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arXiv:2106.02145

Classification of equivariant quasi-local automorphisms on quantum chains

Alex Bols

Published Jun 3, 2021• 0 citations
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Abstract

We classify automorphisms on quantum chains, allowing both spin and fermionic degrees of freedom, that are moreover equivariant with respect to a local symmetry action of a finite symmetry group $G$. The classification is up to equivalence through strongly continuous deformation and stacking with decoupled auxiliary automorphisms. We find that the equivalence classes are uniquely labeled by an index taking values in $\mathbb{Q} \cup \sqrt{2} \mathbb{Q} \times \mathrm{Hom}(G, \mathbb{Z}_2) \times H^2(G, U(1))$. We discuss te relation of this index to the index of one-dimensional symmetry protected topological phases on spin chains, which takes values in $H^2(G, U(1))$.

DOI
10.48550/arxiv.2106.02145

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The equivalence of topological indices for U(1)⋊Z2\mathrm{U}(1)\rtimes\mathbb{Z}_2U(1)⋊Z2​-symmetric quantum spin chains
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Symmetry protected states (SPT's) of quantum spin systems were studied by several authors. For one-dimensional systems (spin chains), there is an essentially complete and rigorous understanding: SPT's corresponding to finite on-site symmetry groups GGG are classified by the second cohomology group H2(G,U(1))H^2(G,U(1))H2(G,U(1)), as established by Kapustin et al. [J. Math. Phys. (2021)]. We extend this result to the case of compact topological symmetry groups GGG. We also strengthen the existing results in the sense that our classification results holds within the class of spin chains with locally bounded on-site dimensions.
Symmetry protected states (SPTs) of quantum spin systems were studied by several authors. For one-dimensional systems (spin chains), there is an essentially complete and rigorous understanding: SPTs corresponding to finite on-site symmetry groups G are classified by the second cohomology group H2(G, U(1)), as established in Kapustin, Sopenko, and Yang. J. Math. Phys. 62(8), 081901 (2021). We extend this result to the case of compact and metrizable topological symmetry groups G. We also strengthen the existing results in the sense that our classification results hold within the class of spin chains with locally bounded on-site dimensions.
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