Topological properties of geometric phases

TL;DR

This paper uses second quantization to analyze geometric phases, revealing that their topological nature can become trivial in practical finite-time scenarios, challenging traditional interpretations.

Contribution

It introduces a second quantization framework to study geometric phases, showing their topological properties depend on the time scale and approximation used.

Findings

Geometric phases can be trivial in finite-time approximations.

Topological proofs like Longuet-Higgins' rule may fail in practical scenarios.

Diagonalization in different limits yields different geometric phase behaviors.

Abstract

The level crossing problem and associated geometric terms are neatly formulated by using the second quantization technique both in the operator and path integral formulations. The analysis of geometric phases is then reduced to the familiar diagonalization of the Hamiltonian. If one diagonalizes the Hamiltonian in one specific limit, one recovers the conventional formula for geometric phases. On the other hand, 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 topological proof of the Longuet-Higgins' phase-change rule, for example, thus 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.