Aimed at graduate physics and chemistry students, this is the first comprechensive monograph covering the concept of the geometric phase in quantum physics from its mathematical foundations to its physical applications and experimental manifestations. It contains all the premises of the adiabatic Berry phase as well as the exact Anandan-Aharonov phase. It discusses quantum systems in a classical time-independent environment (time dependent Hamiltonians) and quantum systems in a changing environment (gauge theory of molecular physics). The mathematical methods used are a combination of differential geometry and the theory of Iinear operators in Hilbert Space. As a result, the monograph demonstrates how non-trivial gauge theories naturally arise and how the consequences can be experimentally observed. Readers benefit by gaining a deep understanding of the long-ignored gauge theoretic effects of quantum methanics and how to measure them.

1. Introduction

2. Quantal Phase Factors for Adiabatic Changes

2.1 Introduction

2.2 Adiabatic Approximation

2.3 Berry's Adiabatic Phase

2.4 Topological Phases and the Aharonov-Bohm Effect

Problems

3. Spinning Quantum System in an External Magnetic Field

3.1 Introduction

3.2 The Parameterization of the Basis Vectors

3.3 Mead-Berry Connection and Berry Phase for Adiabatic Evolutions - Magnetic Monopole Potentials

3.4 The Exact Solution of the SchrSdinger Equation

3.5 Dynamical and Geometrical Phase Factors for Non-Adiabatic Evolution

Problems

4. Quantal Phases for General Cyclic Evolution

4.1 Introduction

4.2 Aharonov-Anandan Phase

4.3 Exact Cyclic Evolution for Periodic Hamiltonians

Problems

5. Fiber Bundles and Gauge Theories

5.1 Introduction

5.2 From Quantal Phases to Fiber Bundles

5.3 An Elementary Introduction to Fiber Bundles

5.4 Geometry of Principal Bundles and the Concept of Holonomy

5.5 Gauge Theories

5.6 Mathematical Foundations of Gauge Theories and Geometry of Vector Bundles

Problems

6. Mathematical Structure of the Geometric Phase I:The Abelian Phase

6.1 Introduction

6.2 Holonomy Interpretations of the Geometric Phase

6.3 Classification of U(1) Principal Bundles and the Relation

    Between the Berry-Simon and Aharonov-Anandan

    Interpretations of the Adiabatic Phase

6.4 Holonomy Interpretation of the Non-Adiabatic Phase

    Using a Bundle over the Parameter Space

6.5 Spinning Quantum System and Topological Aspects of the Geometric Phase

Problems

7. Mathematical Structure of the Geometric Phase II:The Non-Abelian Phase

7.1 Introduction

7.2 The Non-Abelian Adiabatic Phase

7.3 The Non-Abelian Geometric Phase

7.4 Holonomy Interpretations of the Non-Abelian Phase

7.5 Classification of U(N) Principal Bundles and the Relation

    Between the Berry-Simon and Aharonov-Anandan

    Interpretations of Non-Abelian Phase

Problems

8. A Quantum Physical System in a Quantum Environment - The Gauge Theory of Molecular Physics

8.1 Introduction

8.2 The Hamiltonian of Molecular Systems

8.3 The Born-Oppenheimer Method

8.4 The Gauge Theory of Molecular Physics

8.5 The Electronic States of Diatomic Molecule

8.6 The Monopole of the Diatomic Molecule

Problems

9. Crossing of Potential Energy Surfaces

and the Molecular Aharonov-Bohm Effect

9.1 Introduction

9.2 Crossing of Potential Energy Surfaces

9.3 Conical Intersections and Sign-Change of Wave Functions

9.4 Conical Intersections in Jahn-Teller Systems

9.5 Symmetry of the Ground State in Jahn-Teller Systems

9.6 Geometric Phase in Two Kramers Doublet Systems

9.7 Adiabatic-Diabatic Transformation

10. Experimental Detection of Geometric Phases I

Quantum Systems in Classical Environments

10.1 Introduction

10.2 The Spin Berry Phase Controlled by Magnetic Fields .

    10.2.1 Spins in Magnetic Fields: The Laboratory Frame

    10.2.2 Spins in Magnetic Fields: The Rotating Frame

    10.2.3 Adiabatic Reorientation in Zero Field

10.3 Observation of the Aharonov-Anandan Phase

    Through the Cyclic Evolution of Quantum States

Problems

11. Experimental Detection of Geometric Phases II:

Quantum Systems in Quantum Environments

11.1 Introduction

11.2 Internal Rotors Coupled to External Rotors

 11,3 Electronic-Rotational Coupling

 11.4 Vibronic Problems in Jahn-Teller Systems

    11.4.1 Transition Metal Ions in Crystals

    11.4.2 Hydrocarbon Radicals

    11.4.3 Alkali Metal Trimers

 11,5 The Geometric Phase in Chemical Reactions

12. Geometric Phase in Condensed Matter I: Bloch Bands

 12.1 Introduction

 12.2 Bloch Theory

    12.2.1 One-Dimensional Case

    12.2.2 Three-Dimensional Case

    12.2.3 Band Structure Calculation

 12.3 Semiclassical Dynamics

    12.3.1 Equations of Motion

    12.3.2 Symmetry Analysis

    12.3.3 Derivation of the Semiclassical Formulas

    12.3.4 Time-Dependent Bands

 12.4 Applications of Semiclassical Dynamics

    12.4.1 Uniform DC Electric Field

    12,4.2 Uniform and Constant Magnetic Field

    12.4.3 Perpendicular Electric and Magnetic Fields

    12.4.4 Transport

 12.5 Wannier Functions

    12.5.1 General Properties

    12.5.2 Localization Properties

 12.6 Some Issues on Band Insulators

     12.6.1 Quantized Adiabatic Particle Transport

     12.6.2 Polarization

 Problems

13. Geometric Phase in Condensed Matter II:The Quantum Hall Effect

13.1 Introduction

13.2 Basics of the Quantum Hall Effect

    13.2.1 The Halt Effect

    13.2.2 The Quantum Hall Effect

    13.2.3 The Ideal Model

    13.2.4 Corrections to Quantization

13.3 Magnetic Bands in Periodic Potentials

    13.3.1 Single-Band Approximation in a Weak Magnetic Field

    13.3.2 Harper's Equation and Hofstadter's Butterfly

    13.3.3 Magnetic Translations

    13.3.4 Quantized Hall Conductivity

    13.3.5 Evaluation of the Chern Number

    13.3.6 Semiclassical Dynamics and Quantization

    13.3.7 Structure of Magnetic Bands and Hyperorbit Levels

    13.3.8 Hierarchical Structure of the Butterfly

    13.3.9 Quantization of Hyperorbits and Rule of Band Splitting

13.4 Quantization of Hall Conductance in Disordered Systems

    13.4.1 Spectrum and Wave Functions

    13.4.2 Perturbation and Scattering Theory

    13.4.3 Laughlin's Gauge Argument

    13.4.4 Hall Conductance as a Topological Invariant

14. Geometric Phase in Condensed Matter III:Many-Body Systems

14.1 Introduction

14.2 Fractional Quantum Hall Systems

    14.2.1 Laughlin Wave Function

    14.2.2 Fractional Charged Excitations

    14.2.3 Fractional Statistics

    14.2.4 Degeneracy and Fractional Quantization

14.3 Spin-Wave Dynamics in Itinerant Magnets

    14.3.1 General Formulation

    14.3.2 Tight-Binding Limit and Beyond

    14.3.3 Spin Wave Spectrum

14.4 Geometric Phase in Doubly-Degenerate Electronic Bands

Problem

A. An Elementary Introduction to Manifolds and Lie Groups

A.1 Introduction

A.2 Differentiable Manifolds

A.3 Lie Groups

B. A Brief Review of Point Groups of Molecules with Application to Jahn-Teller Systems

References

Index