Topological Aspects of Condensed Matter Physics Lecture Notes of the Les Houches Summer School Volume 103 August 2014 1st Edition by Claudio Chamon, Mark Goerbig, Roderich Moessner, Leticia Cugliandolo ISBN B06WGRK34V 978-0191088797

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ISBN 10:  B06WGRK34V
ISBN 13:  978-0191088797
Author: Claudio Chamon, Mark Goerbig, Roderich Moessner, Leticia Cugliandolo 

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.

Topological Aspects of Condensed Matter Physics: Lecture Notes of the Les Houches Summer School: Volume 103, August 2014 1stTable of contents:

Part I Basic Lectures

1 An introduction to topological phases of electrons

Joel E. Moore

1.1 Introduction

1.2 Basic concepts

1.2.1 Mathematical preliminaries

1.2.2 Berry phases in quantum mechanics

1.3 Topological phases: Thouless phases arising from Berry phases

1.3.1 Bloch states

1.3.2 1D polarization and 2D IQHE

1.3.3 Interactions and disorder: the flux trick

1.3.4 TKNN integers, Chern numbers, and homotopy

1.3.5 Time-reversal invariance in Fermi systems

1.3.6 Experimental status of 2D insulating systems

1.3.7 3D band structure invariants and topological insulators

1.3.8 Axion electrodynamics, second Chern number, and magnetoelectric polarizability

1.3.9 Anomalous Hall effect and Karplus–Luttinger anomalous velocity

1.4 Introduction to topological order

1.4.1 FQHE background

1.4.2 Topological terms in field theories: the Haldane gap and Wess–Zumino–Witten models

1.4.3 Topologically ordered phases: the FQHE

1.A Topological invariants in 2D with time-reversal invariance

1.A.1 An interlude: Wess–Zumino terms in 1D nonlinear σ-models

1.A.2 Topological invariants in time-reversal-invariant Fermi systems

1.A.3 Pumping interpretation of Z2 invariant

References

2 Topological superconductors and category theory

Andrei Bernevig and Titus Neupert

Preface

2.1 Introduction to topological phases in condensed matter

2.1.1 The notion of topology

2.1.2 Classification of non-interacting fermion Hamiltonians: the 10-fold way

2.1.3 The Su–Schrieffer–Heeger model

2.1.4 The 1D p-wave superconductor

2.1.5 Reduction of the 10-fold way classification by interactions:

2.2 Examples of topological order

2.2.1 The toric code

2.2.2 The 2D p-wave superconductor

2.3 Category theory

2.3.1 Fusion category

2.3.2 Braiding category

2.3.3 Modular matrices

2.3.4 Examples: the 16-fold way revisited

Acknowledgements

References

3 Spin liquids and frustrated magnetism

John T. Chalker

3.1 Introduction

3.1.1 Overview

3.1.2 Classical ground-state degeneracy

3.1.3 Order by disorder

3.2 Classical spin liquids

3.2.1 Simple approximations

3.2.2 The triangular lattice Ising antiferromagnet and height models

3.3 Classical dimer models

3.3.1 Introduction

3.3.2 General formulation

3.3.3 Flux sectors, and U(1) and Z2 theories

3.3.4 Excitations

3.4 Spin ices

3.4.1 Materials

3.4.2 Coulomb phase correlations

3.4.3 Monopoles

3.4.4 Dipolar interactions

3.5 Quantum spin liquids

3.5.1 Introduction

3.5.2 Lieb–Schultz–Mattis theorem

3.5.3 Quantum dimer models

3.6 Concluding remarks

3.6.1 Slave particles

3.6.2 Numerics

3.6.3 Summary

Acknowledgements

References

4 Entanglement spectroscopy and its application to the quantum Hall effects

Nicolas Regnault

Preface

4.1 Introduction

4.2 Entanglement spectrum and entanglement entropy

4.2.1 Definitions

4.2.2 A simple example: two spin- 1/2

4.2.3 Entanglement entropy

4.2.4 The AKLT spin chain

4.2.5 Matrix product states and the entanglement spectrum

4.3 Observing an edge mode through the entanglement spectrum

4.3.1 The integer quantum Hall effect

4.3.2 Chern insulators

4.3.3 Entanglement spectrum for a CI

4.4 Fractional quantum Hall effect and entanglement spectra

4.4.1 Fractional quantum Hall effect: overview and notation

4.4.2 Orbital entanglement spectrum

4.4.3 OES beyond model wavefunctions

4.4.4 Particle entanglement spectrum

4.4.5 Real-space entanglement spectrum

4.5 Entanglement spectrum as a tool: probing the fractional Chern insulators

4.5.1 From Chern insulators to fractional Chern insulators

4.5.2 Entanglement spectrum for fractional Chern insulators

4.6 Conclusions

Acknowledgements

References

Part II Topical lectures

5 Duality in generalized Ising models

Franz J. Wegner

Preface

5.1 Introduction

5.2 Kramers–Wannier duality

5.2.1 High-temperature expansion (HTE)

5.2.2 Low-temperature expansion (LTE)

5.2.3 Comparison

5.3 Duality in three dimensions

5.4 General Ising models and duality

5.4.1 General Ising models

5.4.2 Duality

5.5 Lattices and Ising models

5.5.1 Lattices and their dual lattices

5.5.2 Models on the lattice

5.5.3 Euler characteristic and degeneracy

5.6 The models Md,n on hypercubic lattices

5.6.1 Gauge invariance and degeneracy

5.6.2 Self-duality

5.7 Correlations

5.7.1 The model Mdd

5.7.2 Dislocations

5.8 Lattice gauge theories

5.9 Electromagnetic field

References

6 Topological insulators and related phases with strong interactions

Ashvin Vishwanath

6.1 Overview

6.2 Quantum phases of matter. Short-range versus long-range entanglement

6.3 Examples of SRE topological phases

6.3.1 Haldane phase of S = 1 antiferromagnet in d = 1

6.3.2 An exactly soluble topological phase in d = 1

6.4 SRE phase of bosons in two dimensions

6.4.1 Coupled-wire construction

6.4.2 Effective field theory

6.4.3 Implications for IQH state of electrons

6.5 SPT phases of bosons in three dimensions

6.5.1 The m = 0 critical point

6.5.2 Surface topological order of 3D bosonic SRE phases

6.6 Surface topological order of fermionic topological insulators and superconductors

Acknowledgements

References

7 Fractional Abelian topological phases of matter for fermions in two-dimensional space

Christopher MUDRY

7.1 Introduction

7.2 The tenfold way in quasi-one-dimensional space

7.2.1 Symmetries for the case of one one-dimensional channel

7.2.2 Symmetries for the case of two one-dimensional channels

7.2.3 Definition of the minimum rank

7.2.4 Topological spaces for the normalized Dirac masses

7.3 Fractionalization from Abelian bosonization

7.3.1 Introduction

7.3.2 Definition

7.3.3 Chiral equations of motion

7.3.4 Gauge invariance

7.3.5 Conserved topological charges

7.3.6 Quasiparticle and particle excitations

7.3.7 Bosonization rules

7.3.8 From the Hamiltonian to the Lagrangian formalism

7.3.9 Applications to polyacetylene

7.4 Stability analysis for the edge theory in symmetry class AII

7.4.1 Introduction

7.4.2 Definitions

7.4.3 Time-reversal symmetry of the edge theory

7.4.4 Pinning the edge fields with disorder potentials: the Haldane criterion

7.4.5 Stability criterion for edge modes

7.4.6 The stability criterion for edge modes in the FQSHE

7.5 Construction of two-dimensional topological phases from coupled wires

7.5.1 Introduction

7.5.2 Definitions

7.5.3 Strategy for constructing topological phases

7.5.4 Reproducing the tenfold way

7.5.5 Fractionalized phases

7.5.6 Summary

Acknowledgements

References

8 Symmetry-protected topological phases in one-dimensional systems

Frank Pollmann

8.1 Introduction

8.2 Entanglement and matrix product states

8.2.1 Schmidt decomposition and entanglement

8.2.2 Area law

8.2.3 Matrix product states

8.3 Symmetry-protected topological phases

8.3.1 Symmetry transformations of MPS

8.3.2 Classification of projective representations

8.3.3 Symmetry fractionalization

8.3.4 Spin-1 chain and the Haldane phase

8.4 Detection

8.4.1 Degeneracies in the entanglement spectrum

8.4.2 Extraction of projective representations from the mixed transfermatrix

8.4.3 String order parameters

8.5 Summary

Acknowledgement

References

9 Topological superconducting phasesin 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 Majorana 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 4π-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-12 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

10.2 Minimal conductivity close to the Dirac point

10.2.1 Zitterbewegung

10.2.2 Clean large tunnel junction

10.2.3 Minimal conductivity from linear response theory

10.3 Classical conductivity at high Fermi energy

10.3.1 Boltzmann equation

10.3.2 Linear response approach

10.4 Quantum transport of Dirac fermions

10.4.1 Quantum correction to the conductivity: weak antilocalization

10.4.2 Universal conductance fluctuations

10.4.3 Notion of universality class

10.4.4 Effect of a magnetic field

Acknowledgements

References

11 Spin textures in quantum Hall systems

Benoît Douçot

11.1 Introduction

11.2 Physical properties of spin textures

11.2.1 Intuitive picture

11.2.2 Construction of spin textures

11.2.3 Energetics of spin textures

11.2.4 Choice of an effective model

11.2.5 Classical ground states of the CPd−1 model

11.3 Periodic textures

11.3.1 Perturbation theory for degenerate Hamiltonians

11.3.2 Remarks on the Hessian of the exchange energy

11.3.3 Variational procedure for energy minimization

11.3.4 Properties of periodic textures

11.4 Collective excitations around periodic textures

11.4.1 Time-dependent Hartree–Fock equations

11.4.2 Collective-mode spectrum

11.4.3 Towards an effective sigma model description

11.A Coherent states in the lowest Landau level

11.B From covariant symbols on a two-dimensional plane to operators

11.C Single-particle density matrix in a texture Slater determinant

11.D Hamiltonians with quadratic covariant symbol

Acknowledgements

References

12 Out-of-equilibrium behaviour in topologically ordered systems on a lattice: fractionalized excita

Claudio Castelnovo

Preface

12.1 Topological order, broadly interpreted

12.2 Example 1: (classical) spin ice

12.2.1 Thermal quenches

12.2.2 Field quenches

12.3 Example 2: Kitaev’s toric code

12.3.1 The model

12.3.2 Elementary excitations

12.3.3 Dynamics

12.3.4 Intriguing comparison: kinetically constrained models

12.4 Conclusions

Acknowledgements

References

13 What is life?—70 years after Schrödinger

Antti J. Niemi

Preface

13.1 A protein minimum

13.1.1 Why proteins?

13.1.2 Protein chemistry and the genetic code

13.1.3 Data banks and experiments

13.1.4 Phases of proteins

13.1.5 Backbone geometry

13.1.6 Ramachandran angles

13.1.7 Homology modelling

13.1.8 All-atom models

13.1.9 All-atom simulations

13.1.10 Thermostats

13.1.11 Other physics-based approaches

13.2 Bol’she

13.2.1 The importance of symmetry breaking

13.2.2 An optical illusion

13.2.3 Fractional charge

13.2.4 Spin–charge separation

13.2.5 All-atom is Landau liquid

13.3 Strings in three space dimensions

13.3.1 Abelian Higgs model and the limit of slow spatial variations

13.3.2 The Frenet equation

13.3.3 Frame rotation and Abelian Higgs multiplet

13.3.4 The unique string Hamiltonian

13.3.5 Integrable hierarchy

13.3.6 Strings from solitons

13.3.7 Anomaly in the Frenet frames

13.3.8 Perestroika

13.4 Discrete Frenet frames

13.4.1 The Cα trace reconstruction

13.4.2 Universal discretized energy

13.4.3 Discretized solitons

13.4.4 Proteins out of thermal equilibrium

13.4.5 Temperature renormalization

13.5 Solitons and ordered proteins

13.5.1 λ-repressor as a multisoliton

13.5.2 Structure of myoglobin

13.5.3 Dynamical myoglobin

13.6 Intrinsically disordered proteins

13.6.1 Order versus disorder

13.6.2 hIAPP and type 2 diabetes

13.6.3 hIAPP as a three-soliton

13.6.4 Heating and cooling hIAPP

13.7 Beyond Cα

13.7.1 ‘What-you-see-is-what-you-have’

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