# Space-time crystal and space-time group

## TL;DR

This paper introduces the concept of space-time crystals and their symmetry groups, extending static crystal theory to dynamic systems with intertwined space-time periodicities, and classifies their symmetry groups in 1+1D and 2+1D.

## Contribution

It develops the space-time group framework for describing symmetries in dynamic crystals, including new symmetry operations like time-screw rotations and time-glide reflections.

## Key findings

- Classification of 13 space-time groups in 1+1D.
- Kramers-type degeneracy from glide time-reversal symmetry.
- Spectral degeneracies enforced by non-symmorphic space-time symmetries in 2+1D.

## 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.