Dissipative time crystals with long-range Lindbladians

TL;DR

This paper demonstrates the existence of dissipative time crystals in long-range spin systems without the need for spin symmetry, revealing a rich phase diagram and validating mean-field approximations for these models.

Contribution

It introduces a new class of dissipative time crystals using power-law decaying Lindbladians, relaxing previous symmetry constraints and analyzing their phase behavior.

Findings

Time-translation symmetry breaking persists without spin symmetry.

Rich phase diagram including various phase transitions.

Mean-field approximation accurately describes the system in the thermodynamic limit.

Abstract

Dissipative time crystals can appear in spin systems, when the $Z_2$ symmetry of the Hamiltonian is broken by the environment, and the square of total spin operator $S^2$ is conserved. In this manuscript, we relax the latter condition and show that time-translation-symmetry breaking collective oscillations persist, in the thermodynamic limit, even in the absence of spin symmetry. We engineer an \textit{ad hoc} Lindbladian using power-law decaying spin operators and show that time-translation symmetry breaking appears when the decay exponent obeys $0<\eta\leq 1$. This model shows a surprisingly rich phase diagram, including the time-crystal phase as well as first-order, second-order, and continuous transitions of the fixed points. We study the phase diagram and the magnetization dynamics in the mean-field approximation. We prove that this approximation is quantitatively accurate, when $0<\eta\leq1$ and the thermodynamic limit is taken, because the system does not develop sizable quantum fluctuations, if the Gaussian approximation is considered.