Interacting electrons and quantum magnetism
著者
書誌事項
Interacting electrons and quantum magnetism
（Graduate texts in contemporary physics）
Springer-Verlag, c1994
- : us
- : ger
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注記
Includes bibliographical references and index
内容説明・目次
- 巻冊次
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: us ISBN 9780387942865
内容説明
In the excitement and rapid pace of developments, writing pedagogical texts has low priority for most researchers. However, in transforming my lecture l 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. For many years, diagramatic perturbation theory has been the major theoretical 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 the last decade, including heavy fermions, fractional quantum Hall effect, high temperature superconductivity, and quantum spin chains, are not readily accessible from the weak coupling point of view. Therefore, recent years have seen vigorous development of alternative, nonperturbative tools for handling strong electron-electron interactions. I concentrate on two basic paradigms of strongly interacting (or con strained) quantum systems: the Hubbard model and the Heisenberg model. These models are vehicles for fundamental concepts, such as effective Ha miltonians, variational ground states, spontaneous symmetry breaking, and quantum disorder. In addition, they are used as test grounds for various nonperturbative approximation schemes that have found applications in diverse areas of theoretical physics.
目次
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' 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 = 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' 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.
- 巻冊次
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: ger ISBN 9783540942863
内容説明
This book emphasizes the role that electron interactions play in the properties of condensed matter. It teaches the use of the powerful nonperturbative techniques that have become available in the last decades to discuss such topics as mixed valence systems, Kondo systems, heavy electrons, high-temperature copper oxide superconductors, the quantum Hall effect, and low-dimensional isotropic magnets. Mathematical derivations are self contained. Appendices provide standard many-body tools including second quantization, Grassmann variables, generating functionals, Linear response, correlation functions, Fermi and Bose coherent-states path integrals, Matsubara representation, and the method of steepest descents. There are guided bibliographies and exercises at the end of chapters.
目次
- Basic models
- wave functions and correlations
- path integral approximations.
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