Discrete Lorentz symmetry and discrete spacetime translational symmetry in two- and three-dimensional crystals

As is well known, crystals have discrete space translational symmetry. It was recently noticed that one-dimensional crystals possibly have discrete Poincaré symmetry, which contains discrete Lorentz and discrete time translational symmetry as well. In this paper, we classify the discrete Poincaré groups on two- and three- dimensional Bravais lattices. They are the candidate symmetry groups of two- or three-dimensional crystals, respectively. The group is determined by an integer generator g, and it reduces to the space group of crystals at g = 2.