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arXiv:2408.15960

1+1d SPT phases with fusion category symmetry: interface modes and non-abelian Thouless pump

Kansei Inamura, Shuhei Ohyama

Published Aug 28, 2024• 5 citations
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Abstract

We consider symmetry protected topological (SPT) phases with finite non-invertible symmetry $\mathcal{C}$ in 1+1d. In particular, we investigate interfaces and parameterized families of them within the framework of matrix product states. After revealing how to extract the $\mathcal{C}$-SPT invariant, we identify the algebraic structure of symmetry operators acting on the interface of two $\mathcal{C}$-SPT phases. By studying the representation theory of this algebra, we show that there must be a degenerate interface mode between different $\mathcal{C}$-SPT phases. This result generalizes the bulk-boundary correspondence for ordinary SPT phases. We then propose the classification of one-parameter families of $\mathcal{C}$-SPT states based on the explicit construction of invariants of such families. Our invariant is identified with a non-abelian generalization of the Thouless charge pump, which is the pump of a local excitation within a $\mathcal{C}$-SPT phase. Finally, by generalizing the results for one-parameter families of SPT phases, we conjecture the classification of general parameterized families of general gapped phases with finite non-invertible symmetries in both 1+1d and higher dimensions.

DOI
10.48550/arxiv.2408.15960

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We consider symmetry protected topological (SPT) phases with finite non-invertible symmetry C\mathcal{C}C in 1+1d. In particular, we investigate interfaces and parameterized families of them within the framework of matrix product states. After revealing how to extract the C\mathcal{C}C-SPT invariant, we identify the algebraic structure of symmetry operators acting on the interface of two C\mathcal{C}C-SPT phases. By studying the representation theory of this algebra, we show that there must be a degenerate interface mode between different C\mathcal{C}C-SPT phases. This result generalizes the bulk-boundary correspondence for ordinary SPT phases. We then propose the classification of one-parameter families of C\mathcal{C}C-SPT states based on the explicit construction of invariants of such families. Our invariant is identified with a non-abelian generalization of the Thouless charge pump, which is the pump of a local excitation within a C\mathcal{C}C-SPT phase. Finally, by generalizing the results for one-parameter families of SPT phases, we conjecture the classification of general parameterized families of general gapped phases with finite non-invertible symmetries in both 1+1d and higher dimensions.
We construct 2+1-dimensional lattice models of symmetry-protected topological (SPT) phases with non-invertible symmetries and investigate their properties using tensor networks. These models, which we refer to as generalized cluster models, are constructed by gauging a subgroup symmetry H⊂GH \subset GH⊂G in models with a finite group 0-form symmetry GGG. By construction, these models have a non-invertible symmetry described by the group-theoretical fusion 2-category C(G;H)\mathcal{C}(G; H)C(G;H). After identifying the tensor network representations of the symmetry operators, we study the symmetry acting on the interface between two generalized cluster states. In particular, we will see that the symmetry at the interface is described by a multifusion category known as the strip 2-algebra. By studying possible interface modes allowed by this symmetry, we show that the interface between generalized cluster states in different SPT phases must be degenerate. This result generalizes the ordinary bulk-boundary correspondence. Furthermore, we construct parameterized families of generalized cluster states and study the topological charge pumping phenomena, known as the generalized Thouless pump. We exemplify our construction with several concrete cases, and compare them with known phases, such as SPT phases with 2Rep((Z2[1]×Z2[1])⋊Z2[0])2\mathrm{Rep}((\mathbb{Z}_{2}^{[1]}\times\mathbb{Z}_{2}^{[1]})\rtimes\mathbb{Z}_{2}^{[0]})2Rep((Z2[1]​×Z2[1]​)⋊Z2[0]​) symmetry.
We investigate (1+1)d symmetry-protected topological (SPT) phases with fusion category symmetries. We emphasize that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase. We construct an ``onsite'' matrix-product-operator (MPO) version of the Hopf algebra symmetry operators in a lattice model with tensor-product Hilbert space. In particular, we propose a systematic framework for classifying and constructing SPTs with non-invertible symmetries. An SPT phase corresponds to a Q-system in the charge category, such that the Q-system becomes a matrix algebra when the symmetry is forgotten. As an example, we provide an explicit microscopic realization of all three Rep†(D8)\mathsf{Rep}^\dagger(D_8)Rep†(D8​) SPT phases, including a trivial phase, and further demonstrate the S3S_3S3​-duality among these three SPT phases.
Gapless quantum phases can become distinct when internal symmetries are enforced, in analogy with gapped symmetry-protected topological (SPT) phases. However, this distinction does not always lead to protected edge modes, raising the question of how the bulk-boundary correspondence is generalized to gapless cases. We propose that the spatial interface between gapless phases -- rather than their boundaries -- provides a more robust fingerprint. We show that whenever two 1+1d conformal field theories (CFTs) differ in symmetry charge assignments of local operators or twisted sectors, any symmetry-preserving spatial interface between the theories must flow to a non-invertible defect. We illustrate this general result for different versions of the Ising CFT with Z2×Z2T\mathbb{Z}_2 \times \mathbb{Z}_2^TZ2​×Z2T​ symmetry, obtaining a complete classification of allowed conformal interfaces. When the Ising CFTs differ by nonlocal operator charges, the interface hosts 0+1d symmetry-breaking phases with finite-size splittings scaling as 1/L31/L^31/L3, as well as continuous phase transitions between them. For general gapless phases differing by an SPT entangler, the interfaces between them can be mapped to conformal defects with a certain defect 't Hooft anomaly. This classification also gives implications for higher-dimensional examples, including symmetry-enriched variants of the 2+1d Ising CFT. Our results establish a physical indicator for symmetry-enriched criticality through symmetry-protected interfaces, giving a new handle on the interplay between topology and gapless phases.
We explore 1+1 dimensional (1+1D) gapped phases in systems with non-invertible symmetries, focusing on symmetry-protected topological (SPT) phases (defined as gapped phases with non-degenerate ground states), as well as SPT orders (defined as the differences between gapped/gapless phases with identical bulk excitations spectrum). For group-like symmetries, distinct SPT phases share identical bulk excitations and always differ by SPT orders. However, for certain non-invertible symmetries, we discover novel SPT phases that have different bulk excitations and thus do not differ by SPT orders. Additionally, we also study the spontaneous symmetry-breaking (SSB) phases of non-invertible symmetries. Unlike group-like symmetries, non-invertible symmetries lack the concept of subgroups, which complicates the definition of SSB phases as well as their identification. This challenge can be addressed by employing the symmetry-topological-order (symTO) framework for the symmetry. The Lagrangian condensable algebras and automorphisms of the symTO facilitate the classification of gapped phases in systems with such symmetries, enabling the analysis of both SPT and SSB phases (including those that differ by SPT orders). Finally, we apply this methodology to investigate gapless phases in symmetric systems and to study gapless phases differing by SPT orders.
Classification of intrinsically mixed 1+11+11+1D non-invertible Rep(G)×G(G) \times G(G)×G SPT phases
We classify 1+11+11+1d bosonic SPT phases with non-invertible symmetry Rep(G)×G\mathrm{Rep}(G)\times GRep(G)×G, equivalently the fusion-category symmetry H=Rep(G)×VecG\mathcal{H}=\mathrm{Rep}(G)\times\mathrm{Vec}_GH=Rep(G)×VecG​. Focusing on \emph{intrinsically mixed} phases (trivial under either factor alone), we use the correspondence between H\mathcal{H}H-SPTs, H\mathcal{H}H-modules over Vec\mathrm{Vec}Vec, and fiber functors H→Vec\mathcal{H}\to\mathrm{Vec}H→Vec to obtain a complete classification: such phases are parametrized by ϕ∈End⁡(G)ϕ\in\operatorname{End}(G)ϕ∈End(G). For each ϕϕϕ we identify the associated condensable (Lagrangian) algebra Aϕ\mathcal{A}_ϕAϕ​ in the bulk Z(H)≃DG2\mathcal{Z}(\mathcal{H})\simeq\mathcal{D}_G^2Z(H)≃DG2​. We further provide an explicit lattice realization by modifying Kitaev's quantum double model with a domain wall Bϕ\mathcal{B}_ϕBϕ​ and smooth/rough boundaries, and then contracting to a 1D chain, yielding a (possibly twisted) group-based cluster state whose ribbon-generated symmetry operators encode the same ϕϕϕ.
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.