Geometric phase for non-Hermitian Hamiltonian evolution as anholonomy of a parallel transport along a curve

TL;DR

This paper introduces a novel interpretation of geometric phase in non-Hermitian Hamiltonian evolution as an anholonomy in Minkowski space, linking it to stochastic process responses and parallel transport.

Contribution

It provides a new geometric interpretation of the phase in non-Hermitian systems and connects it to stochastic process behavior under parameter changes.

Findings

Geometric phase relates to anholonomy in Minkowski space.

The phase influences stochastic system responses.

Derived implications for periodic parameter variations.

Abstract

We develop a new interpretation of the geometric phase in evolution with a non-Hermitian real value Hamiltonian by relating it to the angle developed during the parallel transport along a closed curve by a unit vector triad in the 3D-Minkovsky space. We also show that this geometric phase is responsible for the anholonomy effects in stochastic processes considered in [N. A. Sinitsyn and I. Nemenman, EPL {\bf 77}, 58001 (2007)], and use it to derive the stochastic system response to periodic parameter variations.