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:2411.11967

Domain walls from SPT-sewing

Yabo Li, Zijian Song, Aleksander Kubica, Isaac H. Kim

Published Nov 18, 2024• 2 citations
arXivHTMLPDF

Abstract

We introduce a systematic method for constructing gapped domain walls of topologically ordered systems by gauging a lower-dimensional symmetry-protected topological (SPT) order. Based on our construction, we propose a correspondence between 1d SPT phases with a non-invertible $G\times \text{Rep}(G)\times G$ symmetry and invertible domain walls in the quantum double associated with the group $G$. We prove this correspondence when $G$ is Abelian and provide evidence for the general case by studying the quantum double model for $G=S_3$. We also use our method to construct \emph{anchoring domain walls}, which are novel exotic domain walls in the 3d toric code that transform point-like excitations to semi-loop-like excitations anchored on these domain walls.

DOI
10.48550/arxiv.2411.11967

Related papers

Domain Wall Decorations, Anomalies and Spectral Sequences in Bosonic Topological Phases

Qing-Rui Wang, Shang-Qiang Ning, Meng Cheng

In this work we investigate the decorated domain wall construction in bosonic group-cohomology symmetry-protected topological (SPT) phases and related quantum anomalies in bosonic topological phases. We first show that a general decorated domain wall construction can be described mathematically as an Atiyah-Hirzebruch spectral sequence, where the terms on the $E_2$ page correspond to decorations by lower-dimensional SPT states at domain wall junctions. For bosonic group-cohomology SPT phases, the spectral sequence becomes the Lyndon-Hochschild-Serre (LHS) spectral sequence for ordinary group cohomology. We then discuss the physical interpretations of the differentials in the spectral sequence, particularly in the context of anomalous SPT phases and symmetry-enriched gauge theories. As the main technical result, we obtain a full description of the LHS spectral sequence concretely at the cochain level. The explicit formulae are then applied to explain Lieb-Schultz-Mattis theorems for SPT phases, and also derive a new LSM theorem for easy-plane spin model in a $π$ flux lattice. We also revisit the classifications of symmetry-enriched 2D and 3D Abelian gauge theories using our results.

arXiv:2104.13233•Apr 27, 2021•10 citations

Symmetry-protected topological interfaces and entanglement sequences

Luiz H. Santos, Jennifer Cano, Michael Mulligan +1 more

Gapped interfaces (and boundaries) of two-dimensional (2D) Abelian topological phases are shown to support a remarkably rich sequence of 1D symmetry-protected topological (SPT) states. We show that such interfaces can provide a physical interpretation for the corrections to the topological entanglement entropy of a 2D state with Abelian topological order found by J. Cano, T. L. Hughes, and M. Mulligan [Phys. Rev. B 92, 075104 (2015)]. The topological entanglement entropy decomposes as $\ensuremath{\gamma}={\ensuremath{\gamma}}_{a}+{\ensuremath{\gamma}}_{s}$, where ${\ensuremath{\gamma}}_{a}>0$ only depends on universal topological properties of the 2D state, while a correction ${\ensuremath{\gamma}}_{s}>0$ signals the emergence of the 1D SPT state that is produced by interactions along the entanglement cut and provides a direct measure of the stabilizing symmetry of the resulting SPT state. A correspondence is established between the possible values of ${\ensuremath{\gamma}}_{s}$ associated with a given interface---which is named the ``boundary topological entanglement sequence''---and classes of 1D SPT states. We show that symmetry-preserving domain walls along such 1D interfaces (or boundaries) generally host localized parafermion-like excitations that are stable to local symmetry-preserving perturbations.

arXiv:1803.04418•Physical review. B./Physical review. B•Aug 17, 2018•30 citations

Lattice model constructions for gapless domain walls between topological phases

Chenfeng Bao, Shuo Yang, Chenjie Wang +1 more

Domain walls between different topological phases are one of the most interesting phenomena that reveal the nontrivial bulk properties of topological phases. Very recently, gapped domain walls between different topological phases have been intensively studied. In this paper, we systematically construct a large class of lattice models for gapless domain walls between twisted and untwisted gauge theories with arbitrary finite group $G$. As simple examples, we numerically study several finite groups (including both Abelian and non-Abelian finite group such as ${S}_{3}$) in $2\mathrm{D}$ using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.

arXiv:1801.00719•Physical Review Research•Apr 14, 2022•30 citations

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

Ömer M. Aksoy, Xiao-Gang Wen

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.

arXiv:2503.21764•Mar 27, 2025•2 citations

Symmetry-enriched topological order from partially gauging symmetry-protected topologically ordered states assisted by measurements

Yabo Li, Hiroki Sukeno, Aswin Parayil Mana +2 more

Symmetry-protected topological (SPT) phases exhibit nontrivial short-ranged entanglement protected by symmetry and cannot be adiabatically connected to trivial product states while preserving the symmetry. In contrast, intrinsic topological phases do not need ordinary symmetry to stabilize them and their ground states exhibit long-range entanglement. It is known that for a given symmetry group $G$, the 2D SPT phase protected by $G$ is dual to the 2D topological phase exemplified by the twisted quantum double model ${D}^{\ensuremath{\omega}}(G)$ via gauging the global symmetry $G$. Recently it was realized that such a general gauging map can be implemented by some local unitaries and local measurements when $G$ is a finite, solvable group. Here, we review the general approach to gauging a $G\text{\ensuremath{-}}\mathrm{SPT}$ starting from a fixed-point ground-state wave function and applying a $N$-step gauging procedure. We provide an in-depth analysis of the intermediate states emerging during the $N$-step gauging and provide tools to measure and identify the emerging symmetry-enriched topological order (SET) of these states. We construct the generic lattice parent Hamiltonians for these intermediate states and show that they form an entangled superposition of a twisted quantum double (TQD) with an SPT-ordered state. Notably, we show that they can be connected to the TQD through a finite-depth, local quantum circuit which does not respect the global symmetry of the SET order. We introduce the so-called symmetry branch line operators and show that they can be used to extract the symmetry fractionalization classes (SFC) and symmetry defectification classes (SDC) of the SET phases with the input data $G$ and $[\ensuremath{\omega}]\ensuremath{\in}{H}^{3}(G,U(1))$ of the pregauged SPT-ordered state. We illustrate the procedure of preparing and characterizing the emerging SET-ordered states for some Abelian and non-Abelian examples such as dihedral groups ${D}_{n}$ and the quaternion group ${Q}_{8}$.

arXiv:2305.09747•Physical review. B./Physical review. B•Sep 21, 2023•20 citations

Anomalies in bosonic symmetry-protected topological edge theories: Connection to F symbols and a method of calculation

Kyle Kawagoe, Michael Levin

We describe a systematic procedure for determining the identity of a 2D bosonic symmetry-protected topological (SPT) phase from the properties of its edge excitations. Our approach applies to general bosonic SPT phases with either unitary or antiunitary symmetries, and with either continuous or discrete symmetry groups, with the only restriction being that the symmetries must be on-site. Concretely, our procedure takes a bosonic SPT edge theory as input, and produces an element $\ensuremath{\omega}$ of the cohomology group ${H}^{3}(G,{U}_{T}(1))$. This element $\ensuremath{\omega}\ensuremath{\in}{H}^{3}(G,{U}_{T}(1))$ can be interpreted as either a label for the bulk 2D SPT phase or a label for the anomaly carried by the SPT edge theory. The basic idea behind our approach is to compute the $F$ symbol associated with domain walls in a symmetry-broken edge theory; this domain-wall $F$ symbol is precisely the anomaly we wish to compute. We demonstrate our approach with several SPT edge theories, including both lattice models and continuum field theories.

arXiv:2105.02909•Physical review. B./Physical review. B•Sep 27, 2021•34 citations
We introduce a systematic method for constructing gapped domain walls of topologically ordered systems by gauging a lower-dimensional symmetry-protected topological (SPT) order. Based on our construction, we propose a correspondence between 1d SPT phases with a non-invertible G×Rep(G)×GG\times \text{Rep}(G)\times GG×Rep(G)×G symmetry and invertible domain walls in the quantum double associated with the group GGG. We prove this correspondence when GGG is Abelian and provide evidence for the general case by studying the quantum double model for G=S3G=S_3G=S3​. We also use our method to construct \emph{anchoring domain walls}, which are novel exotic domain walls in the 3d toric code that transform point-like excitations to semi-loop-like excitations anchored on these domain walls.
In this work we investigate the decorated domain wall construction in bosonic group-cohomology symmetry-protected topological (SPT) phases and related quantum anomalies in bosonic topological phases. We first show that a general decorated domain wall construction can be described mathematically as an Atiyah-Hirzebruch spectral sequence, where the terms on the E2E_2E2​ page correspond to decorations by lower-dimensional SPT states at domain wall junctions. For bosonic group-cohomology SPT phases, the spectral sequence becomes the Lyndon-Hochschild-Serre (LHS) spectral sequence for ordinary group cohomology. We then discuss the physical interpretations of the differentials in the spectral sequence, particularly in the context of anomalous SPT phases and symmetry-enriched gauge theories. As the main technical result, we obtain a full description of the LHS spectral sequence concretely at the cochain level. The explicit formulae are then applied to explain Lieb-Schultz-Mattis theorems for SPT phases, and also derive a new LSM theorem for easy-plane spin model in a πππ flux lattice. We also revisit the classifications of symmetry-enriched 2D and 3D Abelian gauge theories using our results.
Gapped interfaces (and boundaries) of two-dimensional (2D) Abelian topological phases are shown to support a remarkably rich sequence of 1D symmetry-protected topological (SPT) states. We show that such interfaces can provide a physical interpretation for the corrections to the topological entanglement entropy of a 2D state with Abelian topological order found by J. Cano, T. L. Hughes, and M. Mulligan [Phys. Rev. B 92, 075104 (2015)]. The topological entanglement entropy decomposes as $\ensuremath{\gamma}={\ensuremath{\gamma}}_{a}+{\ensuremath{\gamma}}_{s}$, where ${\ensuremath{\gamma}}_{a}>0$ only depends on universal topological properties of the 2D state, while a correction ${\ensuremath{\gamma}}_{s}>0$ signals the emergence of the 1D SPT state that is produced by interactions along the entanglement cut and provides a direct measure of the stabilizing symmetry of the resulting SPT state. A correspondence is established between the possible values of ${\ensuremath{\gamma}}_{s}$ associated with a given interface---which is named the ``boundary topological entanglement sequence''---and classes of 1D SPT states. We show that symmetry-preserving domain walls along such 1D interfaces (or boundaries) generally host localized parafermion-like excitations that are stable to local symmetry-preserving perturbations.
Domain walls between different topological phases are one of the most interesting phenomena that reveal the nontrivial bulk properties of topological phases. Very recently, gapped domain walls between different topological phases have been intensively studied. In this paper, we systematically construct a large class of lattice models for gapless domain walls between twisted and untwisted gauge theories with arbitrary finite group GGG. As simple examples, we numerically study several finite groups (including both Abelian and non-Abelian finite group such as S3{S}_{3}S3​) in 2D2\mathrm{D}2D using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.
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.
Symmetry-protected topological (SPT) phases exhibit nontrivial short-ranged entanglement protected by symmetry and cannot be adiabatically connected to trivial product states while preserving the symmetry. In contrast, intrinsic topological phases do not need ordinary symmetry to stabilize them and their ground states exhibit long-range entanglement. It is known that for a given symmetry group GGG, the 2D SPT phase protected by GGG is dual to the 2D topological phase exemplified by the twisted quantum double model ${D}^{\ensuremath{\omega}}(G)$ via gauging the global symmetry GGG. Recently it was realized that such a general gauging map can be implemented by some local unitaries and local measurements when GGG is a finite, solvable group. Here, we review the general approach to gauging a $G\text{\ensuremath{-}}\mathrm{SPT}$ starting from a fixed-point ground-state wave function and applying a NNN-step gauging procedure. We provide an in-depth analysis of the intermediate states emerging during the NNN-step gauging and provide tools to measure and identify the emerging symmetry-enriched topological order (SET) of these states. We construct the generic lattice parent Hamiltonians for these intermediate states and show that they form an entangled superposition of a twisted quantum double (TQD) with an SPT-ordered state. Notably, we show that they can be connected to the TQD through a finite-depth, local quantum circuit which does not respect the global symmetry of the SET order. We introduce the so-called symmetry branch line operators and show that they can be used to extract the symmetry fractionalization classes (SFC) and symmetry defectification classes (SDC) of the SET phases with the input data GGG and $[\ensuremath{\omega}]\ensuremath{\in}{H}^{3}(G,U(1))$ of the pregauged SPT-ordered state. We illustrate the procedure of preparing and characterizing the emerging SET-ordered states for some Abelian and non-Abelian examples such as dihedral groups Dn{D}_{n}Dn​ and the quaternion group Q8{Q}_{8}Q8​.
We describe a systematic procedure for determining the identity of a 2D bosonic symmetry-protected topological (SPT) phase from the properties of its edge excitations. Our approach applies to general bosonic SPT phases with either unitary or antiunitary symmetries, and with either continuous or discrete symmetry groups, with the only restriction being that the symmetries must be on-site. Concretely, our procedure takes a bosonic SPT edge theory as input, and produces an element $\ensuremath{\omega}$ of the cohomology group H3(G,UT(1)){H}^{3}(G,{U}_{T}(1))H3(G,UT​(1)). This element $\ensuremath{\omega}\ensuremath{\in}{H}^{3}(G,{U}_{T}(1))$ can be interpreted as either a label for the bulk 2D SPT phase or a label for the anomaly carried by the SPT edge theory. The basic idea behind our approach is to compute the FFF symbol associated with domain walls in a symmetry-broken edge theory; this domain-wall FFF symbol is precisely the anomaly we wish to compute. We demonstrate our approach with several SPT edge theories, including both lattice models and continuum field theories.