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:2511.19589

Topological BF Theory construction of twisted dihedral quantum double phases from spontaneous symmetry breaking

Zhi-Qiang Gao, Chunxiao Liu, Joel E. Moore

cond-mat.str-elcond-mat.stat-mechhep-thquant-phSubmitted Nov 24, 2025• Announced Nov 26, 2025• 0 citations• Revised Jan 15, 2026 (v2)
v1 · Nov 24, 2025v2 · Jan 15, 2026
arXivHTMLPDF

Abstract

Nonabelian topological orders host exotic anyons central to quantum computing, yet established realizations rely on case-by-case constructions that are often conceptually involved. In this work, we present a systematic construction of nonabelian dihedral quantum double phases based on a continuous $O(2)$ gauge field. We first formulate a topological $S[O(2)\times O(2)]$ BF theory, and by identifying the Wilson loops and twist operators of this theory with anyons, we show that our topological BF theory reproduces the complete anyon data, and can incorporate all Dijkgraaf--Witten twists. Building on this correspondence, we present a microscopic model with $O(2)$ lattice gauge field coupled to Ising and rotor matter whose Higgsing yields the desired dihedral quantum double phase. A perturbative renormalization group analysis further suggests a direct transition from this phase to a $U(1)$ Coulomb or chiral topological phase at a stable multicritical point with emergent $O(3)$ symmetry. Our proposal offers an alternative route to nonabelian topological order with promising prospects in synthetic gauge field platforms.

Comments
6.5+10 pages, 3+2 figures

Related papers

Twisted quantum double model of topological phases in two dimensions

Yuting Hu, Yidun Wan, Yong-Shi Wu

We propose a discrete model---the twisted quantum double model---of 2D topological phases based on a finite group $G$ and a 3-cocycle $\ensuremath{\alpha}$ over $G$. The detailed properties of the ground states are studied, and we find that the ground-state subspace can be characterized in terms of the twisted quantum double ${D}^{\ensuremath{\alpha}}(G)$ of $G$. When $\ensuremath{\alpha}$ is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group $G$, in which the elementary excitations are known to be classified by the quantum double $D(G)$ of $G$. Our model can be viewed as a Hamiltonian extension of the Dijkgraaf-Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string-net $6j$ symbols to the corresponding 3-cocycles.

arXiv:1211.3695•Physical Review B•Mar 11, 2013•213 citations

Twisted gauge theories in three-dimensional Walker-Wang models

Zitao Wang, Xie Chen

Three-dimensional gauge theories with a discrete gauge group can emerge from spin models as a gapped topological phase with fractional point excitations (gauge charge) and loop excitations (gauge flux). It is known that 3D gauge theories can be ``twisted,'' in the sense that the gauge flux loops can have nontrivial braiding statistics among themselves and such twisted gauge theories are realized in models discovered by Dijkgraaf and Witten. A different framework to systematically construct three-dimensional topological phases was proposed by Walker and Wang and a series of examples have been studied. Can the Walker-Wang construction be used to realize the topological order in twisted gauge theories? This is not immediately clear because the Walker-Wang construction is based on a loop condensation picture while the Dijkgraaf-Witten theory is based on a membrane condensation picture. In this paper, we show that the answer to this question is Yes, by presenting an explicit construction of the Walker-Wang models which realize both the twisted and untwisted gauge theories with gauge group ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$. We identify the topological order of the models by performing modular transformations on the ground-state wave functions and show that the modular matrices exactly match those for the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge theories. By relating the Walker-Wang construction to the Dijkgraaf-Witten construction, our result opens up a way to study twisted gauge theories with fermonic charges, and correspondingly strongly interacting fermionic symmetry protected topological phases and their surface states, through exactly solvable models.

arXiv:1611.09334•Physical review. B./Physical review. B•Mar 24, 2017•79 citations

Field theories for gauged symmetry-protected topological phases: Non-Abelian anyons with Abelian gauge group Z2⊗3

Huan He, Yunqin Zheng, Curt W. von Keyserlingk

Dijkgraaf-Witten (DW) theories are of recent interest to the condensed matter community, in part because they represent topological phases of matter, but also because they characterize the response theory of certain symmetry protected topological (SPT) phases. However, as yet there has not been a comprehensive treatment of the spectra of these models in the field theoretic setting even for Abelian gauge groups, the goal of this work is to fill the gap in the literature, especially for a selection of DW models with Abelian gauge groups but non-Abelian topological order. Particularly, we focus on the appearance of non-Abelian statistics in type-III twisted DW theories with Abelian gauge groups ${\mathbb{Z}}_{2}^{\ensuremath{\bigotimes}3}$. There are only 22 distinguishable line operators, and their fusion rules and correlation functions are calculated. The flux insertion operators have quantum dimension 2, which clearly demonstrates the non-Abelian topological order of type-III twisted DW theories.

arXiv:1608.05393•Physical review. B./Physical review. B•Jan 19, 2017•32 citations

Self-dual $S_3$ gauge theory in 2+1d: lattice model and topological phase transitions

Da-Chuan Lu, Chong Wang, Ashvin Vishwanath

Electric-magnetic self-duality of the $\mathbb{Z}_2$ gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the $S_3$ quantum double $\mathcal{D}(S_3)$ on a tensor product Hilbert space in which the $\mathbb{Z}^{\mathrm{em}}_2$ anyon-permutation symmetry, exchanging the non-Abelian chargeon $C$ and fluxon $F$, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent $\mathbb{Z}_2^{\mathrm{em}}$-preserving bosonic perturbations, driving $\mathcal{D}(S_3)$ into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles $\mathcal{D}(D_{2n})$. Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.

cond-mat.str-elCross-listedarXiv:2608.05294•Aug 7, 2026•0 citations

Dihedral twist liquid models in the orthogonal family

Jeffrey C. Y. Teo, Yichen Hu

We present a family of electron-based coupled-wire models of bosonic orbifold topological phases, referred to as twist liquids, in two spatial dimensions. All local fermion degrees of freedom are gapped and removed from the topological order by many-body interactions. Bosonic chiral spin liquids and anyonic superconductors are constructed on an array of interacting wires, each supports emergent massless Majorana fermions that are non-local (fractional) and constitute the $SO(N)$ Kac-Moody Wess-Zumino-Witten algebra at level 1. We focus on the dihedral $D_k$ symmetry of $SO(2n)_1$, and its promotion to a gauge symmetry by manipulating the locality of fermion pairs. Gauging the symmetry (sub)group generates the $\mathcal{C}/G$ twist liquids, where $G=\mathbb{Z}_2$ for $\mathcal{C}=U(1)_l$, $SU(n)_1$, and $G=\mathbb{Z}_2$, $\mathbb{Z}_k$, $D_k$ for $\mathcal{C}=SO(2n)_1$. We construct exactly solvable models for all of these topological states. We prove the presence of a bulk excitation energy gap and demonstrate the appearance of edge orbifold conformal field theories corresponding to the twist liquid topological orders. We analyze the statistical properties of the anyon excitations, including the non-Abelian metaplectic anyons and a new class of quasiparticles referred to as Ising-fluxons. We show an eight-fold periodic gauging pattern in $SO(2n)_1/G$ by identifying the non-chiral components of the twist liquids with discrete gauge theories.

•Jul 15, 2021•0 citations

Dihedral twist liquid models from emergent Majorana fermions

Jeffrey C. Y. Teo, Yichen Hu

We present a family of electron-based coupled-wire models of bosonic orbifold topological phases, referred to as twist liquids, in two spatial dimensions. All local fermion degrees of freedom are gapped and removed from the topological order by many-body interactions. Bosonic chiral spin liquids and anyonic superconductors are constructed on an array of interacting wires, each supports emergent massless Majorana fermions that are non-local (fractional) and constitute the SO(N) Kac-Moody Wess-Zumino-Witten algebra at level 1. We focus on the dihedral Dk symmetry of SO(2n)1 , and its promotion to a gauge symmetry by manipulating the locality of fermion pairs. Gauging the symmetry (sub)group generates the C/G twist liquids, where G=Z2 for C=U(1)l , SU(n)1 , and G=Z2 , Zk , Dk for C=SO(2n)1 . We construct exactly solvable models for all of these topological states. We prove the presence of a bulk excitation energy gap and demonstrate the appearance of edge orbifold conformal field theories corresponding to the twist liquid topological orders. We analyze the statistical properties of the anyon excitations, including the non-Abelian metaplectic anyons and a new class of quasiparticles referred to as Ising-fluxons. We show an eight-fold periodic gauging pattern in SO(2n)1/G by identifying the non-chiral components of the twist liquids with discrete gauge theories.

arXiv:2107.07242•Quantum•Mar 30, 2023•3 citations
Nonabelian topological orders host exotic anyons central to quantum computing, yet established realizations rely on case-by-case constructions that are often conceptually involved. In this work, we present a systematic construction of nonabelian dihedral quantum double phases based on a continuous O(2)O(2)O(2) gauge field. We first formulate a topological S[O(2)×O(2)]S[O(2)\times O(2)]S[O(2)×O(2)] BF theory, and by identifying the Wilson loops and twist operators of this theory with anyons, we show that our topological BF theory reproduces the complete anyon data, and can incorporate all Dijkgraaf--Witten twists. Building on this correspondence, we present a microscopic model with O(2)O(2)O(2) lattice gauge field coupled to Ising and rotor matter whose Higgsing yields the desired dihedral quantum double phase. A perturbative renormalization group analysis further suggests a direct transition from this phase to a U(1)U(1)U(1) Coulomb or chiral topological phase at a stable multicritical point with emergent O(3)O(3)O(3) symmetry. Our proposal offers an alternative route to nonabelian topological order with promising prospects in synthetic gauge field platforms.
We propose a discrete model---the twisted quantum double model---of 2D topological phases based on a finite group GGG and a 3-cocycle $\ensuremath{\alpha}$ over GGG. The detailed properties of the ground states are studied, and we find that the ground-state subspace can be characterized in terms of the twisted quantum double ${D}^{\ensuremath{\alpha}}(G)$ of GGG. When $\ensuremath{\alpha}$ is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group GGG, in which the elementary excitations are known to be classified by the quantum double D(G)D(G)D(G) of GGG. Our model can be viewed as a Hamiltonian extension of the Dijkgraaf-Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string-net 6j6j6j symbols to the corresponding 3-cocycles.
Three-dimensional gauge theories with a discrete gauge group can emerge from spin models as a gapped topological phase with fractional point excitations (gauge charge) and loop excitations (gauge flux). It is known that 3D gauge theories can be ``twisted,'' in the sense that the gauge flux loops can have nontrivial braiding statistics among themselves and such twisted gauge theories are realized in models discovered by Dijkgraaf and Witten. A different framework to systematically construct three-dimensional topological phases was proposed by Walker and Wang and a series of examples have been studied. Can the Walker-Wang construction be used to realize the topological order in twisted gauge theories? This is not immediately clear because the Walker-Wang construction is based on a loop condensation picture while the Dijkgraaf-Witten theory is based on a membrane condensation picture. In this paper, we show that the answer to this question is Yes, by presenting an explicit construction of the Walker-Wang models which realize both the twisted and untwisted gauge theories with gauge group ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$. We identify the topological order of the models by performing modular transformations on the ground-state wave functions and show that the modular matrices exactly match those for the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge theories. By relating the Walker-Wang construction to the Dijkgraaf-Witten construction, our result opens up a way to study twisted gauge theories with fermonic charges, and correspondingly strongly interacting fermionic symmetry protected topological phases and their surface states, through exactly solvable models.
Dijkgraaf-Witten (DW) theories are of recent interest to the condensed matter community, in part because they represent topological phases of matter, but also because they characterize the response theory of certain symmetry protected topological (SPT) phases. However, as yet there has not been a comprehensive treatment of the spectra of these models in the field theoretic setting even for Abelian gauge groups, the goal of this work is to fill the gap in the literature, especially for a selection of DW models with Abelian gauge groups but non-Abelian topological order. Particularly, we focus on the appearance of non-Abelian statistics in type-III twisted DW theories with Abelian gauge groups ${\mathbb{Z}}_{2}^{\ensuremath{\bigotimes}3}$. There are only 22 distinguishable line operators, and their fusion rules and correlation functions are calculated. The flux insertion operators have quantum dimension 2, which clearly demonstrates the non-Abelian topological order of type-III twisted DW theories.
Self-dual S3S_3S3​ gauge theory in 2+1d: lattice model and topological phase transitions
Electric-magnetic self-duality of the Z2\mathbb{Z}_2Z2​ gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the S3S_3S3​ quantum double D(S3)\mathcal{D}(S_3)D(S3​) on a tensor product Hilbert space in which the Z2em\mathbb{Z}^{\mathrm{em}}_2Z2em​ anyon-permutation symmetry, exchanging the non-Abelian chargeon CCC and fluxon FFF, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent Z2em\mathbb{Z}_2^{\mathrm{em}}Z2em​-preserving bosonic perturbations, driving D(S3)\mathcal{D}(S_3)D(S3​) into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles D(D2n)\mathcal{D}(D_{2n})D(D2n​). Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.
We present a family of electron-based coupled-wire models of bosonic orbifold topological phases, referred to as twist liquids, in two spatial dimensions. All local fermion degrees of freedom are gapped and removed from the topological order by many-body interactions. Bosonic chiral spin liquids and anyonic superconductors are constructed on an array of interacting wires, each supports emergent massless Majorana fermions that are non-local (fractional) and constitute the SO(N)SO(N)SO(N) Kac-Moody Wess-Zumino-Witten algebra at level 1. We focus on the dihedral DkD_kDk​ symmetry of SO(2n)1SO(2n)_1SO(2n)1​, and its promotion to a gauge symmetry by manipulating the locality of fermion pairs. Gauging the symmetry (sub)group generates the C/G\mathcal{C}/GC/G twist liquids, where G=Z2G=\mathbb{Z}_2G=Z2​ for C=U(1)l\mathcal{C}=U(1)_lC=U(1)l​, SU(n)1SU(n)_1SU(n)1​, and G=Z2G=\mathbb{Z}_2G=Z2​, Zk\mathbb{Z}_kZk​, DkD_kDk​ for C=SO(2n)1\mathcal{C}=SO(2n)_1C=SO(2n)1​. We construct exactly solvable models for all of these topological states. We prove the presence of a bulk excitation energy gap and demonstrate the appearance of edge orbifold conformal field theories corresponding to the twist liquid topological orders. We analyze the statistical properties of the anyon excitations, including the non-Abelian metaplectic anyons and a new class of quasiparticles referred to as Ising-fluxons. We show an eight-fold periodic gauging pattern in SO(2n)1/GSO(2n)_1/GSO(2n)1​/G by identifying the non-chiral components of the twist liquids with discrete gauge theories.
We present a family of electron-based coupled-wire models of bosonic orbifold topological phases, referred to as twist liquids, in two spatial dimensions. All local fermion degrees of freedom are gapped and removed from the topological order by many-body interactions. Bosonic chiral spin liquids and anyonic superconductors are constructed on an array of interacting wires, each supports emergent massless Majorana fermions that are non-local (fractional) and constitute the SO(N) Kac-Moody Wess-Zumino-Witten algebra at level 1. We focus on the dihedral Dk symmetry of SO(2n)1 , and its promotion to a gauge symmetry by manipulating the locality of fermion pairs. Gauging the symmetry (sub)group generates the C/G twist liquids, where G=Z2 for C=U(1)l , SU(n)1 , and G=Z2 , Zk , Dk for C=SO(2n)1 . We construct exactly solvable models for all of these topological states. We prove the presence of a bulk excitation energy gap and demonstrate the appearance of edge orbifold conformal field theories corresponding to the twist liquid topological orders. We analyze the statistical properties of the anyon excitations, including the non-Abelian metaplectic anyons and a new class of quasiparticles referred to as Ising-fluxons. We show an eight-fold periodic gauging pattern in SO(2n)1/G by identifying the non-chiral components of the twist liquids with discrete gauge theories.