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arXiv:2403.18573 · Annales Henri Poincaré

Classifying Symmetric and Symmetry-Broken Spin Chain Phases with Anomalous Group Actions

Jose Garre Rubio, András Molnár, Yoshiko Ogata

Published Jul 16, 2026• 1 citation
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Abstract

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.

DOI
10.1007/s00023-026-01717-9

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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.
We classify the different ways in which matrix product states (MPSs) can stay invariant under the action of matrix product operator (MPO) symmetries. This is achieved through a local characterization of how the MPSs, that generate a ground space, remain invariant under a global MPO symmetry. This characterization yields a set of quantities satisfying the coupled pentagon equations, associated with a module category over the fusion category that describes the MPO symmetry. Equivalence classes of these quantities provide complete invariants for an MPO symmetry protected phase: they are robust under continuous deformations of the MPS tensor, and two phases with the same equivalence class can be connected by a symmetric gapped path. Our techniques match and extend the known renormalization fixed point classifications and facilitate the numerical study of these systems. For MPO symmetries described by a group, we recover the symmetry protected topological order classification for unique and degenerate ground states. Moreover, we study the interplay between time reversal symmetry and an MPO symmetry and we also provide examples of our classification, together with explicit constructions based on groups. Finally, we elaborate on the connection between our setup and gapped boundaries of two-dimensional topological systems, where MPO symmetries also play a key role.
Abstract We develop a symmetry action framework for hidden quantum Markov models (HQMMs) tailored to one-dimensional quantum spin systems and symmetry-protected topological (SPT) phases. In our setting, a symmetry group G acts projectively on the hidden (virtual) degrees of freedom and linearly on the physical observation space, yielding a global HQMM state that is invariant under the combined action of G for both conventional and causal (input–output) structures. We show that such symmetry actions are naturally classified by a group-cohomology 2 -cocycle [ ω ] ∈ H 2 ( G , U ( 1 ) ) , in direct analogy with the standard cohomological classification of one-dimensional bosonic SPT phases via projective edge representations. As an explicit example, we apply this construction to the Affleck–Kennedy–Lieb–Tasaki (AKLT) chain, where the hidden layer carries a nontrivial class [ ω ] ∈ H 2 ( SO ( 3 ) , U ( 1 ) ) encoding its SPT order. In this case the HQMM formalism reproduces the known SPT properties of the AKLT state while providing a stochastic, Markovian description of the underlying virtual dynamics. Our results establish HQMMs as a natural bridge between quantum stochastic processes, tensor-network descriptions of many-body systems, and SPT order.
We classify one-dimensional symmetry-protected topological (SPT) phases protected by dipole symmetries. A dipole symmetry comprises two sets of symmetry generators: charge and dipole operators, which together form a nontrivial algebra with translations. Using matrix product states (MPS), we show that for a GGG dipole symmetry with GGG a finite Abelian group, the one-dimensional dipolar SPTs are classified by the group ${H}^{2}[G\ifmmode\times\else\texttimes\fi{}G,U(1)]/{H}^{2}{[G,U(1)]}^{2}$. Because of the symmetry algebra, the MPS tensors exhibit an unusual property, prohibiting the fractionalization of charge operators at the edges. For each phase in the classification, we explicitly construct a stabilizer Hamiltonian to realize the SPT phase and derive the response field theories by coupling the dipole symmetry to background tensor gauge fields. These field theories generalize the Dijkgraaf-Witten theories to twisted finite tensor gauge theories.
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.
Spatially modulated symmetries are one of the new types of symmetries the symmetry actions of which are position dependent. Yet exotic phases resulting from these spatially modulated symmetries are not fully understood and classified. In this work, we systematically classify one-dimensional bosonic symmetry-protected topological phases of multipole symmetries by employing the matrix product state formalism. The symmetry action induces projective representations at the ends of an open chain, which we identify via group cohomology. In particular, for rrr-pole symmetries---for instance, r=0r=0r=0 (global), 1 (dipole), and 2 (quadrupole)---the classification is determined by distinct components of second cohomology groups that encode the boundary projective representations.
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.