Dissipative time crystals originating from parity-time symmetry

TL;DR

This paper demonstrates that dissipative time crystals, specifically boundary time crystals, emerge from parity-time symmetry in collective spin systems with Lindblad dynamics, linking their existence to $ ext{PT}$ symmetry conditions.

Contribution

It proves that boundary time crystals exist only when the stationary state is $ ext{PT}$ symmetric and develops a perturbation theory explaining their emergence under balanced gain and loss.

Findings

BTCs satisfy $ ext{PT}$ symmetry in the large-spin limit

BTCs appear when gain and loss are balanced

Numerical confirmation for multiple BTC models

Abstract

This study aims to provide evidence regarding the emergence of a class of dissipative time crystals when $\mathcal{PT}$ symmetry of the systems is restored in collective spin systems with Lindblad dynamics. First, we show that a standard model of boundary time crystals (BTCs) satisfies the Liouvillian $\mathcal{PT}$ symmetry, and prove that BTC exists only when the stationary state is $\mathcal{PT}$ symmetric in the large-spin limit. Also, a similar statement is confirmed numerically for another BTC model. In addition, the mechanism of the appearance of BTCs is discussed through the development of a perturbation theory for a class of the one-spin models under weak dissipations. Consequently, we show that BTCs appear in the first-order correction when the total gain and loss are balanced. These results strongly suggest that BTCs are time crystals originating from $\mathcal{PT}$ symmetry.