Defining time crystals via representation theory

Abstract

Time crystals are proposed states of matter that spontaneously break time translation symmetry. Despite much recent interest, there is no settled definition for such states and existing definitions only tangentially refer to the well-established foundations of spontaneous symmetry breaking. We offer a definition of time crystals, which treats time translation much like any other symmetry and follows the traditional recipe for defining Wigner symmetries and order parameters. Using our definition, we find: (i) systems with time independent Hamiltonians should not exhibit time translation symmetry breaking, and (ii) the many-body localized $\ensuremath{\pi}$ spin glass/Floquet time crystal can be viewed as breaking both a global internal symmetry and the time translation symmetry, as befits the two aspects of its name.