Space-time crystallography and symmetry-enforced dynamical phenomena in 2+1 dimensions
The concept of space group has long served as a fundamental framework for describing the physical properties of crystalline materials, from electronic bands to photonic dispersions. Recent advances in spatiotemporal control, including laser-driven lattices, dynamic photonic and phononic crystals, and dynamic optical lattices, necessitate the study of symmetries that intrinsically intertwine space and time. Standard Floquet theory primarily organizes temporal periodicity and does not by itself provide a general framework for classifying these intertwined space-time symmetries. Space-time groups include novel intertwined nonsymmorphic spatial-temporal symmetries such as time-glide reflection and time-screw rotation. Here, we establish the complete crystallography of (2+1)-dimensional space-time crystals, systematically classifying all 275 space-time groups into 7 crystal systems, 14 Bravais lattices, and 72 arithmetic crystal classes. We further develop the representation theory of space-time groups and show that these symmetries enforce dynamical phenomena without static counterparts, including handedness-flipping heterodyne responses and a novel "horizontal cone" arising from symmetry-enforced momentum degeneracy in space-time metamaterials. Our results establish space-time crystallography as a general framework for predicting and engineering symmetry-protected phenomena in driven quantum systems and time-modulated metamaterials.