Geometric Phase for Non-Hermitian Hamiltonians and Its Holonomy Interpretation

TL;DR

This paper develops a geometric framework for non-Hermitian Hamiltonians, showing that their adiabatic phases are geometric and not topological, even around exceptional points, with detailed analysis for 2x2 matrices.

Contribution

It constructs a parameter space and line bundle framework for non-Hermitian Hamiltonians, clarifying the nature of geometric phases near exceptional points.

Findings

Geometric phases are generally geometric, not topological.

Constructed a parameter space and line bundle for non-Hermitian Hamiltonians.

Analyzed 2x2 Hamiltonians to illustrate the concepts.

Abstract

For an arbitrary possibly non-Hermitian matrix Hamiltonian H, that might involve exceptional points, we construct an appropriate parameter space M and the lines bundle L^n over M such that the adiabatic geometric phases associated with the eigenstates of the initial Hamiltonian coincide with the holonomies of L^n. We examine the case of 2 x 2 matrix Hamiltonians in detail and show that, contrary to claims made in some recent publications, geometric phases arising from encircling exceptional points are generally geometrical and not topological in nature.