Proof of the absence of long-range temporal orders in Gibbs states

TL;DR

This paper provides a complete proof that Gibbs states and similar stationary states do not exhibit spontaneous breaking of time translation symmetry, thus ruling out the existence of quantum time crystals in such states.

Contribution

It offers a comprehensive proof that long-range temporal order cannot occur in Gibbs states and extends the argument to more general stationary states.

Findings

No temporal long-range order in Gibbs states.

Time translation symmetry cannot be spontaneously broken in these states.

Extension of the no-time-crystal result to broader stationary states.

Abstract

We address the question whether time translation symmetry can be spontaneously broken in a quantum many-body system. One way of detecting such a symmetry breaking is to examine the time-dependence of a correlation function. If the large-distance behavior of the correlation function exhibits a nontrivial time-dependence in the thermodynamic limit, the system would develop a temporal long-range order, realizing a time crystal. In an earlier publication, we sketched a proof for the absence of such time dependence in the thermal equilibrium described by the Gibbs state [H. Watanabe and M. Oshikawa, Phys. Rev. Lett. 114, 251603 (2015)]. Here we present a complete proof and extend the argument to a more general class of stationary states than the Gibbs states.