# Aperiodically driven integrable systems and their emergent steady states

## TL;DR

This paper investigates how breaking periodic driving with aperiodic noise affects the long-term behavior of integrable quantum systems, revealing new steady states and the conditions under which they form.

## Contribution

It introduces the concept of a geometric generalized Gibbs ensemble and analyzes the impact of different types of aperiodic noise on steady state formation in integrable systems.

## Key findings

- Random noise leads to infinite temperature thermalization.
- Self-similar noise results in a novel steady state, the geometric GGE.
- Large time scales are required for the emergence of these steady states.

## Abstract

Does a closed quantum many-body system that is continually driven with a time-dependent Hamiltonian finally reach a steady state? This question has only recently been answered for driving protocols that are periodic in time, where the long time behavior of the local properties synchronize with the drive and can be described by an appropriate periodic ensemble. Here, we explore the consequences of breaking the time-periodic structure of the drive with additional aperiodic noise in a class of integrable systems. We show that the resulting unitary dynamics leads to new emergent steady states in at least two cases. While any typical realization of random noise causes eventual heating to an infinite temperature ensemble for all local properties in spite of the system being integrable, noise which is self-similar in time leads to an entirely different steady state, which we dub as "geometric generalized Gibbs ensemble", that emerges only after an astronomically large time scale. To understand the approach to steady state, we study the temporal behavior of certain coarse-grained quantities in momentum space that fully determine the reduced density matrix for a subsystem with size much smaller than the total system. Such quantities provide a concise description for any drive protocol in integrable systems that are reducible to a free fermion representation.