Space-Time Crystal and Space-Time Group

Crystal structures and the Bloch theorem play a fundamental role in condensed matter physics. We extend the static crystal to the dynamic space-time crystal characterized by the general intertwined spacetime periodicities in D + 1 dimensions, which include both the static crystal and the Floquet crystal as special cases. A new group structure dubbed a space-time group is constructed to describe the discrete symmetries of a space-time crystal. Compared to space and magnetic groups, the space-time group is augmented by time-screw rotations and time-glide reflections involving fractional translations along the time direction. A complete classification of the 13 space-time groups in one-plus-one dimensions (1 + 1D) is performed. The Kramers-type degeneracy can arise from the glide time-reversal symmetry without the half-integer spinor structure, which constrains the winding number patterns of spectral dispersions. In 2 thorn 1D, nonsymmorphic space-time symmetries enforce spectral degeneracies, leading to protected Floquet semimetal states. We provide a general framework for further studying topological properties of the (D + 1)-dimensional space-time crystal.