Classification of locality preserving symmetries on spin chains

• URL: http://arxiv.org/abs/2503.15088v1
• Date: Wed, 19 Mar 2025 10:38:57 GMT
• Title: Classification of locality preserving symmetries on spin chains
• Authors: Alex Bols, Wojciech De Roeck, Michiel De Wilde, Bruno de O. Carvalho,
• Abstract summary: We consider the action of a finite group $G$ by locality preserving automorphisms (quantum cellular automata) on quantum spin chains.<n>We prove that the anomaly of such symmetries provides an isomorphism between the group of stable equivalence classes of symmetries with the cohomology group $H3(G,U(1))$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the action of a finite group $G$ by locality preserving automorphisms (quantum cellular automata) on quantum spin chains. We refer to such group actions as ``symmetries''. The natural notion of equivalence for such symmetries is \emph{stable equivalence}, which allows for stacking with factorized group actions. Stacking also endows the set of equivalence classes with a group structure. We prove that the anomaly of such symmetries provides an isomorphism between the group of stable equivalence classes of symmetries with the cohomology group $H^3(G,U(1))$, consistent with previous conjectures. This amounts to a complete classification of locality preserving symmetries on spin chains. We further show that a locality preserving symmetry is stably equivalent to one that can be presented by finite depth quantum circuits with covariant gates if and only if the slant product of its anomaly is trivial in $H^2(G, U(1)[G])$.

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