Non-Hermitian quantum systems and their geometric phases

TL;DR

This paper explores the theoretical framework of non-Hermitian quantum systems, focusing on their geometric phases, conditions for adiabatic evolution, and illustrating with PT-symmetric models, revealing complex geometric phases linked to wavefunction amplitude.

Contribution

It provides a detailed analysis of the geometric phases in non-Hermitian quantum systems, emphasizing the conditions for real phases and their relation to system symmetries.

Findings

Adiabatic evolution requires real eigen-energies.

Geometric phases in non-Hermitian systems are generally complex.

Illustrations include PT-symmetric Dirac and bosonic models.

Abstract

We discuss the basic theoretical framework for non-Hermitian quantum systems with particular emphasis on the diagonalizability of non-Hermitian Hamiltonians and their $GL(1,\mathbb{C})$ gauge freedom, which are relevant to the adiabatic evolution of non-Hermitian quantum systems. We find that the adiabatic evolution is possible only when the eigen-energies are real. The accompanying geometric phase is found to be generally complex and associated with not only the phase of a wavefunction but also its amplitude. The condition for the real geometric phase is laid out. Our results are illustrated with two examples of non-Hermitian $\mathcal{PT}$ symmetric systems, the two-dimensional non-Hermitian Dirac fermion model and bosonic Bogoliubov quasi-particles.