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Quantum Phase Transitions, Second Edition

Book Description

Describing the physical properties of quantum materials near critical points with long-range many-body quantum entanglement, this book introduces readers to the basic theory of quantum phases, their phase transitions and their observable properties. This second edition begins with a new section suitable for an introductory course on quantum phase transitions, assuming no prior knowledge of quantum field theory. It also contains several new chapters to cover important recent advances, such as the Fermi gas near unitarity, Dirac fermions, Fermi liquids and their phase transitions, quantum magnetism, and solvable models obtained from string theory. After introducing the basic theory, it moves on to a detailed description of the canonical quantum-critical phase diagram at non-zero temperatures. Finally, a variety of more complex models are explored. This book is ideal for graduate students and researchers in condensed matter physics and particle and string theory.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. From the Preface to the first edition
  8. Preface to the second edition
  9. Part I: Introduction
    1. 1. Basic concepts
      1. 1.1 What is a quantum phase transition?
      2. 1.2 Nonzero temperature transitions and crossovers
      3. 1.3 Experimental examples
      4. 1.4 Theoretical models
        1. 1.4.1 Quantum Ising model
        2. 1.4.2 Quantum rotor model
        3. 1.4.3 Physical realizations of quantum rotors
    2. 2. Overview
      1. 2.1 Quantum field theories
      2. 2.2 What’s different about quantum transitions?
  10. Part II: A first course
    1. 3. Classical phase transitions
      1. 3.1 Mean-field theory
      2. 3.2 Landau theory
      3. 3.3 Fluctuations and perturbation theory
        1. 3.3.1 Gaussian integrals
        2. 3.3.2 Expansion for susceptibility
      4. Exercises
    2. 4. The renormalization group
      1. 4.1 Gaussian theory
      2. 4.2 Momentum shell RG
      3. 4.3 Field renormalization
      4. 4.4 Correlation functions
      5. Exercises
    3. 5. The quantum Ising model
      1. 5.1 Effective Hamiltonian method
      2. 5.2 Large-g expansion
        1. 5.2.1 One-particle states
        2. 5.2.2 Two-particle states
      3. 5.3 Small-g expansion
        1. 5.3.1 d = 2
        2. 5.3.2 d = 1
      4. 5.4 Review
      5. 5.5 The classical Ising chain
        1. 5.5.1 The scaling limit
        2. 5.5.2 Universality
        3. 5.5.3 Mapping to a quantum model: Ising spin in a transverse field
      6. 5.6 Mapping of the quantum Ising chain to a classical Ising model
      7. Exercises
    4. 6. The quantum rotor model
      1. 6.1 Large-g[tilde] expansion
      2. 6.2 Small-g[tilde] expansion
      3. 6.3 The classical XY chain and an O(2) quantum rotor
      4. 6.4 The classical Heisenberg chain and an O(3) quantum rotor
      5. 6.5 Mapping to classical field theories
      6. 6.6 Spectrum of quantum field theory
        1. 6.6.1 Paramagnet
        2. 6.6.2 Quantum critical point
        3. 6.6.3 Magnetic order
      7. Exercises
    5. 7. Correlations, susceptibilities, and the quantum critical point
      1. 7.1 Spectral representation
        1. 7.1.1 Structure factor
        2. 7.1.2 Linear response
      2. 7.2 Correlations across the quantum critical point
        1. 7.2.1 Paramagnet
        2. 7.2.2 Quantum critical point
        3. 7.2.3 Magnetic order
      3. Exercises
    6. 8. Broken symmetries
      1. 8.1 Discrete symmetry and surface tension
      2. 8.2 Continuous symmetry and the helicity modulus
        1. 8.2.1 Order parameter correlations
      3. 8.3 The London equation and the superfluid density
        1. 8.3.1 The rotor model
      4. Exercises
    7. 9. Boson Hubbard model
      1. 9.1 Mean-field theory
      2. 9.2 Coherent state path integral
        1. 9.2.1 Boson coherent states
      3. 9.3 Continuum quantum field theories
      4. Exercises
  11. Part III: Nonzero temperatures
    1. 10. The Ising chain in a transverse field
      1. 10.1 Exact spectrum
      2. 10.2 Continuum theory and scaling transformations
      3. 10.3 Equal-time correlations of the order parameter
      4. 10.4 Finite temperature crossovers
        1. 10.4.1 Low T on the magnetically ordered side, Δ > 0, T ≪ Δ
        2. 10.4.2 Low T on the quantum paramagnetic side, Δ < 0, T ≪ |Δ|
        3. 10.4.3 Continuum high T, T ≫ |Δ|
        4. 10.4.4 Summary
    2. 11. Quantum rotor models: large-N limit
      1. 11.1 Continuum theory and large-N limit
      2. 11.2 Zero temperature
        1. 11.2.1 Quantum paramagnet, g > g[sub(c)]
        2. 11.2.2 Critical point, g = g[sub(c)]
        3. 11.2.3 Magnetically ordered ground state, g < g[sub(c)]
      3. 11.3 Nonzero temperatures
        1. 11.3.1 Low T on the quantum paramagnetic side, g > g[sub(c)], T ≪ Δ[sub(+)]
        2. 11.3.2 High T, T ≫ Δ[sub(+)], Δ[sub(−)]
        3. 11.3.3 Low T on the magnetically ordered side, g < g[sub(c)], T ≪ Δ[sub(−)]
      4. 11.4 Numerical studies
    3. 12. The d = 1, O(N ≥ 3) rotor models
      1. 12.1 Scaling analysis at zero temperature
      2. 12.2 Low-temperature limit of the continuum theory, T ≪ Δ[sub(+)]
      3. 12.3 High-temperature limit of the continuum theory, Δ[sub(+)] ≪ T ≪ J
        1. 12.3.1 Field-theoretic renormalization group
        2. 12.3.2 Computation of χ[sub(u)]
        3. 12.3.3 Dynamics
      4. 12.4 Summary
    4. 13. The d = 2, O(N ≥ 3) rotor models
      1. 13.1 Low T on the magnetically ordered side, T ≪ ρ[sub(s)]
        1. 13.1.1 Computation of ξ[sub(c)]
        2. 13.1.2 Computation of τ[sub(φ)]
        3. 13.1.3 Structure of correlations
      2. 13.2 Dynamics of the quantum paramagnetic and high-T regions
        1. 13.2.1 Zero temperature
        2. 13.2.2 Nonzero temperatures
      3. 13.3 Summary
    5. 14. Physics close to and above the upper-critical dimension
      1. 14.1 Zero temperature
        1. 14.1.1 Tricritical crossovers
        2. 14.1.2 Field-theoretic renormalization group
      2. 14.2 Statics at nonzero temperatures
        1. 14.2.1 d < 3
        2. 14.2.2 d > 3
      3. 14.3 Order parameter dynamics in d = 2
      4. 14.4 Applications and extensions
    6. 15. Transport in d = 2
      1. 15.1 Perturbation theory
        1. 15.1.1 σ[sub(I)]
        2. 15.1.2 σ[sub(II)]
      2. 15.2 Collisionless transport equations
      3. 15.3 Collision–dominated transport
        1. 15.3.1 ε expansion
        2. 15.3.2 Large-N limit
      4. 15.4 Physical interpretation
      5. 15.5 The AdS/CFT correspondence
        1. 15.5.1 Exact results for quantum critical transport
        2. 15.5.2 Implications
      6. 15.6 Applications and extensions
  12. Part IV: Other models
    1. 16. Dilute Fermi and Bose gases
      1. 16.1 The quantum X X model
      2. 16.2 The dilute spinless Fermi gas
        1. 16.2.1 Dilute classical gas, k[sub(B)]T ≪ |μ|, μ < 0
        2. 16.2.2 Fermi liquid, k[sub(B)]T ≪ μ, μ > 0
        3. 16.2.3 High-T limit, k[sub(B)]T ≫ |μ|
      3. 16.3 The dilute Bose gas
        1. 16.3.1 d < 2
        2. 16.3.2 d = 3
        3. 16.3.3 Correlators of Z[sub(B)] in d = 1
      4. 16.4 The dilute spinful Fermi gas: the Feshbach resonance
        1. 16.4.1 The Fermi–Bose model
        2. 16.4.2 Large-N expansion
      5. 16.5 Applications and extensions
    2. 17. Phase transitions of Dirac fermions
      1. 17.1 d-wave superconductivity and Dirac fermions
      2. 17.2 Time-reversal symmetry breaking
      3. 17.3 Field theory and RG analysis
      4. 17.4 Ising-nematic ordering
    3. 18. Fermi liquids, and their phase transitions
      1. 18.1 Fermi liquid theory
        1. 18.1.1 Independence of choice of k[arrow][sub(0)]
      2. 18.2 Ising-nematic ordering
        1. 18.2.1 Hertz theory
        2. 18.2.2 Fate of the fermions
        3. 18.2.3 Non-Fermi liquid criticality in d = 2
      3. 18.3 Spin density wave order
        1. 18.3.1 Mean-field theory
        2. 18.3.2 Continuum theory
        3. 18.3.3 Hertz theory
        4. 18.3.4 Fate of the fermions
        5. 18.3.5 Critical theory in d = 2
      4. 18.4 Nonzero temperature crossovers
      5. 18.5 Applications and extensions
    4. 19. Heisenberg spins: ferromagnets and antiferromagnets
      1. 19.1 Coherent state path integral
      2. 19.2 Quantized ferromagnets
      3. 19.3 Antiferromagnets
        1. 19.3.1 Collinear antiferromagnetism and the quantum nonlinear sigma model
        2. 19.3.2 Collinear antiferromagnetism in d = 1
        3. 19.3.3 Collinear antiferromagnetism in d = 2
        4. 19.3.4 Noncollinear antiferromagnetism in d = 2: deconfined spinons and visons
        5. 19.3.5 Deconfined criticality
      4. 19.4 Partial polarization and canted states
        1. 19.4.1 Quantum paramagnet
        2. 19.4.2 Quantized ferromagnets
        3. 19.4.3 Canted and Néel states
        4. 19.4.4 Zero temperature critical properties
      5. 19.5 Applications and extensions
    5. 20. Spin chains: bosonization
      1. 20.1 The X X chain revisited: bosonization
      2. 20.2 Phases of H[sub(12)]
        1. 20.2.1 Sine–Gordon model
        2. 20.2.2 Tomonaga–Luttinger liquid
        3. 20.2.3 Valence bond solid order
        4. 20.2.4 Néel order
        5. 20.2.5 Models with SU (2) (Heisenberg) symmetry
        6. 20.2.6 Critical properties near phase boundaries
      3. 20.3 O(2) rotor model in d = 1
      4. 20.4 Applications and extensions
    6. 21. Magnetic ordering transitions of disordered systems
      1. 21.1 Stability of quantum critical points in disordered systems
      2. 21.2 Griffiths–McCoy singularities
      3. 21.3 Perturbative field-theoretic analysis
      4. 21.4 Metallic systems
      5. 21.5 Quantum Ising models near the percolation transition
        1. 21.5.1 Percolation theory
        2. 21.5.2 Classical dilute Ising models
        3. 21.5.3 Quantum dilute Ising models
      6. 21.6 The disordered quantum Ising chain
      7. 21.7 Discussion
      8. 21.8 Applications and extensions
    7. 22. Quantum spin glasses
      1. 22.1 The effective action
        1. 22.1.1 Metallic systems
      2. 22.2 Mean-field theory
      3. 22.3 Applications and extensions
  13. References
  14. Index