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

## TL;DR

This paper classifies the possible discrete Poincaré symmetry groups in two- and three-dimensional crystals, extending the understanding of symmetry beyond traditional space groups to include Lorentz and time translational symmetries.

## Contribution

It provides a classification of discrete Poincaré groups on 2D and 3D Bravais lattices, identifying their structure and relation to known space groups.

## Key findings

- Discrete Poincaré groups classified for 2D and 3D lattices
- Group structure determined by an integer generator g
- Reduces to space group at g=2

## Abstract

As is well known, crystals have discrete space translational symmetry. It was recently noticed that one-dimensional crystals possibly have discrete Poincar\'{e} symmetry, which contains discrete Lorentz and discrete time translational symmetry as well. In this paper, we classify the discrete Poincar\'{e} 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 $g=2$.