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

Abstract

We present a computational study of the mod-2 cohomology of three-dimensional (3D) space groups, with an eye toward their applications in Lieb--Schultz--Mattis constraints. We prove finite-generation results for the cohomology of crystallographic groups and give ring presentations for $H^{\ast }(G,{\mathbb{Z}}_2)$ for \emph{all} 230 3D space groups, together with explicit inhomogeneous representatives for the degree-$\le 3$ cocycles used in the lattice applications. The all-degree interpretation of the ring presentations is organized through finite LHS-spectral-sequence and Hilbert-series verification checks. We then associate distinguished classes in $H^3(G,{\mathbb{Z}}_2)$ to irreducible Wyckoff positions and use these classes as cohomological representatives of Lieb--Schultz--Mattis anomaly candidates when the on-site projective representations are classified by powers of ${\mathbb{Z}}_2$. Finally, we apply the resulting anomaly data to $\mathrm{U}(1)$ quantum spin liquids on the 3D pyrochlore lattice and compare the symmetry-fractionalization constraints with projective-symmetry-group calculations.

v3: Add the relations for the remaining space group 226, now the cohomology rings for all space groups are obtained from the resolution

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