Description
Book SynopsisI Basic Models.- 1 Electron Interactions in Solids.- 2 Spin Exchange.- 3 The Hubbard Model and Its Descendants.- II Wave Functions and Correlations.- 4 Ground States of the Hubbard Model.- 5 Ground States of the Heisenberg Model.- 6 Disorder in Low Dimensions.- 7 Spin Representations.- 8 Variational Wave Functions and Parent Hamiltonians.- 9 From Ground States to Excitations.- III Path Integral Approximations.- 10 The Spin Path Integral.- 11 Spin Wave Theory.- 12 The Continuum Approximation.- 13 Nonlinear Sigma Model: Weak Coupling.- 14 The Nonlinear Sigma Model: Large N.- 15 Quantum Antiferromagnets: Continuum Results.- 16 SU(N) Heisenberg Models.- 17 The Large N Expansion.- 18 Schwinger Bosons Mean Field Theory.- 19 The Semiclassical Theory of the t J Model.- 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.
Table of ContentsI 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’ 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 ? Integrals.- 7.4 Exercises.- 8 Variational Wave Functions and Parent Hamiltonians.- 8.1 Valence Bond States.- 8.2 S = ½ 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’ 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’ 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’ Renormalization.- 13.4 The ? 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 ? 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 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 Dynamics 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.
