Exact solution of a boundary time-crystal phase transition: time-translation symmetry breaking and non-Markovian dynamics of correlations

TL;DR

This paper provides an exact analytical solution for a boundary time-crystal phase transition in open quantum systems, revealing non-Markovian dynamics, critical fluctuations, and the intrinsic many-body nature of the phase.

Contribution

It offers the first complete analytical characterization of a boundary time-crystal phase transition, including explicit expressions for the order parameter and quantum fluctuation dynamics.

Findings

Boundary time-crystals exhibit non-Markovian quantum fluctuations.

The phase transition is characterized by power-law divergence of fluctuations.

Time-crystalline phase is a genuine many-body phase with unique correlation properties.

Abstract

The breaking of the continuous time-translation symmetry manifests, in Markovian open quantum systems, through the emergence of non-stationary dynamical phases. Systems that display nonequilibrium transitions into these phases are referred to as time-crystals, and they can be realized, for example, in many-body systems governed by collective dissipation and long-ranged interactions. Here, we provide a complete analytical characterization of a boundary time-crystal phase transition. This involves exact expressions for the order parameter and for the dynamics of quantum fluctuations, which, in the time-crystalline phase, remains asymptotically non-Markovian as a consequence of the time-translation symmetry breaking. We demonstrate that boundary time-crystals are intrinsically critical phases, where fluctuations exhibit a power-law divergence with time. Our results show that a dissipative time-crystal phase is far more than merely a classical non-linear and non-stationary (limit cycle) dynamics of a macroscopic order parameter. It is rather a genuine many-body phase where the properties of correlations distinctly differs from that of stationary ones.