Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints
cond-mat.str-el, hep-th, math-ph, math.MP
Submitted: 2024-10-04
Updated: 2026-09-30
Comments: v4: Minor mistakes fixed, full cochain representatives for all 230 space groups obtained
Journal ref: SciPost Phys. 18, 161 (2025)
DOI: 10.21468/SciPostPhys.18.5.161
Code: https://github.com/chxliu/Space-Group-Cohomology-and-LSM
License: http://creativecommons.org/licenses/by/4.0/
The gist: 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.
Terminology
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*(G, Z 2) for all 230 3D space groups, together with explicit inhomogeneous representatives for the degree- at most 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, 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 Z 2. Finally, we apply the resulting anomaly data to U(1) quantum spin liquids on the 3D pyrochlore lattice and compare the symmetry-fractionalization constraints with projective-symmetry-group calculations.
Sources
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- Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems
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