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

arXiv:2410.03607 · cond-mat.str-el, hep-th, math-ph, math.MP · Submitted 2024-10-04 · Read on arXiv

cond-mat.str-el, hep-th, math-ph, math.MP

Submitted: 2024-10-04

Updated: 2026-09-30

Code: https://github.com/chxliu/Space-Group-Cohomology-and-LSM

Terminology

• Verification — 16× in this paper · explained in 2 episodes
• Symmetry Breaking — 8× in this paper · explained in 3 episodes
• Generalization — 6× in this paper · explained in 13 episodes
• Irreducible Wyckoff Positions — 5× in this paper
• No-Go Theorem — 5× in this paper · explained in 2 episodes
• Adam — 4× in this paper · explained in 2 episodes
• International Tables for Crystallography — 3× in this paper
• Entanglement — 2× in this paper · explained in 3 episodes
• Topological Order — 2× in this paper · explained in 2 episodes
• Anisotropy — 1× in this paper · explained in 3 episodes
• Antiferromagnetism — 1× in this paper · explained in 1 episode
• Consistency — 1× in this paper · explained in 3 episodes
• Domain Walls — 1× in this paper · explained in 2 episodes
• Topological Defects — 1× in this paper · explained in 2 episodes
• Trace — 1× in this paper · explained in 2 episodes

Sources

• Triples, Fluxes, and Strings
• Twisted crystallograpic T-duality via the Baum--Connes isomorphism
• On Three-Dimensional Space Groups
• Computing classification of interacting fermionic symmetry-protected topological phases using topological invariants
• Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface
• Topological characterization of Lieb-Schultz-Mattis constraints and applications to symmetry-enriched quantum criticality
• Classification of symmetry-enriched topological quantum spin liquids
• Unifying Description of Competing Orders in Two Dimensional Quantum Magnets
• Dynamical structure factor of the $J_1-J_2$ Heisenberg model on the triangular lattice: magnons, spinons, and gauge fields
• Quantum Electrodynamics in 2+1 Dimensions as the Organizing Principle of a Triangular Lattice Antiferromagnet
• Gauging spatial symmetries and the classification of topological crystalline phases
• Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
• Invertible phases of matter with spatial symmetry
• Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle
• Real-space construction of crystalline topological superconductors and insulators in 2D interacting fermionic systems
• Construction and classification of crystalline topological superconductor and insulators in three-dimensional interacting fermion systems
• Symmetric $\mathrm{U(1)}$ and $\mathbb{Z}_2$ spin liquids on the pyrochlore lattice
• Lattice Homotopy Constraints on Phases of Quantum Magnets
• Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems
• Lieb-Schultz-Mattis Theorem and the Filling Constraint

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