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E-grāmata: Topological Aspects of Condensed Matter Physics: Lecture Notes of the Les Houches Summer School: Volume 103, August 2014

Edited by (Professor of Physics, Physics Department, Boston University, USA), Edited by , Edited by (CNRS Research Director, Laboratoire de Physique des Solides, CNRS UMR, Université Paris-Sud, France), Edited by (Director, Max-Planck-Institute für Physik komplexer Systeme, Ger)
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This book contains lecture notes by world experts on one of the most rapidly growing fields of research in physics. Topological quantum phenomena are being uncovered at unprecedented rates in novel material systems. The consequences are far reaching, from the possibility of carrying currents and performing computations without dissipation of energy, to the possibility of realizing platforms for topological quantum computation.The pedagogical lectures contained in this book are an excellent introduction to this blooming field. The lecture notes are intended for graduate students or advanced undergraduate students in physics and mathematics who want to immerse in this exciting XXI century physics topic.

This Les Houches Summer School presents an overview of this field, along with a sense of its origins and its placement on the map of fundamental physics advancements. The School comprised a set of basic lectures (part 1) aimed at a pedagogical introduction of the fundamental concepts, which was accompanied by more advanced lectures (part 2) covering individual topics at the forefront of today's research in condensed-matter physics.
List of Participants xxvii
Part I: Basic Lectures
1 An introduction to topological phases of electrons
3(60)
Joel E. Moore
1.1 Introduction
5(1)
1.2 Basic concepts
5(13)
1.2.1 Mathematical preliminaries
5(9)
1.2.2 Berry phases in quantum mechanics
14(4)
1.3 Topological phases: Thouless phases arising from Berry phases
18(17)
1.3.1 Bloch states
18(3)
1.3.2 1D polarization and 2D IQHE
21(2)
1.3.3 Interactions and disorder: the flux trick
23(1)
1.3.4 TKNN integers, Chern numbers, and homotopy
24(2)
1.3.5 Time-reversal invariance in Fermi systems
26(3)
1.3.6 Experimental status of 2D insulating systems
29(1)
1.3.7 3D band structure invariants and topological insulators
29(2)
1.3.8 Axion electrodynamics, second Chern number, and magnetoelectric polarizability
31(4)
1.3.9 Anomalous Hall effect and Karplus-Luttinger anomalous velocity
35(1)
1.4 Introduction to topological order
35(20)
1.4.1 FQHE background
35(1)
1.4.2 Topological terms in field theories: the Haldane gap and Wess-Zumino-Witten models
36(9)
1.4.3 Topologically ordered phases: the FQHE
45(10)
1.A Topological invariants in 2D with time-reversal invariance
55(4)
1.A.1 An interlude: Wess-Zumino terms in 1D nonlinear sigma-models
55(1)
1.A.2 Topological invariants in time-reversal-invariant Fermi systems
56(2)
1.A.3 Pumping interpretation of Z2 invariant
58(1)
References
59(4)
2 Topological superconductors and category theory
63(60)
Andrei Bernevig
Titus Neupert
Preface
65(1)
2.1 Introduction to topological phases in condensed matter
65(17)
2.1.1 The notion of topology
65(2)
2.1.2 Classification of non-interacting fermion Hamiltonians: the 10-fold way
67(8)
2.1.3 The Su-Schrieffer-Heeger model
75(2)
2.1.4 The 1D p-wave superconductor
77(3)
2.1.5 Reduction of the 10-fold way classification by interactions: Z Z8 in class BDI
80(2)
2.2 Examples of topological order
82(20)
2.2.1 The toric code
83(7)
2.2.2 The 2D p-wave superconductor
90(12)
2.3 Category theory
102(18)
2.3.1 Fusion category
102(8)
2.3.2 Braiding category
110(4)
2.3.3 Modular matrices
114(4)
2.3.4 Examples: the 16-fold way revisited
118(2)
Acknowledgements
120(1)
References
121(2)
3 Spin liquids and frustrated magnetism
123(42)
John T. Chalker
3.1 Introduction
125(7)
3.1.1 Overview
125(3)
3.1.2 Classical ground-state degeneracy
128(1)
3.1.3 Order by disorder
129(3)
3.2 Classical spin liquids
132(5)
3.2.1 Simple approximations
132(2)
3.2.2 The triangular lattice Ising antiferromagnet and height models
134(3)
3.3 Classical dimer models
137(6)
3.3.1 Introduction
138(1)
3.3.2 General formulation
138(3)
3.3.3 Flux sectors, and U(1) and Z2 theories
141(1)
3.3.4 Excitations
142(1)
3.4 Spin ices
143(7)
3.4.1 Materials
143(1)
3.4.2 Coulomb phase correlations
144(3)
3.4.3 Monopoles
147(1)
3.4.4 Dipolar interactions
148(2)
3.5 Quantum spin liquids
150(10)
3.5.1 Introduction
150(1)
3.5.2 Lieb-Schultz-Mattis theorem
150(2)
3.5.3 Quantum dimer models
152(8)
3.6 Concluding remarks
160(2)
3.6.1 Slave particles
160(1)
3.6.2 Numerics
161(1)
3.6.3 Summary
161(1)
Acknowledgements
162(1)
References
162(3)
4 Entanglement spectroscopy and its application to the quantum Hall effects
165(54)
Nicolas Regnault
Preface
167(1)
4.1 Introduction
167(2)
4.2 Entanglement spectrum and entanglement entropy
169(10)
4.2.1 Definitions
170(1)
4.2.2 A simple example: two spin-1/2
171(1)
4.2.3 Entanglement entropy
172(3)
4.2.4 The AKLT spin chain
175(2)
4.2.5 Matrix product states and the entanglement spectrum
177(2)
4.3 Observing an edge mode through the entanglement spectrum
179(10)
4.3.1 The integer quantum Hall effect
179(5)
4.3.2 Chern insulators
184(1)
4.3.3 Entanglement spectrum for a CI
185(4)
4.4 Fractional quantum Hall effect and entanglement spectra
189(16)
4.4.1 Fractional quantum Hall effect: overview and notation
190(3)
4.4.2 Orbital entanglement spectrum
193(4)
4.4.3 OES beyond model wavefunctions
197(4)
4.4.4 Particle entanglement spectrum
201(3)
4.4.5 Real-space entanglement spectrum
204(1)
4.5 Entanglement spectrum as a tool: probing the fractional Chern insulators
205(4)
4.5.1 From Chern insulators to fractional Chern insulators
205(3)
4.5.2 Entanglement spectrum for fractional Chern insulators
208(1)
4.6 Conclusions
209(1)
Acknowledgements
210(1)
References
210(9)
Part II: Topical lectures
5 Duality in generalized Ising models
219(22)
Franz J. Wegner
Preface
221(1)
5.1 Introduction
221(1)
5.2 Kramers-Wannier duality
221(3)
5.2.1 High-temperature expansion (HTE)
222(1)
5.2.2 Low-temperature expansion (LTE)
223(1)
5.2.3 Comparison
223(1)
5.3 Duality in three dimensions
224(1)
5.4 General Ising models and duality
225(4)
5.4.1 General Ising models
225(1)
5.4.2 Duality
226(3)
5.5 Lattices and Ising models
229(3)
5.5.1 Lattices and their dual lattices
229(1)
5.5.2 Models on the lattice
230(1)
5.5.3 Euler characteristic and degeneracy
230(2)
5.6 The models Mdm on hypercubic lattices
232(2)
5.6.1 Gauge invariance and degeneracy
233(1)
5.6.2 Self-duality
233(1)
5.7 Correlations
234(3)
5.7.1 The model Mdd
234(1)
5.7.2 Dislocations
235(2)
5.8 Lattice gauge theories
237(1)
5.9 Electromagnetic field
237(1)
References
238(3)
6 Topological insulators and related phases with strong interactions
241(24)
Ashvin Vishwanath
6.1 Overview
243(1)
6.2 Quantum phases of matter. Short-range versus long-range entanglement
244(3)
6.3 Examples of SRE topological phases
247(2)
6.3.1 Haldane phase of S = 1 antiferromagnet in d = 1
247(1)
6.3.2 An exactly soluble topological phase in d = 1
247(2)
6.4 SRE phase of bosons in two dimensions
249(6)
6.4.1 Coupled-wire construction
250(2)
6.4.2 Effective field theory
252(2)
6.4.3 Implications for IQH state of electrons
254(1)
6.5 SPT phases of bosons in three dimensions
255(5)
6.5.1 The m = 0 critical point
257(1)
6.5.2 Surface topological order of 3D bosonic SRE phases
257(3)
6.6 Surface topological order of fermionic topological insulators and superconductors
260(2)
Acknowledgements
262(1)
References
262(3)
7 Fractional Abelian topological phases of matter for fermions in two-dimensional space
265(96)
Christopher Mudry
7.1 Introduction
268(9)
7.2 The tenfold way in quasi-one-dimensional space
277(12)
7.2.1 Symmetries for the case of one one-dimensional channel
277(6)
7.2.2 Symmetries for the case of two one-dimensional channels
283(3)
7.2.3 Definition of the minimum rank
286(3)
7.2.4 Topological spaces for the normalized Dirac masses
289(1)
7.3 Fractionalization from Abelian bosonization
289(18)
7.3.1 Introduction
289(1)
7.3.2 Definition
290(1)
7.3.3 Chiral equations of motion
291(1)
7.3.4 Gauge invariance
292(3)
7.3.5 Conserved topological charges
295(2)
7.3.6 Quasiparticle and particle excitations
297(3)
7.3.7 Bosonization rules
300(3)
7.3.8 From the Hamiltonian to the Lagrangian formalism
303(2)
7.3.9 Applications to polyacetylene
305(2)
7.4 Stability analysis for the edge theory in symmetry class AII
307(15)
7.4.1 Introduction
307(5)
7.4.2 Definitions
312(2)
7.4.3 Time-reversal symmetry of the edge theory
314(2)
7.4.4 Pinning the edge fields with disorder potentials: the Haldane criterion
316(1)
7.4.5 Stability criterion for edge modes
317(3)
7.4.6 The stability criterion for edge modes in the FQSHE
320(2)
7.5 Construction of two-dimensional topological phases from coupled wires
322(29)
7.5.1 Introduction
322(4)
7.5.2 Definitions
326(4)
7.5.3 Strategy for constructing topological phases
330(4)
7.5.4 Reproducing the tenfold way
334(10)
7.5.5 Fractionalized phases
344(6)
7.5.6 Summary
350(1)
Acknowledgements
351(1)
References
351(10)
8 Symmetry-protected topological phases in one-dimensional systems
361(26)
Frank Pollmann
8.1 Introduction
363(1)
8.2 Entanglement and matrix product states
364(8)
8.2.1 Schmidt decomposition and entanglement
364(2)
8.2.2 Area law
366(1)
8.2.3 Matrix product states
367(5)
8.3 Symmetry-protected topological phases
372(6)
8.3.1 Symmetry transformations of MPS
372(2)
8.3.2 Classification of projective representations
374(1)
8.3.3 Symmetry fractionalization
375(2)
8.3.4 Spin-1 chain and the Haldane phase
377(1)
8.4 Detection
378(5)
8.4.1 Degeneracies in the entanglement spectrum
378(1)
8.4.2 Extraction of projective representations from the mixed transfer matrix
379(1)
8.4.3 String order parameters
380(3)
8.5 Summary
383. Acknowledgement
383(1)
References
383(4)
9 Topological superconducting phases in one dimension
387(64)
Felix von Oppen
Yang Peng
Falko Pientka
9.1 Introduction
389(5)
9.1.1 Motivation
389(3)
9.1.2 Heuristic arguments
392(2)
9.2 Spinless p-wave superconductors
394(6)
9.2.1 Continuum model and phase diagram
394(3)
9.2.2 Domain walls and Majorana excitations
397(1)
9.2.3 Topological protection and many-body ground states
398(2)
9.2.4 Experimentally accessible systems
400(1)
9.3 Topological insulator edges
400(3)
9.3.1 Model and phases
400(2)
9.3.2 Zero-energy states and Majorana operators
402(1)
9.4 Quantum wires
403(4)
9.4.1 Kitaev limit
405(1)
9.4.2 Topological insulator limit
406(1)
9.5 Chains of magnetic adatoms on superconductors
407(16)
9.5.1 Shiba states
408(2)
9.5.2 Adatom chains
410(10)
9.5.3 Kitaev chain
420(3)
9.6 Non-Abelian statistics
423(7)
9.6.1 Manipulation of Majorana bound states
423(2)
9.6.2 Non-Abelian Berry phase
425(2)
9.6.3 Braiding Majorana zero modes
427(3)
9.7 Experimental signatures
430(7)
9.7.1 Conductance signatures
430(5)
9.7.2 4Pi-periodic Josephson effect
435(2)
9.8 Conclusions
437(1)
9.A Pairing Hamiltonians: BdG and second quantization
438(3)
9.B Proximity-induced pairing
441(3)
9.C Shiba states
444(3)
9.C.1 Adatom as a classical magnetic impurity
444(2)
9.C.2 Adatom as a spin-1/2 Anderson impurity
446(1)
Acknowledgements
447(1)
References
447(4)
10 Transport of Dirac surface states
451(38)
D. Carpentier
10.1 Introduction
453(5)
10.1.1 Purpose of the lectures
453(1)
10.1.2 Dirac surface states of topological insulators
453(2)
10.1.3 Graphene
455(2)
10.1.4 Overview of transport properties
457(1)
10.2 Minimal conductivity close to the Dirac point
458(3)
10.2.1 Zitterbewegung
458(1)
10.2.2 Clean large tunnel junction
459(1)
10.2.3 Minimal conductivity from linear response theory
460(1)
10.3 Classical conductivity at high Fermi energy
461(11)
10.3.1 Boltzmann equation
462(4)
10.3.2 Linear response approach
466(6)
10.4 Quantum transport of Dirac fermions
472(12)
10.4.1 Quantum correction to the conductivity: weak antilocalization
474(3)
10.4.2 Universal conductance fluctuations
477(2)
10.4.3 Notion of universality class
479(4)
10.4.4 Effect of a magnetic field
483(1)
Acknowledgements
484(1)
References
484(5)
11 Spin textures in quantum Hall systems
489(42)
Benoit Doucot
11.1 Introduction
491(2)
11.2 Physical properties of spin textures
493(15)
11.2.1 Intuitive picture
493(4)
11.2.2 Construction of spin textures
497(4)
11.2.3 Energetics of spin textures
501(2)
11.2.4 Choice of an effective model
503(4)
11.2.5 Classical ground states of the CPd-1 model
507(1)
11.3 Periodic textures
508(9)
11.3.1 Perturbation theory for degenerate Hamiltonians
508(3)
11.3.2 Remarks on the Hessian of the exchange energy
511(2)
11.3.3 Variational procedure for energy minimization
513(3)
11.3.4 Properties of periodic textures
516(1)
11.4 Collective excitations around periodic textures
517(5)
11.4.1 Time-dependent Hartree-Fock equations
517(1)
11.4.2 Collective-mode spectrum
518(3)
11.4.3 Towards an effective sigma model description
521(1)
11.A Coherent states in the lowest Landau level
522(1)
11.B From covariant symbols on a two-dimensional plane to operators
523(1)
11.C Single-particle density matrix in a texture Slater determinant
524(2)
11.D Hamiltonians with quadratic covariant symbol
526(1)
Acknowledgements
527(1)
References
527(4)
12 Out-of-equilibrium behaviour in topologically ordered systems on a lattice: fractionalized excitations and kinematic constraints
531(36)
Claudio Castelnovo
Preface
533(1)
12.1 Topological order, broadly interpreted
533(1)
12.2 Example 1: (classical) spin ice
534(18)
12.2.1 Thermal quenches
538(7)
12.2.2 Field quenches
545(7)
12.3 Example 2: Kitaev's toric code
552(12)
12.3.1 The model
553(2)
12.3.2 Elementary excitations
555(2)
12.3.3 Dynamics
557(2)
12.3.4 Intriguing comparison: kinetically constrained models
559(5)
12.4 Conclusions
564(1)
Acknowledgements
564(1)
References
565(2)
13 What is life?-70 years after Schrodinger
567(99)
Antti J. Niemi
Preface
570(1)
13.1 A protein minimum
571(21)
13.1.1 Why proteins?
571(1)
13.1.2 Protein chemistry and the genetic code
572(1)
13.1.3 Data banks and experiments
573(4)
13.1.4 Phases of proteins
577(3)
13.1.5 Backbone geometry
580(2)
13.1.6 Ramachandran angles
582(2)
13.1.7 Homology modelling
584(1)
13.1.8 All-atom models
585(2)
13.1.9 All-atom simulations
587(1)
13.1.10 Thermostats
588(4)
13.1.11 Other physics-based approaches
592(1)
13.2 Bol'she
592(7)
13.2.1 The importance of symmetry breaking
593(1)
13.2.2 An optical illusion
593(1)
13.2.3 Fractional charge
594(2)
13.2.4 Spin-charge separation
596(2)
13.2.5 All-atom is Landau liquid
598(1)
13.3 Strings in three space dimensions
599(13)
13.3.1 Abelian Higgs model and the limit of slow spatial variations
600(2)
13.3.2 The Frenet equation
602(1)
13.3.3 Frame rotation and Abelian Higgs multiplet
603(2)
13.3.4 The unique string Hamiltonian
605(1)
13.3.5 Integrable hierarchy
605(1)
13.3.6 Strings from solitons
606(2)
13.3.7 Anomaly in the Frenet frames
608(2)
13.3.8 Perestroika
610(2)
13.4 Discrete Frenet frames
612(12)
13.4.1 The Calpha trace reconstruction
614(1)
13.4.2 Universal discretized energy
615(3)
13.4.3 Discretized solitons
618(1)
13.4.4 Proteins out of thermal equilibrium
619(1)
13.4.5 Temperature renormalization
620(4)
13.5 Solitons and ordered proteins
624(22)
13.5.1 lambda-repressor as a multisoliton
624(4)
13.5.2 Structure of myoglobin
628(7)
13.5.3 Dynamical myoglobin
635(11)
13.6 Intrinsically disordered proteins
646(13)
13.6.1 Order versus disorder
647(2)
13.6.2 hIAPP and type 2 diabetes
649(2)
13.6.3 hIAPP as a three-soliton
651(4)
13.6.4 Heating and cooling hIAPP
655(4)
13.7 Beyond Calpha
659(7)
13.7.1 'What-you-see-is-what-you-have'
660(6)
Acknowledgements 666(1)
References 666
Claudio Chamon is Professor of Physics at Boston University. He studied at MIT, where he received his BS degree in Aeronautics & Astronautics (1989), MS in Electrical Engineering and Computer Science (1991), and PhD in Physics (1996). He was a postdoctoral fellow at the Univ. of Illinois at Urbana Champaign (1996-1998) and a member at the Institute for Advanced Study (1999). His research focuses on topological states of matter and on systems out-of-equilibrium. He is Fellow of the American Physical Society.

Mark Oliver Goerbig obtained a joint PhD degree from the Universities of Fribourg (Switzerland) and Paris-Sud (France) in 2004 on correlated electronic phases in quantum Hall systems. After a year of postdoctoral research on frustrated quantum magnetism at the University Pierre et Marie Curie, Paris, he was awarded a CNRS researcher position at Laboratoire de Physique des Solides, Orsay, where he started to work on graphene and related topological materials, in addition to his ongoing research on the quantum Hall effect. Since 2012, he is also professor at the École Polytechnique in Palaiseau.

Professor Roderich Moessner is Director at the Max Planck Institute for the Physics of Complex Systems. He has previously held faculty positions at Ecole Normale Superieure in Paris as well as at Oxford University, where he had also done his doctoral studies, which were followed by a postdoctoral appointment at Princeton University. His research interests are in theoretical condensed matter and statistical physics as well as quantum information theory. In particular, he investigates the physics of strong fluctuations in many-body systems, which may be due to frustration, competing degrees of freedom or quantum fluctuations. Among the prizes he has received are the Europhysics Prize of the Condensed Matter Division of the European Physical Society and the Leibniz Prize of the German Science Foundation.

Leticia F. Cugliandolo received her Ph.D. in theoretical physics from the Universidad Nacional de La Plata, Argentina, in 1991. After post-docs in Universita di Roma I, La Sapienza, and CEA/Saclay she joined the Physics Department at Ecole Normale Superieure de Paris in 1997. She is currently a full professor at Universite Pierre et Marie Curie in Paris and the director of Ecole de Physique des Houches since 2007. Her research focuses on statistical physics and condensed matter problems. She received the Marie Curie Excellence Award, the Guggenheim Fellowship and the Prix Langevin of the French Physical Society.
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