Second quantized formulation of geometric phases

TL;DR

This paper introduces a second quantized formulation for geometric phases that does not rely on adiabatic approximation, revealing hidden gauge symmetries and simplifying analysis of level crossing phenomena.

Contribution

It provides an exact formula for geometric terms, including off-diagonal ones, and clarifies the relationship between geometric phases, topological properties, and practical approximations.

Findings

Geometric phases become trivial near level crossing when diagonalized.

The formulation clarifies the failure of topological proofs in finite-time scenarios.

Differences between geometric phases and topological objects like Aharonov-Bohm phase are elucidated.

Abstract

The level crossing problem and associated geometric terms are neatly formulated by the second quantized formulation. This formulation exhibits a hidden local gauge symmetry related to the arbitrariness of the phase choice of the complete orthonormal basis set. By using this second quantized formulation, which does not assume adiabatic approximation, a convenient exact formula for the geometric terms including off-diagonal geometric terms is derived. The analysis of geometric phases is then reduced to a simple diagonalization of the Hamiltonian, and it is analyzed both in the operator and path integral formulations. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singularity) for arbitrarily large but finite time interval $T$. The integrability of Schr\"{o}dinger equation and the appearance of the seemingly non-integrable phases are thus consistent. The topological proof of the Longuet-Higgins' phase-change rule, for example, fails in the practical Born-Oppenheimer approximation where a large but finite ratio of two time scales is involved and $T$ is identified with the period of the slower system. The difference and similarity between the geometric phases associated with level crossing and the exact topological object such as the Aharonov-Bohm phase become clear in the present formulation. A crucial difference between the quantum anomaly and the geometric phases is also noted.