In the excitement and rapid pace of developments, writing pedagogical texts has low priority for most researchers. However, in transforming my lecture notes into this book, I found a personal benefit: the organization of what I understand in a (hopefully simple) logical sequence. Very little in this text is my original contribution. Most of the knowledge was collected from the research literature. Some was acquired by conversations with colleagues; a kind of physics oral tradition passed between disciples of a similar faith.

 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 Ferromsgnetic 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 T=0

 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 θ Integrals

 7.4 Exercises

8 Variational Wave Functions and Parent Hamiltonians

 8.1 Valence Bond States

 8.2 S=1/2 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 Semiclsssical Dynamics

10.2.2 Semiclassieal 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 Haldaue's Mapping

 12.2 The Continuum Hamiltonian

 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 β Function

 13.5 Exercises

14 The Nonlinear Sigma Model: Large N

 14.1 The CP1 Formulation

 14.2 CPN-1 Models at Large N

 14.3 Exercises

15 Quantum Antiferromagnets: Continuum Results

 15.1 One Dimension, the θ Term

 15.2 One Dimension, Integer Spins

 15.3 Two Dimensions

16 SU(N) Helsenberg Models

 16.1 Ferromagnet, Schwinger Bceons

 16.2 Antiferromagnet, Schwinger Bosons

 16.3 Antiferromagnet, Constrsined 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 1/N 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 t-J Model

 19.1 Schwinger Bosons and Slave Fermions

 19.2 Spin-Hole Coherent States

 19.3 The Classical Theory: Small Polarons

 19.4 Polaron Dynanics and Spin Tunneling

 19.5 The t'-J 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

 D.5 Exercises

Appendix E

 The Method of Steepest Descents

 Index