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:1808.09394 · Physical review. B./Physical review. B

Topological nonlinear σ -model, higher gauge theory, and a systematic construction of 3+1D topological orders for boson systems

Chenchang Zhu, Tian Lan, Xiao-Gang Wen

Published Jul 8, 2019• 46 citations
arXivHTMLPDF

Abstract

A discrete nonlinear $\ensuremath{\sigma}$-model is obtained by triangulate both the space-time ${M}^{d+1}$ and the target space $K$. If the path integral is given by the sum of all the simplicial homomorphisms $\ensuremath{\phi}:{M}^{d+1}\ensuremath{\rightarrow}K$ (i.e., maps without any topological defects), with an partition function that is independent of space-time triangulation, then the corresponding nonlinear $\ensuremath{\sigma}$-model will be called topological nonlinear $\ensuremath{\sigma}$-model which is exactly soluble. These exactly soluble models suggest that phase transitions induced by fluctuations with no topological defects usually produce a topologically ordered state and are topological phase transitions. In contrast, phase transitions induced by fluctuations with all topological defects give rise to trivial product states and are not topological phase transitions. Under the classification conjecture of Lan-Kong-Wen [Phys. Rev. X 8, 021074 (2018)], it is shown that, if $K$ is a space with only nontrivial first homotopy group $G$, which is finite, then these topological nonlinear $\ensuremath{\sigma}$-models can already realize all $3+1\text{D}$ bosonic topological orders without emergent fermions, which are described by Dijkgraaf-Witten theory with gauge group ${\ensuremath{\pi}}_{1}(K)=G$. Under the similar conjecture, we show that the $3+1\text{D}$ bosonic topological orders with emergent fermions can be realized by topological nonlinear $\ensuremath{\sigma}$-models with ${\ensuremath{\pi}}_{1}(K)=$ finite groups, ${\ensuremath{\pi}}_{2}(K)={Z}_{2}$, and ${\ensuremath{\pi}}_{n>2}(K)=0$. A subset of these topological nonlinear $\ensuremath{\sigma}$-models corresponds to 2-gauge theories, which realize and may classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections.

DOI
10.1103/physrevb.100.045105

Related papers

Construction of bosonic symmetry-protected-trivial states and their topological invariants via G×SO(∞) nonlinear σ models

Xiao-Gang Wen

It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear $\ensuremath{\sigma}$ models $(\mathrm{NL}\ensuremath{\sigma}\mathrm{Ms})$ of the symmetry group $G$ with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology ${\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$. But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by $G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}) \mathrm{NL}\ensuremath{\sigma}\mathrm{Ms}$ with various topological terms. Through the group cohomology ${\mathcal{H}}^{d}[G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}),\mathbb{R}/\mathbb{Z}]$, we find that the set of our constructed SPT phases in $d$-dimensional space-time are described by ${E}^{d}(G)\ensuremath{\rtimes}{\ensuremath{\bigoplus}}_{k=1}^{d\ensuremath{-}1}{\mathcal{H}}^{k}(G,{\text{iTO}}_{L}^{d\ensuremath{-}k})\ensuremath{\bigoplus}{\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$ where $G$ may contain time reversal. Here ${\text{iTO}}_{L}^{d}$ is the set of the topologically ordered phases in $d$-dimensional space-time that have no topological excitations, and one has ${\text{iTO}}_{L}^{1}={\text{iTO}}_{L}^{2}={\text{iTO}}_{L}^{4}={\text{iTO}}_{L}^{6}=0,\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{3}=\mathbb{Z},\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{5}={\mathbb{Z}}_{2},\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{7}=2\mathbb{Z}$. For $G=U(1)\ensuremath{\rtimes}{Z}_{2}^{T}$ (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond ${\mathcal{H}}^{d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},\mathbb{R}/\mathbb{Z}]$ are described by ${\mathbb{Z}}_{2}$ in 3 + 1D, $\mathbb{Z}$ in 4 + 1D, $3{\mathbb{Z}}_{2}$ in 5 + 1D, and $4{\mathbb{Z}}_{2}$ in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.

arXiv:1410.8477•Physical Review B•May 4, 2015•77 citations

Classification of (3+1)D Bosonic Topological Orders: The Case When Pointlike Excitations Are All Bosons

Liang Kong, Tian Lan, Xiao-Gang Wen

Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems, there is no nontrivial topological order, while in 2+1D bosonic systems, the topological orders are classified by the following pair: a modular tensor category and a chiral central charge. In this paper, following a new line of thinking, we find that in 3+1D the classification is much simpler than it was thought to be; we propose a partial classification of topological orders for 3+1D bosonic systems: If all the pointlike excitations are bosons, then such topological orders are classified by a simpler pair (G,ω_{4}): a finite group G and its group 4-cocycle ω_{4}∈H^{4}[G;U(1)] (up to group automorphisms). Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.

Physical Review Letters•Jun 1, 2018•33 citations

Classification of (3+1)D Bosonic Topological Orders: The Case When Pointlike Excitations Are All Bosons

Tian Lan, Liang Kong, Xiao-Gang Wen

Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group $G$ and its group 4-cocycle $ω_4 \in \mathcal H^4[G;U(1)]$ up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.

arXiv:1704.04221•Physical Review X•Jun 22, 2018•115 citations

Classification of symmetry enriched topological phases with exactly solvable models

Andrej Mesaroš, Ying Ran

Recently, a new class of quantum phases of matter---symmetry protected topological states, such as topological insulators---has attracted much attention. In presence of interactions, group cohomology provides a classification of these [Chen et al., Phys. Rev. B 87, 155114 (2013)]. These phases have short-ranged entanglement and no topological order in the bulk. However, when long-range entangled topological order is present, it is much less understood how to classify quantum phases of matter in the presence of global symmetries. Here we present a classification of bosonic gapped quantum phases with or without long-range entanglement in the presence or absence of on-site global symmetries. In $2+1$ dimensions, the quantum phases in the presence of a global symmetry group $SG$, and with topological order described by a finite gauge group $GG$, are classified by the cohomology group ${H}^{3}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Generally, in $d+1$ dimensions, such quantum phases are classified by ${H}^{d+1}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Although we only partially understand to what extent our classification is complete, we present an exactly solvable local bosonic model, in which the topological order is emergent, for each given class in our classification. When the global symmetry is absent, the topological order in our models is described by the general Dijkgraaf-Witten discrete gauge theories. When the topological order is absent, our models become the exactly solvable models for symmetry protected topological phases [Chen et al., Phys. Rev. B 87, 155114 (2013)]. When both the global symmetry and the topological order are present, our models describe symmetry enriched topological phases. Our classification includes, but goes beyond, the previously discussed projective symmetry group classification. Measurable signatures of these symmetry enriched topological phases and generalizations of our classification are discussed.

arXiv:1212.0835•Physical Review B•Apr 4, 2013•291 citations

Construction of bosonic symmetry-protected-trivial states and their topological invariants via G x SO(∞) nonlinear σ

Xiao-Gang Wen

It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear σ models (NLσMs) of the symmetry group G with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology H[superscript d](G,R/Z). But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by G × SO(∞) NLσMs with various topological terms. Through the group cohomology H[superscript d][G × SO(∞),R/Z], we find that the set of our constructed SPT phases in d-dimensional space-time are described by E[superscript d](G) ⋊ ⊕[d−1 over k=1]H[superscript k](G,iTO[d−k over L]) ⊕ H[superscript d](G,R/Z) where G may contain time reversal. Here iTO[d over L] is the set of the topologically ordered phases in d-dimensional space-time that have no topological excitations, and one has iTO[1 over L] = iTO[2 over L] = iTO[4 over L] = iTO[6 over L] = 0, iTO[3 over L] = Z, iTO[5 over L] = Z[subscript 2], iTO[7 over L] =2Z. For G=U(1) ⋊ Z[ T over 2] (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond H[superscript d][U(1) ⋊ Z[T over 2],R/Z] are described by Z[subscript 2] in 3 + 1D, Z in 4 + 1D, 3Z[subscript 2] in 5 + 1D, and 4Z[subscript 2] in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.

Physical Review Letters•May 1, 2015•4 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
A discrete nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ-model is obtained by triangulate both the space-time Md+1{M}^{d+1}Md+1 and the target space KKK. If the path integral is given by the sum of all the simplicial homomorphisms \ensuremathϕ:Md+1\ensuremath→K\ensuremath{\phi}:{M}^{d+1}\ensuremath{\rightarrow}K\ensuremathϕ:Md+1\ensuremath→K (i.e., maps without any topological defects), with an partition function that is independent of space-time triangulation, then the corresponding nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ-model will be called topological nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ-model which is exactly soluble. These exactly soluble models suggest that phase transitions induced by fluctuations with no topological defects usually produce a topologically ordered state and are topological phase transitions. In contrast, phase transitions induced by fluctuations with all topological defects give rise to trivial product states and are not topological phase transitions. Under the classification conjecture of Lan-Kong-Wen [Phys. Rev. X 8, 021074 (2018)], it is shown that, if KKK is a space with only nontrivial first homotopy group GGG, which is finite, then these topological nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ-models can already realize all 3+1D3+1\text{D}3+1D bosonic topological orders without emergent fermions, which are described by Dijkgraaf-Witten theory with gauge group \ensuremathπ1(K)=G{\ensuremath{\pi}}_{1}(K)=G\ensuremathπ1​(K)=G. Under the similar conjecture, we show that the 3+1D3+1\text{D}3+1D bosonic topological orders with emergent fermions can be realized by topological nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ-models with \ensuremathπ1(K)={\ensuremath{\pi}}_{1}(K)=\ensuremathπ1​(K)= finite groups, \ensuremathπ2(K)=Z2{\ensuremath{\pi}}_{2}(K)={Z}_{2}\ensuremathπ2​(K)=Z2​, and \ensuremathπn>2(K)=0{\ensuremath{\pi}}_{n>2}(K)=0\ensuremathπn>2​(K)=0. A subset of these topological nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ-models corresponds to 2-gauge theories, which realize and may classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections.
It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear $\ensuremath{\sigma}$ models $(\mathrm{NL}\ensuremath{\sigma}\mathrm{Ms})$ of the symmetry group GGG with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology Hd(G,R/Z){\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})Hd(G,R/Z). But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by $G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}) \mathrm{NL}\ensuremath{\sigma}\mathrm{Ms}$ with various topological terms. Through the group cohomology ${\mathcal{H}}^{d}[G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}),\mathbb{R}/\mathbb{Z}]$, we find that the set of our constructed SPT phases in ddd-dimensional space-time are described by ${E}^{d}(G)\ensuremath{\rtimes}{\ensuremath{\bigoplus}}_{k=1}^{d\ensuremath{-}1}{\mathcal{H}}^{k}(G,{\text{iTO}}_{L}^{d\ensuremath{-}k})\ensuremath{\bigoplus}{\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$ where GGG may contain time reversal. Here iTOLd{\text{iTO}}_{L}^{d}iTOLd​ is the set of the topologically ordered phases in ddd-dimensional space-time that have no topological excitations, and one has iTOL1=iTOL2=iTOL4=iTOL6=0,iTOL3=Z,iTOL5=Z2,iTOL7=2Z{\text{iTO}}_{L}^{1}={\text{iTO}}_{L}^{2}={\text{iTO}}_{L}^{4}={\text{iTO}}_{L}^{6}=0,\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{3}=\mathbb{Z},\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{5}={\mathbb{Z}}_{2},\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{7}=2\mathbb{Z}iTOL1​=iTOL2​=iTOL4​=iTOL6​=0,iTOL3​=Z,iTOL5​=Z2​,iTOL7​=2Z. For $G=U(1)\ensuremath{\rtimes}{Z}_{2}^{T}$ (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond ${\mathcal{H}}^{d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},\mathbb{R}/\mathbb{Z}]$ are described by Z2{\mathbb{Z}}_{2}Z2​ in 3 + 1D, Z\mathbb{Z}Z in 4 + 1D, 3Z23{\mathbb{Z}}_{2}3Z2​ in 5 + 1D, and 4Z24{\mathbb{Z}}_{2}4Z2​ in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.
Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems, there is no nontrivial topological order, while in 2+1D bosonic systems, the topological orders are classified by the following pair: a modular tensor category and a chiral central charge. In this paper, following a new line of thinking, we find that in 3+1D the classification is much simpler than it was thought to be; we propose a partial classification of topological orders for 3+1D bosonic systems: If all the pointlike excitations are bosons, then such topological orders are classified by a simpler pair (G,ω_{4}): a finite group G and its group 4-cocycle ω_{4}∈H^{4}[G;U(1)] (up to group automorphisms). Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.
Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group GGG and its group 4-cocycle ω4∈H4[G;U(1)]ω_4 \in \mathcal H^4[G;U(1)]ω4​∈H4[G;U(1)] up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.
Recently, a new class of quantum phases of matter---symmetry protected topological states, such as topological insulators---has attracted much attention. In presence of interactions, group cohomology provides a classification of these [Chen et al., Phys. Rev. B 87, 155114 (2013)]. These phases have short-ranged entanglement and no topological order in the bulk. However, when long-range entangled topological order is present, it is much less understood how to classify quantum phases of matter in the presence of global symmetries. Here we present a classification of bosonic gapped quantum phases with or without long-range entanglement in the presence or absence of on-site global symmetries. In 2+12+12+1 dimensions, the quantum phases in the presence of a global symmetry group SGSGSG, and with topological order described by a finite gauge group GGGGGG, are classified by the cohomology group ${H}^{3}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Generally, in d+1d+1d+1 dimensions, such quantum phases are classified by ${H}^{d+1}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Although we only partially understand to what extent our classification is complete, we present an exactly solvable local bosonic model, in which the topological order is emergent, for each given class in our classification. When the global symmetry is absent, the topological order in our models is described by the general Dijkgraaf-Witten discrete gauge theories. When the topological order is absent, our models become the exactly solvable models for symmetry protected topological phases [Chen et al., Phys. Rev. B 87, 155114 (2013)]. When both the global symmetry and the topological order are present, our models describe symmetry enriched topological phases. Our classification includes, but goes beyond, the previously discussed projective symmetry group classification. Measurable signatures of these symmetry enriched topological phases and generalizations of our classification are discussed.
It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear σ models (NLσMs) of the symmetry group G with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology H[superscript d](G,R/Z). But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by G × SO(∞) NLσMs with various topological terms. Through the group cohomology H[superscript d][G × SO(∞),R/Z], we find that the set of our constructed SPT phases in d-dimensional space-time are described by E[superscript d](G) ⋊ ⊕[d−1 over k=1]H[superscript k](G,iTO[d−k over L]) ⊕ H[superscript d](G,R/Z) where G may contain time reversal. Here iTO[d over L] is the set of the topologically ordered phases in d-dimensional space-time that have no topological excitations, and one has iTO[1 over L] = iTO[2 over L] = iTO[4 over L] = iTO[6 over L] = 0, iTO[3 over L] = Z, iTO[5 over L] = Z[subscript 2], iTO[7 over L] =2Z. For G=U(1) ⋊ Z[ T over 2] (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond H[superscript d][U(1) ⋊ Z[T over 2],R/Z] are described by Z[subscript 2] in 3 + 1D, Z in 4 + 1D, 3Z[subscript 2] in 5 + 1D, and 4Z[subscript 2] in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.
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.