[request_ebook] The Geometric Phase in Quantum Systems Foundations,Mathematical concepts,and Applications in Molecular and Condesed Matter Physics

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physics book .about berry phase. a new approach in condese matter physics for caculating polarization. that would be help to have the least mistake in calculating the polarization in molecules. besides it can be used in classical and Quantum physics too. here is it table of contents:1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Quantal Phase Factors for Adiabatic Changes . . . . . . . . . . . . 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Adiabatic Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Berry’s Adiabatic Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Topological Phases and the Aharonov–Bohm Effect . . . . . . . . . 22 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3. Spinning Quantum System in an External Magnetic Field 31 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 The Parameterization of the Basis Vectors . . . . . . . . . . . . . . . . . 31 3.3 Mead–Berry Connection and Berry Phase for Adiabatic Evolutions – Magnetic Monopole Potentials . . . . 36 3.4 The Exact Solution of the Schr¨odinger Equation. . . . . . . . . . . . 42 3.5 Dynamical and Geometrical Phase Factors for Non-Adiabatic Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4. Quantal Phases for General Cyclic Evolution . . . . . . . . . . . . . 53 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Aharonov–Anandan Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Exact Cyclic Evolution for Periodic Hamiltonians . . . . . . . . . . . 60 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5. Fiber Bundles and Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . 65 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2 From Quantal Phases to Fiber Bundles . . . . . . . . . . . . . . . . . . . . 65 5.3 An Elementary Introduction to Fiber Bundles . . . . . . . . . . . . . . 67 5.4 Geometry of Principal Bundles and the Concept of Holonomy 76 5.5 Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.6 Mathematical Foundations of Gauge Theories and Geometry of Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 XII Table of Contents 6. Mathematical Structure of the Geometric Phase I: The Abelian Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2 Holonomy Interpretations of the Geometric Phase . . . . . . . . . . 107 6.3 Classification of U(1) Principal Bundles and the Relation Between the Berry–Simon and Aharonov–Anandan Interpretations of the Adiabatic Phase . . . . . . . . . . . . . . . . . . . . 113 6.4 Holonomy Interpretation of the Non-Adiabatic Phase Using a Bundle over the Parameter Space . . . . . . . . . . . . . . . . . 118 6.5 Spinning Quantum System and Topological Aspects of the Geometric Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7. Mathematical Structure of the Geometric Phase II: The Non-Abelian Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.2 The Non-Abelian Adiabatic Phase . . . . . . . . . . . . . . . . . . . . . . . . 129 7.3 The Non-Abelian Geometric Phase . . . . . . . . . . . . . . . . . . . . . . . 136 7.4 Holonomy Interpretations of the Non-Abelian Phase . . . . . . . . 139 7.5 Classification of U(N) Principal Bundles and the Relation Between the Berry–Simon and Aharonov–Anandan Interpretations of Non-Abelian Phase . . . . . . . . . . . . . . . . . . . . . 141 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8. A Quantum Physical System in a Quantum Environment – The Gauge Theory of Molecular Physics . . . . . . . . . . . . . . . . . . 147 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.2 The Hamiltonian of Molecular Systems . . . . . . . . . . . . . . . . . . . . 148 8.3 The Born–Oppenheimer Method . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.4 The Gauge Theory of Molecular Physics . . . . . . . . . . . . . . . . . . . 166 8.5 The Electronic States of Diatomic Molecule . . . . . . . . . . . . . . . . 174 8.6 The Monopole of the Diatomic Molecule. . . . . . . . . . . . . . . . . . . 176 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9. Crossing of Potential Energy Surfaces and the Molecular Aharonov–Bohm Effect . . . . . . . . . . . . . . . . 195 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 9.2 Crossing of Potential Energy Surfaces . . . . . . . . . . . . . . . . . . . . . 196 9.3 Conical Intersections and Sign-Change of Wave Functions . . . 198 9.4 Conical Intersections in Jahn–Teller Systems . . . . . . . . . . . . . . . 209 9.5 Symmetry of the Ground State in Jahn–Teller Systems . . . . . . 213 9.6 Geometric Phase in Two Kramers Doublet Systems . . . . . . . . . 219 9.7 Adiabatic–Diabatic Transformation . . . . . . . . . . . . . . . . . . . . . . . 222 Table of Contents XIII 10. Experimental Detection of Geometric Phases I: Quantum Systems in Classical Environments . . . . . . . . . . . . . 225 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 10.2 The Spin Berry Phase Controlled by Magnetic Fields . . . . . . . 225 10.2.1 Spins in Magnetic Fields: The Laboratory Frame . . . . . 225 10.2.2 Spins in Magnetic Fields: The Rotating Frame . . . . . . . 231 10.2.3 Adiabatic Reorientation in Zero Field . . . . . . . . . . . . . . . 237 10.3 Observation of the Aharonov–Anandan Phase Through the Cyclic Evolution of Quantum States . . . . . . . . . . . 248 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 11. Experimental Detection of Geometric Phases II: Quantum Systems in Quantum Environments . . . . . . . . . . . . . 255 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 11.2 Internal Rotors Coupled to External Rotors . . . . . . . . . . . . . . . . 256 11.3 Electronic–Rotational Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 259 11.4 Vibronic Problems in Jahn–Teller Systems . . . . . . . . . . . . . . . . . 260 11.4.1 Transition Metal Ions in Crystals . . . . . . . . . . . . . . . . . . . 261 11.4.2 Hydrocarbon Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 11.4.3 Alkali Metal Trimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 11.5 The Geometric Phase in Chemical Reactions . . . . . . . . . . . . . . . 270 12. Geometric Phase in Condensed Matter I: Bloch Bands . . . 277 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 12.2 Bloch Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 12.2.1 One-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 12.2.2 Three-Dimensional Case. . . . . . . . . . . . . . . . . . . . . . . . . . . 280 12.2.3 Band Structure Calculation . . . . . . . . . . . . . . . . . . . . . . . . 281 12.3 Semiclassical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 12.3.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 12.3.2 Symmetry Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 12.3.3 Derivation of the Semiclassical Formulas . . . . . . . . . . . . 286 12.3.4 Time-Dependent Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 12.4 Applications of Semiclassical Dynamics . . . . . . . . . . . . . . . . . . . . 288 12.4.1 Uniform DC Electric Field. . . . . . . . . . . . . . . . . . . . . . . . . 288 12.4.2 Uniform and Constant Magnetic Field . . . . . . . . . . . . . . 289 12.4.3 Perpendicular Electric and Magnetic Fields . . . . . . . . . . 290 12.4.4 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 12.5 Wannier Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 12.5.1 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 12.5.2 Localization Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 12.6 Some Issues on Band Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . 295 12.6.1 Quantized Adiabatic Particle Transport . . . . . . . . . . . . . 295 12.6.2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 XIV Table of Contents 13. Geometric Phase in Condensed Matter II: The Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 13.2 Basics of the Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . 302 13.2.1 The Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 13.2.2 The Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . 302 13.2.3 The Ideal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 13.2.4 Corrections to Quantization . . . . . . . . . . . . . . . . . . . . . . . 305 13.3 Magnetic Bands in Periodic Potentials . . . . . . . . . . . . . . . . . . . . 307 13.3.1 Single-Band Approximation in a Weak Magnetic Field 307 13.3.2 Harper’s Equation and Hofstadter’s Butterfly . . . . . . . . 309 13.3.3 Magnetic Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 13.3.4 Quantized Hall Conductivity . . . . . . . . . . . . . . . . . . . . . . . 314 13.3.5 Evaluation of the Chern Number . . . . . . . . . . . . . . . . . . . 316 13.3.6 Semiclassical Dynamics and Quantization . . . . . . . . . . . 318 13.3.7 Structure of Magnetic Bands and Hyperorbit Levels . . 321 13.3.8 Hierarchical Structure of the Butterfly . . . . . . . . . . . . . . 325 13.3.9 Quantization of Hyperorbits and Rule of Band Splitting . . . . . . . . . . . . . . . . . . . . . . . . 327 13.4 Quantization of Hall Conductance in Disordered Systems . . . . 329 13.4.1 Spectrum and Wave Functions . . . . . . . . . . . . . . . . . . . . . 329 13.4.2 Perturbation and Scattering Theory . . . . . . . . . . . . . . . . 331 13.4.3 Laughlin’s Gauge Argument . . . . . . . . . . . . . . . . . . . . . . . 332 13.4.4 Hall Conductance as a Topological Invariant . . . . . . . . . 333 14. Geometric Phase in Condensed Matter III: Many-Body Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 14.2 Fractional Quantum Hall Systems . . . . . . . . . . . . . . . . . . . . . . . . 337 14.2.1 Laughlin Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . 337 14.2.2 Fractional Charged Excitations. . . . . . . . . . . . . . . . . . . . . 340 14.2.3 Fractional Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 14.2.4 Degeneracy and Fractional Quantization . . . . . . . . . . . . 344 14.3 Spin-Wave Dynamics in Itinerant Magnets . . . . . . . . . . . . . . . . . 346 14.3.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 14.3.2 Tight-Binding Limit and Beyond . . . . . . . . . . . . . . . . . . . 348 14.3.3 Spin Wave Spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 14.4 Geometric Phase in Doubly-Degenerate Electronic Bands . . . . 353 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 A. An Elementary Introduction to Manifolds and Lie Groups 361 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 A.2 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 A.3 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 Table of Contents XV B. A Brief Review of Point Groups of Molecules with Application to Jahn–Teller Systems . . . . . . . . . . . . . . . . . 407 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

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