Geometric phases and hidden gauge symmetry

TL;DR

This paper reviews the second quantized approach to geometric phases, revealing a hidden gauge symmetry that unifies the understanding of adiabatic and non-adiabatic phases and analyzing the transition between them.

Contribution

It introduces a hidden local gauge symmetry in second quantization that unifies geometric phases and analyzes the adiabatic to non-adiabatic transition quantitatively.

Findings

The hidden gauge symmetry controls all known geometric phases.

The topology of Berry's phase is trivial and fragile against non-adiabatic deformation.

The formulation does not rely on the projective Hilbert space concept.

Abstract

The second quantized approach to geometric phases is reviewed. The second quantization generally induces a hidden local (time-dependent) gauge symmetry. This gauge symmetry defines the parallel transport and holonomy, and thus it controls all the known geometric phases, either adiabatic or non-adiabatic, in a unified manner. The transitional region from the adiabatic to non-adiabatic phases is thus analyzed in a quantitative way. It is then shown that the topology of the adiabatic Berry's phase is trivial in a precise sense and also the adiabatic phase is rather fragile against the non-adiabatic deformation. In this formulation, the notion such as the projective Hilbert space does not appear.