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:1611.09334 · Physical review. B./Physical review. B

Twisted gauge theories in three-dimensional Walker-Wang models

Zitao Wang, Xie Chen

Published Mar 24, 2017• 79 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.95.115142

Related papers

Twisted lattice gauge theory: Membrane operators, three-loop braiding, and topological charge

Joe Huxford, Dung Xuan Nguyen, Yong Baek Kim

$(3+1)$-dimensional topological phases can support loop-like excitations in addition to point-like ones, allowing for nontrivial loop-loop and point-loop braiding statistics not permitted to point-like excitations alone. Furthermore, these loop-like excitations can be linked together, changing their properties. In particular, this can lead to distinct three-loop braiding, involving two loops undergoing an exchange process while linked to a third loop. In this work, we investigate the loop-like excitations in a $(3+1)$-dimensional Hamiltonian realization of Dijkgraaf-Witten theory through direct construction of their membrane operators, for a general finite Abelian group and 4-cocycle twist. Using these membrane operators, we find the braiding relations and fusion rules for the loop-like excitations, including those linked to another loop-like excitation. Furthermore, we use these membrane operators to construct projection operators that measure the topological charge and show that the number of distinct topological charges measured by the 2-torus matches the ground-state degeneracy of the model on the 3-torus, explicitly confirming a general expectation for topological phases. This direct construction of the membrane operators sheds significant light on the key properties of the loop-like excitations in $(3+1)$-dimensional topological phases.

arXiv:2401.13042•Physical review. B./Physical review. B•Jul 8, 2024•0 citations

Twisted gauge theory model of topological phases in three dimensions

Yidun Wan, Juven Wang, Huan He

We propose an exactly solvable lattice Hamiltonian model of topological phases in $3+1$ dimensions, based on a generic finite group $G$ and a 4-cocycle $\ensuremath{\omega}$ over $G$. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the 3-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasiexcitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the $SL(3,\mathbb{Z})$ generators as the modular $S$ and $T$ matrices of the ground states, which yield a set of topological quantum numbers classified by $\ensuremath{\omega}$ and quantities derived from $\ensuremath{\omega}$. Our model fulfills a Hamiltonian extension of the $(3+1)$-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group $G$. This work is presented to be accessible for a wide range of physicists and mathematicians.

arXiv:1409.3216•Physical Review B•Jul 1, 2015•126 citations

Composite particle theory of three-dimensional gapped fermionic phases: Fractional topological insulators and charge-loop excitation symmetry

Peng Ye, Taylor L. Hughes, Joseph Maciejko +1 more

Topological phases of matter are usually realized in deconfined phases of gauge theories. In this context, confined phases with strongly fluctuating gauge fields seem to be irrelevant to the physics of topological phases. For example, the low-energy theory of the two-dimensional (2D) toric code model (i.e., the deconfined phase of ${\mathbb{Z}}_{2}$ gauge theory) is a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ Chern-Simons theory in which gauge charges (i.e., $e$ and $m$ particles) are deconfined and the gauge fields are gapped, while the confined phase is topologically trivial. In this paper, we point out a route to constructing exotic three-dimensional (3D) gapped fermionic phases in a confining phase of a gauge theory. Starting from a parton construction with strongly fluctuating compact $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ gauge fields, we construct gapped phases of interacting fermions by condensing two linearly independent bosonic composite particles consisting of partons and $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ magnetic monopoles. This can be regarded as a 3D generalization of the 2D Bais-Slingerland condensation mechanism. Charge fractionalization results from a Debye-H\"uckel--type screening cloud formed by the condensed composite particles. Within our general framework, we explore two aspects of symmetry-enriched 3D Abelian topological phases. First, we construct a new fermionic state of matter with time-reversal symmetry and $\mathrm{\ensuremath{\Theta}}\ensuremath{\ne}\ensuremath{\pi}$, the fractional topological insulator. Second, we generalize the notion of anyonic symmetry of 2D Abelian topological phases to the charge-loop excitation symmetry ($\mathsf{Charles}$) of 3D Abelian topological phases. We show that line twist defects, which realize $\mathsf{Charles}$ transformations, exhibit non-Abelian fusion properties.

arXiv:1603.02696•Physical review. B./Physical review. B•Sep 2, 2016•40 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

Three-dimensional anomalous twisted gauge theories with global symmetry: Implications for quantum spin liquids

Peng Ye

Topological spin liquids can be described by topological gauge theories with global symmetry. Due to the presence of both nontrivial bulk deconfined gauge fluxes and global symmetry, topological spin liquids are examples of the so-called ``symmetry enriched topological phases'' (SETs). In this paper, we find that, in some twisted versions of topological gauge theories (with discrete Abelian gauge group ${G}_{g}$), implementing a global symmetry (denoted by ${G}_{s}$) is anomalous although symmetry charge carried by topological pointlike excitations is normally fractionalized and classified by the second cohomology group. To demonstrate the anomaly, we fully gauge the global symmetry, rendering a new gauge theory that is not gauge invariant. Therefore, the SET order of the ground state is anomalous, which cannot exist in the three-dimensional system alone. Such an anomalous state construction generalizes the ``2D surface topological order'' to three dimensions. A concrete example with ${G}_{g}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}$ and ${G}_{s}={\mathbb{Z}}_{2}$ is calculated.

arXiv:1610.08645•Physical review. B./Physical review. B•Mar 16, 2018•20 citations

Topological Phases of Matter: Exactly Solvable Models and Classification

Zitao Wang

In this thesis, we study gapped topological phases of matter in systems with strong inter-particle interaction. They are challenging to analyze theoretically, because interaction not only gives rise to a plethora of phases that are otherwise absent, but also renders methods used to analyze non-interacting systems inadequate. By now, people have had a relatively systematic understanding of topological orders in two spatial dimensions. However, less is known about the higher dimensional cases. In Chapter 2, we will explore three dimensional long-range entangled topological orders in the framework of Walker-Wang models, which are a class of exactly solvable models for three-dimensional topological phases that are not known previously to be able to capture these phases. We find that they can represent a class of twisted discrete gauge theories, which were discovered using a different formalism. Meanwhile, a systematic theory of bosonic symmetry protected topological (SPT) phases in all spatial dimensions have been developed based on group cohomology. A generalization of the theory to group supercohomology has been proposed to classify and characterize fermionic SPT phases in all dimensions. However, it can only handle cases where the symmetry group of the system is a product of discrete unitary symmetries. Furthermore, the classification is known to be incomplete for certain symmetries. In Chapter 3, we will construct an exactly solvable model for the two-dimensional time-reversal-invariant topological superconductors, which could be valuable as a first attempt to a systematic understanding of strongly interacting fermionic SPT phases with anti-unitary symmetries in terms of exactly solvable models. In Chapter 4, we will propose an alternative classification of fermionic SPT phases using the spin cobordism theory, which hopefully can capture all the phases missing in the supercohomology classification. We test this proposal in the case of fermionic SPT phases with Z2 symmetry, where Z2 is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known fermionic SPT phases in space dimensions less than or equal to 3.

•Apr 29, 2019•0 citations
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 Z2\ifmmode×\else\texttimes\fiZ2{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}Z2​\ifmmode×\else\texttimes\fiZ2​. 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 Z2\ifmmode×\else\texttimes\fiZ2{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}Z2​\ifmmode×\else\texttimes\fiZ2​ 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.
(3+1)(3+1)(3+1)-dimensional topological phases can support loop-like excitations in addition to point-like ones, allowing for nontrivial loop-loop and point-loop braiding statistics not permitted to point-like excitations alone. Furthermore, these loop-like excitations can be linked together, changing their properties. In particular, this can lead to distinct three-loop braiding, involving two loops undergoing an exchange process while linked to a third loop. In this work, we investigate the loop-like excitations in a (3+1)(3+1)(3+1)-dimensional Hamiltonian realization of Dijkgraaf-Witten theory through direct construction of their membrane operators, for a general finite Abelian group and 4-cocycle twist. Using these membrane operators, we find the braiding relations and fusion rules for the loop-like excitations, including those linked to another loop-like excitation. Furthermore, we use these membrane operators to construct projection operators that measure the topological charge and show that the number of distinct topological charges measured by the 2-torus matches the ground-state degeneracy of the model on the 3-torus, explicitly confirming a general expectation for topological phases. This direct construction of the membrane operators sheds significant light on the key properties of the loop-like excitations in (3+1)(3+1)(3+1)-dimensional topological phases.
We propose an exactly solvable lattice Hamiltonian model of topological phases in 3+13+13+1 dimensions, based on a generic finite group GGG and a 4-cocycle $\ensuremath{\omega}$ over GGG. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the 3-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasiexcitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the SL(3,Z)SL(3,\mathbb{Z})SL(3,Z) generators as the modular SSS and TTT matrices of the ground states, which yield a set of topological quantum numbers classified by $\ensuremath{\omega}$ and quantities derived from $\ensuremath{\omega}$. Our model fulfills a Hamiltonian extension of the (3+1)(3+1)(3+1)-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group GGG. This work is presented to be accessible for a wide range of physicists and mathematicians.
Topological phases of matter are usually realized in deconfined phases of gauge theories. In this context, confined phases with strongly fluctuating gauge fields seem to be irrelevant to the physics of topological phases. For example, the low-energy theory of the two-dimensional (2D) toric code model (i.e., the deconfined phase of Z2{\mathbb{Z}}_{2}Z2​ gauge theory) is a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ Chern-Simons theory in which gauge charges (i.e., eee and mmm particles) are deconfined and the gauge fields are gapped, while the confined phase is topologically trivial. In this paper, we point out a route to constructing exotic three-dimensional (3D) gapped fermionic phases in a confining phase of a gauge theory. Starting from a parton construction with strongly fluctuating compact $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ gauge fields, we construct gapped phases of interacting fermions by condensing two linearly independent bosonic composite particles consisting of partons and $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ magnetic monopoles. This can be regarded as a 3D generalization of the 2D Bais-Slingerland condensation mechanism. Charge fractionalization results from a Debye-H\"uckel--type screening cloud formed by the condensed composite particles. Within our general framework, we explore two aspects of symmetry-enriched 3D Abelian topological phases. First, we construct a new fermionic state of matter with time-reversal symmetry and $\mathrm{\ensuremath{\Theta}}\ensuremath{\ne}\ensuremath{\pi}$, the fractional topological insulator. Second, we generalize the notion of anyonic symmetry of 2D Abelian topological phases to the charge-loop excitation symmetry (Charles\mathsf{Charles}Charles) of 3D Abelian topological phases. We show that line twist defects, which realize Charles\mathsf{Charles}Charles transformations, exhibit non-Abelian fusion properties.
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.
Topological spin liquids can be described by topological gauge theories with global symmetry. Due to the presence of both nontrivial bulk deconfined gauge fluxes and global symmetry, topological spin liquids are examples of the so-called ``symmetry enriched topological phases'' (SETs). In this paper, we find that, in some twisted versions of topological gauge theories (with discrete Abelian gauge group Gg{G}_{g}Gg​), implementing a global symmetry (denoted by Gs{G}_{s}Gs​) is anomalous although symmetry charge carried by topological pointlike excitations is normally fractionalized and classified by the second cohomology group. To demonstrate the anomaly, we fully gauge the global symmetry, rendering a new gauge theory that is not gauge invariant. Therefore, the SET order of the ground state is anomalous, which cannot exist in the three-dimensional system alone. Such an anomalous state construction generalizes the ``2D surface topological order'' to three dimensions. A concrete example with ${G}_{g}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}$ and Gs=Z2{G}_{s}={\mathbb{Z}}_{2}Gs​=Z2​ is calculated.
In this thesis, we study gapped topological phases of matter in systems with strong inter-particle interaction. They are challenging to analyze theoretically, because interaction not only gives rise to a plethora of phases that are otherwise absent, but also renders methods used to analyze non-interacting systems inadequate. By now, people have had a relatively systematic understanding of topological orders in two spatial dimensions. However, less is known about the higher dimensional cases. In Chapter 2, we will explore three dimensional long-range entangled topological orders in the framework of Walker-Wang models, which are a class of exactly solvable models for three-dimensional topological phases that are not known previously to be able to capture these phases. We find that they can represent a class of twisted discrete gauge theories, which were discovered using a different formalism. Meanwhile, a systematic theory of bosonic symmetry protected topological (SPT) phases in all spatial dimensions have been developed based on group cohomology. A generalization of the theory to group supercohomology has been proposed to classify and characterize fermionic SPT phases in all dimensions. However, it can only handle cases where the symmetry group of the system is a product of discrete unitary symmetries. Furthermore, the classification is known to be incomplete for certain symmetries. In Chapter 3, we will construct an exactly solvable model for the two-dimensional time-reversal-invariant topological superconductors, which could be valuable as a first attempt to a systematic understanding of strongly interacting fermionic SPT phases with anti-unitary symmetries in terms of exactly solvable models. In Chapter 4, we will propose an alternative classification of fermionic SPT phases using the spin cobordism theory, which hopefully can capture all the phases missing in the supercohomology classification. We test this proposal in the case of fermionic SPT phases with Z2 symmetry, where Z2 is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known fermionic SPT phases in space dimensions less than or equal to 3.