Anomalous Time Crystals in Higher Dimensions

By regarding the periodicity of Floquet systems as a symmetry, we gain insights into possible non-equilibrium phases. For instance, discrete time crystals can be identified as spontaneously breaking this symmetry. However, the residual action of the broken time translation symmetry - roughly given by the evolution under one Floquet cycle - is very rarely a product of single-site unitaries, as most familiar physical symmetries are. In fact, it may possess a variety of obstructions which prevent this action from being deformed to an on-site action, which thus serve as invariants for distinct out-of-equilibrium phases of matter. Regarding the conjugacy class of the residual action as a representation of a finite group G in the group of locality-preserving unitaries - called quantum cellular automata (QCA) - we explain how this conjugacy class serves as an invariant for time crystalline phases. Further, we identify a family of obstructions associated to these phases which generalize the 't Hooft anomaly of QFT, and explore the implications for these obstructions on many-body localized time crystal phases.