Raffaele Resta
INFM—Dipartimento di Fisica Teorica, Universit`a di Trieste, Strada Costiera 11, I-34014 Trieste,
Italy
Received 28 July 1999
Abstract. Since the appearance of Berry’s seminal paper in 1984, geometric phases have been discovered in virtually all fields of physics. Here we address molecules and solids, and we limit our scope to the Berry’s phases of the many-electron wavefunction. Many advances have occurred in very recent years relating to the theory of such phases and their observable consequences. After discussing the basic features of Berry’s phases in a generic quantum system, we specialize to selected examples taken from molecular physics and condensed matter physics; in each of these cases, a Berry’s phase of the electronic wavefunction leads to measurable effects.
Contents
1. Introduction
2. Fundamentals
2.1. The discrete (Pancharatnam’s) geometric phase
2.2. Berry’s geometric phase
2.3. Connection and curvature
2.4. The paradigm: the Aharonov–Bohm effect
3. Features of the geometric phase
3.1. Parallel transport
3.2. Computing a Berry’s phase
3.3. The open-path geometric phase
3.4. The single-point Berry’s phase
4. The electronic Berry’s phase
4.1. Wavefunctions and density matrices
4.2. Independent electrons
4.3. Bloch orbitals
4.4. Zak’s phase
5. Manifestations of Berry’s phase
5.1. The molecular Aharonov–Bohm effect
5.2. Adiabatic approximation in a magnetic field
5.3. Semiclassical electron dynamics in crystals
5.4. Bloch oscillations and the Wannier–Stark ladder
5.5. Spin-wave dynamics in crystals
6. Macroscopic polarization
6.1. The problem
6.2. Polarization as a Berry’s phase
6.3. A lone electron in a periodic box
6.4. The position operator in extended systems
6.5. A crystalline system of independent electrons
6.6. King-Smith and Vanderbilt’s formula
6.7. Non-crystalline systems
6.8. Correlated electrons and topological phase transitions
7. Conclusions
J. Phys.: Condens. Matter 12 (2000) R107–R143. Printed in the UK
0953-8984_12_9_201.pdf
