Representations of the symmetry groups of infinite crystals
Bachir Bekka, Christian Brouder
TL;DR
This work provides a mathematically rigorous harmonic-analysis framework for the representations of infinite crystallographic groups, bridging the gap from finite toric crystals to infinite crystals used in continuous-k analyses. By showing that crystallographic groups are type I with finite-dimensional irreps bounded by the point-group order, it establishes a Mackey-machine decomposition of irreps into direct integrals over the irreducible Brillouin zone, built from finite projective representations of little groups $G(\mathbf{k})$. The authors extend the machinery to magnetic groups via corepresentations and develop induced-representation theorems (restriction, tensor products, and symmetric/antisymmetric squares) with concrete Halite-example demonstrations, making the theory practically applicable to band and phonon problems without periodic boundary assumptions. The results unify infinite-crystal representation theory with the familiar finite toric-crystal picture, providing tools for analyzing band structures, selection rules, and couplings in a rigorous, boundary-free framework using $\mathbf{k}$-space and projection-valued measures. This framework opens pathways for spin-space extensions and broader symmetry analyses in higher dimensions and magnetic contexts, with potential impact on the study of magnon and phonon spectra in complex crystals.
Abstract
We investigate the representations of the symmetry groups of infinite crystals. Crystal symmetries are usually described as the finite symmetry group of a finite crystal with periodic boundary conditions, for which the Brillouin zone is a finite set of points. However, to deal with the continuous crystal momentum $\mathbf{k}$ required to discuss the continuity, singularity or analyticity of band energies $ε_n(\mathbf{k})$ and Bloch states $ψ_{\mathbf{k}}$, we need to consider infinite crystals. The symmetry groups of infinite crystals belong to the category of infinite non-compact groups, for which many standard tools of group theory break down. For example, character theory is no longer available for these groups and we use harmonic analysis to build the group algebra, the regular representation, the induction of irreducible representations of the crystallographic group from projective representations of the point groups and the decomposition of a representation into its irreducible parts. We deal with magnetic and non-magnetic groups in arbitrary dimensions. In the last part of the paper, we discuss Mackey's restriction of an induced representation to a subgroup, the tensor product of induced representations and the symmetric and antisymmetric squares of induced representations.