimension d, with nn noninteracting and ni interacting (with local Hubbard inter-action) orbitals in each unit cell. Then, we can define a Bloch Hamiltonian hk suchthat the full Hamiltonian is similar to a PAM and can be written as

H Hhop + Honsite + Hint (8.1a)

∑k

c†k hk ck +

∑i

nn+ni∑a1(εa − µ0)(c†i ,a↑ci ,a↑ + c†i ,a↓ci ,a↓)

+

∑i

nn+ni∑ann+1

Ua c†i ,a↑ci ,a↑ c†i ,a↓ci ,a↓ .

(8.1b)

Here, ck is to be understood as a 2(nn + ni) spinor, ck (ck ,1↑, . . . , ck ,nn+ni↓)t, hk isa 2(nn + ni) × 2(nn + ni)matrix, µ0 is the chemical potential, εa are orbital-specificonsite potentials, and Ua is the onsite interaction; in general, we use a as a label fordifferent orbitals. The rational behind the decision to split the Hamiltonian into ahopping term Hhop and an onsite term Honsite will be made clear at the end of thenext section and in Appendix B.4.

8.2 Spin-rotational invariant Kotliar–Ruckenstein slave bosons

Due to the interaction term Hint, the problem cannot be solved analytically ingeneral. In order to overcome this difficulty, we will perform an approximativecalculation using the spin-rotational invariant KR slave bosons [192, 209, 210].

Similar to theprevious chapter,wedefine a set of bosons for eachof the interactingorbitals, e, p0, p (p1 , p2 , p3), and d, labeling empty, singly occupied, and doublyoccupied lattice sites, respectively. The four p bosons replace the two s bosons ofthe previous chapter, where the p0 operator corresponds to an S 0 boson and pto an S 1 boson with according triplet states. The p bosons are always createdtogether with a pseudofermion fa , such that the total spin adds up to S 1/2. This

Chapter8

8.2 Spin-rotational invariant Kotliar–Ruckenstein slave bosons | 91

is most conveniently done by defining the following 2×2 matrix operators

p 12

(p0τ0 + p · τ ) , p

12

(p0τ0 − p · τ) , (8.2)

where τ0 12 and τ (τ1 , τ2 , τ3) are the Pauli matrices in spin space. Now, wereplace the physical electron operators ca by a compound of pseudofermions fa

and slave bosons (note that we have dropped the site index i and the orbital indexa in the interest of readability),

c†σ ≡∑σ′

z†σσ′ f †σ′ , (8.3a)

cσ ≡∑σ′

fσ′zσ′σ , (8.3b)

where z describes the hopping of slave bosons

z (e†p + p†d) . (8.4)

The physical subspace of this extended Fock space is then recovered upon applyingthe following (local) conditions for every interacting orbital [209–211, 215]:

e†e +3∑µ0

p†µpµ + d†d 1 , (8.5a)

3∑µ0

p†µpµ + 2d†d

∑σ↑↓

f †σ fσ , (8.5b)

p†0p + p†p0 − i(p† × p) ∑σσ′

f †σ τσ′σ fσ′ . (8.5c)

The first condition states that each lattice site needs to be either empty, singlyoccupied, or doubly occupied, and corresponds to Equation (7.2a) for the standardKR slave bosons. The second condition connects the presence of physical electronsto the singly and doubly occupied lattice sites, and replaces Equation (7.2b) of theprevious chapter. The last condition ensures that the spin of the physical electrons


matches the spin of the bosons. A detailed derivation of the different conditions isshown in Appendix B.1.

In these notes, we consider arbitrary ni > 0; however, we note that a correcttreatment of multi-orbital effects requires a more elaborated approach using ro-tationally invariant slave bosons in orbital space [216]. Our simplified treatmentis expected to produce incorrect results if multi-orbital effects of the interactionsplay an important role. In our calculations, we focus on the case ni 1 where ourtreatment is indeed correct.

The interaction part of the Hamiltonian gets strongly simplified through thistreatment,

Hint ∑

i

∑a

Ua d†i ,a di ,a , (8.6)

and we can also express the spin density operator through the slave bosons as

S ≡ 12

∑σσ′

c†σ′ τσ′σ cσ 12

p†0 p + p†p0 − i(p† × p) , (8.7)

with p (p1 ,−p2 , p3)t.Note that we have again dropped the site index in the inter-est of readability. As long as the conditions (8.5) are imposed exactly, there againexists a considerable freedom to chose the boson hopping matrix and still leave thephysical solution unchanged. It has been shown that the following choice repro-duces the correctmean-field solutions in the noninteracting limit, see Appendix B.2for details [192, 209, 210, 215]:

z → (e†LMRp + p†LMRd) , (8.8a)

with

L

((1 − d†d)τ0 − 2p†p

)−1/2, (8.8b)

M ©«1 + d†d + e†e +

∑µ

p†µpµª®¬−1/2

, (8.8c)

R

((1 − e†e)τ0 − 2p† p

)−1/2. (8.8d)

Chapter8

8.2 Spin-rotational invariant Kotliar–Ruckenstein slave bosons | 93

Note that in the reduced Fock space of the slave bosons, which is restricted by theconstraints, these additional operators do not change the effect of z: the additionalcontributions of number operators in the power series have zero eigenvalues, sincethere are always two annihilation operators to the right. In order to simplifynotation later-on we will define za τ0, βa 0, and Ua 0 for the noninteractingorbitals, a ≤ nn.

We can now absorb the boson hopping into the Bloch matrix in order to obtain aboson-dependent Hamiltonian

hk[ψ](q1 , q2) Da

(z†q1 ,a

)hk Da

(zq2 ,a

), (8.9)

where we write ψ for the collection of all bosonic fields and Da(xa) is the (block-)diagonal matrix where we iterate over the index a. We now use the bosonic fieldsψ and the fermionic Grassmann variables φ in order to write the Lagrangian of thesystem in imaginary time τ. The Lagrangian is given as

L[φ, ψ] LF[φ, ψ] + LB[ψ] , (8.10a)

where the fermionic contribution is

LF[φ, ψ] ∑

k1 ,k2

φ†k1

[δk1k2

(∂τ − µ0

)+

∑k

hk[ψ](k − k1 , k − k2)

+ Da

(δk1k2 εa + βk1−k2 ,a

) ]φk2

(8.10b)

and the bosonic Lagrangian is given by

LB[ψ] ∑i ,a

[d†i ,a(∂τ + Ua)di ,a

+ αi ,a

(e†i ,a ei ,a + p†0i ,a p0i ,a + p†i ,a · pi ,a + d†i ,a di ,a − 1

)− β0a ,i

(p†0i ,a p0i ,a + p†i ,a · pi ,a + 2d†i ,a di ,a

)− βi ,a ·

(p†0i ,a pi ,a + p†i ,a p0i ,a − ip†i ,a × pi ,a

) ].

(8.10c)


Here, the constraints (8.5) are ensured through the bosonic Lagrange multipliersα, β0, and β (β1 , β2 , β3), and we have defined β β0τ0 + β · τt . Note thatby gauge transformation, one can achieve that there is only a time derivative forthe d fields and not for the other bosonic fields in the Lagrangian and all otherbosons can be chosen to be real [209, 210]. Moreover, all onsite terms, like thechemical potential, do not get an additional z†z in the Lagrangian, because we find∑σ c†σ,i cσ,i

∑σ f †σ,i fσ,i , i.e., the number of fermions is locally and globally equal

to the number of pseudofermions, see Appendix B.4.With this, we can now write down the grand-canonical partition function in the

path-integral formalism,

Z

∫D[φ, φ∗]D[ψ, ψ∗] e−

∫ 1/T0 dτ [L[φ,ψ] , (8.11)

and the free energy,F −T log(Z) + µ0N . (8.12)

8.3 Mean-field solution

First, we are looking for a uniformmean-field solution, replacing all bosons by theirreal mean-field value ψk → δ(k)ψ, ψ ∈ R, and assume a paramagnetic solution.1This simplifies Equations (8.8), (8.9), and (8.10) as

z0 p0(e + d)√

2(1 − d2 − p20/2)(1 − e2 − p2

0/2), (8.13a)

h0k[ψ] Da(τ0z0a)hkDa(τ0z0a) , (8.13b)

L0

∑k

φk[∂τ − µ0 + h0k[ψ] + Da

(τ0(β0a + εa)

) ]φk

+

∑i ,a

Ua d2a + αa(e2

a + p20a + d2

a − 1) − β0a(p20a + 2d2

a ) .(8.13c)

1In the paramagnetic mean-field, the expectation value of the spin operator vanishes on each lattice site,Sphysical state⟩ 0. In addition, the cross product in the spin operator (8.7) vanishes, such that

the desired mean-field solution is found by p 0. The vector constraint in this mean-field is fulfilledautomatically andwe thus can also set β 0. Moreover, time derivatives vanish in the staticmean-field,∂τd 0.

Chapter8

8.3 Mean-field solution | 95

Here, we first diagonalize the matrix h0k[ψ] + Da(τ0(β0a + εa)

) − µ0 for each kby using the unitary and k-dependent matrix Uk and call the energy eigenvaluesεk ,b[ψ], b 1, . . . , 2(nn + ni):

Db(εk ,b[ψ]

) Uk[ψ]

(h0k[ψ] + Da

(τ0(β0a + εa)

) − µ0)

U†k[ψ] . (8.14)

Note that we generally use b as a band index (including spins), while a is used asan index for orbitals.Then, calculating the free energy per unit cell as a function of the remaining

bosons leads to

F0[ψ] −T∑

k

∑b

log(1 + e−εk ,b [ψ]/T

)+

∑a

[Ua d2

a + αa(e2a + p2

0a + d2a − 1) − β0a(p2

0a + 2d2a )

]+ µ0N ,

(8.15)

where 0 < N < 2(nn + ni) is the total filling per lattice site.In order to numerically find a saddle point of the free energy, we first remove

the contribution αa(e2a + p2

0a + d2a − 1) and instead directly use the constraint to

write p0a

√1 − d2

a − e2a . Then, we can numerically minimize the free energy

with respect to ea and da in an iterative process, where at each step we perform amaximization with respect to β0a and µ0, see Appendix B.3 for details. We denotethe resulting mean-field values of the bosons by ψ. In the end, we can calculate αby calculating the derivative of Equation (8.15) w.r.t. e:

α T2e∂e

[∑k

∑b

log(1 + e−εk ,b [ψ]/T

)] ψψ

. (8.16)

The numerical results of this mean-field calculation are in very good agreement,both qualitatively and quantitatively, with those obtained in Chapter 7 using thestandard KR slave-boson scheme. The phase diagram of Figure 7.3 can also be cal-culated with the approach presented here and is shown in Figure 12.1 on page 155.


8.4 Gaussian fluctuations

Now, we want to expand around these mean-field solutions of the bosons andadditionally consider their fluctuations. These fluctuations are not uniform inreal space as the mean-field solution, but have a nontrivial k-dependence. Asdiscussed above, all bosonic fields except one can be chosen real by an appropriategauge transformation. Here, we chose the double occupancy, d d′ + id′′, to bethe only remaining complex field. Then we have the following twelve real bosonicfields: ψ1 e, ψ2 p0, ψ3 d′, ψ4 d′′, ψ5 α, ψ6 β0, ψ7,8,9 p1,2,3, andψ10,11,12 β1,2,3. Please note that the numbering differs slightly from previousworks.

Our goal is to derive a bosonic action which is quadratic in the bosonic fluctua-tions (the first-order correction vanishes as we start from the saddle-point solution)and can be written in the form

δS(2)[δψ] ∑

q($m ,q)

∑µ,ν

δψµ(−q)Mµν(q) δψν(q) , (8.17)

where δψµ are the fluctuations of all the real bosonic fields and $m 2mπT withthe temperature T is the bosonicMatsubara frequency. This action can be separatedinto the fermionic part coming from LF in Equation (8.10b) and the bosonic partcoming from LB in Equation (8.10c).

In order to obtain this bosonic action, we integrate out the fermionic degrees offreedom in Equation (8.11) and expand the then purely bosonic action around themean-field solution. In order to simplify the solution, we define

hk1 ,k2 [ψ] Da

(δk1k2 εa + βk1−k2 ,a

)− δk1 ,k2µ0 +

∑k

hk[ψ](k − k1 , k − k2) ,

(8.18a)

Gk1 ,k2 [ψ] δωn ,ωm

(iωn − hk1 ,k2 [ψ]

)−1, (8.18b)

where k (ωn , k)with the fermionic Matsubara frequency ωn (2n + 1)πT.

Chapter8

8.4 Gaussian fluctuations | 97

Using the identities∫dφ∗ dφ e−φ∗Aφ Det(A) and eTr(A)

Det(eA

)(8.19)

for the integration over the fermionic fields, we obtain

Z

∫D[ψ] eTr log(−iωn+H[ψ]) e−

∫ 1/T0 dτLB . (8.20)

Here, wewriteTrwhenwe consider a summation overmomenta and theMatsubarafrequencies in addition to the simple trace of a matrix,

Tr(A) ≡ T∑ωn ,k

tr[Ak ,k(ωn)

]. (8.21)

Bosonic action

In order to calculate the fluctuations of the bosonic part of the action, we need toexpand the bosonic Lagrangian defined in Equation (8.10c) around its mean-fieldvalue:

LB[ψ] LB[ψ] +∑µ

∂LB[ψ]∂ψµ

ψ

δψµ +12

∑µ,ν

∂2LB[ψ]∂ψµ∂ψν

ψ

δψµδψν + O(δψ3) .

(8.22)Considering only the quadratic termof the expansion and including the correctmo-mentum dependence, we obtain the bosonic contribution to the kernelM definedin Equation (8.33) as

MBµν

12∂2LB[ψ]∂ψµ∂ψν

ψ

. (8.23)

Fermionic action

Now, we need to calculate the fluctuations of the fermionic part of the action. Tothat end, we expand

SF[ψ] −Tr log(−iωn + H[ψ]) (8.24)


around the mean-field value. We obtain

SF[ψ] −Tr log(−iωn + H[ψ] + δH[δψ]

) −Tr log

((−iωn + H[ψ])(1 − G[ψ]δH[δψ])

) −Tr

(log(−iωn + H[ψ]) + log(1 − G[ψ]δH[δψ])

) −Tr log(−iωn + H[ψ]) +

∞∑l1

1l

Tr(G[ψ]δH[δψ])l

≡ SF[ψ] + δSF[δψ] ,

(8.25a)

where the fluctuations δH[ψ] are defined as

δH[δψ] δH(1)[δψ] + δH(2)[δψ] + O(δψ3)

∑q

∑µ

∂H[ψ]∂ψq ,µ

ψ

δψq ,µ +12

∑q ,q′

∑µ,ν

∂2H[ψ]∂ψq ,µ∂ψq′ ,ν

ψ

δψq ,µδψq′ ,ν

+ O(δψ3) .(8.25b)

For the bosonic action (8.17), we expand Equation (8.25a) up to second order in land collect all terms which are of second order in δψµ:

δS(2)F [δψ] 12 Tr

∑q ,q′

∑µ,ν

G[ψ] ∂2H[ψ]∂ψq ,µ∂ψq′ ,ν

ψ

δψq ,µδψq′ ,ν

+

12 Tr

∑q ,q′

∑µ,ν

G[ψ] ∂H[ψ]∂ψq ,µ

ψ

δψq ,µ G[ψ] ∂H[ψ]∂ψq′ ,ν

ψ

δψq′ ,ν

.(8.26a)

This can be simplified to obtain

δS(2)F [δψ] 12

∑q ,q′

∑µ,ν

δψq ,µδψq′ ,ν Tr

[G[ψ] ∂2H[ψ]

∂ψq ,µ∂ψq′ ,ν

ψ

+ G[ψ] ∂H[ψ]∂ψq ,µ

ψ

G[ψ] ∂H[ψ]∂ψq′ ,ν

ψ

].

(8.26b)

Chapter8


Note that there is no relative sign between the two terms which is consistent withReferences [212] and [213].In order to calculate the traces entering the action, we need to find themomentum

and frequency dependence of Gk1 ,k2 [ψ] and δH[δψ]. As we have assumed auniform mean-field solution in Equation (8.13), the Green’s function is given by

Gk1 ,k2 [ψ] δk1 ,k2

[iωn −Da(τ0z0a)hk1 Da(τ0z0a) −Da

(τ0(β0a + εa)

)+ µ0

]−1.

(8.27)We can also use the unitary transformationdefined inEquation (8.14) to diagonalizethe Hamiltonian and rewrite the Green’s function as

Gk1 ,k2 [ψ] Uk[ψ]Gk1 ,k2 [ψ]U†k[ψ] δk1 ,k2 Db

[(iωn − εk1 ,b)−1] . (8.28)

For calculating the derivatives w.r.t. the different bosons, we use the followingrelations which are derived in Appendix B.5:

∂zq

∂ψq′ ,µ δq ,q′

∂z∂ψµ

,∂z†q∂ψq′ ,µ

δq ,−q′∂z†∂ψµ

, (8.29a)

∂2zq

∂ψq1 ,µ∂ψq2 ,ν δq ,q1+q2

∂2z∂ψµ∂ψν

,∂2z†q

∂ψq1 ,µ∂ψq2 ,ν δq ,−q1−q2

∂2z†∂ψµ∂ψν

.

(8.29b)

In addition, we are going to use the following definitions:

z0 Da(z0aτ0

), zµ Da

(∂za∂ψµ

ψ

), zµ Da

©«∂z†a∂ψµ

ψ

ª®¬ , (8.30a)

zµν Da©«

∂2za∂ψµ∂ψν

ψ

ª®¬ , zµν Da©«

∂2z†a∂ψµ∂ψν

ψ

ª®¬ , βµ Da

(∂βa∂ψµ

ψ

).

(8.30b)

The first derivatives of H[ψ] given in Equation (8.18a) at the uniform mean-fieldsolution (where the only non-vanishing bosonic fields are ei e, p0,i p0, d′i d,


αi α, and β0,i β0) are given by[∂H[ψ]∂ψq ,µ

ψ

]k1 ,k2

δk1−k2 ,q βµ

+

∑k

[δk−k1 ,−qδk−k2 ,0 zµhk z0 + δk−k1 ,0δk−k2 ,q z0hk zµ

] δq ,k1−k2

[zµhk2 z0 + z0hk1 zµ + βµ

],

(8.31a)

where we have defined q ($m , q). All second derivatives of βa vanish, and weobtain

∂2H[ψ]∂ψq ,µ∂ψq′ ,ν

ψ

k1 ,k2

∑k

[δk−k1 ,−q−q′δk−k2 ,0 zµνhk z0 + δk−k1 ,0δk−k2 ,q+q′ z0hk zµν

+ δk−k1 ,−qδk−k2 ,q′ zµhk zν + δk−k1 ,−q′δk−k2 ,q zνhk zµ

] δk2 ,k1−q−q′

[zµνhk2 z0 + z0hk1 zµν + zµhk2+q′ zν + zνhk1−q′ zµ

].

(8.31b)

Combining all these results, we can write the fermionic part of the second-ordercorrection of the action as

δS(2)F [δψ] T2

∑q ,q′

∑µ,ν

δψq ,µδψq′ ,ν tr

[δq ,−q′

∑k

Gk[ψ][zµνhk z0 + z0hk zµν + zµhk+q′ zν + zνhk−q′ zµ

]+ δq ,−qδq′ ,q

∑k

Gk[ψ][zµhk+q′ z0 + z0hk zµ + βµ

]Gk+q′[ψ]

[zµhk z0 + z0hk+q′ zµ + βµ

] ].

(8.32)

Chapter8


Then, the kernelM defined in Equation (8.17) is given by

MFµν(q)

T2 tr

∑k

[Gk[ψ]

[zµνhk z0 + z0hk zµν + zµhk+q zν + zνhk−q zµ

]+ Gk[ψ]

[zµhk+q z0 + z0hk zµ + βµ

]Gk+q[ψ]

[zνhk z0 + z0hk+q zν + βν

] ].

(8.33)

For the derivatives of the Hamiltonian, we only need to calculate the derivativesof z and β w.r.t. the different bosonic fields at their mean-field values. For βq , thisis trivially

∂β

∂βµ

ψ

τµ (8.34)

and vanishes for all other bosonic fields. All second derivatives vanish as well,which has already been used in the calculations (8.31).The derivatives of z w.r.t. the constraint fields α, β0, and β vanish, but derivatives

w.r.t. the remaining bosons are nontrivial. In order to calculate the derivatives, werewrite Equation (8.8) in the simple form

z z(+)τ0 + z(−)3∑

i1

pi|p | τi , (8.35a)

where we used the definitions (here, |p | √∑3

i1 p2i )

z± p0(e + d′ + id′′) ± |p |(e − d′ − id′′)√

2[1 − d′2 − d′′2 − (p0 ± |p |)2/2

] [1 − e2 − (p0 ∓ |p |)2/2

] , (8.35b)

z(±) z+ ± z−2 , (8.35c)

see derivation in Appendix B.6. All derivatives are well-defined at the mean-fieldsolution. They can be determined analytically but will be calculated numericallyin the final computation.


Matsubara summations

In the result (8.33), we need to calculate two different types of Matsubara summa-tions,

T∑

nG(ωn ,k)[ψ] and T

∑n

G(ωn ,k)[ψ]M(k , q)G(ωn+$m ,k+q)[ψ] , (8.36)

with a non-ωn-dependent matrix M(k , q), which in this case is given by

M(k , q) [∂H[ψ]∂ψ−q ,µ

ψ

]k ,k+q

. (8.37)

To facilitate the analytic calculation, we use the the diagonal form of the Green’smatrices defined in Equation (8.28). With this, we obtain for the first type ofsummation

T∑

nG(ωn ,k)[ψ] U†k[ψ]

(T

∑n

G(ωn ,k)[ψ])

Uk[ψ]

U†k[ψ]Db[nF(εk ,b)

]Uk[ψ] ,

(8.38)

with the Fermi-Dirac distribution nF(ε) (eε/T + 1

)−1. The second type of Mat-

subara sum can also be calculated analytically, leading to

T∑

nG(ωn ,k)[ψ]M(k , q)G(ωn+$m ,k+q)[ψ]

U†k[ψ](T

∑n

G(ωn ,k)[ψ]Uk M(k , q)U†k+q G(ωn+$m ,k+q)

)Uk+q[ψ]

U†k[ψ][ nF(εk ,b) − nF(εk+q ,b′)

i$m + εk ,b − εk+q ,b′

(Uk M(k , q)U†k+q

)b ,b′

]b ,b′

Uk+q[ψ] .

(8.39)

The details of both calculations are shown in Appendix B.7.

Chapter8

8.5 Spin susceptibility and spin structure factor | 103

Result

Inserting these results and Equation (8.31) into Equation (8.33) and using Equa-tion (8.23), we obtain the final result for the kernel defined in Equation (8.17)

Mµν(q) 12∂2LB[ψ]∂ψµ∂ψν

ψ

+12

∑k

∑b

nF(εk ,b)[Uk[ψ]

(zµνhk z0 + z0hk zµν + zµhk+q zν + zνhk−q zµ

)U†k[ψ]

]b ,b

+

∑b ,b′

nF(εk ,b) − nF(εk+q ,b′)i$m + εk ,b − εk+q ,b′

[Uk

(zµhk+q z0 + z0hk zµ + βµ

)U†k+q

]b ,b′[

Uk+q

(zνhk z0 + z0hk+q zν + βν

)U†k

]b′ ,b

.

(8.40)

Here, we are interested in the continuation of this kernel to the real axis. In orderto perform the numerical calculations, we need to add an imaginary regulationparameter, such that we replace i$m → $ + iη. The parameter η describes theLorentzian broadening of delta functions and we use η 0.01 for our numericalcalculations. In the following we will use only $ in order to simplify our mathe-matical expressions, when we actually mean $ + iη.

8.5 Spin susceptibility and spin structure factor

It has been shown in Reference [212] that the spin susceptibility can be calculatedfrom the inverse of the matrix M given in Equation (8.40). In general, the spinsusceptibility is a 3×3 matrix given by

χs (q) 2p20[S−1(q)] 7,8,9,7,8,9 . (8.41)

Therefore, our calculation enables us to calculate the dynamic spin susceptibilityas a function of momentum and frequency, χs (q , $).2 In the case of an SU(2)-symmetric result, which we find for all models that we considered, the expressionfor χs is significantly simplified. There, we find M77 M88 M99, M7,10

2Remember that we have set q ≡ (q , $).


M8,11 M9,12 M∗10,7 M∗11,8 M∗12,9, andM10,10 M11,11 M12,12, suchthat the spin susceptibility is isotropic and given by

χs (q) 2p20[S−1(q)]77

2p20M10,10(q)

M77(q)M10,10(q) −M7,10(q)

2 . (8.42)

The imaginary part of the spin susceptibility at finite frequency provides infor-mation about the excitation spectrumof the systemandallowsus todetect collectiveexcitations such as spin excitons, see also Chapter 10. We expect to see a clear peakof Im[χs (q , $)] below the particle–hole continuum, which is a clear signature of anexcitonic state.

In order to detect magnetic instabilities, we can instead consider the static spinsusceptibility lim$→0 χs (q , $). By calculating lim$→0 Im

[χs (q , $)

] /$, as a func-tion of the crystal momentum, we find the ordering vector as the peak of the spinsusceptibility. A phase transition to a magnetically ordered phase as a function ofU is indicated by a divergence of the static spin susceptibility at the ordering vector.

For magnetic phases at finite temperature, we have to consider also the effect ofexcitations with finite energies. To this end, we define the spin structure factor

Ss (q ,Ω) −∫ Ω

−∞d$π

Im[χs (q , $)

]1 − e−$/T

, Ss (q) ≡ Ss (q ,∞) . (8.43)

At low temperatures, the calculation of the spin structure factor is not necessary inorder to detect magnetic instabilities and it is more convenient to simply utilize thestatic spin susceptibility.

Conclusion

In this chapter, we have presented an extensive calculation based on the spin-rotational invariant generalization of the KRs slave bosons [192, 209]. We havedescribed how it is possible to obtain response functions, such as the dynamic spinsusceptibility, from a fluctuation calculation for the bosonic fields. This calculation,which canbe applied to avariety ofmodels, provides a route todetect bothmagneticphases and collective excitations in KIs.

Chapter9

Chapter 9

Kondo lattice model and RKKY interaction

An alternative approach for describing magnetism in KIs is the derivation of theRKKY interaction [125, 154, 155]. In the presence of strong interactions of f electrons,the system is driven towards a half-filled f shell, as we have seen in Chapter 7.There, the Kondo lattice model (KLM) is an appropriate description of low-energyexcitations and we can derive the RKKY interaction by performing second-orderperturbation theory, see Section 3.2. In this chapter, we will start from the PAM

that we introduced in Chapter 5 and derive the related KLM by means of theSchrieffer-Wolff transformation [152]. We will continue with a calculation of theRKKY interaction in Section 9.2, which we will discuss in particular with respect tothe impact of different types of hybridization in Section 9.3.

9.1 Schrieffer–Wolff transformation

We start with the model defined in Equation (5.2) for a (topological) Kondo insula-tor,

H H0 + Hhyb , (9.1)

with a hybridization of the form

Hhyb

∑αx ,y ,z

∑〈i , j〉α

[iVc†i σ

α f j+iV f †i σαc j+h.c.

]. (9.2)

Here, the notation 〈i , j〉α again stands for a nearest neighbor (NN) bond in the α-direction and the Pauli matrices in spin space are here denoted with a superscript,σα .

106 | 9 Kondo lattice model and RKKY interaction

9.1.1 Derivation of the Kondo model

Now, we want to consider a single f impurity with this type of hybridization at thepoint R. Following the approach in Reference [112], we want to derive the relatedeffective Kondo model. First, we divide the Hilbert spaceH into three subspaces,H H0 +H1 +H2, in which the impurity is empty, singly occupied, and doublyoccupied, respectively. Then, the Schrödinger equation can be written in the form

©«H00 H01 H02

H10 H11 H12

H20 H21 H22

ª®®®®¬©«ψ0

ψ1

ψ2

ª®®®®¬ E

©«ψ0

ψ1

ψ2

ª®®®®¬, (9.3)

where ψi ∈ Hi , and Hi j Pi HP j , with Pi the projector onto the subspace withimpurity occupation i.

Now, we assume that in the ground state with V 0, the impurity is singlyoccupied and both other configurations have higher energy. We can eliminate ψ0and ψ2 from Equation (9.3) and obtain

Heff ψ1 Eψ1 , with Heff H11 + H12(E − H22)−1H21 + H10(E − H00)−1H01 .

(9.4)In the n f 1 subspace, we find H11 H0 + ε f and define

H+ : H12(E − H22)−1H21 and H− : H10(E − H00)−1H01 , (9.5)

which describe virtual excitations to the doubly occupied and empty impuritysubspace, respectively. These Hamiltonians can be written as

H+

1N

∑k ,k′

−4V2ei(k′−k)RU + ε f − εbk′

(1 −

E − ε f − H0

U + ε f − εbk′

)−1

∑α,α′

sin(kα) sin(k′α′)∑µ,µ′

fµ f †µ′n f −µ′∑ν,ν′

(σαµν

)∗σα′µ′ν′ c

†kνck′ν′ ,

(9.6a)

Chapter9

9.1 Schrieffer–Wolff transformation | 107

Table 9.1: Spin representation for the f -electron operators in the n f 1 subspace.

µ′ ↑ µ′ ↓

µ ↑ Φ+ 12 − Sz , Φ−

12 + Sz Φ+ −S−, Φ− S−

µ ↓ Φ+ −S+, Φ− S+ Φ+ 12 + Sz , Φ−

12 − Sz

H− 1N

∑k ,k′

−4V2ei(k−k′)Rεk′ − ε f

(1 −

E − ε f − H0

εbk′ − ε f

)−1

∑α,α′

sin(kα) sin(k′α′)∑µ,µ′

f †µ fµ′(1 − n f −µ′)∑ν,ν′

σαµν

(σα′µ′ν′

)∗ckνc†k′ν′ .

(9.6b)

At this point, we make the approximations(1 −

E − ε f − H0

U + ε f − εbk′

)≈ 1 and

(1 −

E − ε f − H0

εbk′ − ε f

)≈ 1 . (9.7)

In order to simplify the equations, we exchange k ↔ k′, α ↔ α′, µ ↔ µ′, andν↔ ν′ in Equation (9.6b), and define

C+(k , k′) (U + ε f − εk′

)−1and C−(k , k′)

(εk − ε f

)−1. (9.8)

Furthermore, we define

Φ+µµ′ : fµ f †µ′n f −µ′ and Φ−µµ′ : f †µ′ fµ(1 − n f −µ) . (9.9)

In the n f 1 subspace, these can be replaced by spin operators of the impurity, seeTable 9.1. Using Φ±µµ′ Φ

±µµ′ − 1

2 δµµ′ , we notice that Φ− −Φ+ : Φ. This can beexpressed in terms of the Pauli matrices:

Φµµ′(R) (S(R) · σ)µµ′ . (9.10)

Finally, we use ck′ν′ c†kν δkk′δνν′ − c†kνck′ν′ and ignore the contribution of

δkk′δνν′ , which only leads to a spin-independent potential scattering term. Now


Table 9.2: The term Hα,α′ in terms of Sz , S+, and S−, where we suppressed the dependenceon R, k, and k′ (or r and r′).

α′ x α′ y α′ z

α x Sz (c†↓ c↓−c†↑ c↑)+S+c†↑ c↓+S−c†↓ c↑ −iSz (c†↓ c↓+c†↑ c↑)−iS+c†↑ c↓−iS−c†↓ c↑ Sz (c†↑ c↓+c†↓ c↑)+S+c†↑ c↑−S−c†↓ c↓

α y iSz (c†↓ c↓+c†↑ c↑)+iS+c†↑ c↓+iS−c†↓ c↑ Sz (c†↓ c↓−c†↑ c↑)−S+c†↑ c↓−S−c†↓ c↑ iSz (c†↓ c↑−c†↑ c↓)−iS+c†↑ c↑−iS−c†↓ c↓

α z Sz (c†↑ c↓+c†↓ c↑)−S+c†↓ c↓+S−c†↑ c↑ iSz (c†↓ c↑−c†↑ c↓)+iS+c†↓ c↓+iS−c†↑ c↑ Sz (c†↑ c↑−c†↓ c↓)−S+c†↓ c↑−S−c†↑ c↓

we can write the effective Kondo Hamiltonian as

HK 1N

∑k ,k′

4V2ei(k′−k)R(C+(k , k′) + C−(k , k′))∑α,α′

sin(kα) sin(k′α′)HRα,α′(k , k′) .

(9.11)The term

HRα,α′(k , k′)

∑µ,µ′

∑ν,ν′

(σαµν

)∗σα′µ′ν′ Φµµ′(R)c†kνck′ν′

Sα(R)c†kσα′ck′ + Sα′(R)c†kσαck′ − δαα′ c†k(S(R) · σ)ck′

+ i∑γ

εγαα′Sγ(R)c†k ck′

(9.12)

is shown in Table 9.2 for the 9 different combinations of α and α′. Here we haveused Equation (9.10) and

σασγσα′ iεγαα

′1 + δαγσ

α′+ δα′γσ

α − δαα′σγ . (9.13)

9.1.2 Real-space representation

We can now Fourier transform HK back to real space. Due to the factors C+ and C−,which depend on k and k′, this cannot be done analytically in general. However,we can make the approximations

C+(k , k′) C+

(U + ε f − µ0

)−1and C−(k , k′) C−

(µ0 − ε f

)−1,

(9.14)

Chapter9

9.1 Schrieffer–Wolff transformation | 109

R

x

y

±JHyx

±JHx y

±JHxx

±JHy y

Figure 9.1: Correlated hopping and spin-spin exchange interaction in the x–y plane aroundan impurity at site R. Continuous and dashed lines denote +J and −J, respectively.

with the chemical potential µ0. Analogously to Equation (9.12), we define

HRα,α′(r , r′)

∑µ,µ′

∑ν,ν′

(σαµν

)∗σα′µ′ν′ Φµµ′(R)c†ν(r)cν′(r′) , (9.15)

which is given by the same expressions shown in Equation (9.12) and Table 9.2.With

HRα,α′(k , k′)

1N

∑r ,r′

ei(kr−k′r′)HRα,α′(r , r′) (9.16)

we can perform the Fourier transform and obtain

HK V2(C++ C−)

∑r ,r′

∑α,α′

[δ(r ,R + eα) − δ(r ,R − eα)

][δ(r′,R + eα′) − δ(r′,R − eα′)

]HRα,α′(r , r′)

(9.17)


J∑α,α′

∑λ,λ′±1

λλ′HRα,α′(R + λeα ,R + λ′eα′) .

Therefore, we obtain terms coupling the spin (only for α α′) and spin-dependenthopping between the lattice sites adjacent to the impurity. The different couplingsare shown in Figure 9.1 in the x–y plane. The coefficient

J : V2(C++ C−) (9.18)

is the same for all terms (with different signs).

9.1.3 Spin–spin interaction

In the case α α′, we find local terms in the Hamiltonian (continuous red andgreen arrows in Figure 9.1), where the conduction electron returns to its initialposition after the hybridization with the f electron at site R. These are 6 (for thesix nearest neighbors to the impurity) of the total number of 62 36 terms. Forthese terms, we can use spin operators for conduction electrons in real space,

2sz(r) c†↑(r)c↑(r) − c†↓(r)c↓(r) , (9.19a)

s+(r) c†↑(r)c↓(r) , (9.19b)

s−(r) c†↓(r)c↑(r) , (9.19c)

in order to rewrite the operators Hαα(r , r):

Hxx(r , r) −2Sz(R)sz(r) + S+(R)s+(r) + S−(R)s−(r) (9.20a)

Hy y(r , r) −2Sz(R)sz(r) − S+(R)s+(r) − S−(R)s−(r) (9.20b)

Hzz(r , r) +2Sz(R)sz(r) − S+(R)s−(r) − S−(R)s+(r) (9.20c)

Alternatively, we can start from Equation (9.12) and use

s(r) 12 c†(r)σc(r) , (9.21)

which leads to

Chapter9

9.2 RKKY interaction | 111

HS−sα (r) : Hαα(r , r) 4Sα(R)sα(r) − 2S(R) · s(r) (9.22)

All six terms together can be written as an anisotropic spin–spin interaction,

HS−s J∑

αx ,y ,z

∑λ±

HS−sα (R + λeα) . (9.23)

Whether the interaction is ferromagnetic or antiferromagnetic is determined by theconstants C+ and C− given in Equation (9.14). In the n f 1 subspace, we haveU + ε f > µ0 and ε f < µ0 such that C± > 0 and consequently J > 0.

Note that a non-topological and even-parity nearest-neighbor hybridization ofthe form

Htrivhyb

∑〈i , j〉

V[c†i f j+ f †i c j+h.c.

](9.24)

would lead to an isotropic antiferromagnetic nearest-neighbor spin coupling

HtrivS−s J

∑αx ,y ,z

∑λ±

S(R) · s(R + λeα) . (9.25)

An onsite hybridization, which was usually considered before the emergence ofTKIs, leads to an antiferromagnetic onsite spin coupling.

9.2 RKKY interaction

9.2.1 The Kondo lattice model

Up to now, we have only considered the effect of a single impurity in the system;however, we are interested in the original situation where there is one impurity ateach lattice site. In this case, we can write the Hamiltonian as

H H0 + HKLM , (9.26)

where we define the Kondo lattice Hamiltonian

HKLM

∑R

HK(R) J∑R

∑α,α′

∑λ,λ′±1

λλ′HRα,α′(R + λeα ,R + λ′eα′) . (9.27)


As the conduction Hamiltonian H0 is most easily expressed in momentum space,we rewrite the Kondo Hamiltonian again in momentum space as well:

HKLM 4JN

∑R

∑k ,k′

ei(k′−k)R ∑α,α′

sin(kα) sin(k′α′)HRα,α′(k , k′) . (9.28)

By Fourier transforming the spin, we obtain

S(R) 1√N

∑q

eiqRS(q) and HRα,α′(k , k′)

1√N

∑q

eiqRHqα,α′(k , k′) , (9.29)

where

Hqα,α′(k , k′) Sα(q)c†kσα

′ck′ + Sα′(q)c†kσαck′ − δαα′ c†k(S(q) · σ)ck′

+ i∑γ

εγαα′Sγ(q)c†k ck′ ,

(9.30)

leading to

HKLM 4J√N

∑k ,q

∑α,α′

sin(kα) sin(kα′ − qα′)Hqα,α′(k , k − q) . (9.31)

9.2.2 Calculation of the RKKY interaction

In the following, we use the path-integral formalism [217] to derive the effectiveRKKY interaction for this system [125, 154, 155]. For this, let us first write theLagrangian of the system (in imaginary time τ),

L(τ) Ψ†∂τΨ + H , (9.32)

whereΨ is a vector of Grassmann variables and we have written the Hamiltonianas

H H0 + HKLM Ψ†hΨ +Ψ† ©«∑

q

∑γ

Sγ(q)g(q , γ)ª®¬Ψ . (9.33)

Here, h is a diagonal matrix,

hk ,k′ δkk′(εk − µ0) , (9.34)

Chapter9


where we have added the chemical potential µ0, and the matrices g(q , γ) can bedetermined from Equation (9.31) and Equation (9.12):(

g(q , γ))k ,k′ 4J√Nδk′ ,k−q

∑α,α′

sin(kα) sin(k′α′)[δαγσ

α′+ δα′γσ

α − δαα′σγ + iεγαα′ ].

(9.35)

For simplicity, we write∑i

:∑q ,γ

, Si : Sγ(q) , and gi : g(q , γ) (9.36)

in the following.Now, we can define the action as

S

∫ β

0dτL(τ) , (9.37)

where β 1/T with the temperature T, and write the grand-canonical partitionfunction as

Z

∫D[S]D[Ψ∗ ,Ψ]e−S (9.38)

Here, we again use the identities (8.19) and obtain the partition function

Z

∫D[S] exp

[∫ β

0dτ Tr

(log

(∂τ + h +

∑i

Si gi

))]. (9.39)

We defineM0 : ∂τ + h −i∂t + h and M1 :

∑i

Si gi (9.40)

and perform the expansion (similar to Equation (8.25a))

log(M0 + M1) log(M0(1 + M−10 M1))

log(M0) −∞∑

n1

1n(−M−1

0 M1)n .(9.41)


As the linear term vanishes, we then obtain

log(M0 + M1) log(M0) − 12 (M

−10 M1)2 + O(M4

1) , (9.42)

where the quadratic term is the RKKY interaction:

SRKKY

∫ β

0dτ Tr

[12 (M

−10 M1)2

]

12

∫ β

0dτ Tr

∑i , j

(∂τ + h)−1Si gi (∂τ + h)−1S j g j

.(9.43)

We can then write the partition function as

Z ≈ Z0 ·∫D[S] e−SRKKY , (9.44)

where the term

Z0 : exp[∫ β

0dτ Tr log(∂τ + h)

](9.45)

is the partition function of the conduction electrons which is independent from thelocal spins.

Matsubara summations

We can perform a Fourier transform from time to frequency space and define the(fermionic) Matsubara frequencies ωn (2n +1)π/β, n ∈ Z. Denoting k : (ωn , k),the operator M−1

0 can be written as the noninteracting Green’s function of theconduction electrons,

−(M−1

0

)k ,k′

δkk′Gk δkk′1

iωn − εk + µ0, (9.46)

and we can rewrite M1 using the definitions (9.40) and (9.35),

(M1)k ,k′ δnn′∑γ

Sγ(k − k′) (g(k − k′, γ))k ,k′ . (9.47)

Chapter9


Then, the RKKY interaction (9.43) can be rewritten as

SRKKY T2

∑k ,q

∑γ,γ′

tr[Gk Sγ(−q)(g(−q , γ))k ,k+q Gk+q Sγ′(q)(g(q , γ′))k+q ,k

]:

∑q

∑γ,γ′

Sγ(−q)Mγγ′(q)Sγ′(q) .

(9.48)

Here, we defined the elements of the matrixM as

Mγγ′(q) : T2

∑k

tr[(g(−q , γ))k ,k+q (g(q , γ′))k+q ,k

]∑

n(iωn − εk + µ0)−1(iωn − εk+q + µ0)−1 .

(9.49)

According to the derivation in Appendix B.7, the Matsubara sum in Equa-tion (9.49) can be simplified to

T∑

n(iωn − εk + µ0)−1(iωn − εk+q + µ0)−1

nF(k) − nF(k + q)

εk − εk+q, (9.50)

with the Fermi-Dirac distribution

nF(k) 1eβ(εk−µ0) + 1

. (9.51)

Using the definition (9.35), we can also simplify

tr[(g(−q , γ))k ,k+q (g(q , γ′))k+q ,k

] δγγ′

8J2

N(c1(2k)−3

) (c1(2(k+q))−3

), (9.52)

where we have definedc1(k)

∑α

cos kα . (9.53)

With that, we obtain the diagonal matrix

M(q) 4J2

N13

∑k

[ (c1(2k) − 3

) (c1(2(k + q)) − 3

) nF(k) − nF(k + q)εk − εk+q

], (9.54)


which can now be calculated numerically. As the diagonal elements ofM are equalfor all considered models, we will consider it as a scalar quantity from now on andrewrite Equation (9.48) as

SRKKY

∑qM(q) S(q) · S(−q) . (9.55)

9.2.3 RKKY interaction in real space

We can perform a Fourier transform of the expression (9.55),

SRKKY

∑qM(q) 1

N

∑R1 ,R2

eiq(R1−R2) S(R1) · S(R2)

∑R1 ,R2

S(R1) · S(R2) ©« 1N

∑q

eiq(R1−R2)M(q)ª®¬ ,(9.56)

in order to obtain a real-space expression for the RKKY interaction,

M(R1 − R2) : 1N

∑q

eiq(R1−R2)M(q) . (9.57)

9.3 Model calculations

We now illustrate our analytical derivations with numerical calculations for themodel defined in Equation (5.2). Depending on the model parameters, we ob-tain varying results with generally nonzero ordering vectors, see Figure 9.2 forcalculations corresponding to the band structures of Figure 5.3.

9.3.1 Comparison of different hybridizations

The form of the hybridization between localized and conduction electrons alsoinfluences the effective RKKY coupling. So far, we have assumed a “topological”hybridization defined in Equation (9.2), for which the RKKY interaction can becalculated according to Equation (9.54). On the other hand, for a trivial nearest-

Chapter9

9.3 Model calculations | 117

MJ2

(1)(2)(3)

-0.5 0 0.5

(a) Phase STI(X).

MJ2

(1)(2) (3)

-0.5 0 0.5

(b) Phase TCI(ΓM).

Figure 9.2:Calculationsof theRKKY interaction for aTKI in real (bottom) andmomentumspace(top) at half filling for different model parameters corresponding to the band structuresshown in Figure 5.3. The real-space plot shows the spin–spin interaction for first (1),second (2), and third neighbors (3). Note that negative and positive values, shown by redand blue colors, respectively, correspond to ferromagnetic and antiferromagnetic coupling,respectively.

neighbor hybridization, such as the one defined in Equation (9.24), we obtain(gtriv(q , γ)

)k ,k′

4J√Nδk′ ,k−qσ

γ∑α,α′

cos(kα) cos(k′α′) , (9.58)

leading to

Mtriv(q) 16J2

N

∑k

[ (c1(k) c1(k + q))2 · nF(k) − nF(k + q)

εk − εk+q

]. (9.59)

In addition, we can also consider a common onsite hybridization, for which thecoupling is simply given by(

gonsite(q , γ))

k ,k′

J√Nδk′ ,k−qσ

γ . (9.60)

This leads to a matrix of the form

Monsite(q) J2

N

∑k

nF(k) − nF(k + q)εk − εk+q

. (9.61)


MJ2

(1)(2)(3)

-0.015 0 0.015

(a) Onsite hybridization.

MJ2

(1)(2)(3)

-1 0 1

(b) Topologically trivial NN hybridization.

Figure 9.3: Calculations of the RKKY interaction for different types of hybridizations and thesame band parameters as in Figure 9.2 (a). The plots follow the same conventions as thosein Figure 9.2.

These differences are demonstrated in Figure 9.3 by a calculation of the RKKY

interaction for the two non-topological types of hybridizations discussed above inboth real andmomentumspace. Weobserve significant qualitative andquantitativedifferences of the RKKY interaction when comparing them to each other and theresults for the “topological” hybridization shown in Figure 9.2.

9.3.2 Multiple conduction bands

If there is not a single (spin-degenerate) conduction band but multiple ones, weneed to slightly adjust the derivation of Section 9.2.2 as described in the following.

In general, the Hamiltonian hk is given by a 2n×2n matrix (n is the number oforbitals) instead of just one (degenerate) energy εk . We can diagonalize this matrixfor every k and obtain

hk Da(εk ,a) Uk hkU†k , (9.62)

where Da describes a diagonal matrix where we iterate over the index a. As aconsequence, also the Green’s function is now given by a matrix, which can bewritten in a diagonal form analogously,

Gk ,k′ δk ,k′Da[(iωn − εk ,a)−1]

Uk Gk U†k (9.63)

Chapter9


Assuming the simplest case of an onsite hybridization, we need to calculate theMatsubara sum

T∑

nG(ωn ,k)G(ωn ,k+q) . (9.64)

Using the diagonal form of the Green’s function (9.63), we obtain

T∑

nG(ωn ,k)G(ωn ,k+q) U†k

(T

∑n

G(ωn ,k)UkU†k+q G(ωn ,k+q)

)Uk+q

U†k

[ nF(εk ,a) − nF(εk+q ,a′)εk ,a − εk+q ,a′

(UkU†k+q

)a ,a′

]a ,a′

Uk+q .

(9.65)

We still need to calculate the trace, which leads to the final result

Mmulti(q) J2

2N

∑k

∑a ,a′

nF(εk ,a) − nF(εk+q ,a′)εk ,a − εk+q ,a′

(UkU†k+q

)a ,a′

(Uk+qU†k

)a′ ,a

J2

2N

∑k

∑a ,a′

nF(εk ,a) − nF(εk+q ,a′)εk ,a − εk+q ,a′

(UkU†k+q

)a ,a′

2 .

(9.66)

Conclusion

TKIs are primarily described by a periodic Anderson model (PAM), as this is idealfor modeling mixed-valence materials and the topological properties of the systemare most easily accessible in this form. However, if one is interested in magneticproperties of these materials, which often occur in the limit of a half-filled f level,the Kondo lattice model (KLM) is an appropriate description. Here, we have shownhow theKLM can be derived from the PAM bymeans of the Schrieffer–Wolff transfor-mation for our simplified model of a TKI. Instead of an onsite spin–spin interactionwith localized moments that results for an onsite hybridization, we have foundcorrelated hoppings via the NN hybridization with the localized state. Further-more, we have calculated the RKKY interaction between localized moments by a


second-order perturbation calculation and demonstrated how important a correctmodel of the hybridization can be also for magnetic instabilities.

For a full description of real materials, such as SmB6, more realistic model cal-culations are necessary, e.g., using the full model presented in Chapter 6. Also, wenote that further studies are necessary in order to reconcile the results of the RKKY

interaction with the slave-boson calculations of the previous chapter.

Chapter1

0

Chapter 10

Spin excitons in SmB6

Experimentally, the magnetic properties of bulk SmB6 have been extensively stud-ied using magnetization measurements [40], inelastic neutron scattering [139,140, 144, 145], nuclear magnetic resonance [218–221] and muon spin relaxation(µSR) [146]. In addition to antiferromagnetic ordering under high pressure [43,149], these measurements detected collective magnetic excitations at energies be-low the bulk gap. At the surface, low temperaturemagnetotransportmeasurementsindicate magnetic ordering even under standard pressure below 600 mK. This wasattributed to ferromagnetic [150] or possibly glassy ordering [222] and is claimedto involve Sm3+ magnetic moments which were detected using x-ray absorptionspectroscopy at the surface of SmB6 [223]. Although various experimental andtheoretical studies have now established SmB6 as a true TKI, a number of openquestions remain unanswered. In particular, the source of the magnetic excitationsmentioned above is still unclear. It was suggested that an excitonic state is responsi-ble for these fluctuations [42, 144, 145, 148]. In this context, it is also important tounderstand the interplay between these magnetic excitations and the topologicalsurface sates in order to elucidate the source of reported magnetic ordering at thesurface of SmB6 [150, 222].In this chapter, we address these important aspects by discussing depth-resolved

low-energymuon-spin-relaxation (LE-µSR)measurements on single-crystal samplesof SmB6. After reviewing the experimental data, we will analyze them in order toobtain information about the properties of the collective excitations.

122 | 10 Spin excitons in SmB6

10.1 Measurements

The µSR measurements reported here were performed using the LEM and DOLLYspectrometers at PSI, Switzerland [224, 225]. In these measurements, 100 % spinpolarized positive muons are implanted into the sample. The evolution of thespin polarization, which depends on the local magnetic fields, is monitored via theanisotropic beta-decay positron, which is emitted preferentially in the direction ofthemuon’s spin at the time of decay. Using appropriately positioned detectors, onecan measure the asymmetry, A(t), of the beta decay along the initial polarizationdirection. A(t) is proportional to the time evolution of the spin polarization ofthe ensemble of implanted spin probes [226]. Conventional µSR experiments usesurface muons with an implantation energy of 4.1 MeV, resulting in a stoppingrange in typical-density solids in the rangeof 0.1–1 mm. This limits their applicationto studies of bulk properties, i.e., they cannot provide depth-resolved informationor study extremely thin film samples. Depth-resolved µSR measurement can beperformed at the LEM spectrometer using muons with tunable energies in the 1–30 keV range, corresponding to implantation depths of 10–200 nm. All the µSR datareported here were analyzed using the MUSRFIT package [227].

At 20 K, we observe a Gaussian-like muon spin damping for all four differentenergies. This type of damping is attributed to randomly oriented static magneticfields [226]. A clear change in the shape of the asymmetry is detected uponcooling, which indicates the appearance of additional dilute local magnetic fieldsand/or a change in the internal field distribution. In line with previous bulk-µSR

results [146], we argue that the appearance of additional dilute local magneticfields is most probably due to electronic magnetic moments in SmB6, which aredynamic in nature within the µSR time scale. Most importantly, however, we findthat the difference between the low- and high-temperature asymmetries becomesless pronounced with decreasing E, i.e., as we approach the surface of the SmB6crystals, see Figure 10.1. This indicates that the size and/or fluctuation time of theobserved magnetic fields at low temperatures decreases gradually with decreasingdepth.

We turn now to a quantitative analysis of the µSR data. Following the same anal-ysis procedure used previously for the bulk measurements [146], all zero-field µSR

Chapter1

0

10.1 Measurements | 123

Figure 10.1: The relaxation rates λ at 4 K (red, left axis) and σ (blue, right axis) as a function ofmuon implantation depth in SmB6. The dashed lines are guides to the eye and the dottedvertical lines indicate the different energy values.

spectra can be fitted well using a Kubo–Toyabe relaxation function multiplied by astretched exponential decay function,

A(t) A0

13 +

23

(1 − σ2t2

)e− σ

2 t22

e−(λt)β

+ Abg , (10.1)

where A0 is the initial asymmetry, β is the stretch parameter, Abg is a non-relaxingbackground contribution; σ is the width of static field distribution, e.g., due tonuclear moments, while λ it the muon spin relaxation rates due to the presenceof dynamic local fields. Note that Abg 0 in the bulk µSR measurements, butit is nonzero in the LE-µSR measurements due to muons missing the sample andlanding in the silver backing plate. For consistency with previous bulk-µSR dataanalysis [146], we maintain A0, σ, and Abg as globally common variables for alltemperatures at a particular muon implantation energy. Similarly, we also keepthe value obtained from bulk measurements, β 0.72(1), fixed for all temperaturesand implantation energies.In Figure 10.1, we plot λ at 4 K and σ as a function of the muon implantation

depth in SmB6. The relaxation rate λ decreases rapidly with decreasing depthand may be extrapolated to λ → 0 at the surface of SmB6 (dashed line). This isaccompanied by an increase in σ with decreasing depth. The value of σ in the bulk


λµ

Figure 10.2: The relaxation rate λ at 1.8 K as a function of applied field [146]. The solid lineis the fit described in the text.

is consistent with what we expect from (predominantly boron) nuclear magneticmoments in this system [146]. Therefore, the observed increase near the surfacemust be due to additional sources of relatively small static magnetic fields. Thismay be due to an increased concentration of Sm3+ moments near the surface ofSmB6 which was observed in x-ray-absorption-spectroscopy measurements [223].

Note that λ reflects the spin–lattice relaxation rate of the muon spin in zerofield, which is proportional to ∆B2τ, where ∆B is the size of the fluctuating localfield sensed by the implanted muons and τ is its correlation time. Therefore,the observed decrease in λ at lower implantation energies may be attributed to adecrease in ∆B and/or τ as we approach the surface of SmB6. In order to evaluatethe size of ∆B and τ, we measure the asymmetry as a function of longitudinalmagnetic field, i.e., applied along the direction of initial muon spin polarization.The field dependence of λ follows as [228–230],

λ 2τ(γ∆B)2

1 + (τγB0)2, (10.2)

where γ 2π × 135.5 MHz/T is the gyromagnetic ratio of the muon and B0 is theapplied magnetic field. The experimental results fit well to Equation (10.2) (see

Chapter1

0

10.2 Size and magnetic moments of excitonic state | 125

Figure 10.3: Simulated distribution of the absolute values of magnetic fields affecting themuons, assuming an antiferromagnetic ordering with magnetic moments of 0.01µB. Thedashed gray line marks the experimental value of ∆B ∼ 1.8 mT.

Figure 10.2) giving ∆B 1.8(2)mT and τ 60(10)ns.1 Here we assume that weare in the fast fluctuations limit, τγ∆B 1, which is consistent with the valuesobtained from the fit.

10.2 Size and magnetic moments of excitonic state

We now discuss our data assuming bulk excitons as a source for the observedmagnetic fluctuations. The excitons are believed to be of antiferromagnetic naturewith a wavelength of the order of a few lattice constants [42, 145]. The observeddecay length of magnetic fluctuations near the surface (40–90 nm) should then beinterpreted as the coherence length or “size” of the excitons. This size is muchlarger than the ordering wavelength so that the exciton can be thought of as afluctuating region with antiferromagnetic correlations.The measured value ∆B ∼ 1.8 mT for the width of the distribution of magnetic

fields can be used to estimate the magnitude of the fluctuating magnetic moments.We assume that the muons stop at a random position inside the cubic unit cell ofSmB6 with a lattice constant of 4.13Å. Calculating the distribution of magnetic1The contribution from the level crossing seen at high temperature is subtracted before fitting the data.An additional field independent offset is also needed to fit the data, whichwe attribute to other possiblesources of relaxation.


fields due to the antiferromagnetic correlations in the region of the exciton yieldsan average value of ∼ 0.01µB for the magnetic moments, where µB is the Bohrmagneton, see Figure 10.3. We remark that this magnitude of magnetic momentsis comparable to previously obtained values for (static) antiferromagnetic orderingin other heavy-fermion materials [131, 132].

In addition, we can use a simple hydrogen model for the exciton, describing itas a bound state of an electron and a hole, in order to relate the size to the reducedmass µex of the electron–hole pair via d a0

εrmeµex

. Here, a0 is the Bohr radius andεr is the dielectric constant of SmB6, which is estimated between εr ∼ 600 [135] and1500 [231]. Our measurements then imply a reduced mass of the order of the bareelectronmass me, suggesting that either electrons, holes, or both are relatively lightcompared to reported values for the effective mass in SmB6 of m∗ ∼ 100 me [135].

Conclusion

To summarize, we observe fluctuatingmagnetic fields appearing only below ∼15 Kin the bulk of SmB6. Using LE-µSR measurements we find that these fields arerapidly suppressed with decreasing depth and probably disappear completely atthe surface. We attribute these fluctuating fields to excitonic states, whose extentis limited to the bulk of SmB6 and disappears within ∼ 60 nm of its surface. Thisvalue agrees well with estimates of the “size” of the excitonic state based on themeasured values of the dielectric constant in SmB6, supporting our conclusion thatthe fluctuating magnetic fields stem from such excitons. An estimate of ∼ 0.01µBfor the average magnitude of magnetic moments is obtained from the distributionof fluctuating magnetic fields. We also observe a slight increase in the distributionwidth of staticmagnetic fields near the surface of SmB6, hinting to the appearance ofadditionalmagneticmoments in this region. Our results reveal a complexmagneticbehavior near the surface of the 3D TKI SmB6. We expect that themagnetic nature ofthe near surface region of SmB6mayhave significant implications on the topologicalsurface states at very low temperatures.

IIIProperties of topological invariants

In this part, wewill present connections of topolog-ical invariants to other physical concepts. First, wewill discuss the entanglement spectrum and relateit to topological invariants as well as quantum ge-ometry. Subsequently, we will analyze topologicalphases in the context of critical exponents and therenormalization group.The contents of this part of the thesis are partiallyincluded in References [232] and [233].

Chapter1

1

Chapter 11

Entanglement spectra

Quantum entanglement can be used to classify gapped phases of matter beyondthe paradigm of Ginzburg–Landau-type symmetry breaking. For example, thepresence of long-range entanglement between different parts of a system is inone-to-one correspondence with the notion of its topological order [70, 71]. Thismotivates the important question: How can the “pattern” of long-range entangle-ment of a quantum state be detected and used to study other manifestations oftopological order, topological order, e.g., the fractionalized quasiparticles and theirstatistics. Generically, this question is addressed by studying the entanglementproperties between the two parts 1 and 2 that follow from some bipartitioning ofthe system [234, 235]. The bipartite entanglement entropy can be computed fromthe reduced density matrix ρ1, which is obtained by tracing out the degrees of free-dom in part 2. In a seminal paper, Hui Li and F. Duncan M. Haldane proposed toconsider the full spectrum of the reduced density matrix ρ1 to obtain much richerinformation about the bipartite entanglement of the system [236]. For example,the entanglement spectrum (ES) along a spatial cut reveals the universal countingof low-energy excitations of the boundary conformal field theory of the state [237,238].The abovementioned notion of topological order renders all states without long-

range entanglement topologically trivial, see also Chapter 2. However, such short-range entangled states can still have topological attributes that are protected bysome symmetry of the system, constituting so-called symmetry-protected topolog-ical (SPT) phases, see Section 2.2. A subset of the SPT states that is particularly wellunderstood are the systems of noninteracting fermions with translational symme-

130 | 11 Entanglement spectra

try, i.e., band insulators. The information about their short-range entanglement isentirely contained in themanifold of Bloch states of occupied bands. This manifoldbecomes a metric space upon introducing a quantum distance measure betweenBloch states at different lattice momenta in the Brillouin zone [239, 240]. The cor-responding Hermitian and gauge-invariant quantum metric tensor contains bothuniversal and nonuniversal information about the system. For one, the Chern num-ber as a topological quantum number is obtained by integrating the imaginary partof the quantum metric over the Brillouin zone [26, 98, 241]. On the other hand, thetrace of the quantum metric has been proposed as a measure for the “complexity”of the wave function [240]. Intuitively, it measures by how much the entanglementof local degrees of freedom in the Bloch states changes as a function of momentum.

This descriptionof a band insulator suggests the existence of intimate connectionsbetween geometry, topology, and entanglement. The relation between its topologyand the ES has been studied for spatial bipartitionings of SPT states in variouspublications [242–248]. It has been shown for noninteracting systems that theso-called entanglement Hamiltonian He : − log ρ1 + const is a Hermitian single-particle operator, which is related to the two-point correlation function of theground state [249]. This Hamiltonian then supports gapless protected boundarymodes if the system is topologically nontrivial. In this chapter, we extend theanalysis to bipartitionings that preserve the full translational invariance of thesystem. Using the examples of a Chern insulator and a Z2 topological insulatorin two dimensions, we find that whenever the system is a topologically nontrivialinsulator, the entanglement Hamiltonian describes a metal with an “entanglementFermi sea”. Furthermore, we draw a connection between the quantum geometryand the ES by showing that the trace over the entanglement Hamiltonian is equalto the quantum distance measure between Bloch states. Note that we use a slightlydifferent notation in this chapter compared to the rest of the thesis, e.g., for theBerry curvature.

11.1 The entanglement spectrum

Consider a many-body quantum system with a unique, gapped ground state |G〉that is defined on a Fock space F , built from a single-particle Hilbert space Hsp

Chapter1

1

11.1 The entanglement spectrum | 131

of dimension dim(Hsp) N. A partitioning of the system in two parts 1 and 2is defined by a partitioning Hsp Hsp1 ⊕ Hsp2 inducing a decomposition F

F1 ⊗ F2 of the Fock space. Then, the ground state |G〉 can be written as a Schmidtdecomposition

|G〉 ∑α

1√Z

exp(−Ee

α

2

)|α, 1〉 ⊗ |α, 2〉 , (11.1)

where |α, i〉 ∈ Fi , i 1, 2, and the factor Z−1/2 with Z :∑α e−Ee

α ensures thenormalization. We can define the reduced density matrix of the first subsystem,

ρ1 : Tr2 |G〉〈G| , (11.2)

by performing the partial trace Tr2 over the degrees of freedom in F2. The reduceddensity matrix can be represented as the exponential of a Hermitian operator, theso-called entanglement Hamiltonian He

1 :

ρ1 1

Ze1

e−He1 , Ze

1 : Tr1 e−He1 . (11.3)

The eigenvalues Eeα of this entanglement Hamiltonian are the ES. It has been shown

that for an appropriate spatial cut, gapless edgemodes of the energy spectrum leadto degenerate entanglement eigenvalues [242].

We now specialize to systems of noninteracting fermions with a unique groundstate. Then, theHamiltonian is bilinear in the secondquantized fermionic operatorsand |G〉 is a single Slater determinant. In this case, the entanglement HamiltonianHe

1 is a bilinear in the fermionic operators as well [249]. If we denote with c†n , n

1, · · · ,N, the creation operators for single-particle states in Hsp in some basis, theHamiltonian H and the entanglement Hamiltonian He

1 have the representations

H

N∑m,n1

c†mhm,ncn , He1

N∑m,n1

c†mhe1;m,ncn . (11.4)


It has been shown [249] that the matrix elements he1;m,n of the entanglement Hamil-

tonian are related to the restricted correlation matrix C1 P1CP1 via

C1 1

1 + exp(he

1

) . (11.5)

Here, P1 denotes the projection on the first subsystem Hsp1, represented as anN×N matrix inHsp, and the two-point correlation function C is a N×N matrix withentries Cm,n 〈G| c†mcn |G〉. The correlation matrix C is nothing but the matrixrepresentation of the projector on the single-particle states that are occupied in |G〉in the single-particle Hilbert space Hsp. In systems with translational invariance,when the states can be labeled by some crystal momentum k (see below), we willcall this projection Π(k).

Therefore, if |G〉 is a gapped ground state, we can define a topologically equiva-lent flatband version of the Hamiltonian h by

Q : 1/2 − C , (11.6)

which has eigenvalues −1/2 (+1/2) for the single-particle states that are occupied(empty) in |G〉. The restriction of this matrix to subsystem 1 can be defined as

Q1 : 1/2 − C1 1/2 − P1CP1 , (11.7)

which has eigenvalues λm ∈ [−1/2, 1/2] , m 1, · · · ,N [243].According to Equation (11.5), the eigenvalues λm are monotonously related to

the eigenvalues εm of he1 via

λm 12 tanh

( εm2

), m 1, · · · ,N, (11.8)

and the ES Eeα can be found by

Eeα

N∑m1

n(m)α εm , (11.9)

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1

11.1 The entanglement spectrum | 133

with all possible combinations of occupation numbers n(m)α ∈ 0, 1 of the m-thsingle-particle state [243]. Due to this direct correspondence of λm and Ee

α , we willrefer to the values λm as the single-particle entanglement spectrum (SPES) in thischapter.Note that as tr(P1) < N, the spectrum of C1 will trivially contain the value 0 mul-

tiple times, such that only tr(P1) eigenvalues contain information. The vanishingeigenvalues do not contribute to the ES as they correspond to λ 1/2 and ε ∞.Throughout this chapter they will be omitted by working in the subspace Im(P1).Further, we observe that every vanishing eigenvalue λk 0 leads to a two-folddegeneracy of all entanglement eigenvalues, as it implies εk 0, and Ee

α Eeα′ is

left unchanged by replacing n(k)α 0 with n(k)α′ 1.One strength of the ES lies in the fact that if the Hamiltonian h and the projector

P1 share a symmetry S, i.e., [h , S] [P1 , S] 0, the ES can be ordered by theeigenvalues of S, for S is then also a symmetry of he

1. Here, we will considercases where P1 preserves the translational symmetries of h, either along one or twodirections in two dimensions, so that the latticemomentum k along these directionscan be used to label the eigenstates and the entanglement eigenvalues. In this case,the projector P1 is block diagonal where we call its N×N blocks Π1(k). Now, wecan write the Hamiltonian as

H

∑k

N∑α,β1

c†α(k)hα,β(k)cβ(k) (11.10)

and a spectral decomposition of the Bloch matrix(hα,β

)(k) is given by

h(k) N∑

a1|ua(k)〉 εa(k) 〈ua(k)| , (11.11)

revealing the Bloch states |ua(k)〉, where the bands are labeled by the index a

1, . . . ,N . N is the total number of Bloch bands, and k ∈ BZ is a momentum in thefirst Brillouin zone (BZ). The projector on the n < N occupied bands is then given


by

Π(k) :n∑

a1|ua(k)〉〈ua(k)| . (11.12)

For tr(Π1) m, the spectrum of Π1Π(k)Π1 will contain the value 0 at least N − mtimes. As discussed above, we will omit these values, which do not contain anyuseful information, by just working in the subspace Im(Π1). Note that we use tr forthe trace of matrices to distinguish it from the trace Tr in the Fock space F .

11.2 The sublattice entanglement spectrum

A commonly studied type of partitioning is a cut in position space, separating thesystem into a left and right half. Translational symmetry is only broken orthogonalto the cut, such that the crystal momenta parallel to the cut remain good quantumnumbers. It has been shown that band insulators with nonzero Chern numberfeature a gapless branch of chiral states in the ES, which are localized near theposition-space cut. The single-particle entanglement eigenvalue of this branch ofstates flows from λ ∓1/2 to λ ±1/2 as the momentum parallel to the cut isvaried through the BZ [245].

Here, we want to consider a different class of bipartitionings which leave thetranslational symmetries of the systemunchanged, see Figure 11.1 for a comparisonof the two different types of cuts.1 Wewill refer to the resulting ES as the sublatticeentanglement spectrum (SLES).

To define the SLES, one introduces a partitioning of the internal degrees of free-dom at every lattice site, such as sublattices, orbitals, or spin species, and traces outa subset of these on-site degrees of freedom. As the SLES preserves the translationalsymmetry of the system, all components of k remain good quantum numbers andwe can write

Q1(k) 1/2 −Π1Π(k)Π1 . (11.13)

Here, all operators can be represented by N×N matrices, where the number ofbands N is equal to the number of internal degrees of freedom at each lattice site.

1A similar extensive cut has been considered in Reference [248]

Chapter1

1

11.2 The sublattice entanglement spectrum | 135

(a) Spatial bipartitioning. (b) Sublattice bipartitioning.

Figure 11.1: Comparison of different types of bipartitionings used for the ES.

We find that for a suitable choice of the bipartitioning of internal degrees offreedom, a nontrivial topological phase implies a gapless SLES covering (in thethermodynamic limit) all values in the interval [−1/2, 1/2]. We observe that thelowest many-body entanglement eigenvalue Ee

min, according to Equation (11.9), isobtained by filling all single-particle states with λ < 0. This suggests identifyingthese states with an “entanglement Fermi sea” and defining an “entanglementFermi surface” at λ 0. A topological (Chern) insulator is therefore mappedto an “entanglement metal” (with an entanglement Fermi surface) and we aretrading the topological stability provided by the band insulator’s energy gap forthe topological stability of this Fermi surface [250]. In the following section, wewillexemplify this for 2D systems of symmetry classes A (Chern insulator) and AII (Z2topological insulator) in the Altland–Zirnbauer (AZ) notation, which we discussedin Section 2.2.2.In general, the SLES canbe studied for arbitraryprojections on the internal degrees

of freedom. However, not every choice of partitioning is useful in order to identifytopological properties. For example one could think of two layers of a 2D insulatingsystem coupled weakly (as compared to their gaps) by interlayer hopping. TheES for a partitioning of these two layers will then only depend on the interlayercoupling and not on the topology of the two systems (see also Reference [248]). Asimilar example, using spin degree of freedom, will be presented for the Kane-Melemodel in Section 11.4.2.Also, we assumed the projection Π1 to be k-independent. This is a natural

assumption emphasizing a clear physical interpretation of the bipartitioning but itis not strictly required. However, a necessary condition to a k-dependent projectionis that it can be continuously deformed to a constantΠ1; i.e., it does not carry somenontrivial topology itself. An obvious choice forΠ1 violating this condition would


be the projection Π(k) on the occupied bands. In this case we find λi(k) 1/2everywhere, as [Π(k)]3 Π(k).

Despite these subtleties, the situation here is noworse thanwith the ES of a spatialbipartitioning. Also in that case, there exist bipartitionings which do not containtopological information about the system. For example, a cut between Wannierstates would not reveal the topology of the ground state, for the Wannier states areflowing with the entanglement eigenvalues.

11.2.1 The SLES for Chern insulators

First, we want to consider a 2D model of class A, where for simplicity we assumetwo bands, one of which is occupied. In this case wewill prove that a nonvanishingChern number implies a gapless SLES for any choice of the bipartitioning.

It has been shown that the Chern number of a Bloch band is given by the sumof the vorticities of all vortices of an arbitrarily chosen component α of the two-component vector u(k) (note that we drop the band index a as there is only oneoccupied band) [251, 252]. The vorticity (or charge of a vortex) is there defined asthe winding number of the phase of the uα(k) around the vortex. Note that caremust be taken to choose the gauge such that none of the components of u(k) ismultivalued at the vortices.

Consequently, within the topological phase with nonvanishing Chern numberthere must be at least one vortex at k0

α for each component α of u(k) in the BZ. For atwo-band model (α 1, 2) the vectors u(k0

1) and u(k02) are accordingly orthogonal.

By choosingΠ1 Π(k0

1) (11.14)

as the projector on the occupied band at a vortex of the first component, we find thatthe single-particle entanglement eigenvalue obeys λ(k0

1) −1/2 and λ(k02) +1/2,

proving our statement.In fact, any other nontrivial (Π1 , 0 and Π1 , 1) choice of the bipartitioning

leads to the same result for the two-band model: Any nontrivial projector Π′1 isrelated to Π1 by a unitary transformation

Π′1 UΠ1U† . (11.15)

Chapter1

1


k01

k02

BZ

entanglementFermi surface

u(k)

Figure 11.2: Mapping from the BZ to the Bloch sphere for the bipartitioning defined by theprojector Π1 [Equation (11.14)] in the case of a band with nonvanishing Chern number.The points k0

1 and k02 , where first and the second component of the Bloch spinor vanish, are

mapped to the north and south pole of the Bloch sphere, respectively. The entanglementFermi surface is mapped to the equator. Any other choice ofΠ1 corresponds to a rotation ofthe Bloch sphere, such that themapping from the BZ to the Bloch sphere changes. However,the existence of the entanglement Fermi surface is guaranteed by the nonvanishing Chernnumber for any choice of Π1.

The components of the vectors Uu(k) have vortices at different points in the BZ,where the SLES with the new projector will have the values λ ±1/2.

The reverse statement (a gapless SLES implies topological order) is in generalnot true: For example, it is possible to have two vortices with opposite vorticitiesadding up to a Chern number of zero. Nevertheless in this case we would find agapless SLES.This argument can in principle be generalized to multiple bands. However, in

this case we will find that the topological information within the SLES depends onthe chosen bipartitioning as argued above.

For a two-band model, the relation between a nonvanishing Chern number andthe existence of an entanglement Fermi surface can be visualized using the Blochsphere, parametrized by the polar and azimuthal angles θ and φ, respectively (seeFigure 11.2). The state vector u(k) of the occupied band can be represented by apoint on the unit sphere S2. We define a particular mapping from the BZ to the unitsphere in such a way that the image of the projectorΠ1 is represented by the north


pole θ 0 of the sphere. This can be accomplished by writing the state vector as

u(k) UΠ1

©«cos [θ(k)/2] eiφ(k)/2

sin [θ(k)/2] e−iφ(k)/2ª®¬ , (11.16)

where UΠ1 is a 2×2 unitary matrix that diagonalizes Π1, that is, U†Π1Π1U

Π1

diag(1, 0). Then, the entanglement eigenvalue is given by

λ(k) 1/2 − cos2 [θ(k)/2] −1/2 cos [θ(k)] , (11.17)

which takes the value −1/2, 1/2, and 0 at the north pole, south pole, and equator,respectively. Thus, the entanglement Fermi surface maps to the equator of theBloch sphere.

In two dimensions, the Chern number can be regarded as the winding numberof the mapping u(k) from the BZ to the unit sphere. Thus, a nonzero Chern numberimplies that for every point v on the unit sphere there is a kv in the BZ such thatu(kv) v. In particular, this implies that for every λ ∈ [−1/2, 1/2] there exists apoint in the BZ such that λ(k) λ.

Note that this argument applies to systems with arbitrary Chern numbers C. For|C | > 1, the Bloch vector will wind around the sphere multiple times, leading tomultiple points mapped to the north pole, south pole, and equator. Therefore, thenumber of entanglement Fermi surfaces is equal to or larger than |C | in this case.Again, wefind that a vanishingChern number does not necessarily lead to a gappedSLES: The SLES is gapless if and only if the mapping θ(k) from the BZ to the Blochsphere is surjective. However, surjectivity does not imply a nontrivial winding,such that a gapless SLES can be present even for a vanishing Chern number.

Note that the original choice (11.14) of Π1 is already diagonal, i.e., UΠ1 1.Choosing a different bipartitioning rotates the Bloch sphere in this picture, leadingto different mappings θ(k) and φ(k). However, due to the winding of the Blochvector for a nonzero Chern number, the SLES is gapless for any choice of Π1.

Chapter1

1


11.2.2 The SLES for Z2 topological insulators

Now, we want to analyze topological information in the SLES of a 2D system of classAII. According to the discussion in Section 2.4.1, we can express the Z2 invariant bythe number of zeroes of the Pfaffian p(k) of Equation (2.10). The minimal examplefor a Z2 topological insulator has four bands two of which are occupied2, in whichcase Equation (2.10) simplifies to

p(k) m12(k) 〈u1(k)| T |u2(k)〉 . (11.18)

At k∗where p(k∗) 0, we find that |u2(k∗)〉 andT |u1(k∗)〉 are orthogonal. Further,|u1(k∗)〉 and T |u1(k∗)〉 are orthogonal due to T 2 −1.3 Therefore, we can definethe bipartitioning as the projection

Π1 |u1(k∗)〉〈u1(k∗)| + T |u1(k∗)〉〈u1(k∗)| T −1 . (11.19)

Then, we find at this point k∗

Π(k∗)Π1Π(k∗) |u1(k∗)〉〈u1(k∗)| , (11.20)

as Π1 |u1(k∗)〉 |u1(k∗)〉 and Π1 |u2(k∗)〉 0. For arbitrary projectors P1 and P2,the spectrum of P1P2P1 is equal to that of P2P1P2.4 Therefore, the ES at k∗ is givenby

λ1(k∗) −λ2(k∗) 1/2 . (11.21)

We note that our definition of Π1 is time-reversal invariant due to

TΠ1T −1 T |u1(k∗)〉〈u1(k∗)| T −1

+ |u1(k∗)〉〈u1(k∗)| Π1 . (11.22)

2Kramer’s theorem implies a gapless energy spectrum for a two-band model.3Any antiunitary operator U fulfills 〈Uu |Uv〉〈v |u〉. For v Uu and U2 −1 we find 〈Uu |Uv〉 ⟨

UuU2u

⟩ − 〈Uu |u〉 and 〈v |u〉〈Uu |u〉. This leads to 〈Uu |u〉 − 〈Uu |u〉 implying 〈Uu |u〉 0.

4P1P2P1u λu for λ , 0 implies u ∈ Im(P1) and therefore P1P2u λu as well as P2u , 0. Multiplyingwith P2 and using P2

2 P2 we find P2P1P2(P2u) λ(P2u).


Thus, Kramer’s theorem implies a degeneracy of the eigenvalues of Π1Π(Γi)Π1 atthe time-reversal-invariant momentum (TRIM), leading to

λ1(Γi) λ2(Γi) . (11.23)

Equations (11.21) and (11.23), together with the continuity of the λi as a functionof k, then imply a gapless SLES for the bipartitioning (11.19), if the system is a Z2topological insulator.

Again, the converse is not true in general, as there could be multiple pairs ofpoints with p(k) 0. There are other choices ofΠ1 leading to the same topologicalinformation in the SLES as we will see in the example of the Kane-Mele model inSection 11.4.

11.3 Entanglement spectrum and quantum geometry

The single-particle eigenstates |ua(k)〉 are elements of the complex projective spaceCPN−1. The Fubini–Study distance on CPN−1 can be used to define a distancemeasure

d(k1 , k2) :

√√√n −

n∑a ,b1

|〈ua(k1)|ub(k2)〉|2 (11.24)

between states at momenta k1 and k2 for the lowest n bands occupied, the so-calledquantum distance [239, 240, 253]. The expression (11.24) can be rewritten usingΠ(k) as

d2(k1 , k2) tr [1 −Π(k1)]Π(k2) n − tr [Π(k1)Π(k2)] .

(11.25)

An expansion of the infinitesimal distances

d2(k , k + dk) ∑µν

gµν(k) dkµ dkν , (11.26)

Chapter1

1

11.3 Entanglement spectrum and quantum geometry | 141

defines the Riemannianmetric gµν . This tensor is the symmetric part of the Fubini–Study metric tensor (also called quantum geometric tensor)

qµν gµν − i2 fµν , (11.27)

with its antisymmetric part fµν being the Berry curvature [254].In general, the distance between two projectors can be defined as

d2(P1 , P2) : 12 tr

[(P1 − P2)2

]

12 [tr(P1) + tr(P2)] − tr(P1P2) ,

(11.28)

which is induced by the Frobenius norm on the vector space of N×N matrices.For projectors with equal traces tr P1 tr P2 n this expression reduces to Equa-tion (11.25). Note that for projectors with different traces it is not possible to definea metric tensor, for there exists no smooth interpolation between two projectorswith unequal traces.

We can now investigate specifically the distance between the projection on theoccupied bands Π(k) and the projection Π1 of the bipartitioning in the ES. Wewill assume that we have tr [Π(k)] n occupied bands and that tr(Π1) m. Thedistance is then given by

d2Π1(k) : d2(Π1 ,Π(k)) m + n

2 − tr [Π1Π(k)] . (11.29)

As Π21 Π1 and the trace is invariant under cyclic permutations, we can write

d2Π1(k) m + n

2 − tr [Π1Π(k)Π1] . (11.30)

The trace of an operator is just the sum of all its eigenvalues which in this case aregiven by the ES. Therefore, the distance is

d2Π1(k) m + n

2 −m∑

i1

(12 − λi(k)

)(11.31)


n2 +

m∑i1

λi(k) .

Note that we only sum the m eigenvalues which do not generally vanish, as dis-cussed at the end of Section 11.1.

The quantum distance in this case measures the overlap of the ground state wavefunction with the chosen partitioning (where d 0 for full overlap). The SPES isrelated to the overlap of one single-particle state with the partitioning (λ −1/2for full overlap). It is thus natural that the quantum distance between the occupiedbands and any bipartitioning of a system is defined by the sum of the entanglementeigenvalues.

We can now use the (inverse) triangle inequality to derive a lower bound for themetric tensor in the BZ from the ES. The inequality reads for arbitrary projectors

d(k1 , k2) ≥dΠ1 (k1) − dΠ1 (k2)

, (11.32)

whereweuse thenotation fromabove and d(k1 , k2) : d(Π(k1),Π(k2)). Consideringinfinitesimal distances and the square of this inequality, we find

gµν(k) dkµ dkν d2(k , k + dk)≥ [

dΠ1 (k) − dΠ1 (k + dk)]2,

(11.33a)

where the last expression can be expanded to

©«dΠ1 (k) −

dΠ1 (k)2 −m∑

i1[λi(k + dk) − λi(k)]

1/2ª®¬2

dΠ1 (k) −(dΠ1 (k)2 −

m∑i1

∂µλi(k)dkµ)1/2

2

.

(11.33b)

Assuming dΠ1 (k) , 0, this leads to the estimate

gµν(k) dkµ dkν ≥ gΠ1µν (k) dkµ dkν , (11.34a)

Chapter1

1

11.3 Entanglement spectrum and quantum geometry | 143

for arbitrary dkµ, dkν . Here we defined the lower bound

gΠ1µν (k) : 1

4d2Π1(k)

m∑i , j1

[∂µλi(k)

] [∂νλ j(k)

], (11.34b)

which depends on the projector Π1 representing the bipartitioning of the ES. Inparticular, the independence of thedifferent componentsdkµ implies a lower boundfor the diagonal elements of the metric tensor,

gµµ(k) ≥ gΠ1µµ(k) . (11.35)

For one specific bipartitioning, the estimate (11.34b) will generally not provide agoodapproximation to the quantummetric tensor at everypoint in the BZ.However,the inequalities (11.34a) and (11.35) hold for an arbitrary projection Π1. Rewritingthe expression (11.34b) in terms of the distance dΠ1 (k) hides the relation with theES but produces the simple form

gΠ1µν (k) d2

Π1(k) ∂µ log

[dΠ1 (k)

] ∂ν log

[dΠ1 (k)

]. (11.36)

The relation between quantum geometry and the ES is especially direct for theSLES when we choose a projection on one internal degree of freedom. In this casethe ES is a single real number at every k and we find

d2Π1(k) n

2 + λ(k) . (11.37)

If we consider a two-band model with n 1, this distance takes all values between0 and 1 for a gapless ES.In the case of the spatial ES (with open boundary conditions), there are discon-

tinuities of the eigenvalues in a topological phase; see Section 11.4.1 and Refer-ence [246]. These imply similar discontinuities in the quantum distance (11.31).Using the estimate (11.32), we find that also the quantum distance of arbitrarilyclose points in the BZ is finite. According to Equation (11.26), the metric tensor issingular at those points, reflecting the fact that no smooth gauge can be chosen inbands with nonzero Chern number.


11.4 Model calculations

11.4.1 The π-flux model

The π-flux model is a 2D example of a Chern insulator [32, 255]. It is defined ontwo interpenetrating square lattices and is characterized by a flux of plus/minushalf the flux quantum through each plaquette (half the unit cell). This leads (in ourchosen gauge) to imaginary nearest neighbor (NN) hopping amplitudes of t1e±iπ/4

(with t1 , 0). In addition, one introduces a next-to-nearest neighbor (NNN) hoppingamplitude of ±t2 and a staggered chemical potential of ±η on the two sublattices.

For periodic boundary conditions (PBC) the Hamiltonian can be written as

h(k) b(k) · σ , (11.38)

with the Pauli matrices σ (σ1 , σ2 , σ3)t and

b1(k) + ib2(k) t1[e−i π4

(1 + ei(ky−kx )

)+ ei π4

(e−ikx + eiky

) ], (11.39a)

b3(k) 2t2(cos kx − cos ky

)+ η . (11.39b)

The energy spectrum of this model is gapless for |η| 4|t2 |. The system shows atopologically nontrivial phase for η < 4t2 with Chern number C sign(t1t2).

The SLES for the π-flux model

We nowwant to calculate the SLES for this model using PBC. A natural choice of thebipartitioning is to trace over one sublattice, say sublattice B.5 The related projectorΠ1 on sublattice A reads as

Π1 ©«1 0

0 0ª®¬

12 (σ0 + σ3) . (11.40)

The eigenstates are given by two-component vectors ua(k) (uA

a (k), uBa (k)

) tand

we find a single entanglement eigenvalue which is given by5As discussed in Section 11.2.1, any other choice would lead to the same qualitative results for thistwo-band model.

Chapter1

1


(a) Topological phase for η −0.35. (b) Trivial phase for η −0.45.

Figure 11.3: Plots of the SLES of the π-flux model for different phases using t2 0.1. Theentanglement Fermi sea (λ < 0) is shown in purple color, regions with λ > 0 are shownin red, the thick black line shows the entanglement Fermi surface. The ES is gapless(gapped) in the topological (trivial) case, with a phase transition at η −4t2. According toEquation (11.37) this plot also shows the quantum distance, as λ(k) dΠ1 (k) − 1/2.

λ(k) 12 −

uA1 (k)

2 . (11.41)

This is equal to λ ±1/2 at the vortices of the two components (the poles of theBloch sphere) and vanishes at the equator of the Bloch sphere. Figure 11.3 showsthe SLES for different choices of η. As argued above, the SLES is gapless in thetopological phases (|η| < 4|t2 |) and becomes gapped in the topologically trivialphase. This is due to the change of the occupied state u1(k) at the point k (π, 0)where the energy gap closes for η −4t2.

We note that for the π-flux model (being a two-band model), the quantumdistance d(Π1 ,Π(k)) is given by Equation (11.37); it is just the SLES (being in thiscase a single real number) shifted by +1/2 and therefore contains exactly the sameinformation as the SLES.

The spatial entanglement spectrum for the π-flux model

In addition to the SLES, we can also calculate the SPES and the quantum distanceof a spatial bipartitioning, which is shown in Figure 11.4. The band crossing inthe ES as well as the discontinuity of the quantum distance are clearly visible inthe topological phase and vanish in the trivial phase. We also observe that thediscontinuous jump of one entanglement eigenvalue occurs precisely at the point


(a) Topological phase.

(b) Trivial phase.λ(k) d2(k) − n/2

Figure 11.4: SPES λ(k) (blue dots) and quantum distance d2(k) − n/2 (red continuous line) asa function of the wavevector parallel to the applied spatial cut for the π-flux model. Thecut is applied in the middle of a strip of 100 lattice sites, i.e., N 200 and n m 100.The parameters are the same as in Figure 11.3. The quantum distance is calculated usingEquation (11.31).

where the crossing entanglement eigenvalue λ 0. The spectral flow in the ES

can be understood as a Wannier function moving across the cut introduced by thebipartitioning [245]. However, as λi(2π) λi(0), this spectral flow results in adiscontinuity of at least one eigenvalue λ which jumps from ±1/2 to ∓1/2 and isconnected to the jump of one Wannier state from one to the opposite edge.

In Figure 11.4 we have chosen open boundary conditions orthogonal to the cut:If PBC are also imposed orthogonal to the cut, the bipartitioning results in twodisconnected cuts in position space. This implies a spectral symmetry in the ES,with all single-particle entanglement eigenvalues coming in pairs ±λ [245]. Theuse of open boundary conditions avoids this doubling of entanglement eigenvalues.Note that for a spatial bipartitioning with periodic boundary conditions along bothdirections the symmetry of the SPES implies that the quantum distance is triviallyd2Π1(k) n/2 const.

11.4.2 The Kane-Mele model

In order to study the quantum spin Hall effect in graphene, Kane and Mele in-troduced a tight-binding model on the honeycomb lattice [33, 34]. It consists ofNN hopping t, spin-dependent NNN hopping (spin–orbit coupling (SOC) λSO), a

Chapter1

1


Rashba coupling λR, and a staggered sublattice potential λν , which breaks thesublattice symmetry.If we define the coordinates where k1,2 : 1/2(kx ±

√3ky), the two Dirac points

are at ±k∗ ±(k∗1 , k∗2) ±(2π/3, 2π/3). The energy spectrum at the Dirac points is

εi , j(±k∗) (−1)i 3√

3λSO + (−1) j√λ2ν + (9/4)λ2

R (11.42)

for i , j 1, 2. Note that we replaced the band index a 1, . . . , 4 by the two indicesi and j.The energy gap closes at k∗ when the parameters λν and λR satisfy

1 1

27

(λνλSO

)2+

112

(λRλSO

)2. (11.43a)

and at some k , k∗ forλR/λSO > 2

√3 , λν 0 . (11.43b)

The contour (11.43a) separates the topological quantum spin Hall phase (inside)and the topologically trivial phase (outside). Near λν 0 there exists a region inparameter space, where the bands are separated by a finite direct but no indirectbandgap; see Figure 11.5 (a). In this range the different bands are nondegenerateat all points such that the topological invariant is still meaningful; however, theHamiltonian does not describe an insulator any more for these choices of parame-ters.

We now study the SLES for this model. According to the discussion in Sec-tion 11.2.2 we consider the eigenstate

ui , j(k) (uA↑

i , j (k), uA↓i , j (k), u

B↑i , j (k), u

B↓i , j (k)

) t(11.44)

at the Dirac point k∗. For the four bands with energies given in Equation (11.42),the (unnormalized) eigenstates are

u1, j(k∗) (κ j , 0, 0, 1

) t(11.45a)

u2, j(k∗) (0, κ j , 1, 0

) t, (11.45b)


(a) (b)

Figure 11.5: Regions of gapless energy spectrum (a) and gapless SLES (b) in the Kane-Melemodel for different values of the parameters λν and λR. The plots were obtained bynumerically evaluating the (entanglement) spectrum at random points in the first BZ foreach set of parameters. The color coding shows themaximal distance between eigenvalues;red and yellow colors show values below 0.05. Additional precision with a larger sampleof points was used in a range around the phase transition. The black contour shows theellipse of Equation (11.43a).

withκ j

−i3λR

(2λν + (−1) j

√4λ2

ν + 9λ2R

). (11.46)

The energy ε1,1 is always negative [see Equation (11.42)]; therefore, according toEquation (11.19), the projector Π1 on the states (11.45) is

Π1 ©«|κ1 |2 0 0 κ1

0 |κ1 |2 κ1 00 κ∗1 1 0κ∗1 0 0 1

ª®¬. (11.47)

For general parameter values, this projector does not have a clear physical interpre-tation. However, for λR 0 and λν , 0 the constant κ1 vanishes and Π1 becomesthe projector on the sublattice B,

Π1 diag(0, 0, 1, 1) . (11.48)

Chapter1

1


In the following, we will use the projection (11.48) on sublattice B for all choices ofparameters.The SLES is now given by two eigenvalues λ1,2 of 1/2−Π1Π(k)Π1. Figure 11.5 (b)

shows choices of parameters λν and λR where the SLES fills the whole interval[−1/2, 1/2]. This exactly coincides with the topological phase inside the ellipsedefined in Equation (11.43a).This feature can be understood from an inversion of the energy bands ε1,2 and

ε2,1: While ε1,1 is always negative and ε2,2 is always positive, ε1,2 and ε2,1 changesigns exactly at the phase transition (11.43a). For dominant SOC (inside the topo-logical phase) we have ε1,2 < 0 and ε2,1 > 0, while for dominant Rashba couplingor sublattice potential (in the trivial phase) we find ε1,2 > 0 and ε2,1 < 0. Duringthe phase transition the eigenstates u1,2(k∗) and u2,1(k∗) change from occupied toempty and from empty to occupied, respectively, changing Π(k) qualitatively.

In the topological phase, the two entanglement eigenvalues are λ1(k∗) 1/2 andλ2(k∗) −1/2. As Π1 defined in Equation (11.48) is time-reversal invariant, theKramer’s degeneracy at the TRIM then leads to a gapless SLES. In contrast, in thetopologically trivial phase λ1(k∗) λ2(k∗) 0, such that the SLES is now gapped.

Unlike with the case of the (two-band) Chern insulator, the quantum distance(that is, the sum of the two entanglement eigenvalues of the SLES) admits no conclu-sions about topology of a Z2 topological insulator. For example, we find a spectralsymmetry λ1 −λ2 for λR , λν 0 and therefore a constant quantum distanced(Π1 ,Π(k)) 1.

In order to illustrate the importance of the correct choice of the bipartitioning,we consider a projection on one of the two spin species. Then, the related projectorbreaks time-reversal symmetry (TRS) and therefore we do not find any Kramer’sdegeneracy. In this case the ES does not reveal the topological character of thesystem [248]. Rather it measures the coupling of the two spin species which isrelated to the parameter of the Rashba interaction λR.


Conclusion

We studied the relations between the entanglement properties, the topology, andthe quantumgeometry of noninteracting band insulators. The connections betweenthese quantities are best seen if the entanglement cut preserves the full translationalsymmetry of the system, that is, in the sublattice entanglement spectrum (SLES).First, we extended the known relation that a topologically nontrivial band struc-ture implies a gapless ES to the SLES for both Chern insulators and Z2 topologicalinsulators. We are thus trading the topological stability guaranteed by the gapin the spectrum of Hamiltonian for the topological stability of the “entanglementFermi surface” of the entanglement Hamiltonian. Second, we reinterpreted thetrace over the ES at a given momentum as a quantum distance associated withthis momentum. This allowed us to establish the Fubini–Study metric of a Blochband as an upper bound to the squaredmomentum-derivative of the entanglementeigenvalues.

For this study, we concerned ourselves with the simplest possible case of nonin-teracting fermionic Hamiltonians with full translational symmetry. It is imperativeto ask how our results can be extended to interacting SPT phases and to phaseswith intrinsic topological order. This includes relating their many-body ES to themany-body metric of the ground state which is defined as the function of externalcontrol parameters, such as twists in the boundary conditions.

We close by noting that as the quantum metric tensor of Bloch bands is in partexperimentally accessible via optical susceptibility [256] or current noise measure-ments [257], the relations that we found provide ways of experimentally obtainingsome information about the ES.

Chapter1

2

Chapter 12

Scaling theory of topological phase transitions

Recently, a scaling approach has been suggested for studying phase transitions intopological systems [258]. Akin to stretching a messy string to reveal the numberof knots it contains, the scaling procedure renormalizes the Berry curvature locallywithout changing the topological invariant, through which an renormalizationgroup (RG) flow can be obtained. An intriguing aspect that follows these scalingschemes is that they seem to point to the possibility of formulating a statisticaldescription of topological phase transitions.In this chapter, we advance this statistical description by discussing critical expo-

nents of topological phase transitions in 2D systems of symmetry class A accordingto the AZ classification, see Section 2.2.2. In Section 12.1, we focus on Dirac models,where a topological phase transition can be studied by tuning the mass parame-ter. We find that the Berry curvature and the correlation length of noninteractingDirac Hamiltonians display universal critical exponents regardless what the tuningenergy parameter is. The scaling laws that constrain the critical exponents are iden-tified, and are predicted to be satisfied even in interacting systems. They can alsobe applied to mirror Chern numbers (MCNs) that are calculated in mirror-invariantplanes (MIPs) of a 3D system. In Section 12.2, this is demonstrated for the simplifiedmodel of a topological Kondo insulator (TKI) which we have discussed in Chap-ters 5 and 7. In addition, we will calculate and discuss the RG flow proposed inReference [258] for this model as a function of two model parameters.

152 | 12 Scaling theory of topological phase transitions

12.1 Scaling laws associated with Chern numbers

In the following,we consider 2D systemsof classAaccording to theAZ classification,with rectangular symmetry.

We start from the Berry curvature that can be written in the following form nearthe gap-closing momentum k0 [258]:

F (k0 + δk ,M) ≈ F (k0 ,M)(1 ± ξ2

xδk2x

) (1 ± ξ2

yδk2y

) . (12.1)

First we consider two-band Hamiltonians in two dimensions of the form

H(k) ∑

i

di(k)σi , (12.2)

which have an energy spectrum of E± ±√

d21 + d2

2 + d23 ±d. The most general

parametrization of the three di ’s in a 2×2Diracmodel in terms of the small deviationaway from the gap-closing momentum, k − k0 δk, is that two of them are oddand one is even in δk.For instance, consider a model described by d1 and d2 oddand d3 even in δk, then the lowest order expansion yields

d1 A1xδkx + A1yδky , (12.3a)

d2 A2xδkx + A2yδky , (12.3b)

d3 (M −Mc) + Bxδk2x + Byδk2

y + Bx yδkxδky . (12.3c)

The parameters Aix ,Ai y , By , By , Bx y depend on the details of the model, andthe mass parameter M again has to enter the parity-even channel since it is a scalar.The component d0σ0 is unimportant for the argument below and hence is ignoredfor simplicity. The topological invariant of the Dirac Hamiltonian can be calculatedfrom the Berry curvature that we have discussed in Section 2.3.1 Note that ourchoice is somewhat arbitrary as any rotation in σ space would leave the propertiesof the system unchanged.

1Note that all three components of d are treated equally; therefore, they can be permutatedwithout effectsexcept a possible sign change to the Berry curvature.

Chapter1

2

12.1 Scaling laws associated with Chern numbers | 153

The expansion of the Berry curvature at k0 along the x and y directions (i ∈x , y

) yields Equation (12.1) with

F (k0 ,M) A1xA2y − A1yA2x

2|M −Mc | (M −Mc)∝ sgn(M −Mc)|M −Mc |−2 ,

(12.4a)

ξi

8Bi (M −Mc) + 3A21i + 3A2

2i

2 (M −Mc)2

1/2M→Mc∝ |M −Mc |−1 .

(12.4b)

Thus, the critical exponents of the length scale, ξi ∝ |M −Mc |−1, and the Berrycurvature, F (k0 ,M) ∝ |M−Mc |−2, are always the same regardless the microscopicdetails

Aix ,Ai y , Bx , By , Bx y

of themodel. It is clear that in general the two length

scales do not have to coincide, ξx , ξy , if Aix , Ai y or Bx , By ; nevertheless, thescaling law in Equation (12.7) is always satisfied.We proceed to discuss a scaling law that is expected to be satisfied also in inter-

acting systems, where we denote the critical exponents as

F (k0 ,M) ∝ sgn(M −Mc)|M −Mc |−γ , (12.5a)

ξi ∝ |M −Mc |−νi . (12.5b)

As M → Mc, the Berry curvature at the gap-closing momentum k0 diverges, yetits integration over the range of ξ−1 near k0 converges to a constant:

Cdiv 1

2π

∫ ξ−1x

−ξ−1x

dkx

∫ ξ−1y

−ξ−1y

dky F (k ,M)

F (k0 ,M)

2π

∫ ξ−1x

−ξ−1x

dkx1

1 ± ξ2x k2

x

∫ ξ−1y

−ξ−1y

dky1

1 ± ξ2y k2

y

F (k0 ,M)ξxξy

× O(1) const .

(12.6)


Thus, the critical exponent of F (k0 ,M) is the sum of those of ξx and ξy ,

γ νx + νy , (12.7)

which is indeed satisfied by the result for noninteracting systems given in Equa-tion (12.4b). This also suggests that as M → Mc, the Berry curvature can beparametrized by

F (k0 ,M) ∝ sgn(M −Mc)|M −Mc |−νx−νy . (12.8)

12.2 Scaling laws and RG flow in a TKI

Due to the general nature of the argument, we expect that the scaling laws proposedin Equation (12.7) have to be satisfied even when the topological phase transitionsare driven by interactions. We will demonstrate this by considering interaction-driven phase transitions in a TKI, which we have already discussed in Chapter 7.

There, we have shown that a mean-field treatment of the Kotliar–Ruckenstein(KR) slave-boson scheme can transform the interacting Hamiltonian defined inEquations (5.1) and (5.2) into a noninteracting Hamiltonian with renormalizedparameters. When tuning the onsite-interaction U, the gap can close at the high-symmetry-points Γ, X, M, or R, and one can therefore observe topological phase-transitions. While the system is three-dimensional, the mirror symmetries of thesimple cubic lattice allow the definition of three distinct MCNs, which have beendiscussed in Chapter 5. Close to the high-symmetry point where the gap closes,we can map the states with mirror eigenvalue +i in the different MIPs to 2D Diracmodels, where the mass parameter M is a function of the model parameters.

Keeping all other model parameters fixed, we can consider the phase diagramas a function of ε f and U, see Figure 12.1. We remark that this phase diagram wascalculated using the approach presented in Chapter 8; there is good qualitativeas well as quantitative agreement with the results presented in Chapter 7. Now,we can observe topological phase transitions by varying either ε f or U and fit thecritical exponents γ and ν for different HSPs, see Figures 12.1 and 12.2. Close to thetransition, we can approximate the system by a 2×2 Dirac model, where the mass

Chapter1

2

12.2 Scaling laws and RG flow in a TKI | 155

Γ M

X

R

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 120

2

4

6

8

10

12

ε f

U

Figure 12.1: Phase diagram of the simplemodel for a TKI as a function of ε f and U. The phasediagram is the same as the one presented in Figure 7.3 but calculated using the approachof Chapter 8. Phase transitions occur whenever the gap closes at one of the high-symmetrypoints (HSPs) Γ, X, M, or R. The dashed red line shows the parameters of the plots inFigure 12.2.

term is a function of either ε f orU. In all cases, wefind that M−Mc is approximatelyproportional to ε f − ε f c or U −Uc, such that the critical exponents are close to thevalues of the noninteracting case. In particular, the scaling law (12.7) is alwayssatisfied up to small numerical deviations. In general, the mapping between themass term and the tuning energy parameter (such as U) could be non-linear with(M −Mc) ∝ (U −Uc)ε . In this case, we would obtain the critical exponents γ 2εand ν ε, which differ from the simple situation but still fulfill the scaling law.

In addition, we can also calculate the RG flow [258] for this model. When con-sidering the flow as a function of two parameters M (M1 ,M2), in this caseM (ε f ,U), we would like to implement the RG equation

V ≡ M′ −Mδk2

∂2kF (k ,M)

kk0

2∇MF (k0 ,M)||∇MF (k0 ,M)||2

. (12.9)

Numerically, we need to perform two steps, which fulfill

F (k0 , ε′f ,U) F (k0 + δk , ε f ,U) , (12.10a)


0 2 4 6 8 10 12

-4000

-2000

0

2000

4000

(a)

0 2 4 6 8 10 120

20

40

60

80

100

(b)

Figure 12.2: F (k0) and ξ as a function of U with fit for k0 X in the plane kz 0, ε f −10,and other parameters as in Figure 12.1. The parameters correspond to the dashed red linein Figure 12.1. The phase transition at Γ is not visible, as we are only considering k0 X.

F (k0 , ε′f ,U′) F (k0 + δk , ε′f ,U) , (12.10b)

and are equivalent to

M′i −Mi

δk2

∂2kF (k ,M)

kk0

2∂MiF (k0 ,M) (12.11)

for i 1, 2. In order to obtain the result in Equation (12.9), we can calculate

V (ε f ,U) W (ε f ,U)W (ε f ,U)

2 , (12.12a)

where the vector W is defined as

W (ε f ,U) ©«(ε′f − ε f )−1

(U′ −U)−1ª®¬ . (12.12b)

The RG flow given by V (ε f ,U) is plotted in Figure 12.3 for k0 ∈ Γ,X,M,R andδk π/100. The phase transitions can be identified as the critical lines of the RG

flow, i.e., sources of the RG flow where ||V || → ∞, and are shown as black lines. Itis also possible to find unstable fixed points which are also sources of the RG flow

Chapter1

2

12.2 Scaling laws and RG flow in a TKI | 157U

ε f

Figure 12.3: Plots of the RG flow for the same parameters as in Figure 12.1 for the four HSPs Γ,X, M, and R. The color indicates the flow rate, where blue denotes small and orange highflow rates. The thick black line is the critical line where the gap closes and is the same asthose in Figure 12.1. Unstable fixed points are shown by dashed lines for k0 ∈ M,R.

but where ||V || → 0. At these lines, no topological phase transition takes place.Examples are shown in the plots for M and R as dashed lines.

Conclusion

In summary, we have discussed critical exponents and scaling laws related tophase transitions in 2D systems, as well as in high-symmetry planes of 3D systems.Noninteracting Dirac Hamiltonians are shown to have universal critical exponentsregardless the microscopic details of the Hamiltonian. On the other hand, thesaturation of the integration of Berry curvature near gap-closing momentum leadsto scaling laws. Using a simple model for a TKI, we have shown that the scalinglaws are satisfied even in interacting models. Finally, we have demonstrated thatthe RG flow in a TKI can be used in order to detect topological phase transitions. Aninteresting future project would be the extension of this analysis to systems wherethe interactions play an essential role such as fractional topological insulators.

Chapter1

3

Chapter 13

Conclusions and outlook

The overarching topic of this thesis is the study of the diverse properties that occurin the remarkable material class of topological Kondo insulators. These materialsare located at the interface of two central concepts of modern condensed matterphysics, namely topology and electronic correlations. This gives rise to a uniqueset of features that makes these materials interesting from the point of view offundamental research as well as promising for future applications in technologicaldevelopments. Here, we have discussed effects of both topology and interactions,as well as their interplay. In the following, we will provide a summary of our mostimportant results and a discussion of their relevance, as well as an overview ofopen questions and potential future research directions.

13.1 The significance of the hybridization

A central insight provided by this thesis is the importance of choosing the correcttype of hybridization for an appropriate description of (topological) Kondo insula-tors. Before the theoretical discovery of topological Kondo insulators in 2010 [37],models for Kondo insulators generally used an onsite hybridization and a disper-sionless band for the localized electrons [38, 114]. This leads to a k-independenthybridization gap that cannot close, such that topological phase transitions areimpossible. Choosing an onsite hybridization also disregards the fact that the con-duction and localized electrons in Kondo insulators generally have opposite parity,as they are usually derived from d and f states of rare-earth elements. In order toreflect these different parities, the hybridization in a model for a topological Kondo

160 | 13 Conclusions and outlook

insulator must itself be odd under parity, i.e., fulfill Φ(−k) −Φ(k), which meansthat it must be non-onsite. In turn, this implies that the localized band must itselfhave a non-vanishing dispersion in order to yield a finite (indirect) band gap.

While these facts had been implicitly used in previous theoretical works ontopological Kondo insulators [37, 156, 161], the fundamental significance of thehybridization matrix for the description of topological Kondo insulators had notbeen discussed. In Chapter 5, we have presented a simple two-band model fortopological Kondo insulators in the form of a periodic Anderson model in orderto study the essential features of topological Kondo insulators. We based thedispersion of both d and f electrons on an s-electron-like hopping, which is not anaccurate description of the behavior of these orbitals. Nevertheless, the model wasable to capture the essential topological properties of topological Kondo insulators.This leads to the conclusion that the form of the hybridization is the fundamentalelement of the model that ensures the existence of topological phases. It reflectsboth the opposite parities of thedifferent orbitals and the strong spin–orbit couplingthat is found in the localized states and is essential to the existence of topologicalinsulators. In summary, the hybridization must (i) be odd under parity, (ii) respectcubic symmetry, and (iii) be time-reversal invariant. These conditions imply, asdiscussed above, that it must be non-onsite, and in addition that it has a nontrivialspin structure, leading to the choice of Equation (5.2b).

In contrast to other studies that mainly focused on specific materials such asSmB6, we were able to analyze a variety of different topological phases that may, inprinciple, occur in Kondo insulators. Still, for a specific set of model parameters,the band structure of our simplified Hamiltonian strongly resembles that of SmB6and we obtain results for the topological invariants as well as surface states thatare consistent with those of more realistic models [38, 161]. We found that thetopological invariants are directly related to the high-symmetrypoints of the systemwith band inversions, i.e., those at which the d states are occupied. In addition, wewere able to calculate the topologically protected surface states and directly relatethem to the topological invariants of the system. It is this strongly simplifiedmodelthat most of the subsequent calculations of this thesis are based on.

Chapter1

3

13.2 Interplay of topology and interactions | 161

The importance of correctly modeling the hybridization also became apparentin Chapter 6, where we related the mirror Chern numbers in a topological Kondoinsulators to the hybridization matrix at the bulk high-symmetry points. We foundthat, depending on the form of the hybridization as well as the relative magnitudeof different-range hybridization parameters, multiple sets ofmirror Chern numberscan be realized, which in turn define the spin texture of the topologically protectedsurface states. Therefore, for an accurate description of a specific material, itis imperative to know which f orbital is highest in energy, i.e., relevant for theband inversion. Furthermore, due to the sensitivity towards the precise values ofhybridization parameters, care must be taken when basing a theoretical calculationon ab initio results.In Chapter 9, we reproduced the well-known relationship between the periodic

Anderson model and the Kondo lattice model [152] and derived the RKKY interac-tion using the path-integral formalism. In our calculations, we observed significantdifferences of both the Kondo lattice model and the RKKY interaction for varioustypes of hybridizations. Again,we showed that the “topological” hybridization thatwe introduced in Chapter 5 leads to very different results from the traditionallyused onsite hybridization.

13.2 Interplay of topology and interactions

In Chapter 7, we discussed the effects that the electron–electron interactions haveon the topological properties in a Kondo insulator. In previous studies, this hasonly been taken into account qualitatively by generally arguing that parametersare renormalized or by performing a U → ∞ approximation [37, 156, 161]. Here,we also assumed a local Fermi liquid and only considered the renormalizationof the microscopic parameters due to interactions. However, by performing amean-field calculation using the Kotliar–Ruckenstein slave-boson scheme, we wereable to study the properties of the system at finite U and derive a noninteractingmodel from the interacting one. This enabled us to discuss how the renormaliza-tion of hopping, hybridization, and energy parameters depends on the interactionstrength.


Considering only mean-field effects, the most significant consequence of theinteractions is a shift of the onsite energy of localized states. This changes thepoints with band inversions, which in turn were shown to be directly linked to thevalues of topological invariants in Chapter 5. Therefore, we were able to discussinteraction-driven topological phase transitions in Kondo insulators. In the limit ofstrong interactions, the system is driven towards the local-moment regime with ahalf-filled f band, which is related to the phase of a weak topological insulator. Incontrast, a strong topological insulator is favored in a mixed-valence regime, whichis relevant for SmB6. In general, the filling of the f band provides a relatively robustcriterion to change the topological phase in Kondo insulators.

Interaction-driven phase transitions have also been analyzed using dynamicalmean-field theory in a two-dimensional model for a topological Kondo insulator,yielding results consistent with our work [207]. In a related study, the topologicalproperties have also been discussed at finite temperature for this two-dimensionalmodel [208]. It would be an interesting future project to calculate a similartemperature-dependent phase diagram for our simplified three-dimensionalmodelof a topological Kondo insulator, or even a more realistic one, such as those pre-sented in Chapter 6.

13.3 Unconventional spin texture of surface states

Motivated by recent spin- and angle-resolved photoemission spectroscopy mea-surements for SmB6 [136] and conflicting theoretical predictions [194, 197, 198],we performed a detailed analysis of the spin texture of the topologically protectedsurface states of SmB6 in Chapter 6. We argued that the spin texture at the Xpoints on the surface of SmB6 can have two different winding numbers, which arerelated to the mirror Chern numbers of the system. These in turn can be calculatedfrom the form of the hybridization matrix at the bulk high-symmetry points, as wehave briefly discussed above. Performing explicit calculations for various modelsdescribing SmB6, we were able to show that depending on the relative energies ofdifferent f orbitals as well as the ratio of first- and second-neighbor hybridizationparameters, different topological invariants and winding numbers are possible.

Chapter1

3

13.4 Magnetic phases and collective excitations | 163

Our findings are consistent with similar calculations by other authors [199], aswell aswith the results of the spin- and angle-resolved photoemission spectroscopymeasurements for SmB6 thatmotivated our study [136]. The data of Reference [136]suggest that the spin textures of all topologically protected surface states in SmB6have awindingnumber of+1 and therefore resemble those in conventional topolog-ical insulators. Recently, however, it has been suggested that transitions betweenphases with different surface-state spin textures might be accessible experimen-tally by applying pressure to the material in order to shift the energies of the forbitals relative to each other [201]. This may lead to the first experimental ob-servation of an unconventional spin texture of a topological surface state, i.e., onewith negative winding number. Furthermore, in a preceding work it was shownthat the spin texture of surface states has significant consequences on the patternof quasiparticle interference, such that it may also be experimentally accessible viascanning-tunneling microscopy [198]. The analysis presented in this thesis thenallows us to draw conclusions about the microscopic properties of materials, i.e.,the energies of different orbitals and the structure of the hybridization, from theseexperimental results.

13.4 Magnetic phases and collective excitations

It has been demonstrated experimentally that SmB6 develops long-range (antiferro-magnetic) order when subjected to high pressure [43, 149] and many other heavy-fermion materials have magnetic phases at low temperatures [112]. In Chapters 8and 9, we presented two different approaches that enable us to study magneticphases: a spin-rotational-invariant generalization of the Kotliar–Ruckenstein slavebosons [209] and a calculation of the Ruderman–Kittel–Kasuya–Yosida (RKKY) in-teraction after performing a Schrieffer-Wolff transformation [152].Our slave-boson calculations in Chapter 8 are kept as general as possible to

allow for a very simple adaptation to various different models, which is a majorsimplification over previous studies [212, 213]. In Chapter 9, we derived the RKKY

interaction for various choices of hybridizations, including the one we introducedto model topological Kondo insulators. However, both calculations are basedon strongly simplified models for topological Kondo insulators and so far there


is a lack of detailed studies of magnetism in real materials such as SmB6, bothexperimentally and theoretically. Extending the calculations of our simplifiedmodel, it would be desirable to obtain a more realistic description of magnetism,as well as a phase diagram as a function of experimentally accessible parameters,such as temperature and pressure.

In Chapter 10, we presented an experimental study of the magnetic propertiesof SmB6 using low-energy muon spin relaxation. The data are consistent withspin excitons in SmB6 and a tendency to magnetic ordering on the surface [42, 144,145, 148, 150, 223]. The spin excitons can be pictured as fluctuating regions withantiferromagnetic ordering and a detailed interpretation of the data allowed us toextract both the size of this region and the magnitude of the fluctuating magneticmoments. The size of 40–90 nm that we obtained from the experimental data is invery good agreement with what can be expected from previously reported valuesfor the dielectric constant [135, 231]. The magnitude of fluctuating moments wasdetermined to be ∼ 0.01µB, which is consistent with the magnetic moments inantiferromagnetically ordered phases of other heavy-fermion materials [112, 131,132]. Overall, this analysis confirmed previous studies of both excitonic states andmagnetic ordering at the surface and additionally provided valuable new insightsinto the spatial behavior of the excitonic states close to the surface.

13.5 Properties of topological invariants

The last part of this thesiswas concernedwith relationships between topological in-variants and other important concepts of condensed matter physics. In Chapter 11,we related two-dimensional topological phases to the entanglement of quantumstates and quantum geometry. We reviewed established results on the entangle-ment spectrum of noninteracting fermionic systems and proposed a new variant,where the system is cut in a way that conserves the translational invariance, thesublattice entanglement spectrum. We were also able to show that this in turn isrelated to the quantum geometry, which may also be experimentally accessible tosome extent.

Finally, we studied how topological invariants, in particular the Chern number,are related to concepts of statistical physics. We showed that at topological phase

Chapter1

3

13.6 Real materials and applications | 165

transitions, the Berry curvature obeys certain scaling laws and has critical expo-nents independent on microscopic model parameters. We complemented theseanalytical results with calculations in the mirror-invariant planes of topologicalKondo insulators and discussed the idea of a renormalization-group flow.

13.6 Real materials and applications

While in this thesis we were able to illuminate multiple different and interrelatedeffects in topological Kondo insulators, we remark that most calculations presentedhere are based on the strongly simplified model for this material class that wasintroduced inChapter 5. A complete theoretical understanding of the details of realmaterials belonging to the class of topological Kondo insulators will require furtherresearch efforts in the future. Even the material samarium hexaboride (SmB6),which has been studied for over 50 years [39], still retains unsolved mysteries andopen questions. For example, there remains some controversy concerning theinterpretation of quantum-oscillation measurements in this material [179].In general, different effects in SmB6 and other potential topological Kondo insula-

tors havemostly been studied separately so far; for amore thoroughunderstanding,it is necessary to also explore the details of their interrelations. In particular, theimplications of magnetism for its topological properties, both in the bulk and atthe surface, are largely unexplored. Furthermore, the detailed relationships ofcollective excitations (see Chapter 10 and above), magnetism and the topologicallyprotected surface states havenot been studiedyet. Anotherpossible futuredirectionmight be the search for an even more direct connection of the strong correlations inSmB6 with its topology, e.g., in the form of highly-entangled topologically orderedstates.With regard to future applications, engineered phases are a particularly promis-

ing field of research, e.g., heterostructures of topological insulators with othermaterials. An interesting question for future studies would be how the multiplesurface states in topological Kondo insulators and their potentially unconventionalspin texture impacts existing proposals for heterostructures. This concerns ideasfor constructing topological superconductors by utilizing the proximity effect atthe surface of topological insulators [53], as well as other proposals, such as the


engineering of a Weyl semimetal by layering topological insulators with trivialinsulators [259].

Finally, the research on topological Kondo insulators has been driven mostlyby studies of a single material, SmB6, in recent years. Other proposed topologi-cal Kondo insulators have not yet been confirmed experimentally, such as PuB6,or have later been found to be trivial insulators under standard conditions, e.g.,YbB6 [160]. This poses the important question: Is SmB6 an anomaly in the widefield of Kondo insulators as the only one to feature topological properties? Orare there other materials that have not been considered yet, which belong to theclass of topological Kondo insulators as well and exhibit similar features comparedto SmB6? Considering additional materials may also lead to the observation ofthe unconventional spin texture that we discussed in this thesis under standardconditions.

Overall, the remarkable field of topological Kondo insulators provides a largenumber of opportunities for future experimental and theoretical research and willlikely contribute novel insights for condensed matter physics.

IVAppendices

App

endixA

Appendix A

Topological invariants and surface states

This appendix includes detailed calculations related to the content of Part I.

A.1 Calculation of surface states

We can explicitly calculate the surface energy spectrum by considering slabs whichhave periodic boundary conditions along two directions and are finite (with openboundary conditions) along the third. To simplify the formulas we define

h0 ε f1 − τz

2 , (A.1a)

h1r −td1 + τz

2 − t f1 − τz

2 , (A.1b)

hα1i Vτzσα , (A.1c)

h2 −t′d1 + τz

2 − t′f1 − τz

2 , (A.1d)

h3 −t′′d1 + τz

2 − t′′f1 − τz

2 . (A.1e)

Then, we can write the Hamiltonian (5.3) as

Hni ∑

i

Ψ†i h0Ψi +

[ ∑αx ,y ,z

∑〈i , j〉α

Ψ†i(h1r + ihα1i

)Ψ j

+

∑〈〈i , j〉〉

Ψ†i h2Ψ j +∑〈〈〈i , j〉〉〉

Ψ†i h3Ψ j + h.c.

].

(A.2)

170 | A Topological invariants and surface states

For the (100) surface, we can now do a Fourier transform of this Hamiltonianalong the y and z directions. This leads to

Hni ∑

x ,ky ,kz

[Ψ†(x , ky , kz) h(100)

0 (ky , kz)Ψ(x , ky , kz)

+

(Ψ†(x , ky , kz) h(100)

1 (ky , kz)Ψ(x+1, ky , kz) + h.c.) ],

(A.3a)

where the matrices h(100)0 and h(100)

1 are given by

h(100)0 (ky , kz) h0 + 2(cy + cz)h1r − 2

(sy h y

1i + sz hz1i

)+ 4cyz h2 , (A.3b)

h(100)1 (ky , kz) h1r + ihx

1i + 2(cy + cz)h2 + 4cyz h3. (A.3c)

For the (110) surface, we first have to define a new variable y : 1/√2(y − x),which lies in the surface-plane. Now, we can again do a Fourier transform of (A.2)along the y and z directions. This leads to

Hni ∑

x ,k y ,kz

[Ψ†(x , k y , kz) h(110)

0 (k y , kz)Ψ(x , k y , kz)

+

(Ψ†(x , k y , kz) h(110)

1 (k y , kz)Ψ(x+1, k y , kz) + h.c.)

+

(Ψ†(x , k y , kz) h(110)

2 (k y , kz)Ψ(x+2, k y , kz) + h.c.) ]

,

(A.4a)

where the matrices h(110)0 , h(110)

1 , and h(110)2 are given by

h(110)0 (k y , kz) h0 + 2cz h1r − 2sz hz

1i + 2 cos(√

2k y

)(h2 + 2cz h3) , (A.4b)

h(110)1 (k y , kz) h1r + ihx

1i + ei√

2k y(h1r + ih y

1i

)+ 2cz

(1 + ei

√2k y

)h2 , (A.4c)

h(110)2 (k y , kz) ei

√2k y (h2 + 2cz h3) . (A.4d)

Now, the Hamiltonians (A.3) and (A.4) can be diagonalized numerically in orderto obtain the surface energy spectrum.

App

endixA

A.2 The k·p theory on the (110) surface | 171

A.2 The k·p theory on the (110) surface

In this section, we derive the result (5.22) for the velocity of the surface statescrossing the Y point on the (110) surface in theWTI(ΓR) phase. For this purpose, wederive the k · p model by expanding around one of the three, by symmetry related,X points. We choose X (0, 0, π), which is projected onto the Y point on the (110)surface. For notational simplicity, we assume that t′′d t′′f 0 in the following. Thegap closes at the X point if

ε f ε f (X) ≡ −2(td − 2t′d − t f + 2t′f ) . (A.5)

Denoting 2M ε f − ε f (X) and expanding around the X point yields the followingmatrix:

HX(k) ©«−M+td k2−2(td+2t′d )k2

z 0 2Vkz −2Vk−0 −M+td k2−2(td+2t′d )k2

z −2Vk+ −2Vkz

2Vkz −2Vk− M+t f k2−2(t f +2t′f )k2z 0

−2Vk+ −2Vkz 0 M+t f k2−2(t f +2t′f )k2z

ª®®®¬ .(A.6)

Here, we have introduced k± kx ± iky . For the (110) surface, the mirror planekz π is projected onto the line Y–S, along which we observe a second crossingin the XM-inverted phase. We therefore use the eigenbasis of the mirror operatorMz iσzτz . Furthermore, we introduce the new coordinates

u 1√2(−x + y) , w

1√2(x + y) , (A.7a)

for which we can also calculate the related crystal momenta,

ku 1√2(−kx + ky) , kw

1√2(kx + ky) . (A.7b)

Then, along the Y–S line on the (110) surface Brillouin zone (SBZ), kz 0, and theHamiltonian decouples into the eigensectors of the mirror operator Mz . For the +i


eigensector, we find

H(+i)

©«M + t f (k2

u + k2w) 2V(ku − ikw)

2V∗(ku + ikw) −M + td(k2u + k2

w)ª®¬ , (A.8)

where V Ve−iπ/4 and a similar block for the (−i) eigensector:

H(−i)

©«M + t f (k2

u + k2w) 2V∗(ku + ikw)

2V(ku − ikw) −M + td(k2u + k2

w)ª®¬ . (A.9)

In the following, we focus on the +i sector. To obtain the edge states, we replacekw → −i∂w and use an exponentially decaying ansatz in the −w direction

ψλ(w , ku) eλ(ku )w ©«φλ(ku)χλ(ku)

ª®¬ . (A.10)

The secular equation yields four solutions βλα , with β ±1, α 1, 2, and [260]

λ21 k2

u + F −√

F2 − (M2 − E2)/(t2− − t2+) , (A.11a)

λ22 k2

u + F +

√F2 − (M2 − E2)/(t2− − t2

+) . (A.11b)

Here, we have defined

F Et+ −Mt− + 2V2

t2− − t2+

, (A.12a)

t f t+ − t− , (A.12b)

td t+ + t− . (A.12c)

The corresponding spinor wave functions are

ψβ,α(w) eβλαw©«

1E−M−t f (k2

u−λ2α)

2V(ku−βλα)

ª®®¬ . (A.13)

App

endixA

A.3 Mirror operators and sign choice of the mirror Chern numbers | 173

Surface states at the w 0 surface are exponentially decaying solutions in the −wdirection. We therefore make a linear superposition of the β +1 states:

Ψ(w) aψ+,1(w) + bψ+,2(w) (A.14)

and impose the boundary condition

Ψ(w 0) 0 . (A.15)

For a nontrivial solution, we have λ1 , λ2, and the energy of the surface states hasto satisfy

M − E (−t f )(λ1 − ku)(λ2 − ku) , (A.16a)

M + E td(ku + λ1)(ku + λ2) . (A.16b)

If M > 0, td > 0 and t f < 0, the solution to these equations is given by

E(+i)(ku) Mt+t−

+

2|V |√

t2− − t2+

t−ku . (A.17)

It follows that the edge state velocity is

v

2|V |√

t2− − t2+

t−

4|V |√−t f td

td − t f> 0 . (A.18)

This is what we wanted to show. In the XM-inverted phase, the mirror Chernnumber (MCN) for eigensector (+i) is −1. But because v > 0, there must be twoadditional crossings along the line S-Y–S.

A.3 Mirror operators and sign choice of the mirror Chern numbers

In general, the mirror operator for a plane with normal vector n can be written as

Mn IRn IRorbn Rps

n , (A.19)


where I is the inversion operator and Rn denotes a rotation by π around n. Thespin part of the rotation is given by Rps

n −i(n · σ), where σ are the Pauli ma-trices acting in spin space. Rorb

n denotes the orbital part of the rotation and de-pends on the symmetry of the considered orbitals. For example, for the orbitalsdx2−y2 , d3z2−r2 , fΓ7 , f

Γ(1)8, fΓ(2)8, the orbital part is Rorb

n 1 if n eα (α x , y , z);but for the mirror planes kα ±kβ this operator contains additional nontrivialsigns. Specifically, for n

1√2(ey − ex) we obtain Rorb

n ψi νiψi with ν 1 in thesubspaces d3z2−r2 , f

Γ(2)8 and ν −1 in the subspace dx2−y2 , f

Γ(1)8, fΓ7 .

We note that the signs of the MCNs are not uniquely determined in general: Onecan choose the sign of the mirror operator M and the orientation of the mirrorinvariant plane for the calculation of the MCNs, nmp. We fix the signs of the MCNsby the convention

Rpsnmp −i(nmp · σ) . (A.20)

The choice of orientation nmp and the sign convention during the calculation of theChern numbers still affects the positive directions in the SBZ. We use the conventionwhere the Berry connection, Berry curvature, and Chern number are defined asAµ i〈ψ(k)|∂µ |ψ(k)〉, Fµν ∂µAν − ∂νAµ, and C

12π

∫dk1 dk2F12, respec-

tively. Note that this convention leads to an additional factor of −1 compared toReference [100], which we use for our numerical calculations of the MCNs.

For the given conventions used in the definition of the mirror operator and theChern number, the positive directions in the SBZ can be defined as

npos nsf × nmp , (A.21)

where nsf is the outward pointing normal vector of the surface. In this work, weuse the conventions nmp eα for a mirror-invariant plane kα 0 or kα π andnmp

1√2(ey − ex) for the plane kx ky . This leads to the positive directions on

the top (nsf +ez) (001) surface as shown in Figure 6.1 (b).

App

endixA

A.4 Spin operator for f orbitals | 175

A.4 Spin operator for f orbitals

While the spin operator for the d orbitals is a block-diagonal matrix Sd 1⊗ σ, thisis not the case for the f orbitals [198]. The spin operator for the f orbitals can beobtained by calculating its matrix elements for Γ7 and Γ8 states. Writing the spinor

of f electrons as ψ

(fΓ(1)8 ,± , f

Γ(2)8 ,± , f

Γ7

) t, we obtain the spin operators

S fx

121 σx

©«−5 −2

√3 2

√5

−2√

3 −9 2√

15

2√

5 2√

15 5

ª®®®®¬, (A.22a)

S fy

121 σy

©«−5 2

√3 2

√5

2√

3 −9 −2√

15

2√

5 −2√

15 5

ª®®®®¬, (A.22b)

S fz

121 σz

©«−11 0 −4

√5

0 −3 0

−4√

5 0 5

ª®®®®¬. (A.22c)

Again, the spin matrix for the reduced models is obtained by removing the matrixelements that include either the Γ7 or Γ8 orbitals.

App

endixB

Appendix B

Details of slave-boson calculation

This appendix includes detailed calculations related to the content of Chapter 8.

B.1 Constraints in the spin-rotational-invariant slave-bosonrepresentation

The constraints which are to be shown and can be motivated with physical argu-ments are

1 e†e + d†d +

3∑µ0

p†µpµ , (B.1a)

f †σ′ fσ 2∑σ1

p†σ1σpσ′σ1

+ δσσ′d†d , (B.1b)

with the matrices

p†σσ′ 12

3∑µ0

p†µτµ,σσ′ 12©«

p†0 + p†z p†x − ip†yp†x + ip†y p†0 − p†z

ª®¬ , (B.2a)

pσσ′ 12

3∑µ0

pµτµ,σσ′ 12©«

p0 + pz px − ipy

px + ipy p0 − pz

ª®¬ . (B.2b)

178 | B Details of slave-boson calculation

Here, τ is the vector of Pauli matrices including the identity matrix as zero element.From Equation (B.1a) it follows that

ni n j niδi , j , for any number operator ni ∈ e†e , d†d , p†µpµ , (B.3a)

b†i b†j bi b j 0 , for any boson bi ∈ e , d , p . (B.3b)

Equation (B.1b) can be rewritten to the following:

∑σ

f †σ fσ

3∑µ0

p†µpµ + 2d†d , (B.4a)∑σσ′τσσ′ f †σ′ fσ p†0p + p†p0 − ip† × p . (B.4b)

Here, τ is the vector of Pauli matrices without the identity matrix. To map thefermionic creation and annihilation operators, c† and c, of the initial system to theslave-boson picture, we define

c†σ

∑σ′

z†σσ′ f †σ′ , (B.5a)

cσ

∑σ′

fσ′zσ′σ , (B.5b)

where we have defined

zσσ′ e†pσσ′ + p†σσ′d , (B.6a)

pσσ′ σσ′p−σ′−σ

(T pT −1

)σσ′

. (B.6b)

In our notation, we use σ, σ′ ∈ 1,−1 corresponding to ↑↓ and T denotes thetime-reversal operator. To clarify the notation, it is

p†σσ′ ©«

p1,1 p1,−1

p−1,1 p−1,−1

ª®¬ . (B.7)

Consequently, a minus in front of the index is an inversion of rows or columns, andσσ′ gives a minus sign for off-diagonal elements.

App

endixB

B.1 Constraints in the spin-rotational-invariant slave-boson representation | 179

For the rest of this section, we will remove the underlining of τ, p, p, and z in theinterest of readability. Using the commutation relation[

pσ1σ2 , p†σ3σ4

]

14

∑µµ′

τµ, σ1σ2τµ′, σ3σ4

[pµ , p

†µ′

]

14

∑µµ′

τµ, σ1σ2τµ′, σ3σ4δµµ′

12 δσ1σ4δσ2σ3

(B.8)

and the canonical (anti-)commutation relations for f , e, and d, the correct rela-tions for the c operators should be recovered, provided the above constraints aresatisfied [210]. The anticommutator for the c operators reads

cσ′ , c†σ

∑σ1σ2

zσ1σ fσ1 , z

†σ′σ2

f †σ2 ,

∑σ1σ2

(zσ1σ fσ1 z†σ′σ2

f †σ2 + z†σ′σ1f †σ1 zσ1σ fσ1

).

(B.9)

Using the anticommutation relation

f †σ1 , fσ2

δσ1σ2 and imposing that the

fermionic fields commute with the bosonic ones we findcσ′ , c

†σ

∑σ1σ2

( [z†σ′σ2

, zσ1σ

]f †σ2 fσ1 + zσ1σz†σ′σ2

δσ2σ1

)

∑σ1σ2

[e p†σ′σ2

+ pσ′σ2d† , e†pσ1σ + p†σ1σd

]f †σ2 fσ1︸¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨︷︷¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨︸

(I)

+

∑σ1

(e†pσ1σ + p†σ1σd

) (e p†σ′σ1

+ pσ′σ1d†

)︸¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨︷︷¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨︸

(II)

.

(B.10)

First we compute the commutator (I)1 in term (I)

(I)1

[e p†σ′σ2

+ pσ′σ2d† , e†pσ1σ + p†σ1σd

](B.11)


[e p†σ′σ2

, e†pσ1σ

]+

[e p†σ′σ2

, p†σ1σd]+

[pσ′σ1

d† , e†pσ1σ

]+

[pσ′σ2

d† , p†σ1σd].

We immediately see that the second and third commutator have to vanish, imposedby constraint Equation (B.3b). The remaining part then reads, using the linearityof the commutator

(I)1

[e p†σ′σ2

, e†pσ1σ

]+

[pσ′σ2

d† , p†σ1σd]

e[p†σ′σ2

, e†pσ1σ

]+

[e , e†pσ1σ

]p†σ′σ2

+ p†σ1σ

[pσ′σ2

d† , d]+

[pσ′σ2

d† , p†σ1σ

]d

e e†[p†σ′σ2

, pσ1σ

]+ e

[p†σ′σ2

, e†]

pσ1σ + e†[e , pσ1σ

]p†σ′σ2

+

[e , e†

]pσ1σp†σ′σ2

+ p†σ1σ pσ′σ2

[d† , d

]+ p†σ1σ

[pσ′σ2

, d]

d†

+ pσ′σ2

[d† , p†σ1σ

]d +

[pσ′σ2

, p†σ1σ

]d†d .

(B.12)

Different bosonic fields commute, so we are left with

(I)1 e e†[p†σ′σ2

, pσ1σ

]+

[e , e†

]pσ1σp†σ′σ2

+ p†σ1σ pσ′σ2

[d† , d

]+

[pσ′σ2

, p†σ1σ

]d†d

−e e† 12 δσ1σ2δσ′σ + pσ1σp†σ′σ2

− p†σ1σ pσ′σ2+ d†dσ1σ2σσ

′ 12 δσ1σ2δσ′σ

−e†e 12 δσ1σ2δσ′σ −

12 δσ1σ2δσ′σ + p†σ′σ2

pσ1σ +12 δσ2σ1δσ′σ − p†σ1σ pσ′σ2

+ d†dσ1σ2σσ′ 12 δσ1σ2δσ′σ

12 δσ1σ2δσ′σ

(d†d − e†e

)+ p†σ′σ2

pσ1σ − σ′σσ1σ2p†−σ−σ1 p−σ2−σ′ .

(B.13)

Consequently, we obtain

(I) 12 δσσ

′∑σ1

f †σ1 fσ1

(d†d − e†e

)+

∑σ1σ2

f †σ2 fσ1

(p†σ′σ2

pσ1σ − σσ′σ1σ2p†−σ−σ1 p−σ2−σ′).

(B.14)

App

endixB


For term (II), we find

(II) ∑σ1

(e†pσ1σe p†σ′σ1

+ e†pσ1σ pσσ1 d† + p†σ1σde p†σ′σ1+ p†σ1σdpσ′σ1

d†). (B.15)

Again, we can use Equation (B.3b) to set the second and third term to zero and arethen left with

(II) ∑σ1

(e†pσ1σe p†σ′σ1

+ p†σ1σdpσ′σ1d†

)

∑σ1

e†e(p†σ′σ1

pσ1σ +12 δσ1σ1δσ′σ

)+

∑σ1

p†σ1σ pσ′σ1

(d†d + 1

) δσ′σe†e +

∑σ1

e†e(p†σ′σ1

pσ1σ

)+

∑σ1

σσ′p†−σ−σ1 p−σ1−σ′(d†d + 1

).

(B.16)

Using Equations (B.14) and (B.16), we finally find for the fermionic anticommutator

c†σ , cσ′ δσσ′ e†e +∑σ1

[e†ep†σ′σ1

pσ1σ +

(1 + d†d

)σσ′p†−σσ1 pσ1−σ′

+12 δσσ

′ f †σ1 fσ1

(d†d − e†e

)+

∑σ2

f †σ2 fσ1

(p†σ′σ2

pσ1σ − σσ′σ1σ2p†−σ−σ1 p−σ2−σ′) ]

.

(B.17)We will now calculate the diagonal and off-diagonal terms in (B.17) separately.

First, we look at the diagonal terms, σ σ′ ±1, where we use

Q ≡ ip†x py − ip†y px and P ≡ p†0pz + p†z p0 (B.18)

for readability:

∑σ1

p†±1σ1pσ1±1

14©«

3∑µ0

p†µpµ ± (P + Q)ª®¬ , (B.19a)

12 δ±1±1

∑σ1

f †σ1 fσ1

(d†d − e†e

)

12©«

3∑µ0

p†µpµ + 2d†dª®¬(d†d − e†e

) d†d . (B.19b)


In order to calculate the last term with the double sum, we need a side calculation.With (B.1b), one finds

f †±1 f±1 d†d +12©«

3∑µ0

p†µpµ ± (P −Q)ª®¬ , (B.20a)

f †±1 f∓1 12

((p†0 ∓ p†z)(px ∓ ipy) + (p†x ∓ ip†y)(pz ± p0)

). (B.20b)

Subtracting Equation (B.20a) with minus sign from Equation (B.20a) with plus signand taking the square yields

(P −Q)2 f †−1 f−1 f †−1 f−1 + f †1 f1 f †1 f1 − f †−1 f−1 f †1 f1 − f †1 f1 f †−1 f−1

f †−1 f−1 + f †1 f1 − 2d†d

3∑µ0

p†µpµ .

(B.21)

Moreover, we can explicitly verify

f †1 f1 f †−1 f−1 ©«d†d +

12©«

3∑µ0

p†µpµ + (P −Q)ª®¬ª®¬ ©«d†d +12©«

3∑µ0

p†µpµ − (P −Q)ª®¬ª®¬ d†d +

14©«

3∑µ0

p†µpµ − (P −Q)2ª®¬ d†d .

(B.22)

Now, we calculate the missing term∑σ1σ2

f †σ2 fσ1

(p†±1σ2

pσ1±1 − σ1σ2p†∓1−σ1p−σ2∓1

)

±12 f †±1 f±1P +

14 f †−1 f1

((p†0 − p†z)(px − ipy) + (p†x − ip†y)(pz + p0)

)+

14 f †1 f−1

((p†x + ip†y)(p0 − pz) + (p†0 + p†z)(px + ipy)

)± 1

2 f †∓1 f∓1Q .

(B.23)

App

endixB


Using Equation (B.20b), this can be rewritten to only contain fermionic operatorsas well as P and Q:∑

σ1σ2

f †σ2 fσ1

(p†±1σ2

pσ1±1 − σ1σ2p†∓1−σ1p−σ2∓1

) (B.24)

±12 f †±1 f±1P +

12 f †−1 f1 f †1 f−1 +

12 f †1 f−1 f †−1 f1 ±

12 f †∓1 f∓1Q

±12 f †±1 f±1P +

12 f †−1 f−1

(1 − f †1 f1

)f−1Q

+12

(1 − f †−1 f−1

)f †1 f1 ±

12 f †∓1 f∓1Q

12

(f †−1 f−1 + f †1 f1 − 2d†d

)± 1

2 f †±1 f±1P ± 12 f †∓1 f∓1Q .

Now, we use Equations (B.4a) and (B.20a), which leads to∑σ1σ2

f †σ2 fσ1

(p†±1σ2

pσ1±1 − σ1σ2p†∓1−σ1p−σ2∓1

)

12

3∑µ0

p†µpµ ±14©«2d†d +

3∑µ0

p†µpµ ± (P −Q)ª®¬ P

± 14©«2d†d +

3∑µ0

p†µpµ ∓ (P −Q)ª®¬ Q

12

3∑µ0

p†µpµ ±12

(d†d(P + Q)

)± 1

4©«

3∑µ0

p†µpµ(P + Q)ª®¬ + 14 (P −Q)2

34

3∑µ0

p†µpµ ±12

(d†d(P + Q)

)± 1

4©«

3∑µ0

p†µpµ(P + Q)ª®¬ .(B.25)

Using all of the previous results and Equation (B.1a), we can calculate the diagonalmatrix elements of Equation (B.17):

c†±1 , c±1 e†e + e†e ©«14©«

3∑µ0

p†µpµ ± (P + Q)ª®¬ª®¬ (B.26)


+ (1 + d†d) ©«14©«

3∑µ0

p†µpµ ∓ (P + Q)ª®¬ª®¬+ d†d +

34

3∑µ0

p†µpµ ±12 d†d(P + Q) ± 1

4©«

3∑µ0

p†µpµ(P + Q)ª®¬ e†e + d†d +

3∑µ0

p†µpµ ±14 (P + Q) ©«−1 + e†e + d†d +

3∑µ0

p†µpµª®¬

1 .

Now, we calculate the off-diagonal matrix elements σ ±1, σ′ ∓1. It is againconvenient to define

R ≡ 14

(p†x p0 + p†0px + ip†y pz − ip†z py

), (B.27a)

S ≡ 14

(ip†y p0 + ip†0py + p†x pz − p†z px

). (B.27b)

The nonzero terms in Equation (B.17) yield∑σ1

p†±1σ1pσ1∓1 R ∓ S , (B.28a)∑

σ1σ2

f †σ2 fσ1

(p†±1σ2

pσ1∓ − (+1)(−1)σ1σ2p†±−σ1 p−σ2∓) ( f †1 f1 + f †−1 f−1)(R ∓ S) .

(B.28b)

Inserting in Equation (B.17), one finds

c†±1 , c∓1 (R ∓ S)(e†e − 1 − d†d + f †−1 f−1 + f †1 f1

) (R ∓ S) ©«−1 + e†e + d†d +

3∑µ0

p†µpµª®¬

0 .

(B.29)

Equations (B.26) and (B.29) are a confirmation that the constraints guarantee thefermionic creation and annihilation operators still fulfilling the canonic anticom-

App

endixB

B.2 Correct mean-field results in the noninteracting limit | 185

mutation relationsc†σ , cσ′ ≡ δσσ′ . (B.30)

B.2 Correct mean-field results in the noninteracting limit

In thenoninteracting case, orbitals are occupied randomly. Assuming that a fractionx with 0 ≤ x ≤ 1 of states are occupied, a random distribution leads to d2 x2

doubly occupied, p20 2x − 2x2 2x(1− x) singly occupied and e2 1− p2

0 − d2

(1− x)2 empty sites. In a paramagnetic mean-field calculation the matrix z definedin Equation (8.8) is reduced to Equation (8.13a), where we use Equation (8.5a) tofind M → 2−1/2. Inserting here the values for the random distribution, we obtaina trivial renormalization z0 1, which we expect in the noninteracting case.

B.3 Maximization with respect to µ0 and β0

Consider the free energy defined in Equation (8.12). Calculating the derivativew.r.t. µ0 leads to

∂µ0 F ∂µ0 (−T log(Z) + µ0N) −T∂µ0 Z

Z+ N . (B.31)

If we plug in the partition function, we find

∂µ0 F −T∂µ0

∑k ,b(1 + exp(−εk ,b/T)∑

k ,b(1 + exp(−εk ,b/T)) + N −∑k ,b

nF(εk ,b) + N , (B.32)

with ∂µ0 εk ,b −1 and nF(ε) denoting the Fermi distribution. This of course hasto be zero, since it is simply the constraint for the electron filling. The secondderivative w.r.t. µ0 can than be performed easily and yields

∂2µ0 F −∂µ0

∑k ,b

nF(εk ,b) ∑k ,b

∂εk ,b nF(εk ,b) < 0 , (B.33)

since nF(ε) is monotonically falling for all ε. Therefore, there must be a maximumwith respect to µ0.


In our case, the β0 constraint can be seen as a renormalization of µ0 and thereforehas to follow the same mathematics and physical interpretation.

B.4 Slave-boson number operator

The following calculation is to show the equivalence of number operators of phys-ical electrons and pseudofermions.∑

σ

c†σcσ

∑σσ1σ2

z†σσ1 f †σ1 fσ2 zσ2σ

∑σσ1σ2

(ep†σσ1+ pσσ1 d†)(e†pσ2σ

+ p†σ2σd) f †σ1 fσ2

∑σσ1σ2

[(1 + e†e)p†σσ1

pσ2σ+ d†d

(12 δσσδσ2σ1 + p†σ2σ pσσ1

)]f †σ1 fσ2

∑σσ1σ2

p†σσ1pσ2σ

f †σ1 fσ2 +

∑σ

d†d f †σ fσ

∑σ1σ2

12

(f †σ2 fσ1 − δσ1σ2 d†d

)f †σ1 fσ2 +

∑σ

d†d f †σ fσ

∑σ1σ2

12 f †σ2 fσ1 f †σ1 fσ2 +

12

∑σ

d†d f †σ fσ

∑σ1σ2

12

(− f †σ2 fσ2 f †σ1 fσ1 + δσ2σ1 f †σ2 fσ1 + δσ1σ1 f †σ2 fσ2

)+

12 d†d ©«

3∑µ0

p†µpµ + 2d†dª®¬

12

(−2d†d +

∑σ

(3 f †σ fσ − f †σ fσ))+ d†d

∑σ

f †σ fσ

(B.34)

Two bosonic annihilators to the very right annihilate any physical state on thereduced Fock space, and such terms consequently vanish. Moreover, we have usedthe canonical (anti-)commutation relations as well as Equations (B.1b), (8.3), (B.4a),and (B.22).

App

endixB

B.5 Calculation of derivatives in momentum space | 187

B.5 Calculation of derivatives in momentum space

We calculate the first-order variation of zq :

δzq

∑r

e−iqr δzr

∑r

e−iqr∑µ

∂zr∂ψr ,µ

δψr ,µ .(B.35)

We know that ∂zr∂ψr ,µ

≡ ∂z∂ψµ

is independent of r . Therefore, we obtain

δzq

∑µ

∂z∂ψµ

∑r

e−iqr δψr ,µ

∑µ

∂z∂ψµ

δψq ,µ ,

(B.36)

which leads to∂zq

∂ψq′ ,µ δq ,q′

∂z∂ψµ

. (B.37)

The same calculation for z†q yields

δz†q

∑r

eiqr δz†r (B.38)

∑µ

∂z†∂ψµ

∑r

e−i(−q)r δψr ,µ

∑µ

∂z†∂ψµ

δψ−q ,µ ,

which is equivalent to∂z†q∂ψq′ ,µ

δq ,−q′∂z∂ψµ

. (B.39)


Calculating the second-order variation leads to

δ2zq

∑r

e−iqr δ2zr (B.40)

∑r

e−iqr∑µ,ν

∂2z∂ψµ∂ψν

δψr ,µδψr ,ν

∑q1 ,q2

∑µ,ν

∂2z∂ψµ∂ψν

δψq1 ,µδψq2 ,ν

∑r

ei(q1+q2−q)r︸¨ ¨ ¨ ¨ ¨ ¨ ¨︷︷¨ ¨ ¨ ¨ ¨ ¨ ¨︸δq ,q1+q2

∑q1

∑µ,ν

∂2z∂ψµ∂ψν

δψq1 ,µδψq−q1 ,ν ,

such that we obtain

∂2zq

∂ψq1 ,µ∂ψq2 ,ν δq ,q1+q2

∂2z∂ψµ∂ψν

. (B.41)

The same calculation for z† leads to

δ2zq

∑q1

∑µ,ν

∂2z∂ψµ∂ψν

δψq1 ,µδψ−q−q1 ,ν , (B.42)

which is equivalent to

∂2z†q∂ψq1 ,µ∂ψq2 ,ν

δq ,−q1−q2∂2z†

∂ψµ∂ψν. (B.43)

B.6 Derivation of Equation (8.35)

For any 2×2 matrix A a0σ0 + a · σ,1 it is possible to find a unitary matrix Ua ,which diagonalizes A:

U†aAUa a0σ0 + |a |σ3 . (B.44)

1Note thatwewrite σ to denote the Paulimatrices and dismiss the underlining in the interest of readability

App

endixB

B.6 Derivation of Equation (8.35) | 189

It is clear that Ua can only depend on a a/|a |: The part proportional to theidentity is left invariant by any unitary transformation and we can write

U†(a · σ)U |a |U†(a · σ)U . (B.45)

Note that if we use instead a −a/|a |, the sign of the second summand in Equa-tion (B.44) would be reversed.We now consider the definition of z in Equation (8.8). We first calculate

4p†p (p0σ0 + piσi)(p0σ0 + p jσ j) p2

0σ0 + 2p0(piσi) + pi p jσiσ j

p20σ0 + 2p0(p · σ) + p2σ0 ,

(B.46)

as the products for i , j vanish due to σiσ j iεi jkσk . Similarly, 4p† p p20σ0 −

2p0(p · σ) + p2σ0. Now, Equation (8.8) can be written as

z

[(1 − |d |2 − p2

0+|p |22

)σ0 − p0(p · σ)

]−1/2· 1√

2

[(e + d)p0σ0 + (e − d)(p · σ)]·[(

1 − e2 − p20+p2

2

)σ0 + p0(p · σ)

]−1/2,

(B.47)where we used the constraint 1+ e2 + d2 + p2 2. We immediately see that we canuse the same unitary transformationUp to diagonalize the threematrices (althoughwe will obtain a different sign for the first matrix), such that all unitary matricescancel except the ones on the outside. The product of diagonal matrices can simplybe obtained by multiplying the diagonal entries separately. This leads to

z Up diag(z+ , z−)U†p , (B.48)

where z± are defined as in Equation (8.35b). We can rewrite this matrix as

diag(z+ , z−) z+ + z−2 σ0 +

z+ − z−2 σ3 z(+)σ0 + z(−)σ3 . (B.49)


Re

Im

Re

Im

Figure B.1: Paths in the complex plane used for the calculation of Matsubara sums in Equa-tion (B.51). The poles on the imaginary axis come from nF(z) and the single pole on thereal axis from (z − εk ,b)−1.

The unitary transformation now just rotates back the part proportional to σ3 to p ·σ:

z z(+)σ0 + z(−)(

p|p | · σ

), (B.50)

which is the same as Equation (8.35a).

B.7 Details of Matsubara summations

The derivation of Equation (8.38) is a standard calculation using complex analysis.For the diagonalized Green’s function Gk Db

((iωn − εk ,b)−1) of Equation (8.28),we can rewrite the sum as a complex integral using the theorem of residua anddeform the path of integration (see Figure B.1):

T∑

nG(ωn ,k)[ψ] T

∑n

Db

((iωn − εk ,b)−1

)(B.51)

− 12πi

∮dz nF(z)Db

((z − εk ,b)−1

) Db

(nF(εk ,b)

).

App

endixB

B.7 Details of Matsubara summations | 191

Re

Im

Re

Im

Figure B.2: Paths in the complex plane used for the calculation of Matsubara sums in Equa-tion (B.53). The two poles which do not lie on the imaginary axis come from (iωn − εk ,b)−1

and (iωn + i$ − εk+q ,b′ )−1. Note that they can also lie in the same half plane.

This is the result quoted in Equation (8.38). Here, the first sign change (and factorof T−1) comes from the negative residue of nF at z iωn , while the second followsfrom the negative (clockwise) direction around the pole at z εk ,b .

The derivation of Equation (8.38)works in a similarmannerwith some additionalsubtleties. We consider a single element of the matrix, where we can rewrite thematrix multiplication due to the diagonal Green’s matrices,

Sb ,b′ ≡[T

∑n

G(ωn ,k)[ψ]Uk M(k , q)U†k+q G(ωn+$m ,k+q)

]b ,b′

(Uk M(k , q)U†k+q

)b ,b′

[T

∑n(iωn − εk ,b)−1 (iωn + i$ − εk+q ,b′)−1

].

(B.52)

Again we can use the theorem of residua in order to replace the sum over n by acomplex integral. In this case, there are two additional poles, see Figure B.2. Ateach of the poles we pick up the residue which includes the value of the other


Green’s function:

Sb ,b′ (Uk M(k , q)U†k+q

)b ,b′−12πi

∮dz nF(z)

(z − εk ,b

)−1(z + i$ − εk+q ,b′

)−1

(Uk M(k , q)U†k+q

)b ,b′

[nF(εk ,b)

εk ,b + i$ − εk+q ,b′+

nF(εk+q ,b′ − i$)εk+q ,b′ − i$ − εk ,b

]

(Uk M(k , q)U†k+q

)b ,b′

nF(εk ,b) − nF(εk+q ,b′)i$m + εk ,b − εk+q ,b′

.

(B.53)

In the last step we used that nF(ε − i$m) nF(ε) for any bosonic Matsubarafrequency $.

App

endixC

Appendix C

Dispersion of excitonic states

In this appendix, wewant to complement the analysis discussed in Chapter 10 withmodel calculations for SmB6 leading to the dispersion of excitonic states as well astheir size and distribution in real and momentum space.The dielectric constant for SmB6 at low temperatures (T ≈ 3 K) was estimated

to be between εr 600 and εr 1500 by different measurements [135, 231]. Inrecent inelastic-neutron-scattering experiments, a collective mode was found atE0 ≈ 14 meV, which is interpreted as a spin exciton [145].

In order to describe the Kondo insulator SmB6, we use the simplified two-orbitalmodel defined in Chapter 5 with up to second-neighbor hopping. Due to the spin-degeneracy of the system, we can further simplify the model by removing the spindegree-of-freedom leading to H0

∑k h(k), where the BlochHamiltonian h(k) has

the form of Equation (5.6) and is now given by a 2×2 matrix with

hd(k) [−2td (c1 + c2 + c3) − 4t′d (c1c2 + c2c3 + c1c3)

], (C.1a)

h f (k) [ε f − 2t f (c1 + c2 + c3) − 4t′f (c1c2 + c2c3 + c1c3)

], (C.1b)

Φ(k) −2Φ0

√s2

x + s2y + s2

z , (C.1c)

where we use cα ≡ cos(kα), sα ≡ sin(kα), and the spinor ψ (c , f )t.We use the parameters td 180 meV, t′d 70 meV, t f −0.05 td −9 meV,

t′f −0.05t′d 3.5 meV, V 18 meV, and ε f −450 meV, leading to a bandinversion at the X point and a band gap of 19.5 meV, see Figure C.1. This is a goodfit for experimental values for the band gap in the range of 15−20 meV [135–137].

194 | C Dispersion of excitonic states

Γ X M Γ R X M R−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

E(k

)/eV

Figure C.1: Band structure of the Hamiltonian defined by Equation (C.1).

C.1 Ansatz for the excitonic state

As the ground state of the system, we consider a situation where the valence bandis completely filled and the conduction band is empty,

|G〉 ∏

k

c†v(k) |0〉 , (C.2)

and study particle–hole excitations

|k ,K〉 c†c (k)cv(k − K) |G〉 . (C.3)

In addition to the kinetic energy, we now also consider the interaction among theelectrons leading to

H H0 + Hint , (C.4a)

where the interaction is given by

Hint 12

∑r ,r′

V(|r − r′ |)n(r)n(r′) , (C.4b)

App

endixC

C.2 Matrix elements | 195

with the number operators

n(r) ∑αd , f

c†αcα (C.4c)

and the Coulomb potential (for orbitals approximated as pointlike)

V(r)

e2

4πε0εr

1r, r , 0 ,

V0 , r 0 .(C.4d)

C.2 Matrix elements

In order to calculate the energies of an exciton ofmomentum K, we have to calculatethe matrix elements

Mk1 ,k2 (K) 〈k1 ,K |H | k2 ,K〉 − δk1 ,k2 〈G |H |G〉 . (C.5)

We candiagonalizeH0 bydefiningvalence and conductionbandoperators (a v, c)

ca(k) ∑αd , f

uαa (k)cα(k) . (C.6)

After rewriting the kinetic term as

H0

∑k

[εc(k)nc(k) + εv(k)nv(k)] , (C.7)

we immediately obtain

M0k1 ,k2(K) δk1 ,k2 [εc(k1) − εv(k1 − K)] . (C.8)

In the interest of legibility, we will use the conventions∑a≡

∑av,c

,∑α

≡∑αd , f

(C.9)

for the remainder of the calculations.


In order to evaluate the interaction part of the matrix elements, we first have torewrite Hint in momentum space. We insert

n(r) 1N

∑α

∑k ,k′

eir ·(k−k′)c†α(k)cα(k′)

1N

∑q

eiq·r ∑k

∑α

∑a ,b

uαa∗(k + q)uαb (k)c†a(k + q)cb(k) ,

(C.10)

where N L3 is the number of lattice sites into Equation (C.4b) and obtain

Hint 12

∑q

V(q)∑k ,k

∑a ,ba ,b

[∑α

uαa∗(k + q)uαb (k)

] [∑α

u αa∗(k − q)u α

b(k)

]c†a(k + q)cb(k)c†a(k − q)cb(k)

(C.11)

with the Fourier-transformed Coulomb potential

V(q) 1N

∑r

eiq·r V(r) . (C.12)

This can be calculated numerically by taking r (r1 , r2 , r3) with ri ∈ [−L/2, L/2[.We define

Σ(q , k , k , a , b , a , b) ≡ V(q)2

[∑α

uαa∗(k + q)uαb (k)

] [∑α

u αa∗(k − q)u α

b(k)

](C.13)

in order to simplify the following terms. After performing the commutations, weobtain

MVk1 ,k2(K) Σ(K , k1 − K , k2 , c, v, v, c) − Σ(k1 − k2 , k2 , k1 − K , c, c, v, v)

− Σ(k2 − k1 , k1 − K , k2 , v, v, c, c) + Σ(−K , k2 , k1 − K , v, c, c, v)+ δk1 ,k2

∑q

[Σ(q , k1 − K , q + k1 − K , v, v, v, v)− Σ(q , k1 − K − q , k1 − K , v, c, c, v)+ Σ(q , k1 − q , k1 , c, c, c, c) − Σ(q , k1 , k1 + q , v, c, c, v)] .

(C.14)

App

endixC

C.3 Energy and size | 197

C.3 Energy and size

For n lattice sites in each direction, M is an N×N matrix. The eigenvalues E i andeigenvectors Ui of this matrix are the exciton energies and states, respectively. Inparticular, we can numerically compute the lowest exciton energy E0 as a functionof εr, V0, and K. We define ∆E as the difference of E and the bottom of theparticle–hole continuum.As we can perform our calculations only for small systems up to n ∼ 30, it is

difficult to calculate the size of the exciton in real space. Instead, we estimate thestandard deviation of the state in momentum space σ(K) and obtain

σ(K) ≈ 12σ(K) (C.15)

in units of the lattice constant a ≈ 4.13Å.For the estimation of σ we use the following approach: We search for the

maximum of the state U0(K) and assume a Gauss-like distribution around it.Then, with m0(K) maxk |U0,k(K)|2 and the maximum of its nearest neighbors,m1 max〈k ,k〉 |U0,k(K)|2, the Gauss distribution leads to an estimation of

σ(K) ≈ πL

[12 log

(m0m1

)]−1/2. (C.16)

C.4 Calculations

We are now looking for parameters V0 and εr ∈ [600, 1500] which yield an excitonenergy of ∼ 14 meV and a size of 10 ∼ 100 nm and calculate their dispersion.We find reasonable agreement for the parameters εr 1000 and V0 −40 meV.

The dispersion of the excitonic states is shown in Figure C.2 and the lowest excitonstate at q X in is shown in Figures C.3 (a) and C.3 (b) in real and momentumspace, respectively. The size of the lowest-energy state at the X point (calculatedfor L 20) in real space is d ≈ 50a ≈ 20 nm which is in reasonable agreement withthe implantation-depth measurements discussed in Chapter 10.


Γ X M Γ R X M R10

20

30

40

50

60

70

80

E/m

eV

Figure C.2:Dispersionof the lowestfive exciton states for L 14, εr 1000, andV0 −40 meV.The gray area is the particle–hole continuum, its bottom is shown by the thick black line.

−10 −5 0 5 10−10

−5

0

5

10

(a) Exciton in real space.

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

(b) Exciton in momentum space.

Figure C.3: Plot of log(|U0(r)|2

)(a) and log

(|U0(k)|2

)(b) for r and k in the x y and kx ky

plane, respectively at K X. The hole is placed at the origin and the parameters are L 20,εr 1000, and V0 −40 meV. The size of the exciton is estimated at d ≈ 50a.

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List of acronyms

ARPES angle-resolved photoemission spectroscopy.AZ Altland–Zirnbauer.

BI band insulator.BZ Brillouin zone.

CMP condensed matter physics.

ES entanglement spectrum.

FQHE fractional quantum Hall effect.

HSL high-symmetry line.HSP high-symmetry point.

KI Kondo insulator.KLM Kondo lattice model.KR Kotliar–Ruckenstein.

LE-µSR low-energy muon spin relaxation.

MCN mirror Chern number.MIL mirror-invariant line.MIP mirror-invariant plane.

216 | List of acronyms

NN nearest neighbor.NNN next-to-nearest neighbor.NNNN next-to-next-to-nearest neighbor.

PAM periodic Anderson model.PBC periodic boundary conditions.PHS particle–hole symmetry.

QAHE quantum anomalous Hall effect.QHE quantum Hall effect.QSHE quantum spin Hall effect.

RG renormalization group.RKKY Ruderman–Kittel–Kasuya–Yosida.

SBZ surface Brillouin zone.SLES sublattice entanglement spectrum.SOC spin–orbit coupling.SPES single-particle entanglement spectrum.SPT symmetry-protected topological.µSR muon spin relaxation.STI strong topological insulator.

TCI topological crystalline insulator.TI topological insulator.TKI topological Kondo insulator.TKNN Thouless–Kohmoto–Nightingale–den Nĳs.TRIM time-reversal-invariant momentum.TRS time-reversal symmetry.TSC topological superconductor.TSS topologically protected surface state.

List of materials

The following list gives an overview over all materials mentioned in this thesis, thematerial class they belong to, as well as the pages on which they are discussed.

Bi2Se3 bismuth selenide (topological insulator). 9Bi2Te3 bismuth telluride (topological insulator). 9Bi1−xSbx bismuth antimony (topological insulator). 9

CeAl3 cerium aluminium (heavy-fermion metal). 26, 28CeCoIn5 cerium cobalt indium 5 (heavy-fermion superconductor). 28CeCu2Si2 cerium copper silicon (heavy-fermion superconductor). 27CeNiSn cerium nickel tin (heavy-fermion metal). 27CeRhAs cerium rhodium arsenide (heavy-fermion metal). 27CeRhSb cerium rhodium antimony (heavy-fermion metal). 27

HgTe mercury telluride (quantum spin Hall effect). 9

Pb1−xSnxSe lead tin selenide (topological insulator). 23Pb1−xSnxTe lead tin telluride (topological insulator). 23PuB6 plutonium hexaboride (topological Kondo insulator). 27, 33, 168

Sb2Te3 antimony telluride (topological insulator). 9SmB6 samarium hexaboride (topological Kondo insulator). v–viii, 4–6, 25–

27, 29, 30, 33–36, 41, 43, 44, 47, 48, 60–62, 65, 69–71, 73, 75–79, 89,122–128, 162, 164–168, 195

218 | List of materials

SmS samarium sulfide (Kondo insulator). 26SnTe tin telluride (topological crystalline insulator). 23Sr2RuO4 strontium ruthenate (topological superconductor). 10

TaAs tantalum arsenide (Weyl semimetal). 11

UPt3 uranium platinum (heavy-fermion superconductor). 28

YbB12 ytterbium dodecaboride (topological crystalline insulator). 27, 33YbB6 ytterbium hexaboride (considered as a topological Kondo insulator).

27, 33, 168

List of publications

1. Markus Legner and Titus Neupert, “Relating the entanglement spectrum of nonin-teracting band insulators to their quantum geometry and topology”, Physical ReviewB 88, 115114 (2013).

2. Markus Legner, Andreas Rüegg, and Manfred Sigrist, “Topological invariants,surface states, and interaction-driven phase transitions in correlated Kondo insulatorswith cubic symmetry”, Physical Review B 89, 085110 (2014).

3. Markus Legner, Andreas Rüegg, andManfred Sigrist, “Surface-state spin texturesand mirror Chern numbers in topological Kondo insulators”, Physical Review Letters115, 156405 (2015).

4. Pabitra K. Biswas,Markus Legner, Geetha Balakrishnan, M. Ciomaga Hatnean,Martin R. Lees, Don McK. Paul, Ekaterina Pomjakushina, Thomas Prokscha,Andreas Suter, Titus Neupert, and Zaher Salman, “Suppression of magnetic excita-tions near the surface of the topological Kondo insulator SmB6”, submitted to PhysicalReview B (2016).

5. Wei Chen, Markus Legner, Andreas Rüegg, and Manfred Sigrist, “Correlationlength, universality classes, and scaling laws associated with topological phase transi-tions”, in preparation (2016).

6. Markus Legner, Michael Klett, David Riegler, Seulgi Ok, Titus Neupert, andRonny Thomale, “Excitons and magnetism in topological Kondo insulators from spin-rotation invariant slave-boson calculations with fluctuations”, in preparation (2016).










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