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International Journal of Modern Physics A

Noninvertible symmetry protected topological phases from the duality method: Classification and lattice construction

Weiguang Cao

Published Aug 30, 2025• 1 citation
Publisher

Abstract

As the understanding of global symmetry has been expanded, quantum phases of matter can have a refined classification, protected or enriched by noninvertible symmetry, a type of generalized symmetry whose transformation may not have an inverse. In this paper, I use the duality method, proposed in Ref. 1 , to classify noninvertible symmetry protected topological (NISPT) phases. Under the duality transformation, different NISPT phases are mapped to different classes of spontaneous symmetry breaking phases with ordinary group symmetry. I demonstrate this method with the simplest Kramers–Wannier duality transformation, also known as gauging [Formula: see text] symmetry, to classify Rep([Formula: see text]) NISPT in [Formula: see text]. I also construct lattice models realizing these NISPT on tensor product Hilbert space and study the projective algebras at the interfaces between different classes of NISPTs.

DOI
10.1142/s0217751x25480021

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As the understanding of global symmetry has been expanded, quantum phases of matter can have a refined classification, protected or enriched by noninvertible symmetry, a type of generalized symmetry whose transformation may not have an inverse. In this paper, I use the duality method, proposed in Ref. 1 , to classify noninvertible symmetry protected topological (NISPT) phases. Under the duality transformation, different NISPT phases are mapped to different classes of spontaneous symmetry breaking phases with ordinary group symmetry. I demonstrate this method with the simplest Kramers–Wannier duality transformation, also known as gauging [Formula: see text] symmetry, to classify Rep([Formula: see text]) NISPT in [Formula: see text]. I also construct lattice models realizing these NISPT on tensor product Hilbert space and study the projective algebras at the interfaces between different classes of NISPTs.
Recent advancements in generalized symmetries have drawn significant attention to gapped phases of matter exhibiting novel symmetries, such as noninvertible symmetries. By leveraging the duality transformations, the classification and construction of gapped phases with noninvertible symmetry can be mapped to those involving conventional group symmetries. We demonstrate this approach by classifying symmetry-protected-topological phases with a broad class of noninvertible symmetries in arbitrary spacetime dimensions. Our results reveal new classifications that extend beyond those based on group symmetries. Additionally, we construct lattice models in (1+1)D and (2+1)D that realize these new phases and explore their anomalous interfaces.
Noninvertible symmetry generalizes traditional group symmetries, advancing our understanding of quantum matter, especially one-dimensional gapped quantum systems. In critical lattice models, it is usually realized as emergent symmetries in the corresponding low-energy conformal field theories. In this work, we study critical lattice models with the noninvertible Rep(D8D_8D8​) symmetry in one dimension. This leads us to a new class of quantum critical points (QCP), noninvertible symmetry-enriched QCPs, as a generalization of known group symmetry-enriched QCPs. They are realized as phase transitions between one noninvertible symmetry-protected topological (SPT) phase and another different one or spontaneous symmetry breaking (SSB) phase. We identify their low-energy properties and topological features through the Kennedy-Tasaki (KT) duality transformation. We argue that distinct noninvertible symmetry-enriched QCPs can not be smoothly connected without a phase transition or a multi-critical point.
We explicitly realize the Rep(Q8Q_8Q8​) non-invertible symmetry-protected topological (SPT) state as a 1+1d cluster state on a tensor product Hilbert space of qubits. Using the Kramers-Wannier operator, we construct the lattice models for the phases of all the symmetries in the Rep(Q8Q_8Q8​) duality web. We further show that we can construct a class of lattice models with Rep(GGG) symmetry including non-invertible SPT phases if they have a dual anomalous abelian symmetry. Upon dualizing, there is a rich interplay between onsite symmetries, non-onsite symmetries, non-abelian symmetries, and non-invertible symmetries. We show that these interplay can be explained using the symmetry fractionalization in the 2+1d bulk SET.
Decoherence in realistic quantum platforms motivates a mixed-state notion of topological phases of matter, including average symmetry-protected topological (ASPT) phases. Alongside this progress, generalized symmetries--notably noninvertible and dipole symmetries--have become powerful organizing principles for exotic quantum phases, yet their implications for mixed states remain less explored. In this work, we bridge these directions through a gauging correspondence between mixed-state phases with generalized symmetries and mixed-state phases with ordinary group symmetries, recasting the classification of noninvertible and dipole ASPT phases into familiar classifications of symmetry breaking and ASPT phases with dual symmetries. Using this approach, we classify and construct a subclass of ASPT phases with non-invertible and dipole symmetries in (1+1)d(1+1)d(1+1)d, including phases that are intrinsic to mixed states, and characterize them via string order parameters and protected edge modes.
A bosonic non-invertible Symmetry Protected Topological (NISPT) phase in (1+1)-dim is referred to as intrinsic\textit{intrinsic}intrinsic if it cannot be mapped, under discrete gauging, to a gapped phase with any invertible symmetry, that is, if it is protected by a non-group-theoretical fusion category symmetry. We construct the intrinsic NISPT phases by performing discrete gauging in a partial SSB phase with a fusion category symmetry that has a certain mixed anomaly. Sometimes, the anomaly of that symmetry category can be alternatively understood as a self-anomaly of a proper categorical sub-symmetry; when this is the case, the same gauging provides an anomaly resolution of this anomalous categorical sub-symmetry. This allows us to construct intrinsic gapless SPT (igSPT) phases, where the anomalous faithfully acting symmetry is non-invertible; and we refer to such igSPT phases as igNISPT phases. We provide two concrete lattice models realizing an intrinsic NISPT phase and an igNISPT phase, respectively. We also generalize the construction of intrinsic NISPT phases to (3+1)-dim.
We give an explicit operator representation (via a sequential circuit and projection to symmetry subspaces) of Kramers-Wannier duality transformation in higher-dimensional subsystem symmetric models, generalizing the construction in the 1D transverse-field Ising model. Using the Kramers-Wannier duality operator, we also construct the Kennedy-Tasaki transformation that maps subsystem symmetry-protected topological phases to spontaneous subsystem symmetry-breaking phases, where the symmetry group for the former is either ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ or Z2{\mathbb{Z}}_{2}Z2​. This also generalizes the recently proposed picture of the one-dimensional Kennedy-Tasaki transformation as a composition of manipulations involving gauging and stacking symmetry-protected topological phases to higher dimensions.