Applications of Group Cohomology to the Classification of Fourier-Space Quasicrystals

TL;DR

This paper uses group cohomology to classify Fourier-space quasicrystals, providing a unified algebraic framework that extends previous classifications and links physical phenomena to topological invariants.

Contribution

It reformulates the classification of Fourier-space quasicrystals using group cohomology, generalizing prior results and introducing new invariants for understanding quasicrystal symmetries.

Findings

Classifies space groups for arbitrary rank quasilattices in 2D

Links diffraction extinction and electronic degeneracy to homology classes

Provides a duality framework for gauge invariants and space group classification

Abstract

In 1962, Bienenstock and Ewald described the classification of crystalline space groups algebraically in the dual, or Fourier, space. Recently, the method has been applied to quasicrystals and modulated crystals. This paper phrases Bienenstock and Ewald's definitions in terms of group cohomology. A \textit{Fourier quasicrystal} is defined, along with its space group, without requiring that it come from a quasicrystal in real (direct) space. A certain cohomology group classifies the space groups associated to a given point group and quasilattice, and the dual homology group gives all gauge invariants. This duality is exploited to prove several results that were previously known only in special cases, including the classification of space groups for quasilattices of arbitrary rank in two dimensions. Extinctions in X-ray diffraction patterns and degeneracy of electronic levels are interpreted as physical manifestations of non-zero homology classes.