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

TL;DR

This work provides a complete construction of the mod-2 cohomology rings for 3D space groups and establishes a precise map from irreducible Wyckoff positions to elements of H^3(G, Z_2), enabling a full set of 3D Lieb–Schultz–Mattis constraints. It develops two computational frameworks (GAP-based resolutions and bar resolutions) to generate and verify 1-, 2-, and 3-cocycle data, and proves finiteness and periodicity properties essential for tractable classification. The authors demonstrate that these cohomological data yield concrete LSM constraints and enable anomaly matching in 3D U(1) quantum spin liquids on pyrochlore lattices, aligning with PSG results and informing crystalline SPT analyses. The methods and data support a program toward a comprehensive crystalline SPT/LSM classification across all 3D space groups, with potential extensions to higher dimensions and fermionic systems.

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.

Figures (3)

• Figure 1: Panel (a) shows the generators of the wallpaper group $p6m$. In panel (b), the hexagon is a translation unit cell of the wallpaper group $p6m$. It has three IWPs, conventionally labeled by $a$, $b$ and $c$ in crystallography, and they form the sites of the triangular, honeycomb and kagome lattices, respectively.
• Figure 2: Panel (a) shows the generators of the space group \ref{['subsub:sg227']}. In panel (b), the cube is a conventional unit cell of the $Fd\bar{3}m$. It has four IWPs, conventionally labeled by $a$, $b$, $c$, and $d$ in crystallography, where $a$ and $b$ form two sets of diamond lattices and $c$ and $d$ form two sets of pyrochlore lattices.
• Figure 3: Bulk-edge correspondence for $k$-dimensional spin-1/2 lattice (the figure illustrates $k=2$). The on-site spin-1/2 lives on the boundary of a fictitious $(k+1)$-dimensional bulk. (a): the $(k+1)$-dimensional bulk. The bulk SPT consists of $S=1$ Haldane chains forming an $k$-dimensional array. The boundary of each Haldane chain has an $S=1/2$ edge mode (red dots). (b): the $k$-dimensional boundary on which the on-site $S=1/2$ degrees of freedom live.

Theorems & Definitions (19)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• proof
• Theorem 5
• Theorem 6