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arXiv:2502.20435 · Physical Review Letters

Duality Viewpoint of Noninvertible Symmetry-Protected Topological Phases

Weiguang Cao, Masahito Yamazaki, Linhao Li

Published Dec 15, 2025• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/4zfz-x9xh

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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.
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.
Higher-order topological phases with invertible symmetries have been extensively studied in recent years, revealing gapless modes localized on boundaries of higher codimension. In this work, we extend the framework of higher-order symmetry-protected topological (SPT) phases to include noninvertible symmetries. We construct a concrete model of a second-order SPT phase in 2+12+12+1 dimensions that hosts symmetry-protected corner modes protected by a noninvertible symmetry. This construction is then generalized to a dthd^{th}dth-order SPT phase in d+1d+1d+1 dimensions, featuring similarly protected corner modes. Additionally, we demonstrate a second-order SPT phase in 3+13+13+1 dimensions exhibiting hinge modes protected by a noninvertible symmetry.
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.
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.
A novel kind of generalized global symmetries is uncovered in a large class of familiar 3+1-dimensional gauge theories, including the free Maxwell theory and Yang-Mills gauge theories. These new symmetries, known as the non-invertible global symmetries, do not have inverses and thus go beyond the traditional paradigm of (anti-)unitary transformations. They are implemented by topological duality defects, generalizing the Kramers-Wannier duality defects in 1+1 dimensions. Remarkably, the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase and hence gives new constraints on renormalization group flows.
Despite great success we have achieved in the past two decades on systematically understanding gapped topological phases, a firm foundation for understanding gapless topological phases is still lacking. In this dissertation I develop a general framework to describe, classify and construct gapped and gapless phases with or without symmetry. I then apply this framework to construct new gapless phases, and classify some classes of them completely. The first chapter is a non-technical summary of the subjects, ideas and results of the rest of the dissertation. In the first chapter I introduce the notion of gapless topological phases, and briefly discuss their general structures. We also discuss how topological holography can be useful for classifying and constructing gapless phases with generalized symmetries. In chapter two I develop a framework for understanding general quantum phases and phase transitions with generalized symmetries. After reviewing the established theory of gapped phases, we introduce two key notions: quantum order and symmetry-enriched quantum order. These concepts are the correct generalized of topological order and symmetry-enriched topological orders. I show that these new notions subsume conventional phase transitions, gapless spin liquids, and gapless symmetry-protected topological phases. For a class of symmetry-enriched quantum orders with finite many topological defects, we present a general classification using higher tensor category theory. The rest of the dissertation provides more detailed studies of specific gapless topological phases, including classification and lattice model construction of gapless symmetry-protected topological phases and the connection to the theory of topological holography.