Part Ⅰ Introduction

1 Basic concepts

1.1 What is a quantum phase transition?

1.2 Nonzero temperature transitions and crossovers

1.3 Experimental examples

1.4 Theoretical models

1.4.1 Quantum Ising model

1.4.2 Quantum rotor model

1.4.3 Physical realizations of quantum rotors

2 Overview

2.1 Quantum field theories

2.2 What's different about quantum transitions?

Part Ⅱ A first course

3 Classical phase transitions

3.1 Mean-field theory

3.2 Landau theory

3.3 Fluctuations and perturbation theory

3.3.1 Gaussian integrals

3.3.2 Expansion for susceptibility

Exercises

4 The renormalization group

4.1 Gaussian theory

4.2 Momentum shell RG

4.3 Field renormalization

4.4 Correlation functions

Exercises

5 The quantum Ising model

5.1 Effective Hamiltonian method

5.2 Large-g expansion

5.2.1 One-particle states

5.2.2 Two-particle states

5.3 Small-g expansion

5.3.1 d=2

5.3.2 d=1

5.4 Review

5.5 The classical Ising chain

5.5.1 The scaling limit

5.5.2 Universality

5.5.3 Mapping to a quantum model:Ising spin in atransversefield

5.6 Mapping of the quantum Ising chain to a classical Ising model

Exercises

6 The quantum rotor model

6.1 Large-？expansion

6.2 Small-？expansion

6.3 The classical X Y chain and an O(2)quantum rotor

6.4 The classical Heisenberg chain and an O(3)quantum rotor

6.5 Mapping to classical field theories

6.6 Spectrum of quantum field theory

6.6.1 Paramagnet

6.6.2 Quantum critical point

6.6.3 Magnetic order

Exercises

7 Correlations,susceptibilities,and the quantum critical point

7.1 Spectral representation

7.1.1 Structure factor

7.1.2 Linear response

7.2 Correlations across the quantum critical point

7.2.1 Paramagnet

7.2.2 Quantum critical point

7.2.3 Magnetic order

Exercises

8 Broken symmetries

8.1 Discrete symmetry and surface tension

8.2 Continuous symmetry and the helicity modulus

8.2.1 Order parameter correlations

8.3 The London equation and the superfluid density

8.3.1 The rotor model

Exercises

9 Boson Hubbard model

9.1 Mean-field theory

9.2 Coherent state path integral

9.2.1 Boson coherent states

9.3 Continuum quantum field theories

Exercises

Part Ⅲ Nonzero temperatures

10 The Ising chainin atransversefield

10.1 Exact spectrum

10.2 Continuum theory and scaling transformations

10.3 Equal-time correlations of the order parameter

10.4 Finite temperature crossovers

10.4.1 Low T on the magnetically ordered side,△＞0,T《△

10.4.2 Low T on the quantum paramagnetic side,△＜0,T《|△|

10.4.3 Continuum high T,T》| △|

10.4.4 Summary

11 Quantum rotor models:large-N limit

11.1 Continuum theory and large-N limit

11.2 Zero temperature

11.2.1 Quantum paramagnet,g＞gc

11.2.2 Critical point,g=gc

11.2.3 Magnetically ordered ground state,g＜gc

11.3 Nonzero temperatures

11.3.1 Low T on the quantum paramagnetic side,g＞gc,T《△+

11.3.2 High T,T》△+,△-

11.3.3 Low T on the magnetically ordered side,g＜gc,T《△-

11.4 Numerical studies

12 Thed=1,0(N≥3)rotormodels

12.1 Scaling analysis at zero temperature

12.2 Low-temperature limit ofthe continuum theory,T《△+

12.3 High-temperature limit of the continuum theory,△+《T《J

12.3.1 Field-theoretic renormalization group

12.3.2 Computation of xu

12.3.3 Dynamics

12.4 Summary

13 The d=2,0(N≥3)rotor models

13.1 Low T on the magnetically ordered side,T《ρs

13.1.1 Computation of ξc

13.1.2 Computation of τψ

13.1.3 Structure of correlations

13.2 Dynamics of the quantum paramagnetic and high-T regions

13.2.1 Zero temperature

13.2.2 Nonzero temperatures

13.3 Summary

14 Physics close to and above the upper-critical dimension

14.1 Zero temperature

14.1.1 Tricritical crossovers

14.1.2 Field-theoretic renormalization group

14.2 Statics at nonzero temperatures

14.2.1 d＜3

14.2.2 d＞3

14.3 Order parameter dynamics in d=2

14.4 Applications and extensions

15 Transportind=2

15.1 Perturbation theory

15.1.1 σI

15.1.2 σII

15.2 Collisionless transport equations

15.3 Collision-dominated transport

15.3.1 ∈expansion

15.3.2 Large-N limit

15.4 Physical interpretation

15.5 The AdS/CFT correspondence

15.5.1 Exact results for quantum critical transport

15.5.2 Implications

15.6 Applications and extensions

Part Ⅳ Other models

16 Dilute Fermi and Bose gases

16.1 The quantum XX model

16.2 The dilute spinless Fermi gas

16.2.1 Dilute classical gas,kBT《|μ|,μ＜0

16.2.2 Fermi liquid,kBT《μ,μ＞0

16.2.3 High-T limit,kBT》|μ|

16.3 The dilute Bose gas

16.3.1 d＜2

16.3.2 d=3

16.3.3 Correlators of ZB in d=1

16.4 The dilute spinful Fermi gas:the Feshbach resonance

16.4.1 The Fermi-Bose model

16.4.2 Large-N expansion

16.5 Applications and extensions

17 Phase transitions of Dirac fermions

17.1 d-wave superconductivity and Dirac fermions

17.2 Time-reversal symmetry breaking

17.3 Field theory and RG analysis

17.4 Ising-nematic ordering

18 Fermi liquids,and their phase transitions

18.1 Fermi liquid theory

18.1.1 Independence of choice of？0

18.2 Ising-nematic ordering

18.2.1 Hertz theory

18.2.2 Fate of the fermions

18.2.3 Non-Fermi liquid criticality in d=2

18.3 Spin density wave order

18.3.1 Mean-field theory

18.3.2 Continuum theory

18.3.3 Hertz theory

18.3.4 Fate of the fermions

18.3.5 Critical theory in d=2

18.4 Nonzero temperature crossovers

18.5 Applications and extensions

19 Heisenberg spins:ferromagnets and antiferromagnets

19.1 Coherent state path integral

19.2 Quantized ferromagnets

19.3 Antiferromagnets

19.3.1 Collinear antiferromagnetism and the quantum nonlinear sigma model

19.3.2 Collinear antiferromagnetism in d=1

19.3.3 Collinear antiferromagnetism in d=2

19.3.4 Noncollinear antiferromagnetism in d=2:deconfined spinons and visons

19.3.5 Deconfined criticality

19.4 Partial polarization and canted states

19.4.1 Quantum paramagnet

19.4.2 Quantized ferromagnets

19.4.3 Canted and Néel states

19.4.4 Zero temperature critical properties

19.5 Applications and extensions

20 Spin chains:bosonization

20.1 The XX chain revisited:bosonization

20.2 Phases of H12

20.2.1 Sine-Gordon model

20.2.2 Tomonaga-Luttinger liquid

20.2.3 Valence bond solid order

20.2.4 Néel order

20.2.5 Models with SU(2)(Heisenberg)symmetry

20.2.6 Critical properties near phase boundaries

20.3 O(2)rotor modelin d=1

20.4 Applications and extensions

21 Magnetic ordering transitions of disordered systems

21.1 Stability of quantum critical points in disordered systems

21.2 Griffiths-McCoy singularities

21.3 Perturbative field-theoretic analysis

21.4 Metallic systems

21.5 Quantum lsing models near the percolation transition

21.5.1 Percolation theory

21.5.2 Classieal dilute Ising models

21.5.3 Quantum dilute Ising models

21.6 The disordered quantum Ising chain

21.7 Discussion

21.8 Applications and extensions

22 Quantum spin glasses

22.1 The effective action

22.1.1 Metallic systems

22.2 Mean-feld theory

22.3 Applications and extensions

References

Index