Geometric phase for non-Hermitian Hamiltonians and its holonomy interpretation

Abstract

For an arbitrary possibly non-Hermitian matrix Hamiltonian H that might involve exceptional points, we construct an appropriate parameter space m and line 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.