Table Of ContentSpringer Series in Solid-State Sciences 190
Dario Bercioux · Jérôme Cayssol
Maia G. Vergniory · M. Reyes Calvo
Editors
Topological
Matter
Lectures from the Topological Matter
School 2017
Springer Series in Solid-State Sciences
Volume 190
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é ô
Dario Bercioux J r me Cayssol
(cid:129)
Maia G. Vergniory M. Reyes Calvo
(cid:129)
Editors
Topological Matter
Lectures from the Topological Matter
School 2017
123
Editors
DarioBercioux Maia G.Vergniory
Theoretical Mesoscopic Physics DonostiaInternational Physics Center
DonostiaInternational Physics Center Donostia/San Sebastián, Gipuzkoa,Spain
Donostia/San Sebastián, Gipuzkoa,Spain
M.Reyes Calvo
Jérôme Cayssol CIC nanoGUNE
Laboratoire Ondeset Matière d’aquitaine Donostia/San Sebastián, Gipuzkoa,Spain
Talence,France
ISSN 0171-1873 ISSN 2197-4179 (electronic)
SpringerSeries inSolid-State Sciences
ISBN978-3-319-76387-3 ISBN978-3-319-76388-0 (eBook)
https://doi.org/10.1007/978-3-319-76388-0
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Preface
In the past few years, the Donostia International Physics Center (DIPC) has laun-
chedanextensiveresearchefforttotheinvestigationoftopologicalstatesofmatter
(TSM). The education of graduate students and young postdoctoral fellows is an
essential aspect of this effort. In this spirit, we have organised on a yearly basis a
summer school which gathers worldwide experts on the subject—the Topological
Matter School series. Our goal is to provide students with a pedagogical but
comprehensive and up-to-date presentation of this quickly growing field. The lec-
turesfromthepast2017editionoftheschoolarenowcollectedinthisbook,which
aims to serve as an educational introduction to those newly approaching the study
oftopologyincondensedmattersystems.Thevolumeincludeschaptersbothonthe
fundamental theoretical aspects and on some of the latest experimental break-
throughs in the field.
Topology is a field of mathematics that has fed almost all the domains of
physics, ranging from high-energy to condensed matter systems, from the
pioneering work of Dirac on magnetic monopoles to modern gauge theories,
classification of topological defects in ordered phases, Berezinskii–Kosterlitz–
Thouless transition, spin chains, quantum Hall effects, and in the last decade
topological insulators and semimetals. The topological effects discussed in this
book are related to the particular winding properties of electronic Berry phase of
Blochelectronicstatesinspecificinsulatingandconductingmaterials.Therefore,it
can be seen as a refinement of standard band structure theory where the primary
interest was focused only on the energy level dispersion and gaps while ignoring
the subtle properties of the quantum mechanical phases.
In condensed matter, topological effects are mainly related to the non-trivial
winding of the phase of bulk Bloch states around the whole Brillouin zone. TSM
manifests themselves in several new features: (i) the emergence of gapless con-
ducting states confined at the external boundary of the system or between two
systems with different windings, like the chiral edge states of quantum Hall
(QH) insulators and Chern insulators, or the helical edge states or surface states of
time-reversal invariant topological insulators (TIs); (ii) the existence of quantised
responsefunctionsthataretopologicallyprotectedfromlocalperturbations,likethe
v
vi Preface
polarisationin1DinsulatorsorthetransverseconductivityinQHinsulatorsand2D
Cherninsulators;(iii)quantumanomaliesinWeylsemimetals(WSs);(iv)Majorana
quasi-particle in hybrid systems combining topological insulators with supercon-
ductors, or even nanowires with strong spin–orbit, Zeeman coupling and
proximity-induced superconductivity. Interestingly all these effects can be tested
experimentally, and most of them have already been (or on the way to be) con-
firmed in real materials.
The implications of TSM go beyond condensed matter physics and have given
rise to a highly multidisciplinary field of research that includes chemistry, pho-
tonics,atomic,polymerphysics,etc.Besidesthefundamentalinterestthatthefield
haspeakedinthescientificcommunity,therealisationoftopologicalmaterialsalso
has important technological consequences. A crucial advantage common to all
topological materials is the robustness of specific features (like the existence of
metallic edge states) against local perturbations or details of sample preparation.
The properties of topological materials are expected to lead to technological
applications in electronics, spintronics and optoelectronics. For example, the
spin-momentum locking of edge or surface states may serve to generate
spin-polarised currentsor thelarge quantisednonlinear optical effects can promote
technologies like solar cells or photodetectors beyond their current limits. Another
relevant aspect of TSM for electronics is related to quantum confinement (QC): to
thedate,QChasbeentypicallyobtainedbycomplicatedengineeringprocesses.The
advent of topological matter has changed the rules of the game; confinement
spontaneously occurs at the surface of a topological insulator and can be thus
optimised by the right choice of material, even if the available energy window for
the quasi-particles is limited compared to the nanofabrication platforms.
The book is composed of tencontributions. The first three chaptersare devoted
to the general characterisation of the topology of condensed matter systems.
Chapters4 and5 aredevoted, respectively, totheoretical andexperimental aspects
of electronic transport in low-dimensional hybrid systems made of TIs and super-
conductors. Chapters 6 and 7 are devoted to the physics of 3D Weyl semimetal.
Chapters 8, 9 and 10 describe various aspect of growth and characterisation of
topological materials.
Chapter 1 “Band Theory Without Any Hamiltonians or ‘The way Band Theory
ShouldbeTaught’”introducesthetheoryofTopologicalQuantumChemistry.This
new formalism predicts the presence or absence of topological phases by studying
thebehaviouroforbitalslyinginsomespecialpositionsofthecrystallatticeinreal
space. Throughout the chapter, the main concepts of the theory will be analysed
following a well-known example: graphene.
In Chap. 2 “Topological Crystalline Insulators”, Titus Neupert and Frank
SchindlerintroducetheconceptofWilsonloop,whichisBerryphasediagnosticfor
systemswithbanddegeneracy.Theauthorsshowtheconnectionofthistopological
quantitytotheeigenvaluesofthepositionoperator,thustothegeneralisationofthe
problem of polarisation in solids. The concept of Wilson loop is employed for
investigating topological crystalline insulators and a new class of topological sys-
tems named higher-order topological insulators.
Preface vii
Dominik Gresch and Alexey Soluyanov in their contribution entitled
“Calculating Topological Invariants with Z2Pack” (Chap. 3) present a general
introductiontotheconceptofChernnumberinnon-interactingbandsystems.They
describeanefficientprocedureforextractingtheChernnumberinconnectionwith
the Berry phase and the Wilson loop. This efficient method is at the core of the
PythoncodeZ2Pack:throughtheuseofseveralexamples,theauthorsexplainhow
to use this package for the evaluation of topological properties.
In the contribution “Transport in Topological Insulator Nanowires” by Jens H.
Bardarson and Roni Ilan (Chap. 4), the authors make an in-depth analysis of the
quantum transport properties of quasi-one-dimensional topological insulator
quantum wire. They present the effect of magnetic field and disorder on the
transportpropertiesofthesewires.Thesecondpartisdevotedtoproximity-induced
superconductivity in topological insulator nanowires, with an emphasis on the
emergence and possible detection of Majorana fermions in these hybrid junctions.
In his contribution to this book (Chap. 5), Erwann Bocquillon reviews the
consequences of induced superconductivity in the surface states of a topological
insulator, long predicted as a path for the generation of topological superconduc-
tivity and Majorana states. By using microwave excitation and detection tech-
niques, Bocquillon and collaborators have detected the elusive signatures of
Majorana bound states in Josephson junctions using HgTe as a weak link. In this
chapter, the theoretical and technical aspects of their experiments are first intro-
duced to provide the reader with the necessary background to understand the fol-
lowing detailed review or their results and prospective work in the field.
In Chap. 6, Adolfo G. Grushin presents how field theoretical tools borrowed
from high-energy physics can be used to study low-energy/effective models of
topological matter. First, a generic model for Weyl semimetals is interpreted as a
Lorentz breaking theory for fermions in the continuum. Then, three different pos-
sibilitiestopromotesuchafieldtheorytoliveonalatticearediscussedtouchingon
the importance of the Nielsen–Ninomiya (or fermion doubling) theorem. Finally,
AdolfoG.GrushinemphasisesthatWeylsemi-metallicphasesofmatterandrelated
systems be described by ambiguous field theories (theories predicting observable
quantities that are finite but depend on the regularisation procedure), which high-
light interesting aspects of their responses to external fields and make contact with
quantum anomalies.
In Chap. 7, Alberto Cortijo explains how quantum anomalies, and in particular
the chiral anomaly, arise in the recently discovered Weyl semimetals. In such
semimetals, Weyl nodes appear in pairs with opposite chiralities (left-handed and
right-handedWeylfermions).Foragivenchirality,thedensityofWeylfermionsis
not conserved in the presence of collinear electric and magnetic fields: this is the
chiral anomaly. After reviewing the role of symmetries in field theory (both clas-
sical and quantum) briefly and defining quantum anomalies, Alberto Cortijo uses
the semiclassical Boltzmann theory to derive the formula for the rate of change of
left (respectively, right)-handed fermionic densities. The crucial ingredient is the
introduction of Berry phase terms in the semiclassical equations of motion since
eachWeylnodebehavesasamagneticmonopoleink-space,namelyasource/sink
viii Preface
ofBerrycurvatureflux.Thechapterisself-containedandalsoaddressesthecrucial
role of internode relaxation processes and the positive magnetoconductivity of
Weyl semimetals arising from the quantum anomaly.
InChap.8“TopologicalMaterialsinHeuslerCompounds”,FelserandSunwill
presenttheHeuslerscompoundsandallthedifferenttopologicalmaterialstheycan
realise. The interplay of symmetry, spin–orbit coupling and magnetic structure
allows for the realisation of a wide variety of topological phases through Berry
curvature design, from Weyl semimetals to nodal lines or the recently discovered
antiskyrmions.
In Chap. 9, Schoop and Topp introduce some basic concepts of solid-state
chemistry and how they can help identify new topological materials, providing a
short overview of common crystal growth methods and the most significant char-
acterisation techniques available to identify topological properties. This chapter
aims to provide a guide for implementing simple chemical principles in the search
for new topological materials, as well as giving a basic introduction to the steps
necessary to experimentally verify the electronic structure of a material.
In Chap. 10, Haim Beidenkopf presents an intuitive analogy between the real
space topological screw dislocation in solids and the momentum space Weyl node
structure of topological semimetals. Bulk-boundary correspondence results in
unique surface features in the form of step edges at the surface of bulk with screw
dislocations and surface Fermi arcs in the surface for Weyl semimetals. In both
cases,Beidenkopfandhisteamapply scanningtunnellingmicroscopytothestudy
ofthesephenomena.Therealspacecaseofdislocationscanbecharacterisedbyjust
topographical images of step edges in the surface. The detection of momentum
surface Fermi arcs requires more advanced techniques such as quasi-particle
interference(QPI),whichallowextractingvaluableinformationonthepropertiesof
Weyl fermions in materials such as TaAs.
The editors thank the authors of each contribution for making this volume
possibleandsuccessful.WearealsogratefultothestaffoftheDIPCandofCursos
de Verano of the University of Basque Country for the support during the running
of the different editions of the TMS school: from basic to advanced.
Gipuzkoa, Spain Dario Bercioux
Gipuzkoa, Spain M. Reyes Calvo
Talence, France Jérôme Cayssol
Gipuzkoa, Spain Maia G. Vergniory
Contents
1 Band Theory Without Any Hamiltonians or “The Way Band
Theory Should Be Taught”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I. Robredo, B. A. Bernevig and Juan L. Mañes
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Hexagonal Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Orbits for the Different q Points. . . . . . . . . . . . . . . . . . 6
1.2.2 Adding Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Adding p Orbitals at 2b Positions . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Spinless p Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Spinful p Orbitals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Inducing a Band Representation. . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Little Groups at k Points in the First BZ . . . . . . . . . . . . . . . . . 13
1.6 Example of Band Representation . . . . . . . . . . . . . . . . . . . . . . . 15
1.6.1 Spinful Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6.2 Spinless Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 Subducing the Band Representation . . . . . . . . . . . . . . . . . . . . . 19
1.7.1 C Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7.2 K Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.7.3 M Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.7.4 High-Symmetry Lines. . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Topological Crystalline Insulators. . . . . . . . . . . . . . . . . . . . . . . . . . 31
Titus Neupert and Frank Schindler
2.1 Wilson Loops and the Bulk-Boundary Correspondence. . . . . . . 31
2.1.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . 31
2.1.2 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.3 Wilson Loop and Position Operator. . . . . . . . . . . . . . . . 33
2.1.4 Bulk-Boundary Correspondence . . . . . . . . . . . . . . . . . . 41
ix
