Topological aspects of condensed matter physics: École de physique des Houches, Session CIII, 4-29 August 2014
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| Format: | Conference Proceeding Book |
| Language: | English |
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Oxford
Oxford University Press
2017
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| Edition: | First edition |
| Series: | Session
103 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references |
| Physical Description: | xxxi, 671 Seiten Illustrationen 26 cm |
| ISBN: | 9780198785781 |
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| 245 | 1 | 0 | |a Topological aspects of condensed matter physics |b École de physique des Houches, Session CIII, 4-29 August 2014 |c edited by Claudio Chamon (Physics Department, Boston University, Boston, Massachusetts, 02215, USA), Mark O. Goerbig (Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris-Sud, Université Paris-Saclay F-91405, France), Roderich Moessner (Max-Planck-Institut für Physikl komplexer Systeme, 901187 Dresden, Germany), Leticia F. Cugliandolo (Sorbornn Universités, Universités Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France) |
| 250 | |a First edition | ||
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| adam_text | Contents
List of Participants xxvii
Part I Basic Lectures
1 An introduction to topological phases of electrons 3
Joel E. MOORE
1.1 Introduction 5
1.2 Basic concepts 5
1.2.1 Mathematical preliminaries 5
1.2.2 Berry phases in quantum mechanics 14
1.3 Topological phases: Thouless phases arising from Berry phases 18
1.3.1 Bloch states 18
1.3.2 ID polarization and 2D IQHE 21
1.3.3 Interactions and disorder; the flux trick 23
1.3.4 TKNN integers, Chern numbers, and homotopy 24
1.3.5 Time-reversal invariance in Fermi systems 26
1.3.6 Experimental status of 2D insulating systems 29
1.3.7 3D band structure invariants and topological insulators 29
1.3.8 Axion electrodynamics, second Chern number,
and magnetoelectric polarizability 31
1.3.9 Anomalous Hall effect and Karplus—Luttinger
anomalous velocity 35
1.4 Introduction to topological order 35
1.4.1 FQHE background 35
1.4.2 Topological terms in field theories: the Haldane gap
and Wess-Zumino-Witten models 36
1.4.3 Topologically ordered phases: the FQHE 45
l.A Topological invariants in 2D with time-reversal invariance 55
l.A.l An interlude: Wess—Zumino terms in ID
nonlinear T-models 55
l.A.2 Topological invariants in time-reversal-invariant Fermi systems 56
1.A.3 Pumping interpretation of Z2 invariant 58
References 59
2 Topological superconductors and category theory 63
Andrei BERNEVIG and Titus NEUPERT
Preface 65
2.1 Introduction to topological phases in condensed matter 65
2.1.1 The notion of topology 65
2.1.2 Classification of non-interacting fermion Hamiltonians:
the 10-fold way
67
xviii Contents
2.1.3 The Su-Schrieffer-Heeger model 75
2.1.4 The ID p-wave superconductor 77
2.1.5 Reduction of the 10֊fold way classification by
interactions: Z ---► Zg in class BDI 80
2.2 Examples of topological order 82
2.2.1 The toric code 83
2.2.2 The 2D p-wave superconductor 90
2.3 Category theory 102
2.3.1 Fusion category 102
2.3.2 Braiding category 110
2.3.3 Modular matrices 114
2.3.4 Examples: the 16-fold way revisited 118
Acknowledgements 120
References 121
Spin liquids and frustrated magnetism 123
John T. CHALKER
3.1 Introduction 125
3.1.1 Overview 125
3.1.2 Classical ground-state degeneracy 128
3.1.3 Order by disorder 129
3.2 Classical spin liquids 132
3.2.1 Simple approximations 132
3.2.2 The triangular lattice Ising antiferromagnet and height models 134
3.3 Classical dimer models 137
3.3.1 Introduction 138
3.3.2 General formulation 138
3.3.3 Flux sectors, and i7(l) and Z2 theories 141
3.3.4 Excitations 142
3.4 Spin ices 143
3.4.1 Materials 143
3.4.2 Coulomb phase correlations 144
3.4.3 Monopoles 147
3.4.4 Dipolar interactions 148
3.5 Quantum spin liquids 150
3.5.1 Introduction 150
3.5.2 Lieb-Schultz-Mattis theorem 150
3.5.3 Quantum dimer models 152
3.6 Concluding remarks 160
3.6.1 Slave particles 160
3.6.2 Numerics 161
3.6.3 Summary 161
Acknowledgements 162
References 162
Contents хіх
4 Entanglement spectroscopy and its application to the
quantum Hall effects 165
Nicolas REGNAULT
Preface 167
4.1 Introduction 167
4.2 Entanglement spectrum and entanglement entropy 169
4.2.1 Definitions 170
4.2.2 A simple example: two spin-·^ 171
4.2.3 Entanglement entropy 172
4.2.4 The AKLT spin chain 175
4.2.5 Matrix product states and the entanglement spectrum 177
4.3 Observing an edge mode through the entanglement spectrum 179
4.3.1 The integer quantum Hall effect 179
4.3.2 Chern insulators 184
4.3.3 Entanglement spectrum for a Cl 185
4.4 Fractional quantum Hall effect and entanglement spectra 189
4.4.1 Fractional quantum Hall effect: overview and notation 190
4.4.2 Orbital entanglement spectrum 193
4.4.3 OES beyond model wavefunctions 197
4.4.4 Particle entanglement spectrum 201
4.4.5 Real-space entanglement spectrum 204
4.5 Entanglement spectrum as a tool: probing
the fractional Chern insulators 205
4.5.1 From Chern insulators to fractional Chern insulators 205
4.5.2 Entanglement spectrum for fractional Chern insulators 208
4.6 Conclusions 209
Acknowledgements 210
References 210
Part II Topical lectures
5 Duality in generalized Ising models 219
Franz J. WEGNER
Preface 221
5.1 Introduction 221
5.2 Kramers-Wannier duality 221
5.2.1 High-temperature expansion (HTE) 222
5.2.2 Low-temperature expansion (LTE) 223
5.2.3 Comparison 223
5.3 Duality in three dimensions 224
5.4 General Ising models and duality 223
5.4.1 General Ising models 223
5.4.2 Duality 223
5.5 Lattices and Ising models 229
5.5.1 Lattices and their dual lattices 229
xx Contents
5.5.2 Models on the lattice 230
5.5.3 Euler characteristic and degeneracy 230
5.6 The models on hypercubic lattices 232
5.6.1 Gauge invariance and degeneracy 233
5.6.2 Self-duality 233
5.7 Correlations 234
5.7.1 The model Mdd 234
5.7.2 Dislocations 235
5.8 Lattice gauge theories 237
5.9 Electromagnetic field 237
References 238
6 Topological insulators and related phases with strong
interactions 241
Ashvin VISHWANATH
6.1 Overview 243
6.2 Quantum phases of matter. Short-range versus
long-range entanglement 244
6.3 Examples of SRE topological phases 247
6.3.1 Haldane phase of S ~ 1 antiferromagnet in d = 1 247
6.3.2 An exactly soluble topological phase in d = 1 247
6.4 SRE phase of bosons in two dimensions 249
6.4.1 Coupled-wire construction 250
6.4.2 Effective field theory 252
6.4.3 Implications for IQH state of electrons 254
6.5 SPT phases of bosons in three dimensions 255
6.5.1 The m = 0 critical point 257
6.5.2 Surface topological order of 3D bosonic SRE phases 257
6.6 Surface topological order of fermionic topological insulators
and superconductors 260
Acknowledgements 262
References 262
7 Fractional Abelian topological phases of matter for
fermions in two-dimensional space 265
Christopher MUDRY
7.1 Introduction 268
7.2 The tenfold way in quasi-one-dimensional space 277
7.2.1 Symmetries for the case of one one-dimensional channel 277
7.2.2 Symmetries for the case of two
one-dimensional channels 283
7.2.3 Definition of the minimum rank 286
7.2.4 Topological spaces for the normalized Dirac masses 289
7.3 Fractionalization from Abelian bosonization 289
7.3.1 Introduction 289
7.3.2 Definition 290
Contents xxi
7.3.3 Chiral equations of motion 291
7.3.4 Gauge invariance 292
7.3.5 Conserved topological charges 295
7.3.6 Quasiparticle and particle excitations 297
7.3.7 Bosonization rules 300
7.3.8 From the Hamiltonian to the Lagrangian formalism 303
7.3.9 Applications to polyacetylene 305
7.4 Stability analysis for the edge theory in symmetry
class AII 307
7.4.1 Introduction 307
7.4.2 Definitions 312
7.4.3 Time-reversal symmetry of the edge theory 314
7.4.4 Pinning the edge fields with disorder
potentials: the Haldane criterion 316
7.4.5 Stability criterion for edge modes 317
7.4.6 The stability criterion for edge modes in the FQSHE 320
7.5 Construction of two-dimensional topological
phases from coupled wires 322
7.5.1 Introduction 322
7.5.2 Definitions 326
7.5.3 Strategy for constructing topological phases 330
7.5.4 Reproducing the tenfold way 334
7.5.5 Fractionalized phases 344
7.5.6 Summary 350
Acknowledgements 351
References 351
8 Symmetry-protected topological phases in one-dimensional
systems 361
Frank POLLMANN
8.1 Introduction 363
8.2 Entanglement and matrix product states 364
8.2.1 Schmidt decomposition and entanglement 364
8.2.2 Area law 366
8.2.3 Matrix product states 367
8.3 Symmetry-protected topological phases 372
8.3.1 Symmetry transformations of MPS 372
8.3.2 Classification of projective representations 374
8.3.3 Symmetry fractionalization 375
8.3.4 Spin-1 chain and the Haldane phase 377
8.4 Detection 378
8.4.1 Degeneracies in the entanglement spectrum 378
8.4.2 Extraction of projective representations from the mixed
transfer matrix 379
8.4.3 String order parameters 380
383
383
383
387
389
389
392
394
394
397
398
400
400
400
402
403
405
406
407
408
410
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425
427
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430
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438
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453
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457
xxii Contents
8.5 Summary
Acknowledgement
References
9 Topological superconducting phases in one dimension
Felix von OPPEN, Yang PENG, and Falko PIENTKA
9.1 Introduction
9.1.1 Motivation
9.1.2 Heuristic arguments
9.2 Spinless p-wave superconductors
9.2.1 Continuum model and phase diagram
9.2.2 Domain walls and Major ana excitations
9.2.3 Topological protection and many-body ground states
9.2.4 Experimentally accessible systems
9.3 Topological insulator edges
9.3.1 Model and phases
9.3.2 Zero-energy states and Majorana operators
9.4 Quantum wires
9.4.1 Kitaev limit
9.4.2 Topological insulator limit
9.5 Chains of magnetic adatoms on superconductors
9.5.1 Shiba states
9.5.2 Adatom chains
9.5.3 Kitaev chain
9.6 Non-Abelian statistics
9.6.1 Manipulation of Majorana bound states
9.6.2 Non-Abelian Berry phase
9.6.3 Braiding Majorana zero modes
9.7 Experimental signatures
9.7.1 Conductance signatures
9.7.2 4TT-periodic Josephson effect
9.8 Conclusions
9.A Pairing Hamiltonians: BdG and second quantization
9.B Proximity-induced pairing
9.C Shiba states
9.C.1 Adatom as a classical magnetic impurity
9.C.2 Adatom as a spin-֊ Anderson impurity
Acknowledgements
References
10 Transport of Dirac surface states
D. Carpentier
10.1 Introduction
10.1.1 Purpose of the lectures
10.1.2 Dirac surface states of topological insulators
10.1.3 Graphene
10.1.4 Overview of transport properties
Contents xxiii
10.2 Minimal conductivity close to the Dirac point 458
10.2.1 Zitterbewegung 458
10.2.2 Clean large tunnel junction 459
10.2.3 Minimal conductivity from linear response theory 460
10.3 Classical conductivity at high Fermi energy 461
10.3.1 Boltzmann equation 462
10.3.2 Linear response approach 466
10.4 Quantum transport of Dirac fermions 472
10.4.1 Quantum correction to the conductivity: weak
antilocalization 474
10.4.2 Universal conductance fluctuations 477
10.4.3 Notion of universality class 479
10.4.4 Effect of a magnetic field 483
Acknowledgements 484
References 484
11 Spin textures in quantum Hall systems 489
Benoît DOUÇOT
11.1 Introduction 491
11.2 Physical properties of spin textures 493
11.2.1 Intuitive picture 493
11.2.2 Construction of spin textures 497
11.2.3 Energetics of spin textures 501
11.2.4 Choice of an effective model 503
11.2.5 Classical ground states of the CPd~l model 507
11.3 Periodic textures 508
11.3.1 Perturbation theory for degenerate Hamiltonians 508
11.3.2 Remarks on the Hessian of the exchange energy 511
11.3.3 Variational procedure for energy minimization 513
11.3.4 Properties of periodic textures 516
11.4 Collective excitations around periodic textures 517
11.4.1 Time-dependent Hartree-Fock equations 517
11.4.2 Collective-mode spectrum 518
11.4.3 Towards an effective sigma model description 521
11.A Coherent states in the lowest Landau level 522
ll.B From covariant symbols on a two-dimensional plane to operators 523
ll.C Single-particle density matrix in a texture Slater determinant 524
ll.D Hamiltonians with quadratic covariant symbol 526
Acknowledgements 527
References 527
12 Out-of-equilibrium behaviour in topologically ordered
systems on a lattice: fractionalized excitations
and kinematic constraints 531
Claudio CASTELNOVO
Preface 533
12.1 Topological order, broadly interpreted 533
xxiv Contents
12.2 Example 1: (classical) spin ice 534
12.2.1 Thermal quenches 538
12.2.2 Field quenches 545
12.3 Example 2: Kitaev’s toric code 552
12.3.1 The model 553
12.3.2 Elementary excitations 555
12.3.3 Dynamics 557
12.3.4 Intriguing comparison: kinetically constrained models 559
12.4 Conclusions 564
Acknowledgements 564
References 565
13 What is life?—70 years after Schrödinger 567
Antti J. NIEMI
Preface 570
13.1 A protein minimum 571
13.1.1 Why proteins? 571
13.1.2 Protein chemistry and the genetic code 572
13.1.3 Data banks and experiments 573
13.1.4 Phases of proteins 577
13.1.5 Backbone geometry 580
13.1.6 Ramachandran angles 582
13.1.7 Homology modelling 584
13.1.8 All-atom models 585
13.1.9 All-atom simulations 587
13.1.10 Thermostats 588
13.1.11 Other physics-based approaches 592
13.2 Bol’she 592
13.2.1 The importance of symmetry breaking 593
13.2.2 An optical illusion 593
13.2.3 Fractional charge 594
13.2.4 Spin-charge separation 596
13.2.5 All-atom is Landau liquid 598
13.3 Strings in three space dimensions 599
13.3.1 Abelian Higgs model and the limit of slow spatial variations 600
13.3.2 The Frenet equation 602
13.3.3 Frame rotation and Abelian Higgs multiplet 603
13.3.4 The unique string Hamiltonian 605
13.3.5 Integrable hierarchy 605
13.3.6 Strings from solitons 606
13.3.7 Anomaly in the Frenet frames 608
13.3.8 Perestroika 610
13.4 Discrete Frenet frames 612
13.4.1 The Cct trace reconstruction 614
13.4.2 Universal discretized energy 615
Contents XXV
13.4.3 Discretized solitons 618
13.4.4 Proteins out of thermal equilibrium 619
13.4.5 Temperature renormalization 620
13.5 Solitons and ordered proteins 624
13.5.1 A-repressor as a multisoliton 624
13.5.2 Structure of myoglobin 628
13.5.3 Dynamical myoglobin 635
13.6 Intrinsically disordered proteins 646
13.6.1 Order versus disorder 647
13.6.2 hlAPP and type 2 diabetes 649
13.6.3 hlAPP as a three-soliton 651
13.6.4 Heating and cooling hlAPP 655
13.7 Beyond Ca 659
13.7.1 ‘What-you-see-is-what-you-have5 660
Acknowledgements 666
References 666
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| author_facet | Chamon, Claudio Goerbig, Mark Oliver Moessner, Roderich Cugliandolo, Leticia F. 1965- École de physique des Houches Les Houches |
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| discipline | Physik |
| edition | First edition |
| format | Conference Proceeding Book |
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| genre | (DE-588)1071861417 Konferenzschrift 04.08.2014-29.08.2014 Les Houches gnd-content |
| genre_facet | Konferenzschrift 04.08.2014-29.08.2014 Les Houches |
| id | DE-604.BV044551265 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:55:40Z |
| institution | BVB |
| institution_GND | (DE-588)1126607487 |
| isbn | 9780198785781 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029950116 |
| oclc_num | 994216999 |
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| owner | DE-355 DE-BY-UBR DE-91G DE-BY-TUM |
| owner_facet | DE-355 DE-BY-UBR DE-91G DE-BY-TUM |
| physical | xxxi, 671 Seiten Illustrationen 26 cm |
| publishDate | 2017 |
| publishDateSearch | 2017 |
| publishDateSort | 2017 |
| publisher | Oxford University Press |
| record_format | marc |
| series | Session |
| series2 | Session |
| spelling | École de physique des Houches 103. 2014 Les Houches Verfasser (DE-588)1126607487 aut Topological aspects of condensed matter physics École de physique des Houches, Session CIII, 4-29 August 2014 edited by Claudio Chamon (Physics Department, Boston University, Boston, Massachusetts, 02215, USA), Mark O. Goerbig (Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris-Sud, Université Paris-Saclay F-91405, France), Roderich Moessner (Max-Planck-Institut für Physikl komplexer Systeme, 901187 Dresden, Germany), Leticia F. Cugliandolo (Sorbornn Universités, Universités Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France) First edition Oxford Oxford University Press 2017 xxxi, 671 Seiten Illustrationen 26 cm txt rdacontent n rdamedia nc rdacarrier Session 103 Includes bibliographical references Quantentheorie Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Kondensierte Materie (DE-588)4132810-3 gnd rswk-swf Condensed matter Quantum theory (DE-588)1071861417 Konferenzschrift 04.08.2014-29.08.2014 Les Houches gnd-content Kondensierte Materie (DE-588)4132810-3 s Mathematische Physik (DE-588)4037952-8 s Topologie (DE-588)4060425-1 s DE-604 Chamon, Claudio (DE-588)1130347516 edt Goerbig, Mark Oliver edt Moessner, Roderich (DE-588)1154526275 edt Cugliandolo, Leticia F. 1965- (DE-588)1126058149 edt Session 103 (DE-604)BV000022608 103 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029950116&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Topological aspects of condensed matter physics École de physique des Houches, Session CIII, 4-29 August 2014 Session Quantentheorie Mathematische Physik (DE-588)4037952-8 gnd Topologie (DE-588)4060425-1 gnd Kondensierte Materie (DE-588)4132810-3 gnd |
| subject_GND | (DE-588)4037952-8 (DE-588)4060425-1 (DE-588)4132810-3 (DE-588)1071861417 |
| title | Topological aspects of condensed matter physics École de physique des Houches, Session CIII, 4-29 August 2014 |
| title_auth | Topological aspects of condensed matter physics École de physique des Houches, Session CIII, 4-29 August 2014 |
| title_exact_search | Topological aspects of condensed matter physics École de physique des Houches, Session CIII, 4-29 August 2014 |
| title_full | Topological aspects of condensed matter physics École de physique des Houches, Session CIII, 4-29 August 2014 edited by Claudio Chamon (Physics Department, Boston University, Boston, Massachusetts, 02215, USA), Mark O. Goerbig (Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris-Sud, Université Paris-Saclay F-91405, France), Roderich Moessner (Max-Planck-Institut für Physikl komplexer Systeme, 901187 Dresden, Germany), Leticia F. Cugliandolo (Sorbornn Universités, Universités Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France) |
| title_fullStr | Topological aspects of condensed matter physics École de physique des Houches, Session CIII, 4-29 August 2014 edited by Claudio Chamon (Physics Department, Boston University, Boston, Massachusetts, 02215, USA), Mark O. Goerbig (Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris-Sud, Université Paris-Saclay F-91405, France), Roderich Moessner (Max-Planck-Institut für Physikl komplexer Systeme, 901187 Dresden, Germany), Leticia F. Cugliandolo (Sorbornn Universités, Universités Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France) |
| title_full_unstemmed | Topological aspects of condensed matter physics École de physique des Houches, Session CIII, 4-29 August 2014 edited by Claudio Chamon (Physics Department, Boston University, Boston, Massachusetts, 02215, USA), Mark O. Goerbig (Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris-Sud, Université Paris-Saclay F-91405, France), Roderich Moessner (Max-Planck-Institut für Physikl komplexer Systeme, 901187 Dresden, Germany), Leticia F. Cugliandolo (Sorbornn Universités, Universités Pierre et Marie Curie, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Paris, France) |
| title_short | Topological aspects of condensed matter physics |
| title_sort | topological aspects of condensed matter physics ecole de physique des houches session ciii 4 29 august 2014 |
| title_sub | École de physique des Houches, Session CIII, 4-29 August 2014 |
| topic | Quantentheorie Mathematische Physik (DE-588)4037952-8 gnd Topologie (DE-588)4060425-1 gnd Kondensierte Materie (DE-588)4132810-3 gnd |
| topic_facet | Quantentheorie Mathematische Physik Topologie Kondensierte Materie Konferenzschrift 04.08.2014-29.08.2014 Les Houches |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029950116&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000022608 |
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