Two-Dimensional Space-Time Groups: Classification and Applications

TL;DR

This paper classifies 2+1D space-time groups using group cohomology, revealing new symmetries and phenomena in spatiotemporal materials, including non-symmorphic crystals and unique optical responses.

Contribution

It provides a complete classification of 2+1D space-time groups and uncovers novel physical phenomena arising from these symmetries.

Findings

Identified all 275 space-time crystals, including 203 non-symmorphic ones.

Discovered a chirality-selective response rule linked to space-time symmetry.

Predicted a 'horizontal cone' structure in space-time metamaterials.

Abstract

The concept of space group has long served as the fundamental framework to describe the physical properties of crystalline materials, from electronic bands to photonic dispersions. The recent progress of spatiotemporal control, such as laser-driven lattices, dynamic photonic and phononic crystals, and dynamic optical lattices, necessitates the study of a new framework, space-time group, beyond that based on the Floquet theorem. Space-time group includes novel intertwined non-symmorphic spatial-temporal symmetries such as time-glide reflection and time-screw rotation. Here, we perform a complete classification of the 2+1D space-time groups based on the method of group cohomology, leading to the identification of all 275 space-time crystals, including 203 non-symmorphic ones. Under this formalism, unique physical phenomena are uncovered: A chirality-selective response rule with specific space-time symmetry is fully investigated and a novel ``horizontal cone" structure is predicted in space-time metamaterials as a direct consequence of non-symmorphic space-time symmetry. This work serves as a starting point for predicting and engineering a wide range of novel spatiotemporal phenomena across condensed matter and metamaterials.