Piecewise Adiabatic Following in Non-Hermitian Cycling

TL;DR

This paper explores the phenomenon of piecewise adiabatic following in non-Hermitian systems under periodic driving, revealing critical behavior, phase boundaries, and a new perspective on the Aharonov-Anandan phase in non-unitary dynamics.

Contribution

It introduces the concept of piecewise adiabatic following interrupted by eigenstate hopping, explains it via the Stokes phenomenon, and characterizes the AA phase in non-Hermitian cyclic dynamics.

Findings

Piecewise adiabatic following can occur in non-Hermitian systems.

The phase boundary for this behavior is unrelated to exceptional points.

The AA phase can be real and reduces to Berry phase in certain limits.

Abstract

The time evolution of periodically driven non-Hermitian systems is in general non-unitary but can be stable. It is hence of considerable interest to examine the adiabatic following dynamics in periodically driven non-Hermitian systems. We show in this work the possibility of piecewise adiabatic following interrupted by hopping between instantaneous system eigenstates. This phenomenon is first observed in a computational model and then theoretically explained, using an exactly solvable model, in terms of the Stokes phenomenon. In the latter case, the piecewise adiabatic following is shown to be a genuine critical behavior and the precise phase boundary in the parameter space is located. Interestingly, the critical boundary for piecewise adiabatic following is found to be unrelated to the domain for exceptional points. To characterize the adiabatic following dynamics, we also advocate a simple definition of the Aharonov-Anandan (AA) phase for non-unitary cyclic dynamics, which always yields real AA phases. In the slow driving limit, the AA phase reduces to the Berry phase if adiabatic following persists throughout the driving without hopping, but oscillates violently and does not approach any limit in cases of piecewise adiabatic following. This work exposes the rich features of non-unitary dynamics in cases of slow cycling and should stimulate future applications of non-unitary dynamics.