Title: Space-time crystal and space-time group,
Abstract: 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 space-time periodicities in
$D+1$ dimensions, which include both the static crystal and the Floquet crystal
as special cases. A new group structure dubbed "space-time" group is
constructed to describe the discrete symmetries of space-time crystal. Compared
to space and magnetic groups, 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
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+1D,
non-symmorphic space-time symmetries enforce spectral degeneracies, leading to
protected Floquet semi-metal states. Our work provides a general framework for
further studying topological properties of the $D+1$ dimensional space-time
crystal.