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arXiv:1812.09183 · Physical Review Letters

Classification of Matrix-Product Unitaries with Symmetries

Zongping Gong, Christoph Sünderhauf, Norbert Schuch, J. I. Cirac

Published Mar 13, 2020• 36 citations
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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.

DOI
10.1103/physrevlett.124.100402

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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.
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Recently, it has been shown that two-dimensional bosonic symmetry-protected topological(SPT) phases with on-site unitary symmetry GGG can be completely classified by the group cohomology class H3(G,U(1))H^3(G, \mathrm{U}(1))H3(G,U(1)). Later, group super-cohomology class was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the mathematical framework of GGG-extension of unitary braided tensor category(UBTC) theory. We first reproduce the partial classifications given by group super-cohomology, then we show that with an additional H1(G,Z2)H^1(G, \mathbb{Z}_2)H1(G,Z2​) structure, a complete classification of SPT phases for two-dimensional interacting fermion systems for a total symmetry group G×Z2fG\times\mathbb{Z}_2^fG×Z2f​ can be achieved. We also discuss the classification of interacting fermionic SPT phases protected by time-reversal symmetry.
Large classes of matter have been systematized before by using a pair of groups to classify all spontaneous symmetry breaking phases or by using a pair of groups plus projective representations to classify all gapped quantum phases of one-dimensional boson and fermion systems with various symmetries. In this paper, the authors go beyond this and use a pair of groups plus three braided fusion categories and a chiral central charge in order to classify gapped quantum phases of two-dimensional boson and fermion systems with various finite symmetries. This allows, among other things, to generate many group tables of two-dimensional gapped phases.
We use low-depth quantum circuits, a specific type of tensor networks, to classify two-dimensional symmetry-protected topological many-body localized phases. For (anti)unitary onsite symmetries we show that the (generalized) third cohomology class of the symmetry group is a topological invariant; however, our approach leaves room for the existence of additional topological indices. We argue that our classification applies to quasiperiodic systems in two dimensions and systems with true random disorder within times which scale superexponentially with the inverse interaction strength. Our technique might be adapted to supply arguments suggesting the same classification for two-dimensional symmetry-protected topological ground states with a rigorous proof.
Abstract We consider the classification problem of quantum spin chains invariant under local-decomposable group actions, covering matrix product unitaries (MPUs), using an operator algebraic approach. We focus on finite group symmetries hosting both symmetric and symmetry-broken phases. The local-decomposable group actions we consider have a 3-cocycle class of the symmetry group associated with them. We derive invariants for our classification that naturally cover one-dimensional symmetry-protected topological (SPT) phases. We prove that these invariants coincide with the ones of [3] using matrix product state (MPS) techniques, by explicitly working out the GNS representation of MPSs and MPUs, resulting in a useful dictionary between both approaches that could be of independent interest.