Anomalous Time Crystals

We study the classification problem of whether two Floquet many-body localized (MBL) time crystals can be deformed into each other without delocalization. The Floquet cycle can be decomposed into a local part, which is the exponential of a Floquet Hamiltonian, times a global ℤ_N symmetry action. In one dimension, we show that the anomaly class H³(ℤ_N,U(1)) of the symmetry is an invariant of the time crystal. In particular, there exist time crystals whose local integrals of motion (LIOMs) are isomorphic but cannot be deformed into each other. An anomalous time crystal, one in a nontrivial class, has boundary observables with a period longer than N times the Floquet period. More generally, we argue that, under some conditions for the LIOMs, the problem of classifying MBL time crystals can be reduced to classifying the conjugacy classes of quantum cellular automata (QCA)-representation of the ℤ_N symmetry.