The Geometric Phase in Quantum Systems: Foundations, Mathematical Concepts, and Applications

Table of Contents......Page 9
1. Introduction......Page 15
2.1 Introduction......Page 19
2.2 Adiabatic Approximation......Page 24
2.3 Berry\'s Adiabatic Phase......Page 28
2.4 Topological Phases and the Aharonov-Bohm Effect......Page 36
Problems......Page 43
3.2 The Parameterization of the Basis Vectors......Page 45
3.3 Mead-Berry Connection and Berry Phase for Adiabatic Evolutions - Magnetic Monopole Potentials......Page 50
3.4 The Exact Solution of the Schrodinger Equation......Page 56
3.5 Dynamical and Geometrical Phase Factors for Non-Adiabatic Evolution......Page 62
Problems......Page 66
4.2 Aharonov-Anandan Phase......Page 67
4.3 Exact Cyclic Evolution for Periodic Hamiltonians......Page 74
Problems......Page 78
5.2 From Quantal Phases to Fiber Bundles......Page 79
5.3 An Elementary Introduction to Fiber Bundles......Page 81
5.4 Geometry of Principal Bundles and the Concept of Holonomy......Page 90
5.5 Gauge Theories......Page 101
5.6 Mathematical Foundations of Gauge Theories and Geometry of Vector Bundles......Page 109
Problems......Page 116
6.2 Holonomy Interpretations of the Geometric Phase......Page 121
6.3 Classification of U(1) Principal Bundles and the Relation Between the Berry-Simon and Aharonov-Anandan Interpretations of the Adiabatic Phase......Page 127
6.4 Holonomy Interpretation of the Non-Adiabatic Phase Using a Bundle over the Parameter Space......Page 132
6.5 Spinning Quantum System and Topological Aspects of the Geometric Phase......Page 137
Problems......Page 140
7.2 The Non-Abelian Adiabatic Phase......Page 143
7.3 The Non-Abelian Geometric Phase......Page 150
7.4 Holonomy Interpretations of the Non-Abelian Phase......Page 153
7.5 Classification of U(N) Principal Bundles and the Relation Between the Berry-Simon and Aharonov-Anandan Interpretations of Non-Abelian Phase......Page 155
Problems......Page 159
8.1 Introduction......Page 161
8.2 The Hamiltonian of Molecular Systems......Page 162
8.3 The Born-Oppenheimer Method......Page 171
8.4 The Gauge Theory of Molecular Physics......Page 180
8.5 The Electronic States of Diatomic Molecule......Page 188
8.6 The Monopole of the Diatomic Molecule......Page 190
Problems......Page 205
9.1 Introduction......Page 209
9.2 Crossing of Potential Energy Surfaces......Page 210
9.3 Conical Intersections and Sign-Change of Wave Functions......Page 212
9.4 Conical Intersections in Jahn-Teller Systems......Page 223
9.5 Symmetry of the Ground State in Jahn-Teller Systems......Page 227
9.6 Geometric Phase in Two Kramers Doublet Systems......Page 233
9.7 Adiabatic-Diabatic Transformation......Page 236
10.2.1 Spins in Magnetic Fields: The Laboratory Frame......Page 239
10.2.2 Spins in Magnetic Fields: The Rotating Frame......Page 245
10.2.3 Adiabatic Reorientation in Zero Field......Page 251
10.3 Observation of the Aharonov-Anandan Phase Through the Cyclic Evolution of Quantum States......Page 262
Problems......Page 266
11.1 Introduction......Page 269
11.2 Internal Rotors Coupled to External Rotors......Page 270
11.3 Electronic-Rotational Coupling......Page 273
11.4 Vibronic Problems in Jahn-Teller Systems......Page 274
11.4.1 Transition Metal Ions in Crystals......Page 275
11.4.2 Hydrocarbon Radicals......Page 278
11.4.3 Alkali Metal Trimers......Page 279
11.5 The Geometric Phase in Chemical Reactions......Page 284
12.1 Introduction......Page 291
12.2.1 One-Dimensional Case......Page 292
12.2.2 Three-Dimensional Case......Page 294
12.2.3 Band Structure Calculation......Page 295
12.3.1 Equations of Motion......Page 297
12.3.2 Symmetry Analysis......Page 299
12.3.3 Derivation of the Semiclassical Formulas......Page 300
12.3.4 Time-Dependent Bands......Page 301
12.4.1 Uniform DC Electric Field......Page 302
12.4.2 Uniform and Constant Magnetic Field......Page 303
12.4.4 Transport......Page 304
12.5.1 General Properties......Page 306
12.5.2 Localization Properties......Page 307
12.6.1 Quantized Adiabatic Particle Transport......Page 309
12.6.2 Polarization......Page 311
Problems......Page 313
13.1 Introduction......Page 315
13.2.2 The Quantum Hall Effect......Page 316
13.2.3 The Ideal Model......Page 318
13.2.4 Corrections to Quantization......Page 319
13.3.1 Single-Band Approximation in a Weak Magnetic Field......Page 321
13.3.2 Harper\'s Equation and Hofstadter\'s Butterfly......Page 323
13.3.3 Magnetic Translations......Page 325
13.3.4 Quantized Hall Conductivity......Page 328
13.3.5 Evaluation of the Chern Number......Page 330
13.3.6 Semiclassical Dynamics and Quantization......Page 332
13.3.7 Structure of Magnetic Bands and Hyperorbit Levels......Page 335
13.3.8 Hierarchical Structure of the Butterfly......Page 339
13.3.9 Quantization of Hyperorbits and Rule of Band Splitting......Page 341
13.4.1 Spectrum and Wave Functions......Page 343
13.4.2 Perturbation and Scattering Theory......Page 345
13.4.3 Laughlin\'s Gauge Argument......Page 346
13.4.4 Hall Conductance as a Topological Invariant......Page 347
14.2.1 Laughlin Wave Function......Page 351
14.2.2 Fractional Charged Excitations......Page 354
14.2.3 Fractional Statistics......Page 355
14.2.4 Degeneracy and Fractional Quantization......Page 358
14.3.1 General Formulation......Page 360
14.3.2 Tight-Binding Limit and Beyond......Page 362
14.3.3 Spin Wave Spectrum......Page 364
14.4 Geometric Phase in Doubly-Degenerate Electronic Bands......Page 367
Problem......Page 373
A.1 Introduction......Page 375
A.2 Differentiate Manifolds......Page 385
A.3 Lie Groups......Page 402
B. A Brief Review of Point Groups of Molecules with Application to Jahn-Teller Systems......Page 421
References......Page 443
Index......Page 451