Qian Niu

The Geometric Phase in Quantum Systems

1. Introduction.- 2. Quantal Phase Factors for Adiabatic Changes.- 3. Spinning Quantum System in an External Magnetic Field.- 4. Quantal Phases for General Cyclic Evolution.- 5. Fiber Bundles and Gauge Theories.- 6. Mathematical Structure of the Geometric Phase I: The Abelian Phase.- 7. Mathematical Structure of the Geometric Phase II: The Non-Abelian Phase.- 8. A Quantum Physical System in a Quantum Environment - The Gauge Theory of Molecular Physics.- 9. Crossing of Potential Energy Surfaces and the Molecular Aharonov-Bohm Effect.- 10. Experimental Detection of Geometric Phases I: Quantum Systems in Classical Environments.- 11. Experimental Detection of Geometric Phases II: Quantum Systems in Quantum Environments.- 12. Geometric Phase in Condensed Matter I: Bloch Bands.- 13. Geometric Phase in Condensed Matter II: The Quantum Hall Effect.- 14. Geometric Phase in Condensed Matter III: Many-Body Systems.- A. An Elementary Introduction to Manifolds and Lie Groups.- B. A Brief Review of Point Groups of Molecules with Application to Jahn-Teller Systems.- References.

An Elementary Introduction to Manifolds and Lie Groups

The following is an elementary introduction to some basic concepts in modern differential geometry. The aim is to provide the reader with a clear understanding of the key ideas and to motivate the concepts rather than to promote the rigor and generality pursued in mathematical texts. Specifically, our main objective is to arrive at a comprehensive description of smooth manifolds, Lie groups, and their most basic properties.

Geometric Phase in Condensed Matter I: Bloch Bands

The Schrodinger equation in a spatially periodic system has eigenstates in the form of Bloch waves, plane waves with amplitudes modulated periodically in space. We will consider in this chapter geometric phases associated with the Bloch waves. Crystalline solids are naturally occuring periodic systems in which an electron sees a periodic potential due to the nuclei and other electrons. Periodic systems also include artificial structures such as superlattices, networks of microwires, and arrays of quantum dots or antidots. A perfect periodic potential can also be produced on cold atoms by an interference pattern of laser beams of suitable frequency. Throughout this chapter, we will treat the periodic potentials as given and focus on the properties of particles in the Bloch states under external forces.

Spinning Quantum System in an External Magnetic Field

In this chapter we will discuss in detail a quantum particle with magnetic moment m = µ B g J in an external magnetic field B(t) = (math) whose direction (math) is changing periodically. In particular we will consider the case in which the direction of the magnetic field precesses around a fixed axis which we take as the 3-axis (z-axis) of our (laboratory) coordinate frame in space (ℝ3). If the direction rotates slowly (“adiabatically”) this system provides an application of the general ideas developed in Chap. 2. The Schrodinger equation for a magnetic moment in a precessing magnetic field has been solved exactly in [216]. Therefore we need not restrict ourselves to the adiabatic approximation and will obtain the non-adiabatic geometric phase. The latter is also known as the Aharonov-Anandan phase [7, 9] for their pioneering work. With the help of this example we will then introduce in Sect. 3.5 the non-adiabatic geometric phase for a general cyclic evolution.

Geometric Phase in Condensed Matter II: The Quantum Hall Effect

In this chapter we review the application of the geometric phase in the context of the quantum Hall effect, a striking quantum phenomenon involving two-dimensional electrons in strong magnetic fields and low temperatures. We will focus our attention on the case of the integer quantum Hall effect in this chapter, and show how the quantization of the Hall conductance observed in experiments may be explained as a topological invariant represented by the (first) Chern number.

A Brief Review of Point Groups of Molecules with Application to Jahn—Teller Systems

In this appendix we present a brief discussion of the point groups of molecules and their applications to Jahn-Teller systems. Our aim is to provide the reader with the minimum background necessary for following the related discussions in the text. Therefore we shall often sacrifice rigor for brevity.

Quantal Phase Factors for Adiabatic Changes

In the quantum mechanical description of a physical system, one has a finite or infinite dimensional Hilbert space of state vectors and a set of linear operators acting on these state vectors. The linear operators are interpreted as the observables. If a quantum system is not isolated from its environment, the observables are described by operators that depend on a set of parameters, R = (R 1 , R 2, ... ). Each value of R characterizes a particular configuration of the environment. In particular, a changing environment is described by time-dependent parameters, R = R(t).

Experimental Detection of Geometric Phases I: Quantum Systems in Classical Environments

In the previous chapters of this book we have shown that quantum states undergoing either adiabatic or exact time evolution can acquire phase factors that reflect the geometry of the spaces in which they evolve. Knowledge of these phases can be helpful in understanding the way in which a system is embedded in an environment.

Experimental Detection of Geometric Phases II: Quantum Systems in Quantum Environments

In the previous chapter we described a variety of experiments that explore geometric phases under controlled conditions, in other words, when the (classical) environment can be manipulated by the experimenter. In this case the exact solution and adiabatic solutions are both known, thus, the experimental challenge is whether we can control the quantum state so that the geometric phase effect can be observable.

Fiber Bundles and Gauge Theories

In the preceding chapter we introduced the concept of the geometric phase and showed its natural emergence from the basic principles of quantum mechanics. We emphasized the geometric nature of the phase and briefly outlined a holonomy interpretation of the geometric phase using the Aharonov-Anandan principal fiber bundle. This involved several differential geometric concepts such as those of a fiber bundle, a horizontal lift, a connection, a parallel transport and a holonomy.