Abstract
We prove that matrix-product unitaries with on-site unitary symmetries are completely classified by the (chiral) index and the cohomology class of the symmetry group G, provided that we can add trivial and symmetric ancillas with arbitrary on-site representations of G. If the representations in both system and ancillas are fixed to be the same, we can define symmetry-protected indices (SPIs) which quantify the imbalance in the transport associated to each group element and greatly refines the classification. These SPIs are stable against disorder and measurable in interferometric experiments. Our results lead to a systematic construction of two-dimensional Floquet symmetry-protected topological phases beyond the standard classification, and thus shed new light on understanding nonequilibrium phases of quantum matter.
Highlights
We prove that matrix-product unitaries with on-site unitary symmetries are completely classified by the index and the cohomology class of the symmetry group G, provided that we can add trivial and symmetric ancillas with arbitrary on-site representations of G
We prove that the combination of the index and the second cohomology class completely classifies all the matrix-product unitaries (MPUs) with given symmetries
Having in mind that symmetry-protected topological (SPT) phases with nontrivial cohomology classes usually exhibit exotic edge physics [62], we are naturally led to think about a similar situation for symmetry-protected indices (SPIs), which depend on g
Summary
We prove that the combination of the index and the second cohomology class completely classifies all the MPUs with given symmetries. Equivalence and complete classification.—We classify the MPUs according to Definition 1.—(Equivalence) Two G-symmetric MPUs U0 and U1 are equivalent if we allow for blocking (i.e., treat multiple sites as a single site), and the addition of local ancillas with the identity operator, such that the MPUs can be continuously connected within the manifold of symmetric MPUs. Both xg and yÃg belong to the same cohomology class as the associated MPS [45].
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