Topological aspects of condensed matter physics : lecture notes of the Les Houches Summer School. Volume 103, August 2014 / Claudio Chamon, Mark O. Goerbig, Roderich Moessner, and Leticia F. Cugliandolo
- Published:
- Oxford : Oxford University Press, 2017.
- Edition:
- First edition.
- Physical Description:
- 1 online resource : illustrations (black and white)
- Additional Creators:
- Chamon, Claudio, Goerbig, Mark O., Moessner, Roderich, and Cugliandolo, L. F. (Leticia F.)
Access Online
- Oxford scholarship online: ezaccess.libraries.psu.edu
- Contents:
- Machine generated contents note: 1.An introduction to topological phases of electrons -- 1.1.Introduction / Joel E. Moore -- 1.2.Basic concepts / Joel E. Moore -- 1.2.1.Mathematical preliminaries / Joel E. Moore -- 1.2.2.Berry phases in quantum mechanics / Joel E. Moore -- 1.3.Topological phases: Thouless phases arising from Berry phases / Joel E. Moore -- 1.3.1.Bloch states / Joel E. Moore -- 1.3.2.1D polarization and 2D IQHE / Joel E. Moore -- 1.3.3.Interactions and disorder: the flux trick / Joel E. Moore -- 1.3.4.TKNN integers, Chern numbers, and homotopy / Joel E. Moore -- 1.3.5.Time-reversal invariance in Fermi systems / Joel E. Moore -- 1.3.6.Experimental status of 2D insulating systems / Joel E. Moore -- 1.3.7.3D band structure invariants and topological insulators / Joel E. Moore -- 1.3.8.Axion electrodynamics, second Chern number, and magnetoelectric polarizability / Joel E. Moore -- 1.3.9.Anomalous Hall effect and Karplus-Luttinger anomalous velocity / Joel E. Moore -- 1.4.Introduction to topological order / Joel E. Moore -- 1.4.1.FQHE background / Joel E. Moore -- 1.4.2.Topological terms in field theories: the Haldane gap and Wess-Zumino-Witten models / Joel E. Moore -- 1.4.3.Topologically ordered phases: the FQHE / Joel E. Moore -- 1.A.Topological invariants in 2D with time-reversal invariance / Joel E. Moore -- 1.A.1.An interlude: Wess-Zumino terms in 1D nonlinear sigma-models / Joel E. Moore -- 1.A.2.Topological invariants in time-reversal-invariant Fermi systems / Joel E. Moore -- 1.A.3.Pumping interpretation of Z2 invariant / Joel E. Moore -- References / Joel E. Moore -- 2.Topological superconductors and category theory / Joel E. Moore -- Preface / Andrei Bernevig / Titus Neupert -- 2.1.Introduction to topological phases in condensed matter / Andrei Bernevig / Titus Neupert -- 2.1.1.The notion of topology / Andrei Bernevig / Titus Neupert -- 2.1.2.Classification of non-interacting fermion Hamiltonians: the 10-fold way / Andrei Bernevig / Titus Neupert -- 2.1.3.The Su-Schrieffer-Heeger model / Andrei Bernevig / Titus Neupert -- 2.1.4.The 1D p-wave superconductor / Andrei Bernevig / Titus Neupert -- 2.1.5.Reduction of the 10-fold way classification by interactions: Z Z8 in class BDI / Andrei Bernevig / Titus Neupert -- 2.2.Examples of topological order / Andrei Bernevig / Titus Neupert -- 2.2.1.The toric code / Andrei Bernevig / Titus Neupert -- 2.2.2.The 2D p-wave superconductor / Andrei Bernevig / Titus Neupert -- 2.3.Category theory / Andrei Bernevig / Titus Neupert -- 2.3.1.Fusion category / Andrei Bernevig / Titus Neupert -- 2.3.2.Braiding category / Andrei Bernevig / Titus Neupert -- 2.3.3.Modular matrices / Andrei Bernevig / Titus Neupert -- 2.3.4.Examples: the 16-fold way revisited / Andrei Bernevig / Titus Neupert -- Acknowledgements / Andrei Bernevig / Titus Neupert -- References / Andrei Bernevig / Titus Neupert -- 3.Spin liquids and frustrated magnetism / Andrei Bernevig / Titus Neupert -- 3.1.Introduction / John T. Chalker -- 3.1.1.Overview / John T. Chalker -- 3.1.2.Classical ground-state degeneracy / John T. Chalker -- 3.1.3.Order by disorder / John T. Chalker -- 3.2.Classical spin liquids / John T. Chalker -- 3.2.1.Simple approximations / John T. Chalker -- 3.2.2.The triangular lattice Ising antiferromagnet and height models / John T. Chalker -- 3.3.Classical dimer models / John T. Chalker -- 3.3.1.Introduction / John T. Chalker -- 3.3.2.General formulation / John T. Chalker -- 3.3.3.Flux sectors, and U(1) and Z2 theories / John T. Chalker -- 3.3.4.Excitations / John T. Chalker -- 3.4.Spin ices / John T. Chalker -- 3.4.1.Materials / John T. Chalker -- 3.4.2.Coulomb phase correlations / John T. Chalker -- 3.4.3.Monopoles / John T. Chalker -- 3.4.4.Dipolar interactions / John T. Chalker -- 3.5.Quantum spin liquids / John T. Chalker -- 3.5.1.Introduction / John T. Chalker -- 3.5.2.Lieb-Schultz-Mattis theorem / John T. Chalker -- 3.5.3.Quantum dimer models / John T. Chalker -- 3.6.Concluding remarks / John T. Chalker -- 3.6.1.Slave particles / John T. Chalker -- 3.6.2.Numerics / John T. Chalker -- 3.6.3.Summary / John T. Chalker -- Acknowledgements / John T. Chalker -- References / John T. Chalker -- 4.Entanglement spectroscopy and its application to the quantum Hall effects / John T. Chalker -- Preface / Nicolas Regnault -- 4.1.Introduction / Nicolas Regnault -- 4.2.Entanglement spectrum and entanglement entropy / Nicolas Regnault -- 4.2.1.Definitions / Nicolas Regnault -- 4.2.2.A simple example: two spin-1/2 / Nicolas Regnault -- 4.2.3.Entanglement entropy / Nicolas Regnault -- 4.2.4.The AKLT spin chain / Nicolas Regnault -- 4.2.5.Matrix product states and the entanglement spectrum / Nicolas Regnault -- 4.3.Observing an edge mode through the entanglement spectrum / Nicolas Regnault -- 4.3.1.The integer quantum Hall effect / Nicolas Regnault -- 4.3.2.Chern insulators / Nicolas Regnault -- 4.3.3.Entanglement spectrum for a CI / Nicolas Regnault -- 4.4.Fractional quantum Hall effect and entanglement spectra / Nicolas Regnault -- 4.4.1.Fractional quantum Hall effect: overview and notation / Nicolas Regnault -- 4.4.2.Orbital entanglement spectrum / Nicolas Regnault -- 4.4.3.OES beyond model wavefunctions / Nicolas Regnault -- 4.4.4.Particle entanglement spectrum / Nicolas Regnault -- 4.4.5.Real-space entanglement spectrum / Nicolas Regnault -- 4.5.Entanglement spectrum as a tool: probing the fractional Chern insulators / Nicolas Regnault -- 4.5.1.From Chern insulators to fractional Chern insulators / Nicolas Regnault -- 4.5.2.Entanglement spectrum for fractional Chern insulators / Nicolas Regnault -- 4.6.Conclusions / Nicolas Regnault -- Acknowledgements / Nicolas Regnault -- References / Nicolas Regnault -- 5.Duality in generalized Ising models / Nicolas Regnault -- Preface / Franz J. Wegner -- 5.1.Introduction / Franz J. Wegner -- 5.2.Kramers-Wannier duality / Franz J. Wegner -- 5.2.1.High-temperature expansion (HTE) / Franz J. Wegner -- 5.2.2.Low-temperature expansion (LTE) / Franz J. Wegner -- 5.2.3.Comparison / Franz J. Wegner -- 5.3.Duality in three dimensions / Franz J. Wegner -- 5.4.General Ising models and duality / Franz J. Wegner -- 5.4.1.General Ising models / Franz J. Wegner -- 5.4.2.Duality / Franz J. Wegner -- 5.5.Lattices and Ising models / Franz J. Wegner -- 5.5.1.Lattices and their dual lattices / Franz J. Wegner -- 5.5.2.Models on the lattice / Franz J. Wegner -- 5.5.3.Euler characteristic and degeneracy / Franz J. Wegner -- 5.6.The models Mdm on hypercubic lattices / Franz J. Wegner -- 5.6.1.Gauge invariance and degeneracy / Franz J. Wegner -- 5.6.2.Self-duality / Franz J. Wegner -- 5.7.Correlations / Franz J. Wegner -- 5.7.1.The model Mdd / Franz J. Wegner -- 5.7.2.Dislocations / Franz J. Wegner -- 5.8.Lattice gauge theories / Franz J. Wegner -- 5.9.Electromagnetic field / Franz J. Wegner -- References / Franz J. Wegner -- 6.Topological insulators and related phases with strong interactions / Franz J. Wegner -- 6.1.Overview / Ashvin Vishwanath -- 6.2.Quantum phases of matter. Short-range versus long-range entanglement / Ashvin Vishwanath -- 6.3.Examples of SRE topological phases / Ashvin Vishwanath -- 6.3.1.Haldane phase of S = 1 antiferromagnet in d = 1 / Ashvin Vishwanath -- 6.3.2.An exactly soluble topological phase in d = 1 / Ashvin Vishwanath -- 6.4.SRE phase of bosons in two dimensions / Ashvin Vishwanath -- 6.4.1.Coupled-wire construction / Ashvin Vishwanath -- 6.4.2.Effective field theory / Ashvin Vishwanath -- 6.4.3.Implications for IQH state of electrons / Ashvin Vishwanath -- 6.5.SPT phases of bosons in three dimensions / Ashvin Vishwanath -- 6.5.1.The m = 0 critical point / Ashvin Vishwanath -- 6.5.2.Surface topological order of 3D bosonic SRE phases / Ashvin Vishwanath -- 6.6.Surface topological order of fermionic topological insulators and superconductors / Ashvin Vishwanath -- Acknowledgements / Ashvin Vishwanath -- References / Ashvin Vishwanath -- 7.Fractional Abelian topological phases of matter for fermions in two-dimensional space / Ashvin Vishwanath -- 7.1.Introduction / Christopher Mudry -- 7.2.The tenfold way in quasi-one-dimensional space / Christopher Mudry -- 7.2.1.Symmetries for the case of one one-dimensional channel / Christopher Mudry -- 7.2.2.Symmetries for the case of two one-dimensional channels / Christopher Mudry -- 7.2.3.Definition of the minimum rank / Christopher Mudry -- 7.2.4.Topological spaces for the normalized Dirac masses / Christopher Mudry -- 7.3.Fractionalization from Abelian bosonization / Christopher Mudry -- 7.3.1.Introduction / Christopher Mudry -- 7.3.2.Definition / Christopher Mudry -- 7.3.3.Chiral equations of motion / Christopher Mudry -- 7.3.4.Gauge invariance / Christopher Mudry -- 7.3.5.Conserved topological charges / Christopher Mudry -- 7.3.6.Quasiparticle and particle excitations / Christopher Mudry -- 7.3.7.Bosonization rules / Christopher Mudry -- 7.3.8.From the Hamiltonian to the Lagrangian formalism / Christopher Mudry -- 7.3.9.Applications to polyacetylene / Christopher Mudry -- 7.4.Stability analysis for the edge theory in symmetry class AII / Christopher Mudry -- 7.4.1.Introduction / Christopher Mudry -- 7.4.2.Definitions / Christopher Mudry -- 7.4.3.Time-reversal symmetry of the edge theory / Christopher Mudry -- 7.4.4.Pinning the edge fields with disorder potentials: the Haldane criterion / Christopher Mudry -- 7.4.5.Stability criterion for edge modes / Christopher Mudry -- 7.4.6.The stability criterion for edge modes in the FQSHE / Christopher Mudry -- 7.5.Construction of two-dimensional topological phases from coupled wires / Christopher Mudry -- 7.5.1.Introduction / Christopher Mudry -- 7.5.2.Definitions / Christopher Mudry -- 7.5.3.Strategy for constructing topological phases / Christopher Mudry -- 7.5.4.Reproducing the tenfold way / Christopher Mudry -- 7.5.5.Fractionalized phases / Christopher Mudry -- 7.5.6.Summary / Christopher Mudry -- Acknowledgements / Christopher Mudry -- and Contents note continued: References / Christopher Mudry -- 8.Symmetry-protected topological phases in one-dimensional systems / Christopher Mudry -- 8.1.Introduction / Frank Pollmann -- 8.2.Entanglement and matrix product states / Frank Pollmann -- 8.2.1.Schmidt decomposition and entanglement / Frank Pollmann -- 8.2.2.Area law / Frank Pollmann -- 8.2.3.Matrix product states / Frank Pollmann -- 8.3.Symmetry-protected topological phases / Frank Pollmann -- 8.3.1.Symmetry transformations of MPS / Frank Pollmann -- 8.3.2.Classification of projective representations / Frank Pollmann -- 8.3.3.Symmetry fractionalization / Frank Pollmann -- 8.3.4.Spin-1 chain and the Haldane phase / Frank Pollmann -- 8.4.Detection / Frank Pollmann -- 8.4.1.Degeneracies in the entanglement spectrum / Frank Pollmann -- 8.4.2.Extraction of projective representations from the mixed transfer matrix / Frank Pollmann -- 8.4.3.String order parameters / Frank Pollmann -- 8.5.Summary 383. Acknowledgement / Frank Pollmann -- References / Frank Pollmann -- 9.Topological superconducting phases in one dimension / Frank Pollmann -- 9.1.Introduction / Felix von Oppen / Yang Peng / Falko Pientka -- 9.1.1.Motivation / Felix von Oppen / Falko Pientka / Yang Peng -- 9.1.2.Heuristic arguments / Felix von Oppen / Yang Peng / Falko Pientka -- 9.2.Spinless p-wave superconductors / Yang Peng / Felix von Oppen / Falko Pientka -- 9.2.1.Continuum model and phase diagram / Felix von Oppen / Falko Pientka / Yang Peng -- 9.2.2.Domain walls and Majorana excitations / Felix von Oppen / Yang Peng / Falko Pientka -- 9.2.3.Topological protection and many-body ground states / Felix von Oppen / Yang Peng / Falko Pientka -- 9.2.4.Experimentally accessible systems / Felix von Oppen / Yang Peng / Falko Pientka -- 9.3.Topological insulator edges / Felix von Oppen / Yang Peng / Falko Pientka -- 9.3.1.Model and phases / Yang Peng / Felix von Oppen / Falko Pientka -- 9.3.2.Zero-energy states and Majorana operators / Felix von Oppen / Yang Peng / Falko Pientka -- 9.4.Quantum wires / Felix von Oppen / Falko Pientka / Yang Peng -- 9.4.1.Kitaev limit / Felix von Oppen / Falko Pientka / Yang Peng -- 9.4.2.Topological insulator limit / Felix von Oppen / Falko Pientka / Yang Peng -- 9.5.Chains of magnetic adatoms on superconductors / Yang Peng / Felix von Oppen / Falko Pientka -- 9.5.1.Shiba states / Felix von Oppen / Falko Pientka / Yang Peng -- 9.5.2.Adatom chains / Felix von Oppen / Falko Pientka / Yang Peng -- 9.5.3.Kitaev chain / Felix von Oppen / Falko Pientka / Yang Peng -- 9.6.Non-Abelian statistics / Felix von Oppen / Falko Pientka / Yang Peng -- 9.6.1.Manipulation of Majorana bound states / Felix von Oppen / Falko Pientka / Yang Peng -- 9.6.2.Non-Abelian Berry phase / Felix von Oppen / Falko Pientka / Yang Peng -- 9.6.3.Braiding Majorana zero modes / Felix von Oppen / Falko Pientka / Yang Peng -- 9.7.Experimental signatures / Felix von Oppen / Falko Pientka / Yang Peng -- 9.7.1.Conductance signatures / Felix von Oppen / Falko Pientka / Yang Peng -- 9.7.2.4Pi-periodic Josephson effect / Felix von Oppen / Falko Pientka / Yang Peng -- 9.8.Conclusions / Felix von Oppen / Falko Pientka / Yang Peng -- 9.A.Pairing Hamiltonians: BdG and second quantization / Felix von Oppen / Falko Pientka / Yang Peng -- 9.B.Proximity-induced pairing / Felix von Oppen / Falko Pientka / Yang Peng -- 9.C.Shiba states / Felix von Oppen / Falko Pientka / Yang Peng -- 9.C.1.Adatom as a classical magnetic impurity / Felix von Oppen / Falko Pientka / Yang Peng -- 9.C.2.Adatom as a spin-1/2 Anderson impurity / Felix von Oppen / Falko Pientka / Yang Peng -- Acknowledgements / Felix von Oppen / Falko Pientka / Yang Peng -- References / Yang Peng / Felix von Oppen / Falko Pientka -- 10.Transport of Dirac surface states / Felix von Oppen / Falko Pientka / Yang Peng -- 10.1.Introduction / D. Carpentier -- 10.1.1.Purpose of the lectures / D. Carpentier -- 10.1.2.Dirac surface states of topological insulators / D. Carpentier -- 10.1.3.Graphene / D. Carpentier -- 10.1.4.Overview of transport properties / D. Carpentier -- 10.2.Minimal conductivity close to the Dirac point / D. Carpentier -- 10.2.1.Zitterbewegung / D. Carpentier -- 10.2.2.Clean large tunnel junction / D. Carpentier -- 10.2.3.Minimal conductivity from linear response theory / D. Carpentier -- 10.3.Classical conductivity at high Fermi energy / D. Carpentier -- 10.3.1.Boltzmann equation / D. Carpentier -- 10.3.2.Linear response approach / D. Carpentier -- 10.4.Quantum transport of Dirac fermions / D. Carpentier -- 10.4.1.Quantum correction to the conductivity: weak antilocalization / D. Carpentier -- 10.4.2.Universal conductance fluctuations / D. Carpentier -- 10.4.3.Notion of universality class / D. Carpentier -- 10.4.4.Effect of a magnetic field / D. Carpentier -- Acknowledgements / D. Carpentier -- References / D. Carpentier -- 11.Spin textures in quantum Hall systems / D. Carpentier -- 11.1.Introduction / Benoit Doucot -- 11.2.Physical properties of spin textures / Benoit Doucot -- 11.2.1.Intuitive picture / Benoit Doucot -- 11.2.2.Construction of spin textures / Benoit Doucot -- 11.2.3.Energetics of spin textures / Benoit Doucot -- 11.2.4.Choice of an effective model / Benoit Doucot -- 11.2.5.Classical ground states of the CPd-1 model / Benoit Doucot -- 11.3.Periodic textures / Benoit Doucot -- 11.3.1.Perturbation theory for degenerate Hamiltonians / Benoit Doucot -- 11.3.2.Remarks on the Hessian of the exchange energy / Benoit Doucot -- 11.3.3.Variational procedure for energy minimization / Benoit Doucot -- 11.3.4.Properties of periodic textures / Benoit Doucot -- 11.4.Collective excitations around periodic textures / Benoit Doucot -- 11.4.1.Time-dependent Hartree-Fock equations / Benoit Doucot -- 11.4.2.Collective-mode spectrum / Benoit Doucot -- 11.4.3.Towards an effective sigma model description / Benoit Doucot -- 11.A.Coherent states in the lowest Landau level / Benoit Doucot -- 11.B.From covariant symbols on a two-dimensional plane to operators / Benoit Doucot -- 11.C.Single-particle density matrix in a texture Slater determinant / Benoit Doucot -- 11.D.Hamiltonians with quadratic covariant symbol / Benoit Doucot -- Acknowledgements / Benoit Doucot -- References / Benoit Doucot -- 12.Out-of-equilibrium behaviour in topologically ordered systems on a lattice: fractionalized excitations and kinematic constraints / Benoit Doucot -- Preface / Claudio Castelnovo -- 12.1.Topological order, broadly interpreted / Claudio Castelnovo -- 12.2.Example 1: (classical) spin ice / Claudio Castelnovo -- 12.2.1.Thermal quenches / Claudio Castelnovo -- 12.2.2.Field quenches / Claudio Castelnovo -- 12.3.Example 2: Kitaev's toric code / Claudio Castelnovo -- 12.3.1.The model / Claudio Castelnovo -- 12.3.2.Elementary excitations / Claudio Castelnovo -- 12.3.3.Dynamics / Claudio Castelnovo -- 12.3.4.Intriguing comparison: kinetically constrained models / Claudio Castelnovo -- 12.4.Conclusions / Claudio Castelnovo -- Acknowledgements / Claudio Castelnovo -- References / Claudio Castelnovo -- 13.What is life?-70 years after Schrodinger / Claudio Castelnovo -- Preface / Antti J. Niemi -- 13.1.A protein minimum / Antti J. Niemi -- 13.1.1.Why proteins? / Antti J. Niemi -- 13.1.2.Protein chemistry and the genetic code / Antti J. Niemi -- 13.1.3.Data banks and experiments / Antti J. Niemi -- 13.1.4.Phases of proteins / Antti J. Niemi -- 13.1.5.Backbone geometry / Antti J. Niemi -- 13.1.6.Ramachandran angles / Antti J. Niemi -- 13.1.7.Homology modelling / Antti J. Niemi -- 13.1.8.All-atom models / Antti J. Niemi -- 13.1.9.All-atom simulations / Antti J. Niemi -- 13.1.10.Thermostats / Antti J. Niemi -- 13.1.11.Other physics-based approaches / Antti J. Niemi -- 13.2.Bol'she / Antti J. Niemi -- 13.2.1.The importance of symmetry breaking / Antti J. Niemi -- 13.2.2.An optical illusion / Antti J. Niemi -- 13.2.3.Fractional charge / Antti J. Niemi -- 13.2.4.Spin-charge separation / Antti J. Niemi -- 13.2.5.All-atom is Landau liquid / Antti J. Niemi -- 13.3.Strings in three space dimensions / Antti J. Niemi -- 13.3.1.Abelian Higgs model and the limit of slow spatial variations / Antti J. Niemi -- 13.3.2.The Frenet equation / Antti J. Niemi -- 13.3.3.Frame rotation and Abelian Higgs multiplet / Antti J. Niemi -- 13.3.4.The unique string Hamiltonian / Antti J. Niemi -- 13.3.5.Integrable hierarchy / Antti J. Niemi -- 13.3.6.Strings from solitons / Antti J. Niemi -- 13.3.7.Anomaly in the Frenet frames / Antti J. Niemi -- 13.3.8.Perestroika / Antti J. Niemi -- 13.4.Discrete Frenet frames / Antti J. Niemi -- 13.4.1.The Calpha trace reconstruction / Antti J. Niemi -- 13.4.2.Universal discretized energy / Antti J. Niemi -- 13.4.3.Discretized solitons / Antti J. Niemi -- 13.4.4.Proteins out of thermal equilibrium / Antti J. Niemi -- 13.4.5.Temperature renormalization / Antti J. Niemi -- 13.5.Solitons and ordered proteins / Antti J. Niemi -- 13.5.1.lambda-repressor as a multisoliton / Antti J. Niemi -- 13.5.2.Structure of myoglobin / Antti J. Niemi -- 13.5.3.Dynamical myoglobin / Antti J. Niemi -- 13.6.Intrinsically disordered proteins / Antti J. Niemi -- 13.6.1.Order versus disorder / Antti J. Niemi -- 13.6.2.hIAPP and type 2 diabetes / Antti J. Niemi -- 13.6.3.hIAPP as a three-soliton / Antti J. Niemi -- 13.6.4.Heating and cooling hIAPP / Antti J. Niemi -- 13.7.Beyond Calpha / Antti J. Niemi -- 13.7.1.'What-you-see-is-what-you-have' / Antti J. Niemi.
- Summary:
- This work 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.
- Subject(s):
- ISBN:
- 9780191827600 (ebook)
- Audience Notes:
- Specialized.
- Note:
- This edition previously issued in print: 2017.
- Bibliography Note:
- Includes bibliographical references.
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