Adiabatic Perturbation Theory and Geometry of Periodically-Driven Systems

TL;DR

This paper reviews adiabatic perturbation theory in Floquet systems, highlighting non-adiabatic corrections, their geometric implications, and the conditions under which the theory accurately describes driven many-body systems.

Contribution

It provides a systematic analysis of non-adiabatic corrections in Floquet systems and explores the validity of Floquet Adiabatic Perturbation Theory across various models.

Findings

Non-adiabatic corrections relate to Berry curvature and Chern numbers.

FAPT breaks down at ultra slow ramp rates due to avoided crossings.

Large ramp rate windows exist where FAPT remains valid.

Abstract

We give a systematic review of the adiabatic theorem and the leading non-adiabatic corrections in periodically-driven (Floquet) systems. These corrections have a two-fold origin: (i) conventional ones originating from the gradually changing Floquet Hamiltonian and (ii) corrections originating from changing the micro-motion operator. These corrections conspire to give a Hall-type linear response for non-stroboscopic (time-averaged) observables allowing one to measure the Berry curvature and the Chern number related to the Floquet Hamiltonian, thus extending these concepts to periodically-driven many-body systems. The non-zero Floquet Chern number allows one to realize a Thouless energy pump, where one can adiabatically add energy to the system in discrete units of the driving frequency. We discuss the validity of Floquet Adiabatic Perturbation Theory (FAPT) using five different models covering linear and non-linear few and many-particle systems. We argue that in interacting systems, even in the stable high-frequency regimes, FAPT breaks down at ultra slow ramp rates due to avoided crossings of photon resonances, not captured by the inverse-frequency expansion, leading to a counter-intuitive stronger heating at slower ramp rates. Nevertheless, large windows in the ramp rate are shown to exist for which the physics of interacting driven systems is well captured by FAPT.