Topological space-time crystal

We introduce a new class of out-of-equilibrium noninteracting topological phases, the topological space-time crystals. These are time-dependent quantum systems which do not have discrete spatial translation symmetries, but instead are invariant under discrete space-time translations. Similar to the Floquet-Bloch systems, the space-time crystals can be described by a frequency-domain-enlarged Hamiltonian, which is used to classify topologically distinct space-time crystals. We show that these space-time crystals can be engineered from conventional crystals with an additional time-dependent drive that behaves like a traveling wave moving across the crystal. Interestingly, we are able to construct 1D and 2D examples of topological space-time crystals based on tight-binding models which involve only one orbital, in contrast to the two-orbital minimal models for any previous discovered static or Floquet topological phases with crystalline structures.