內容簡介

In the excitement and rapid pace of developments, writing pedagogical textshas low priority for most researchers. However, in transforming my lecturenotesI into this book, I found a personal benefit: the organization of what Iunderstand in a （hopefully simple） logical sequence. Very little in this textis my original contribution. Most of the knowledge was collected from theresearch literature. Some was acquired by conversations with colleagues; akind of physics oral tradition passed between disciples of a similar faith.For many years, diagramatic perturbation theory has been the majortheoretical tool for treating interactions in metals, semiconductors, itiner-ant magnets, and superconductors.

Preface
Ⅰ Basic Models
1 Electron Interactions in Solids
1.1 Single Electron Theory
1.2 Fields and Interactions
1.3 Magnitude of Interactions in Metals
1.4 Effective Models
1.5 Exercises
2 Spin Exchange
2.1 Ferromagnetic Exchange
2.2 Antiferromagnetic Exchange
2.3 Superexchange
2.4 Exercises
3 The Hubbard Model and Its Descendants
3.1 Truncating the Interactions
3.2 At Large U: The t-J Model
3.3 The Negative-U Model
3.3.1 The Pseudo-spin Model and Superconductivity
3.4 Exercises

Ⅱ Wave Functions and Correlations
4 Ground States of the Hubbard Model
4.1 Variational Magnetic States
4.2 Some Ground State Theorems
4.3 Exercises
5 Ground States of the Heisenberg Model
5.1 The Antiferromagnet
5.2 Half-Odd Integer Spin Chains
5.3 Exercises
6 Disorder in Low Dimensions
6.1 Spontaneously Broken Symmetry
6.2 Mermin and Wagner’’s Theorem
6.3 Quantum Disorder at
6.4 Exercises
7 Spin Representations
7.1 Holstein-Primakoff Bosons
7.2 Schwinger Bosons
7.2.1 Spin Rotations
7.3 Spin Coherent States
7.3.1 The 0 Integrals
7.4 Exercises
8 Variational Wave Functions and Parent Hamiltonians
8.1 Valence Bond States
8.2 States
8.2.1 The Majumdar-Ghosh Hamiltonian
8.2.2 Square Lattice RVB States
8.3 Valence Bond Solids and AKLT Models
8.3.1 Correlations in Valence Bond Solids
8.4 Exercises
9 From Ground States to Excitations
9.1 The Single Mode Approximation
9.2 Goldstone Modes
9.3 The Haldane Gap and the SMA

Ⅲ Path Integral Approximations
10 The Spin Path Integral
10.1 Construction of the Path Integral
10.1.1 The Green’’s Function
10.2 The Large S Expansion
10.2.1 Semiclassical Dynamics
10.2.2 Semiclassical Spectrum
10.3 Exercises
11 Spin Wave Theory
11.1 Spin Waves: Path Integral Approach
11.1.1 The Ferromagnet
11.1.2 The Antiferromagnet
11.2 Spin Waves: Holstein-Primakoff Approach
11.2.1 The Ferromagnet
11.2.2 The Antiferromagnet
11.3 Exercises
12 The Continuum Approximation
12.1 Haldane’’s Mapping
12.2 The Continuum Harniltonian
12.3 The Kinetic Term
12.4 Partition Function and Correlations
12.5 Exercises
13 Nonlinear Sigma Model: Weak Coupling
13.1 The Lattice Regularization
13.2 Weak Coupling Expansion
13.3 Poor Man’’s Renormalization
13.4 The/3 Function
13.5 Exercises
14 The Nonlinear Sigma Model: Large N
14.1 The CPI Formulation
14.2 CPU Models at Large N
14.3 Exercises
15 Quantum Antiferromagnets: Continuum Results
15.1 One Dimension, the e Term
15.2 One Dimension, Integer Spins
15.3 Two Dimensions
16 SU(N) Heisenberg Models
16.1 Ferromagnet, Schwinger Bosons
16.2 Antiferromagnet, Schwinger Bosons
16.3 Antiferromagnet, Constrained Fermions
16.4 The Generating Functional
16.5 The Hubbard-Stratonovich Transformation
16.6 Correlation Functions
17 The Large N Expansion
17.1 Fluctuations and Gauge Fields
17.2 1IN Expansion Diagrams
17.3 Sum Rules
17.3.1 Absence of Charge Fluctuations
17.3.2 On-Site Spin Fluctuations
17.4 Exercises
18 Schwinger Bosons Mean Field Theory
18.1 The Case of the Ferromagnet
18.1.1 One Dimension
18.1.2 Two Dimensions
18.2 The Case of the Antiferromagnet
18.2.1 Long-Range Antiferromagnetic Order
18.2.2 One Dimension
18.2.3 Two Dimensions
18.3 Exercises
19 The Semiclassical Theory of the Model
19.1 Schwinger Bosons and Slave Fermions
19.2 Spin-Hole Coherent States
19.3 The Classical Theory: Small Polarons
19.4 Polaron Dynamics and Spin Tunneling
19.5 The Model
19.5.1 Superconductivity?
19.6 Exercises

Ⅳ Mathematical Appendices
Appendix A
Second Quantization
A.1 Fock States
A.2 Normal Bilinear Operators
A.3 Noninteracting Hamiltonians
A.4 Exercises
Appendix B
Linear Response and Generating Functionals
B. 1 Spin Response Function
B.2 Fluctuations and Dissipation
B.3 The Generating Functional
Appendix C
Bose and Fermi Coherent States
C.1 Complex Integration
C.2 Grassmann Variables
C.3 Coherent States
C.4 Exercises
Appendix D
Coherent State Path Integrals
D.1 Constructing the Path Integral
D.2 Normal Bilinear Hamiltonians
D.3 Matsubara Representation
D.4 Matsubara Sums

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序

In the excitement and rapid pace of developments, writing pedagogical textshas low priority for most researchers. However, in transforming my lecturenotesI into this book, I found a personal benefit: the organization of what Iunderstand in a （hopefully simple） logical sequence. Very little in this textis my original contribution. Most of the knowledge was collected from theresearch literature. Some was acquired by conversations with colleagues; akind of physics oral tradition passed between disciples of a similar faith.For many years, diagramatic perturbation theory has been the majortheoretical tool for treating interactions in metals, semiconductors, itiner-ant magnets, and superconductors. It is in essence a weak coupling expan-sion about free quasiparticles. Many experimental discoveries during thelast decade, including heavy fermions, fractional quantum Hall effect, high-temperature superconductivity, and quantum spin chains, are not readilyaccessible from the weak coupling point of view.

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