Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints

TL;DR

This paper computes the mod-2 cohomology of 3D space groups to derive Lieb-Schultz-Mattis constraints, linking crystal symmetry to quantum spin liquid properties and symmetry fractionalization patterns.

Contribution

It provides a complete set of cohomology-based constraints for 3D lattice magnets and connects them to quantum spin liquid phenomena, extending previous classification methods.

Findings

Computed the mod-2 cohomology ring for 3D space groups.

Established a link between cohomology and Lieb-Schultz-Mattis constraints.

Analyzed symmetry fractionalization in 3D quantum spin liquids.

Abstract

We compute the mod-2 cohomology ring for three-dimensional (3D) space groups and establish a connection between them and the lattice structure of crystals with space group symmetry. This connection allows us to obtain a complete set of Lieb-Schultz-Mattis constraints, specifying the conditions under which a unique, symmetric, gapped ground state cannot exist in 3D lattice magnets. We associate each of these constraints with an element in the third mod-2 cohomology of the space group, when the internal symmetry acts on-site and its projective representations are classified by powers of $\mathbb{Z}_2$. We demonstrate the relevance of our results to the study of $\mathrm{U}(1)$ quantum spin liquids on the 3D pyrochlore lattice. We determine, through anomaly matching, the symmetry fractionalization patterns of both electric and magnetic charges, extending previous results from projective symmetry group classifications.