Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2408.05585 · SciPost Physics

SymTFT for (3+1)d gapless SPTs and obstructions to confinement

Andrea Antinucci, Christian Copetti, Sakura Schäfer‐Nameki

Published Mar 31, 2025• 17 citations
arXivHTMLPDF

Abstract

We study gapless phases in (3+1)d in the presence of 1-form and non-invertible duality symmetries. Using the Symmetry Topological Field Theory (SymTFT) approach, we classify the gapless symmetry-protected (gSPT) phases in these setups, with particular focus on intrinsically gSPTs (igSPTs). These are symmetry protected critical points which cannot be deformed to a trivially gapped phase without spontaneously breaking the symmetry. Although these are by now well-known in (1+1)d, we demonstrate their existence in (3+1)d gauge theories. Here, they have a clear physical interpretation in terms of an obstruction to confinement, even though the full 1-form symmetry does not suffer from ’t Hooft anomalies. These igSPT phases provide a new way to realize 1-form symmetries in CFTs, that has no analog for gapped phases. The SymTFT approach allows for a direct generalization from invertible symmetries to non-invertible duality symmetries, for which we study gSPT and igSPT phases as well. We accompany these theoretical results with concrete physical examples realizing such phases and explain how obstruction to confinement is detected at the level of symmetric deformations.

DOI
10.21468/scipostphys.18.3.114

Related papers

Classification of 1+1D gapless symmetry protected phases via topological holography

Wen, Rui, Andrew C. Potter

Symmetry topological field theory (SymTFT) gives a holographic correspondence between systems with a global symmetry and a higher-dimensional topological field theory. In this framework, classification of gapped phases of matter in spacetime dimension 1+1D correspond to classifications of mechanisms to confine the SymTFT by condensing anyons. In this work, we extend these results to characterize gapless symmetry-protected topological states: symmetry-enriched gapless phases or critical points that exhibit edge modes protected by symmetry and topology. We establish a one-to-one correspondence between 1+1D bosonic gSPTs, and partially-confined boundaries of 2+1D SymTFTs. From general physical considerations, we determine the set of data and consistency conditions required to define a 1+1D gSPT, and show that this data precisely matches that of symmetry-preserving partial confinement (or partially gapped boundaries) of 2+1D quantum double models. We illustrate this correspondence through a dimensional reduction (thin-slab) construction, which enables a physically-intuitive derivation of how properties of the gSPT such as edge modes, emergent anomalies, and stability to perturbations arise from the SymTFT perspective.ditions required to define a 1+1D gSPT and show that they fully determine the physics of the gSPT including edge modes and emergent anomaly.

arXiv:2311.00050•Oct 31, 2023•7 citations

Hasse diagrams for gapless SPT and SSB phases with non-invertible symmetries

Lakshya Bhardwaj, Daniel Pajer, Sakura Schäfer‐Nameki +1 more

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.

arXiv:2403.00905•SciPost Physics•Oct 28, 2025•31 citations

Intrinsic NISPT Phases, igNISPT Phases, and Mixed Anomalies of Non-Invertible Symmetries

Da-Chuan Lu, Zhengdi Sun

A bosonic non-invertible Symmetry Protected Topological (NISPT) phase in (1+1)-dim is referred to as $\textit{intrinsic}$ 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.

arXiv:2511.01965•Nov 3, 2025•0 citations

Classification of 1 + 1 D gapless symmetry protected phases via topological holography

Rui Wen, Andrew C. Potter

Classifying gapless phases of matter remains a fundamental challenge in condensed matter theory. The authors pioneer here a systematic classification of 1+1D bosonic gapless symmetry-protected topological phases via the Symmetry Topological Field Theory (SymTFT) framework. They derive, from physical considerations, the algebra of symmetry operators in a 1+1D bosonic gSPT and show that it precisely matches the structure of condensable algebras in the SymTFT. This study marks the first systematic classification of gapless quantum phases in the SymTFT paradigm.

Physical review. B./Physical review. B•Mar 31, 2025•26 citations

Gapless phases in (2+1)d with non-invertible symmetries

Lakshya Bhardwaj, Yuhan Gai, Sheng Jie Huang +4 more

The study of gapless phases with categorical (or so-called non-invertible) symmetries is a formidable task, in particular in higher than two space-time dimensions. In this paper we build on previous works [1,2] on gapped phases in (2+1)d and provide a system¬atic framework to study phase transitions with categorical symmetries. The Symmetry Topological Field Theory (SymTFT) is, as often in these matters, the central tool. Applied to gapless theories, we need to consider the extension of the SymTFT to interfaces be¬tween topological orders, so-called “club sandwiches”, which realize generalizations of so-called Kennedy-Tasaki (KT) transformations. This requires an input phase transition for a smaller symmetry, such as the Ising transition for Z2, and the SymTFT constructs a transformation to a gapless phase with a larger categorical symmetry. We carry this out for categorical symmetries whose SymTFT is a (3+1)d Dijkgraaf-Witten (DW) theory for a finite group G with twist – so-called all bosonic fusion 2-categories. We classify such interfaces using a physically motivated picture of generalized gauging, as well as with a complementary analysis using (bi-)module 2-categories. This is exemplified in numer¬ous Abelian and non-Abelian DW theories, giving rise to interesting gapless phases such as intrinsically gapless symmetry protected phases (igSPTs) and spontaneous symmetry breaking phases (igSSBs) from Abelian, S3, and D8 DW theories.

•Jul 1, 2026•1 citation

Non-Invertible Interfaces Between Symmetry-Enriched Critical Phases

Saranesh Prembabu, Shu-Heng Shao, Ruben Verresen

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 $\mathbb{Z}_2 \times \mathbb{Z}_2^T$ 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 $1/L^3$, 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.

cond-mat.str-elCross-listedarXiv:2512.23706•Dec 30, 2025•0 citations
We study gapless phases in (3+1)d in the presence of 1-form and non-invertible duality symmetries. Using the Symmetry Topological Field Theory (SymTFT) approach, we classify the gapless symmetry-protected (gSPT) phases in these setups, with particular focus on intrinsically gSPTs (igSPTs). These are symmetry protected critical points which cannot be deformed to a trivially gapped phase without spontaneously breaking the symmetry. Although these are by now well-known in (1+1)d, we demonstrate their existence in (3+1)d gauge theories. Here, they have a clear physical interpretation in terms of an obstruction to confinement, even though the full 1-form symmetry does not suffer from ’t Hooft anomalies. These igSPT phases provide a new way to realize 1-form symmetries in CFTs, that has no analog for gapped phases. The SymTFT approach allows for a direct generalization from invertible symmetries to non-invertible duality symmetries, for which we study gSPT and igSPT phases as well. We accompany these theoretical results with concrete physical examples realizing such phases and explain how obstruction to confinement is detected at the level of symmetric deformations.
Symmetry topological field theory (SymTFT) gives a holographic correspondence between systems with a global symmetry and a higher-dimensional topological field theory. In this framework, classification of gapped phases of matter in spacetime dimension 1+1D correspond to classifications of mechanisms to confine the SymTFT by condensing anyons. In this work, we extend these results to characterize gapless symmetry-protected topological states: symmetry-enriched gapless phases or critical points that exhibit edge modes protected by symmetry and topology. We establish a one-to-one correspondence between 1+1D bosonic gSPTs, and partially-confined boundaries of 2+1D SymTFTs. From general physical considerations, we determine the set of data and consistency conditions required to define a 1+1D gSPT, and show that this data precisely matches that of symmetry-preserving partial confinement (or partially gapped boundaries) of 2+1D quantum double models. We illustrate this correspondence through a dimensional reduction (thin-slab) construction, which enables a physically-intuitive derivation of how properties of the gSPT such as edge modes, emergent anomalies, and stability to perturbations arise from the SymTFT perspective.ditions required to define a 1+1D gSPT and show that they fully determine the physics of the gSPT including edge modes and emergent anomaly.
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.
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.
Classifying gapless phases of matter remains a fundamental challenge in condensed matter theory. The authors pioneer here a systematic classification of 1+1D bosonic gapless symmetry-protected topological phases via the Symmetry Topological Field Theory (SymTFT) framework. They derive, from physical considerations, the algebra of symmetry operators in a 1+1D bosonic gSPT and show that it precisely matches the structure of condensable algebras in the SymTFT. This study marks the first systematic classification of gapless quantum phases in the SymTFT paradigm.
The study of gapless phases with categorical (or so-called non-invertible) symmetries is a formidable task, in particular in higher than two space-time dimensions. In this paper we build on previous works [1,2] on gapped phases in (2+1)d and provide a system¬atic framework to study phase transitions with categorical symmetries. The Symmetry Topological Field Theory (SymTFT) is, as often in these matters, the central tool. Applied to gapless theories, we need to consider the extension of the SymTFT to interfaces be¬tween topological orders, so-called “club sandwiches”, which realize generalizations of so-called Kennedy-Tasaki (KT) transformations. This requires an input phase transition for a smaller symmetry, such as the Ising transition for Z2, and the SymTFT constructs a transformation to a gapless phase with a larger categorical symmetry. We carry this out for categorical symmetries whose SymTFT is a (3+1)d Dijkgraaf-Witten (DW) theory for a finite group G with twist – so-called all bosonic fusion 2-categories. We classify such interfaces using a physically motivated picture of generalized gauging, as well as with a complementary analysis using (bi-)module 2-categories. This is exemplified in numer¬ous Abelian and non-Abelian DW theories, giving rise to interesting gapless phases such as intrinsically gapless symmetry protected phases (igSPTs) and spontaneous symmetry breaking phases (igSSBs) from Abelian, S3, and D8 DW theories.
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.