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:2507.05335

The Symmetry Taco: Equivalences between Gapped, Gapless, and Mixed-State SPTs

Marvin Qi, Ramanjit Sohal, Xie Chen, David T. Stephen, Abhinav Prem

Published Jul 7, 2025• 1 citation
arXivHTMLPDF

Abstract

Symmetry topological field theory (SymTFT), or topological holography, offers a unifying framework for describing quantum phases of matter and phase transitions between them. While this approach has seen remarkable success in describing gapped and gapless pure-state phases in $1+1$d, its applicability to open quantum systems remains entirely unexplored. In this work, we propose a natural extension of the SymTFT framework to mixed-state phases by introducing the \textit{symmetry taco}: a bilayer topological order in $2+1$d whose folded geometry naturally encapsulates both strong and weak symmetries of the $1+1$d theory. We use this perspective to identify a series of correspondences, including a one-to-one map between intrinsically gapless SPTs (igSPTs) and certain gapped SPTs, and a mapping between igSPTs and intrinsically average SPTs (iASPTs) arising in $1+1$d mixed states. More broadly, our framework yields a classification of short-range correlated $G$-symmetric Choi states in $1+1$d, provides a route for systematically generating mixed-state SPTs via local decoherence of igSPTs, and allows us to identify a new mixed-state ``anomaly". Besides folding in mixed-state phases into the SymTFT paradigm, the symmetry taco opens new avenues for exploring dualities, anomalies, and non-equilibrium criticality in mixed-state quantum matter.

DOI
10.48550/arxiv.2507.05335

Related papers

Topological holography for fermions

Wen, Rui, Weicheng Ye, Andrew C. Potter

Topological holography is a conjectured correspondence between the symmetry charges and defects of a $D$-dimensional system with the anyons in a $(D+1)$-dimensional topological order: the symmetry topological field theory (SymTFT). Topological holography is conjectured to capture the topological aspects of symmetry in gapped and gapless systems, with different phases corresponding to different gapped boundaries (anyon condensations) of the SymTFT. This correspondence was previously considered primarily for bosonic systems, excluding many phases of condensed matter systems involving fermionic electrons. In this work, we extend the SymTFT framework to establish a topological holography correspondence for fermionic systems. We demonstrate that this fermionic SymTFT framework captures the known properties of $1+1D$ fermion gapped phases and critical points, including the classification, edge-modes, and stacking rules of fermionic symmetry-protected topological phases (SPTs), and computation of partition functions of fermionic conformal field theories (CFTs). Beyond merely reproducing known properties, we show that the SymTFT approach can additionally serve as a practical tool for discovering new physics, and use this framework to construct a new example of a fermionic intrinsically gapless SPT phase characterized by an emergent fermionic anomaly.

arXiv:2404.19004•Apr 29, 2024•5 citations

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

Topological Holography for Mixed-State Phases and Phase Transitions

Ran Luo, Yi-Nan Wang, Zhen Bi

We extend the symmetry topological field theory (SymTFT) framework to open quantum systems. Using canonical purification, we embed mixed states into a doubled (2+1)-dimensional topological order and employ the slab construction to study (1+1)-dimensional mixed-state phases through condensable algebras in the doubled SymTFT. Hermiticity and positivity of the density matrix impose additional constraints on allowable anyon condensations, enabling a systematic classification of mixed-state phases—including strong-to-weak symmetry breaking (SWSSB) and average symmetry-protected topological (ASPT) phases. We present examples of mixed-state phase transitions involving SWSSB and show how gauging within the open SymTFT framework reveals connections among different mixed-state phases.

arXiv:2507.06218•PRX Quantum•Nov 5, 2025•3 citations

Topological holography, quantum criticality, and boundary states

Sheng-Jie Huang, Meng Cheng

Topological holography is a holographic principle that describes the generalized global symmetry of a local quantum system in terms of a topological order in one higher dimension. This framework separates the topological data from the local dynamics of a theory and provides a unified description of the symmetry and duality in gapped and gapless phases of matter. In this work, we develop the topological holographic picture for (1+1) d d quantum phases, including both gapped phases as well as a wide range of quantum critical points, including phase transitions between symmetry protected topological (SPT) phases, symmetry enriched quantum critical points, deconfined quantum critical points, and intrinsically gapless SPT phases. Topological holography puts a strong constraint on the emergent symmetry and the anomaly for these critical theories. We show how the partition functions of these critical points can be obtained from dualizing (orbifolding) more familiar critical theories. The topological responses of the defect operators are also discussed in this framework. We further develop a topological holographic picture for conformal boundary states of (1+1) d d rational conformal field theories. This framework provides a simple physical picture to understand conformal boundary states and also uncovers the nature of the gapped phases corresponding to the boundary states.

arXiv:2310.16878•SciPost Physics•Jun 30, 2025•46 citations

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

Andrea Antinucci, Christian Copetti, Sakura Schäfer‐Nameki

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.

arXiv:2408.05585•SciPost Physics•Mar 31, 2025•17 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
Symmetry topological field theory (SymTFT), or topological holography, offers a unifying framework for describing quantum phases of matter and phase transitions between them. While this approach has seen remarkable success in describing gapped and gapless pure-state phases in 1+11+11+1d, its applicability to open quantum systems remains entirely unexplored. In this work, we propose a natural extension of the SymTFT framework to mixed-state phases by introducing the \textit{symmetry taco}: a bilayer topological order in 2+12+12+1d whose folded geometry naturally encapsulates both strong and weak symmetries of the 1+11+11+1d theory. We use this perspective to identify a series of correspondences, including a one-to-one map between intrinsically gapless SPTs (igSPTs) and certain gapped SPTs, and a mapping between igSPTs and intrinsically average SPTs (iASPTs) arising in 1+11+11+1d mixed states. More broadly, our framework yields a classification of short-range correlated GGG-symmetric Choi states in 1+11+11+1d, provides a route for systematically generating mixed-state SPTs via local decoherence of igSPTs, and allows us to identify a new mixed-state ``anomaly". Besides folding in mixed-state phases into the SymTFT paradigm, the symmetry taco opens new avenues for exploring dualities, anomalies, and non-equilibrium criticality in mixed-state quantum matter.
Topological holography is a conjectured correspondence between the symmetry charges and defects of a DDD-dimensional system with the anyons in a (D+1)(D+1)(D+1)-dimensional topological order: the symmetry topological field theory (SymTFT). Topological holography is conjectured to capture the topological aspects of symmetry in gapped and gapless systems, with different phases corresponding to different gapped boundaries (anyon condensations) of the SymTFT. This correspondence was previously considered primarily for bosonic systems, excluding many phases of condensed matter systems involving fermionic electrons. In this work, we extend the SymTFT framework to establish a topological holography correspondence for fermionic systems. We demonstrate that this fermionic SymTFT framework captures the known properties of 1+1D1+1D1+1D fermion gapped phases and critical points, including the classification, edge-modes, and stacking rules of fermionic symmetry-protected topological phases (SPTs), and computation of partition functions of fermionic conformal field theories (CFTs). Beyond merely reproducing known properties, we show that the SymTFT approach can additionally serve as a practical tool for discovering new physics, and use this framework to construct a new example of a fermionic intrinsically gapless SPT phase characterized by an emergent fermionic anomaly.
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 extend the symmetry topological field theory (SymTFT) framework to open quantum systems. Using canonical purification, we embed mixed states into a doubled (2+1)-dimensional topological order and employ the slab construction to study (1+1)-dimensional mixed-state phases through condensable algebras in the doubled SymTFT. Hermiticity and positivity of the density matrix impose additional constraints on allowable anyon condensations, enabling a systematic classification of mixed-state phases—including strong-to-weak symmetry breaking (SWSSB) and average symmetry-protected topological (ASPT) phases. We present examples of mixed-state phase transitions involving SWSSB and show how gauging within the open SymTFT framework reveals connections among different mixed-state phases.
Topological holography is a holographic principle that describes the generalized global symmetry of a local quantum system in terms of a topological order in one higher dimension. This framework separates the topological data from the local dynamics of a theory and provides a unified description of the symmetry and duality in gapped and gapless phases of matter. In this work, we develop the topological holographic picture for (1+1) d d quantum phases, including both gapped phases as well as a wide range of quantum critical points, including phase transitions between symmetry protected topological (SPT) phases, symmetry enriched quantum critical points, deconfined quantum critical points, and intrinsically gapless SPT phases. Topological holography puts a strong constraint on the emergent symmetry and the anomaly for these critical theories. We show how the partition functions of these critical points can be obtained from dualizing (orbifolding) more familiar critical theories. The topological responses of the defect operators are also discussed in this framework. We further develop a topological holographic picture for conformal boundary states of (1+1) d d rational conformal field theories. This framework provides a simple physical picture to understand conformal boundary states and also uncovers the nature of the gapped phases corresponding to the boundary states.
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.
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.