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Field Theories of Condensed Matter Physics, Second Edition

Book Description

Presenting the physics of the most challenging problems in condensed matter using the conceptual framework of quantum field theory, this book is of great interest to physicists in condensed matter and high energy and string theorists, as well as mathematicians. Revised and updated, this second edition features new chapters on the renormalization group, the Luttinger liquid, gauge theory, topological fluids, topological insulators and quantum entanglement. The book begins with the basic concepts and tools, developing them gradually to bring readers to the issues currently faced at the frontiers of research, such as topological phases of matter, quantum and classical critical phenomena, quantum Hall effects and superconductors. Other topics covered include one-dimensional strongly correlated systems, quantum ordered and disordered phases, topological structures in condensed matter and in field theory and fractional statistics.

Table of Contents

  1. Coverpage
  2. Field Theories of Condensed Matter Physics
  3. Title page
  4. Copyright page
  5. Contents
  6. Preface to the second edition
  7. Preface to the first edition
  8. 1 Introduction
    1. 1.1 Field theory and condensed matter physics
    2. 1.2 What has been included in this book (first edition)
    3. 1.3 What was left out of the first edition
    4. 1.4 What has been included in the second edition
  9. 2 The Hubbard model
    1. 2.1 Introduction
    2. 2.2 Symmetries of the Hubbard model
    3. 2.3 The strong-coupling limit
    4. 2.4 The weak-coupling limit
    5. 2.5 Correlation functions
  10. 3 The magnetic instability of the Fermi system
    1. 3.1 Mean-field theory
    2. 3.2 Path-integral representation of the Hubbard model
    3. 3.3 Path integrals and mean-field theory
    4. 3.4 Fluctuations: the non-linear sigma model
    5. 3.5 The Néel state and the non-linear sigma model
  11. 4 The renormalization group and scaling
    1. 4.1 Scale invariance
    2. 4.2 Examples of fixed points
    3. 4.3 Scaling behavior of physical observables
    4. 4.4 General consequences of scale invariance
    5. 4.5 Perturbative renormalization group about a fixed point
    6. 4.6 The Kosterlitz renormalization group
  12. 5 One-dimensional quantum antiferromagnets
    1. 5.1 The spin-1/2 Heisenberg chain
    2. 5.2 Fermions and the Heisenberg model
    3. 5.3 The quantum Ising chain
    4. 5.4 Duality
    5. 5.5 The quantum Ising chain as a free-Majorana-fermion system
    6. 5.6 Abelian bosonization
    7. 5.7 Phase diagrams and scaling behavior
  13. 6 The Luttinger liquid
    1. 6.1 One-dimensional Fermi systems
    2. 6.2 Dirac fermions and the Luttinger model
    3. 6.3 Order parameters of the one-dimensional electron gas
    4. 6.4 The Luttinger model: bosonization
    5. 6.5 Spin and the Luttinger model
    6. 6.6 Scaling and renormalization in the Luttinger model
    7. 6.7 Correlation functions of the Luttinger model
    8. 6.8 Susceptibilities of the Luttinger model
  14. 7 Sigma models and topological terms
    1. 7.1 Generalized spin chains: the Haldane conjecture
    2. 7.2 Path integrals for spin systems: the single-spin problem
    3. 7.3 The path integral for many-spin systems
    4. 7.4 Quantum ferromagnets
    5. 7.5 The effective action for one-dimensional quantum antiferromagnets
    6. 7.6 The role of topology
    7. 7.7 Quantum fluctuations and the renormalization group
    8. 7.8 Asymptotic freedom and Haldane’s conjecture
    9. 7.9 Hopf term or no Hopf term?
    10. 7.10 The Wess–Zumino–Witten model
    11. 7.11 A (brief) introduction to conformal field theory
    12. 7.12 The Wess–Zumino–Witten conformal field theory
    13. 7.13 Applications of non-abelian bosonization
  15. 8 Spin-liquid states
    1. 8.1 Frustration and disordered spin states
    2. 8.2 Valence bonds and disordered spin states
    3. 8.3 Spinons, holons, and valence-bond states
    4. 8.4 The gauge-field picture of the disordered spin states
    5. 8.5 Flux phases, valence-bond crystals, and spin liquids
    6. 8.6 Is the large-N mean-field theory reliable?
    7. 8.7 SU(2) gauge invariance and Heisenberg models
  16. 9 Gauge theory, dimer models, and topological phases
    1. 9.1 Fluctuations of valence bonds: quantum-dimer models
    2. 9.2 Bipartite lattices: valence-bond order and quantum criticality
    3. 9.3 Non-bipartite lattices: topological phases
    4. 9.4 Generalized quantum-dimer models
    5. 9.5 Quantum dimers and gauge theories
    6. 9.6 The Ising gauge theory
    7. 9.7 The 2 confining phase
    8. 9.8 The Ising deconfining phase: the 2 topological fluid
    9. 9.9 Boundary conditions and topology
    10. 9.10 Generalized 2 gauge theory: matter fields
    11. 9.11 Compact quantum electrodynamics
    12. 9.12 Deconfinement and topological phases in the U(1) gauge theory
    13. 9.13 Duality transformation and dimer models
    14. 9.14 Quantum-dimer models and monopole gases
    15. 9.15 The quantum Lifshitz model
  17. 10 Chiral spin states and anyons
    1. 10.1 Chiral spin liquids
    2. 10.2 Mean-field theory of chiral spin liquids
    3. 10.3 Fluctuations and flux phases
    4. 10.4 Chiral spin liquids and Chern–Simons gauge theory
    5. 10.5 The statistics of spinons
    6. 10.6 Fractional statistics
    7. 10.7 Chern–Simons gauge theory: a field theory of anyons
    8. 10.8 Periodicity and families of Chern–Simons theories
    9. 10.9 Quantization of the global degrees of freedom
    10. 10.10 Flux phases and the fractional quantum Hall effect
    11. 10.11 Anyons at finite density
    12. 10.12 The Jordan–Wigner transformation in two dimensions
  18. 11 Anyon superconductivity
    1. 11.1 Anyon superconductivity
    2. 11.2 The functional-integral formulation of the Chern–Simons theory
    3. 11.3 Correlation functions
    4. 11.4 The semi-classical approximation
    5. 11.5 Effective action and topological invariance
  19. 12 Topology and the quantum Hall effect
    1. 12.1 Quantum mechanics of charged particles in magnetic fields
    2. 12.2 The Hofstadter wave functions
    3. 12.3 The quantum Hall effect
    4. 12.4 The quantum Hall effect and disorder
    5. 12.5 Linear-response theory and correlation functions
    6. 12.6 The Hall conductance and topological invariance
    7. 12.7 Quantized Hall conductance of a non-interacting system
    8. 12.8 Quantized Hall conductance of Hofstadter bands
  20. 13 The fractional quantum Hall effect
    1. 13.1 The Laughlin wave function
    2. 13.2 Composite particles
    3. 13.3 Landau–Ginzburg theory of the fractional quantum Hall effect
    4. 13.4 Fermion field theory of the fractional quantum Hall effect
    5. 13.5 The semi-classical excitation spectrum
    6. 13.6 The electromagnetic response and collective modes
    7. 13.7 The Hall conductance and Chern–Simons theory
    8. 13.8 Quantum numbers of the quasiparticles: fractional charge
    9. 13.9 Quantum numbers of the quasiparticles: fractional statistics
  21. 14 Topological fluids
    1. 14.1 Quantum Hall fluids on a torus
    2. 14.2 Hydrodynamic theory
    3. 14.3 Hierarchical states
    4. 14.4 Multi-component abelian fluids
    5. 14.5 Superconductors as topological fluids
    6. 14.6 Non-abelian quantum Hall states
    7. 14.7 The spin-singlet Halperin states
    8. 14.8 Moore–Read states and their generalizations
    9. 14.9 Topological superconductors
    10. 14.10 Braiding and fusion
  22. 15 Physics at the edge
    1. 15.1 Edge states of integer quantum Hall fluids
    2. 15.2 Hydrodynamic theory of the edge states
    3. 15.3 Edges of general abelian quantum Hall states
    4. 15.4 The bulk–edge correspondence
    5. 15.5 Effective-field theory of non-abelian states
    6. 15.6 Tunneling conductance at point contacts
    7. 15.7 Noise and fractional charge
    8. 15.8 Quantum interferometers
    9. 15.9 Topological quantum computation
  23. 16 Topological insulators
    1. 16.1 Topological insulators and topological band structures
    2. 16.2 The integer quantum Hall effect as a topological insulator
    3. 16.3 The quantum anomalous Hall effect
    4. 16.4 The quantum spin Hall effect
    5. 16.5 2 topological invariants
    6. 16.6 Three-dimensional topological insulators
    7. 16.7 Solitons in polyacetylene
    8. 16.8 Edge states in the quantum anomalous Hall effect
    9. 16.9 Edge states and the quantum spin Hall effect
    10. 16.10 2 topological insulators and the parity anomaly
    11. 16.11 Topological insulators and interactions
    12. 16.12 Topological Mott insulators and nematic phases
    13. 16.13 Topological insulators and topological phases
  24. 17 Quantum entanglement
    1. 17.1 Classical and quantum criticality
    2. 17.2 Quantum entanglement
    3. 17.3 Entanglement in quantum field theory
    4. 17.4 The area law
    5. 17.5 Entanglement entropy in conformal field theory
    6. 17.6 Entanglement entropy in the quantum Lifshitz universality class
    7. 17.7 Entanglement entropy in 4 theory
    8. 17.8 Entanglement entropy and holography
    9. 17.9 Quantum entanglement and topological phases
    10. 17.10 Outlook
  25. References
  26. Index