# Relaxation of dynamically prepared out-of-equilibrium initial states within and beyond linear response theory

## TL;DR

This paper investigates how quantum systems out of equilibrium relax back to equilibrium or nonthermal states depending on the strength of external forces, revealing new behaviors beyond linear response and ETH violations.

## Contribution

It demonstrates that systems violating ETH can still thermalize within linear response but relax to nonthermal states beyond it, supported by theoretical analysis and numerical simulations.

## Key findings

- Weak external forces lead to thermalization despite ETH violation
- Beyond linear response, systems relax to nonthermal long-time states
- Numerical simulations confirm theoretical predictions in quantum spin models

## Abstract

We consider a realistic nonequilibrium protocol, where a quantum system in thermal equilibrium is suddenly subjected to an external force. Due to this force, the system is driven out of equilibrium and the expectation values of certain observables acquire a dependence on time. Eventually, upon switching off the external force, the system unitarily evolves under its own Hamiltonian and, as a consequence, the expectation values of observables equilibrate towards specific constant long-time values. Summarizing our main results, we show that, in systems which violate the eigenstate thermalization hypothesis (ETH), this long-time value exhibits an intriguing dependence on the strength of the external force. Specifically, for weak external forces, i.e., within the linear response regime, we show that expectation values thermalize to their original equilibrium values, despite the ETH being violated. In contrast, for stronger perturbations beyond linear response, the quantum system relaxes to some nonthermal value which depends on the previous nonequilibrium protocol. While we present theoretical arguments which underpin these results, we also numerically demonstrate our findings by studying the real-time dynamics of two low-dimensional quantum spin models.