Classification of locality preserving symmetries on spin chains
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912367104753664 |
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| author | Bols, Alex De Roeck, Wojciech De Wilde, Michiel Carvalho, Bruno de O. |
| author_facet | Bols, Alex De Roeck, Wojciech De Wilde, Michiel Carvalho, Bruno de O. |
| contents | 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 $H^2(G, U(1)[G])$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15088 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of locality preserving symmetries on spin chains Bols, Alex De Roeck, Wojciech De Wilde, Michiel Carvalho, Bruno de O. Quantum Physics Strongly Correlated Electrons Mathematical Physics 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 $H^2(G, U(1)[G])$. |
| title | Classification of locality preserving symmetries on spin chains |
| topic | Quantum Physics Strongly Correlated Electrons Mathematical Physics |
| url | https://arxiv.org/abs/2503.15088 |