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arXiv:2503.21764

Phases with non-invertible symmetries in 1+1D $\unicode{x2013}$ symmetry protected topological orders as duality automorphisms

Ömer M. Aksoy, Xiao-Gang Wen

Published Mar 27, 2025• 2 citations
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Abstract

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.

DOI
10.48550/arxiv.2503.21764

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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.
We discuss (1+1)d gapless phases with non-invertible global symmetries, also referred to as categorical symmetries. This includes gapless phases showing properties analogous to gapped symmetry protected topological (SPT) phases, known as gapless SPT (or gSPT) phases; and gapless phases showing properties analogous to gapped spontaneous symmetry broken (SSB) phases, that we refer to as gapless SSB (or gSSB) phases. We fit these gapless phases, along with gapped SPT and SSB phases, into a phase diagram describing possible deformations connecting them. This phase diagram is partially ordered and defines a so-called Hasse diagram. Based on these deformations, we identify gapless phases exhibiting symmetry protected criticality, that we refer to as intrinsically gapless SPT (igSPT) and intrinsically gapless SSB (igSSB) phases. This includes the first examples of igSPT and igSSB phases with non-invertible symmetries. Central to this analysis is the Symmetry Topological Field Theory (SymTFT), where each phase corresponds to a condensable algebra in the Drinfeld center of the symmetry category. On a mathematical note, gSPT phases are classified by functors between fusion categories, generalizing the fact that gapped SPT phases are classified by fiber functors; and gSSB phases are classified by functors from fusion to multi-fusion categories. Finally, our framework can be applied to understand gauging of trivially acting non-invertible symmetries, including possible patterns of decomposition arising due to such gaugings.
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.
1+1D symmetry-protected topological (SPT) phases with invertible symmetry can be characterized by the symmetric entanglers: globally symmetric finite-depth local unitary circuits that generate representative ground states from a product state. For non-invertible symmetries, however, distinct SPT phases, such as the three Rep(D8)\mathrm{Rep}(D_8)Rep(D8​) SPT phases, may not be connected by any symmetric entangler. We show that the three Rep(D8)\mathrm{Rep}(D_8)Rep(D8​) SPT phases can nevertheless be connected by \emph{symmetry-permuting} finite-depth local unitary circuits, realizing dualities predicted by topological quantum field theory (TQFT). We construct an explicit matrix product unitary connecting two of the Rep(D8)\mathrm{Rep}(D_8)Rep(D8​) SPT phases, and show that its action on string order parameters reproduces the anyon permutation predicted by the TQFT duality.
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.
Generalized symmetries often appear in the form of emergent symmetries in low energy effective descriptions of quantum many-body systems. Non-invertible symmetries are a particularly exotic class of generalized symmetries, in that they are implemented by transformations that do not form a group. Such symmetries appear in large families of gapless states of quantum matter and constrain their low-energy dynamics. To provide a UV-complete description of such symmetries, it is useful to construct lattice models that respect these symmetries exactly. In this paper, we discuss two families of one-dimensional lattice Hamiltonians with finite on-site Hilbert spaces: one with (invertible) S_3 S 3 symmetry and the other with non-invertible \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) symmetry. Our models are largely analytically tractable and demonstrate all possible spontaneous symmetry breaking patterns of these symmetries. Moreover, we use numerical techniques to study the nature of continuous phase transitions between the different symmetry-breaking gapped phases associated with both symmetries. Both models have self-dual lines, where the models are enriched by so-called intrinsically non-invertible symmetries generated by Kramers-Wannier-like duality transformations. We provide explicit lattice operators that generate these non-invertible self-duality symmetries. We show that the enhanced symmetry at the self-dual lines is described by a 2+1d symmetry-topological-order (SymTO) of type JK_4\boxtimes \overline{JK}_4 J K 4 ⊠ J K ¯ 4 . The condensable algebras of the SymTO determine the allowed gapped and gapless states of the self-dual S_3 S 3 -symmetric and \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) -symmetric models.
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.