Defining Time Crystals via Representation Theory

TL;DR

This paper proposes a new, representation theory-based definition of time crystals, clarifying their symmetry-breaking properties and distinguishing between different types of time translation symmetry breaking.

Contribution

It introduces a formal definition of time crystals using traditional symmetry and order parameter concepts, providing a clearer framework for understanding their properties.

Findings

Systems with time-independent Hamiltonians should not exhibit TTSB

Floquet time crystals can be viewed as breaking a global internal symmetry

The new definition clarifies the symmetry-breaking nature of time crystals

Abstract

Time crystals are proposed states of matter which spontaneously break time translation symmetry. There is no settled definition of such states. We offer a new definition which follows the traditional recipe for Wigner symmetries and order parameters. Supplementing our definition with a few plausible assumptions we find that a) systems with time independent Hamiltonians should not exhibit TTSB while b) the recently studied $\pi$ spin glass/Floquet time crystal can be viewed as breaking a global internal symmetry and as breaking time translation symmetry as befits its two names.