# Driven-dissipative Ising model: Dynamical crossover at weak dissipation

## TL;DR

This paper studies a driven-dissipative Ising model revealing a dynamical crossover from relaxational to underdamped critical behavior in the weak dissipation limit, showing this phenomenon persists beyond mean-field approximations.

## Contribution

It identifies a novel dynamical crossover in a driven-dissipative quantum model and demonstrates its robustness beyond mean-field theory using an exact diagrammatic approach.

## Key findings

- Dynamical crossover from relaxational to underdamped critical dynamics.
- Distinct critical exponents $6=1/2$ and $6=1/4$ in different regimes.
- Crossover persists even with short-range perturbations.

## Abstract

Driven quantum systems coupled to an environment typically exhibit effectively thermal behavior with relaxational dynamics near criticality. However, a different qualitative behavior might be expected in the weakly dissipative limit due to the competition between coherent dynamics and weak dissipation. In this work, we investigate a driven-dissipative infinite-range Ising model in the presence of individual atomic dissipation, a model that emerges from the paradigmatic open Dicke model in the large-detuning limit. We show that the system undergoes a dynamical crossover from relaxational dynamics, with a characteristic dynamical exponent $\zeta=1/2$, to underdamped critical dynamics governed by the exponent $\zeta=1/4$ in the weakly dissipative regime; a behavior that is markedly distinct from that of equilibrium. Finally, utilizing an exact diagrammatic representation, we demonstrate that the dynamical crossover to underdamped criticality is not an artifact of the mean-field nature of the model and persists even in the presence of short-range perturbations.