Real energies and Berry phases in all PT-regimes in time-dependent non-Hermitian theories

TL;DR

This paper shows that in time-dependent non-Hermitian quantum systems, the existence of a Hermitian intertwining operator ensures real energies across all PT-regimes and introduces a modified adiabatic approximation that guarantees real Berry phases.

Contribution

It introduces a Hermitian intertwining operator framework for real energies in all PT-regimes and proposes a modified adiabatic approximation ensuring real Berry phases in time-dependent non-Hermitian systems.

Findings

Real energies are guaranteed across all PT-regimes using the intertwining operator.

A modified adiabatic approximation yields always real Berry phases.

Validated with explicit examples of a non-Hermitian spin model.

Abstract

We demonstrate that the existence of a Hermitian time-dependent intertwining operator that maps the non-Hermitian time-dependent energy operator to its Hermitian conjugate and its right to its left eigenstates guarantees the reality of the instantaneous energies. This property holds throughout all three $\cal{PT}$-regimes, in the time-independent scenario referred to as the $\cal{PT}$-symmetric regime, the exceptional point and the spontaneously broken $\cal{PT}$-regime. We also propose a modified adiabatic approximation consisting of an expansion of the wavefunctions in terms the instantaneous eigenstates of the energy operator, instead of the usually used eigenfunctions of the Hamiltonian. We show that this proposal always leads to real Berry phases. We illustrate the working of our general proposals with two explicit examples for a time-dependent non-Hermitian spin model.