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:2109.10913 · Physical review. B./Physical review. B

Fermionic symmetry fractionalization in ( 2 + 1 ) dimensions

Daniel Bulmash, Maissam Barkeshli

Published Mar 11, 2022• 44 citations
arXivHTMLPDF

Abstract

We develop a systematic theory of symmetry fractionalization for fermionic topological phases of matter in $(2+1)\mathrm{D}$ with a general fermionic symmetry group ${G}_{f}$. In general, ${G}_{f}$ is a central extension of the bosonic symmetry group ${G}_{b}$ by fermion parity, ${(\ensuremath{-}1)}^{F}$, characterized by a nontrivial cohomology class $[{\ensuremath{\omega}}_{2}]\ensuremath{\in}{\mathcal{H}}^{2}({G}_{b},{\mathbb{Z}}_{2})$. We show how the presence of local fermions places a number of constraints on the algebraic data that defines the action of the symmetry on the supermodular tensor category that characterizes the anyon content. We find two separate obstructions to defining symmetry fractionalization, which we refer to as the bosonic and fermionic symmetry localization obstructions. The former is valued in ${\mathcal{H}}^{3}({G}_{b},K(\mathcal{C}))$, while the latter is valued in either ${\mathcal{H}}^{3}({G}_{b},\mathcal{A}/{1,\ensuremath{\psi}})$ or ${Z}^{2}({G}_{b},{\mathbb{Z}}_{2})$ depending on additional details of the theory. $K(\mathcal{C})$ is the Abelian group of functions from anyons to $\mathrm{U}(1)$ phases obeying the fusion rules, $\mathcal{A}$ is the Abelian group defined by fusion of Abelian anyons, and $\ensuremath{\psi}$ is the fermion. When these obstructions vanish, we show that distinct symmetry fractionalization patterns form a torsor over ${\mathcal{H}}^{2}({G}_{b},\mathcal{A}/{1,\ensuremath{\psi}})$. We study a number of examples in detail; in particular, we provide a characterization of fermionic Kramers degeneracy arising in symmetry class DIII within this general framework, and we discuss fractional quantum Hall and ${\mathbb{Z}}_{2}$ quantum spin liquid states of electrons.

DOI
10.1103/physrevb.105.125114

Related papers

Characterization and Classification of Fermionic Symmetry Enriched Topological Phases

David Aasen, Parsa Bonderson, Christina Knapp

We examine the interplay of symmetry and topological order in $2+1$ dimensional fermionic topological phases of matter. We define fermionic topological symmetries acting on the emergent topological effective theory described using braided tensor category theory. Connecting this to the ${\cal G}^{\rm f}$ fermionic symmetry of the microscopic physical system, we characterize and classify symmetry fractionalization in fermionic topological phases. We find that the physical fermion provides constraints that result in a tiered structure of obstructions and classification of fractionalization with respect to the physical fermions, the quasiparticles, and the vortices. The fractionalization of the (bosonic) symmetry $G= {\cal G}^{\rm f}/\mathbb{Z}_2^{\rm f}$ on the physical fermions is essentially the central extension of $G$ by the $\mathbb{Z}_2^{\rm f}$ fermion parity conservation that yields the fermionic symmetry ${\cal G}^{\rm f}$. We develop an algebraic theory of ${\cal G}^{\rm f}$ symmetry defects for fermionic topological phases using $G$-crossed braided tensor category theory. This formalism allows us to fully characterize and classify $2+1$ dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group ${\cal G}^{\rm f}$. We first apply this formalism to extract the minimal data specifying a general fermionic symmetry protected topological phase, and demonstrate that such phases with fixed ${\cal G}^{\rm f}$ form a group under fermionic stacking. Then we analyze general fermionic symmetry enriched topological phases and find their classification is given torsorially by the classification of the symmetry fractionalization of quasiparticles combined with the classification of fermionic symmetry protected topological phases. We illustrate our results by detailing a number of examples, including all the invertible fermionic topological phases.

arXiv:2109.10911•Sep 22, 2021•27 citations

Anomaly cascade in ( 2+1 )-dimensional fermionic topological phases

Daniel Bulmash, Maissam Barkeshli

We develop a theory of anomalies of fermionic topological phases of matter in ($2+1$)D with a general fermionic symmetry group ${G}_{f}$. In general, ${G}_{f}$ can be a nontrivial central extension of the bosonic symmetry group ${G}_{b}$ by fermion parity ${(\ensuremath{-}1)}^{F}$. We encounter four layers of obstructions to gauging the ${G}_{f}$ symmetry, which we dub the anomaly cascade: (i) An ${\mathcal{H}}^{1}({G}_{b},{\mathbb{Z}}_{\mathbf{T}})$ obstruction to extending the symmetry permutations on the anyons to the fermion parity gauged theory, (ii) An ${\mathcal{H}}^{2}({G}_{b},kerr)$ obstruction to extending the ${G}_{b}$ group structure of the symmetry permutations to the fermion parity gauged theory, where $r$ is a map that restricts symmetries of the fermion parity gauged theory to the anyon theory, (iii) An ${\mathcal{H}}^{3}({G}_{b},{\mathbb{Z}}_{2})$ obstruction to extending the symmetry fractionalization class to the fermion parity gauged theory, and (iv) the well-known ${\mathcal{H}}^{4}({G}_{b},U(1[))]$ obstruction to developing a consistent theory of ${G}_{b}$ symmetry defects for the fermion parity gauged theory. We describe how the ${\mathcal{H}}^{2}$ obstruction can be canceled by anomaly inflow from a bulk ($3+1$)D symmetry-protected topological state (SPT) and also its relation to the Arf invariant of spin structures on a torus. If any anomaly in the above sequence is nontrivial, the subsequent ones become relative anomalies. A number of conjectures regarding symmetry actions on super-modular categories, guided by general expectations of anomalies in physics, are also presented.

arXiv:2109.10922•Physical review. B./Physical review. B•Apr 15, 2022•19 citations

Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries

Tian Lan, Liang Kong, Xiao-Gang Wen

We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry $G$. The key is to use the so-called symmetric fusion category $\mathcal{E}$ to describe the symmetry. Here, $\mathcal{E}=\mathrm{sRep}({\mathbb{Z}}_{2}^{f})$ describing particles in a fermionic product state without symmetry, or $\mathcal{E}=\mathrm{sRep}({G}^{f}) [\mathcal{E}=\mathrm{Rep}(G)]$ describing particles in a fermionic (bosonic) product state with symmetry $G$. Then, topological orders with symmetry $\mathcal{E}$ are classified by nondegenerate unitary braided fusion categories over $\mathcal{E}$, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to $p+\phantom{\rule{1.0pt}{0ex}}\mathrm{i}\phantom{\rule{1.0pt}{0ex}}p$ fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the $K=\left(\begin{array}{cc}\hfill \ensuremath{-}1& \hfill 0\\ \hfill 0& \hfill 2\end{array}\right)$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge $c=\frac{1}{4}$, whose only topological excitation has non-Abelian statistics with spin $s=\frac{1}{4}$ and quantum dimension $d=1+\sqrt{2}$.

arXiv:1507.04673•Physical review. B./Physical review. B•Oct 10, 2016•126 citations

Symmetry-preserving boundary of (2+1)D fractional quantum Hall states

Ryohei Kobayashi

We investigate symmetry-preserving gapped boundary of $(2+1)$-dimensional $[(2+1)\mathrm{D}]$ topological phases with global symmetry, which can be either bosonic or fermionic. We develop a general algebraic description for gapped boundary condition for symmetry-enriched or fermionic topological phases, extending the framework of Lagrangian algebra anyon for bosonic phases without symmetry. We then focus on application to the case with U(1) symmetry. We derive new obstructions to symmetry-preserving gapped boundary for U(1)${}^{f}$-symmetric $(2+1)\mathrm{D}$ fermionic topological phases, which are beyond chiral central charge ${c}_{\ensuremath{-}}$ and electric Hall conductivity ${\ensuremath{\sigma}}_{H}$. These obstructions are given by a simple Gauss-Milgram-type formula valid for supermodular category and regarded as a higher version of ${c}_{\ensuremath{-}}$ and ${\ensuremath{\sigma}}_{H}$.

arXiv:2203.08156•Physical Review Research•Aug 19, 2022•11 citations

Symmetry fractionalization and twist defects

Nicolas Tarantino, Netanel H Lindner, Lukasz Fidkowski

Topological order in two-dimensions can be described in terms of deconfined quasiparticle excitations—anyons—and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization—termed symmetry enriched topological order. When the global symmetry group G , which we take to be discrete, does not change topological superselection sectors—i.e. does not change one type of anyon into a different type of anyon—one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with the second cohomology group H 2 ( G , A a b e l i a n ) being the relevant group. In this paper, we treat the general case of a symmetry group G possibly permuting anyon types. We show that despite the lack of a local action of G , one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic ‘twist’ defects of the symmetry. Furthermore, building on work of Hermele (2014 Phys. Rev. B 90 184418 ), we construct a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.

arXiv:1506.06754•New Journal of Physics•Mar 7, 2016•145 citations

Fractionalizing global symmetry on looplike topological excitations

Shang-Qiang Ning, Zheng-Xin Liu, Peng Ye

Symmetry fractionalization on topological excitations is one of the most remarkable quantum phenomena in topological orders with symmetry, i.e., symmetry-enriched topological phases. While much progress has been theoretically and experimentally made in two dimensions (2D), the understanding on symmetry fractionalization in 3D is far from complete. A long-standing challenge is to understand symmetry fractionalization on looplike topological excitations, which are spatially extended objects. In this paper, we construct a powerful topological-field-theoretical framework approach for 3D topological orders, which leads to a systematic characterization and classification of symmetry fractionalization. For systems with Abelian gauge groups (${G}_{g}$) and Abelian symmetry groups (${G}_{s}$), we successfully establish equivalence classes that lead to a finite number of patterns of symmetry fractionalization, although there are notoriously infinite number of Lagrangian-descriptions of the system. Based on this, we compute topologically distinct types of fractional symmetry charges carried by particles. Then, for each type, we compute topologically distinct statistical phases of braiding processes among loop excitations and external symmetry fluxes. As a result, we are able to unambiguously list all physical observables for each pattern of symmetry fractionalization. We present detailed calculations on many concrete examples. As an example, we find that the symmetry fractionalization in an untwisted ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological order with ${\mathbb{Z}}_{2}$ symmetry is classified by ${({\mathbb{Z}}_{2})}^{6}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}$. If the topological order is twisted, the classification reduces to ${({\mathbb{Z}}_{2})}^{6}$ in which particle excitations always carry integer charge. Inspired by the field-theoretical analysis, we find that our classification of symmetry fractionalization with twist $\ensuremath{\omega}$ can be formally organized into an algebraic formalism: ${⨁}_{{\ensuremath{\nu}}_{i}}{\mathcal{H}}^{4}({G}_{g}{\ensuremath{\leftthreetimes}}_{{\ensuremath{\nu}}_{i}}{G}_{s},U(1))/{\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i}),$ where anomaly-free symmetry fractionalization on particles ${\ensuremath{\nu}}_{i}\ensuremath{\in}{\mathcal{H}}_{non}^{2}({G}_{s},{G}_{g})$. Despite the lack of detailed dependence of ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$ on $\ensuremath{\omega}$ and ${\ensuremath{\nu}}_{i}$, the present field-theoretical approach allows us to efficiently calculate and understand ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$. Several future directions are presented at the end of this paper.

arXiv:1801.01638•Physical review. B./Physical review. B•May 27, 2022•12 citations
We develop a systematic theory of symmetry fractionalization for fermionic topological phases of matter in (2+1)D(2+1)\mathrm{D}(2+1)D with a general fermionic symmetry group Gf{G}_{f}Gf​. In general, Gf{G}_{f}Gf​ is a central extension of the bosonic symmetry group Gb{G}_{b}Gb​ by fermion parity, (\ensuremath−1)F{(\ensuremath{-}1)}^{F}(\ensuremath−1)F, characterized by a nontrivial cohomology class [\ensuremathω2]\ensuremath∈H2(Gb,Z2)[{\ensuremath{\omega}}_{2}]\ensuremath{\in}{\mathcal{H}}^{2}({G}_{b},{\mathbb{Z}}_{2})[\ensuremathω2​]\ensuremath∈H2(Gb​,Z2​). We show how the presence of local fermions places a number of constraints on the algebraic data that defines the action of the symmetry on the supermodular tensor category that characterizes the anyon content. We find two separate obstructions to defining symmetry fractionalization, which we refer to as the bosonic and fermionic symmetry localization obstructions. The former is valued in H3(Gb,K(C)){\mathcal{H}}^{3}({G}_{b},K(\mathcal{C}))H3(Gb​,K(C)), while the latter is valued in either H3(Gb,A/1,\ensuremathψ){\mathcal{H}}^{3}({G}_{b},\mathcal{A}/{1,\ensuremath{\psi}})H3(Gb​,A/1,\ensuremathψ) or Z2(Gb,Z2){Z}^{2}({G}_{b},{\mathbb{Z}}_{2})Z2(Gb​,Z2​) depending on additional details of the theory. K(C)K(\mathcal{C})K(C) is the Abelian group of functions from anyons to U(1)\mathrm{U}(1)U(1) phases obeying the fusion rules, A\mathcal{A}A is the Abelian group defined by fusion of Abelian anyons, and \ensuremathψ\ensuremath{\psi}\ensuremathψ is the fermion. When these obstructions vanish, we show that distinct symmetry fractionalization patterns form a torsor over H2(Gb,A/1,\ensuremathψ){\mathcal{H}}^{2}({G}_{b},\mathcal{A}/{1,\ensuremath{\psi}})H2(Gb​,A/1,\ensuremathψ). We study a number of examples in detail; in particular, we provide a characterization of fermionic Kramers degeneracy arising in symmetry class DIII within this general framework, and we discuss fractional quantum Hall and Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquid states of electrons.
We examine the interplay of symmetry and topological order in 2+12+12+1 dimensional fermionic topological phases of matter. We define fermionic topological symmetries acting on the emergent topological effective theory described using braided tensor category theory. Connecting this to the Gf{\cal G}^{\rm f}Gf fermionic symmetry of the microscopic physical system, we characterize and classify symmetry fractionalization in fermionic topological phases. We find that the physical fermion provides constraints that result in a tiered structure of obstructions and classification of fractionalization with respect to the physical fermions, the quasiparticles, and the vortices. The fractionalization of the (bosonic) symmetry G=Gf/Z2fG= {\cal G}^{\rm f}/\mathbb{Z}_2^{\rm f}G=Gf/Z2f​ on the physical fermions is essentially the central extension of GGG by the Z2f\mathbb{Z}_2^{\rm f}Z2f​ fermion parity conservation that yields the fermionic symmetry Gf{\cal G}^{\rm f}Gf. We develop an algebraic theory of Gf{\cal G}^{\rm f}Gf symmetry defects for fermionic topological phases using GGG-crossed braided tensor category theory. This formalism allows us to fully characterize and classify 2+12+12+1 dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group Gf{\cal G}^{\rm f}Gf. We first apply this formalism to extract the minimal data specifying a general fermionic symmetry protected topological phase, and demonstrate that such phases with fixed Gf{\cal G}^{\rm f}Gf form a group under fermionic stacking. Then we analyze general fermionic symmetry enriched topological phases and find their classification is given torsorially by the classification of the symmetry fractionalization of quasiparticles combined with the classification of fermionic symmetry protected topological phases. We illustrate our results by detailing a number of examples, including all the invertible fermionic topological phases.
We develop a theory of anomalies of fermionic topological phases of matter in (2+12+12+1)D with a general fermionic symmetry group Gf{G}_{f}Gf​. In general, Gf{G}_{f}Gf​ can be a nontrivial central extension of the bosonic symmetry group Gb{G}_{b}Gb​ by fermion parity ${(\ensuremath{-}1)}^{F}$. We encounter four layers of obstructions to gauging the Gf{G}_{f}Gf​ symmetry, which we dub the anomaly cascade: (i) An H1(Gb,ZT){\mathcal{H}}^{1}({G}_{b},{\mathbb{Z}}_{\mathbf{T}})H1(Gb​,ZT​) obstruction to extending the symmetry permutations on the anyons to the fermion parity gauged theory, (ii) An H2(Gb,kerr){\mathcal{H}}^{2}({G}_{b},kerr)H2(Gb​,kerr) obstruction to extending the Gb{G}_{b}Gb​ group structure of the symmetry permutations to the fermion parity gauged theory, where rrr is a map that restricts symmetries of the fermion parity gauged theory to the anyon theory, (iii) An H3(Gb,Z2){\mathcal{H}}^{3}({G}_{b},{\mathbb{Z}}_{2})H3(Gb​,Z2​) obstruction to extending the symmetry fractionalization class to the fermion parity gauged theory, and (iv) the well-known H4(Gb,U(1[))]{\mathcal{H}}^{4}({G}_{b},U(1[))]H4(Gb​,U(1[))] obstruction to developing a consistent theory of Gb{G}_{b}Gb​ symmetry defects for the fermion parity gauged theory. We describe how the H2{\mathcal{H}}^{2}H2 obstruction can be canceled by anomaly inflow from a bulk (3+13+13+1)D symmetry-protected topological state (SPT) and also its relation to the Arf invariant of spin structures on a torus. If any anomaly in the above sequence is nontrivial, the subsequent ones become relative anomalies. A number of conjectures regarding symmetry actions on super-modular categories, guided by general expectations of anomalies in physics, are also presented.
We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry GGG. The key is to use the so-called symmetric fusion category E\mathcal{E}E to describe the symmetry. Here, E=sRep(Z2f)\mathcal{E}=\mathrm{sRep}({\mathbb{Z}}_{2}^{f})E=sRep(Z2f​) describing particles in a fermionic product state without symmetry, or E=sRep(Gf)[E=Rep(G)]\mathcal{E}=\mathrm{sRep}({G}^{f}) [\mathcal{E}=\mathrm{Rep}(G)]E=sRep(Gf)[E=Rep(G)] describing particles in a fermionic (bosonic) product state with symmetry GGG. Then, topological orders with symmetry E\mathcal{E}E are classified by nondegenerate unitary braided fusion categories over E\mathcal{E}E, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to p+ipp+\phantom{\rule{1.0pt}{0ex}}\mathrm{i}\phantom{\rule{1.0pt}{0ex}}pp+ip fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the $K=\left(\begin{array}{cc}\hfill \ensuremath{-}1& \hfill 0\\ \hfill 0& \hfill 2\end{array}\right)$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge c=14c=\frac{1}{4}c=41​, whose only topological excitation has non-Abelian statistics with spin s=14s=\frac{1}{4}s=41​ and quantum dimension d=1+2d=1+\sqrt{2}d=1+2​.
We investigate symmetry-preserving gapped boundary of (2+1)(2+1)(2+1)-dimensional [(2+1)D][(2+1)\mathrm{D}][(2+1)D] topological phases with global symmetry, which can be either bosonic or fermionic. We develop a general algebraic description for gapped boundary condition for symmetry-enriched or fermionic topological phases, extending the framework of Lagrangian algebra anyon for bosonic phases without symmetry. We then focus on application to the case with U(1) symmetry. We derive new obstructions to symmetry-preserving gapped boundary for U(1)f{}^{f}f-symmetric (2+1)D(2+1)\mathrm{D}(2+1)D fermionic topological phases, which are beyond chiral central charge ${c}_{\ensuremath{-}}$ and electric Hall conductivity ${\ensuremath{\sigma}}_{H}$. These obstructions are given by a simple Gauss-Milgram-type formula valid for supermodular category and regarded as a higher version of ${c}_{\ensuremath{-}}$ and ${\ensuremath{\sigma}}_{H}$.
Topological order in two-dimensions can be described in terms of deconfined quasiparticle excitations—anyons—and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization—termed symmetry enriched topological order. When the global symmetry group G , which we take to be discrete, does not change topological superselection sectors—i.e. does not change one type of anyon into a different type of anyon—one can imagine a local version of the action of G around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with the second cohomology group H 2 ( G , A a b e l i a n ) being the relevant group. In this paper, we treat the general case of a symmetry group G possibly permuting anyon types. We show that despite the lack of a local action of G , one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic ‘twist’ defects of the symmetry. Furthermore, building on work of Hermele (2014 Phys. Rev. B 90 184418 ), we construct a wide class of exactly-solvable models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.
Symmetry fractionalization on topological excitations is one of the most remarkable quantum phenomena in topological orders with symmetry, i.e., symmetry-enriched topological phases. While much progress has been theoretically and experimentally made in two dimensions (2D), the understanding on symmetry fractionalization in 3D is far from complete. A long-standing challenge is to understand symmetry fractionalization on looplike topological excitations, which are spatially extended objects. In this paper, we construct a powerful topological-field-theoretical framework approach for 3D topological orders, which leads to a systematic characterization and classification of symmetry fractionalization. For systems with Abelian gauge groups (Gg{G}_{g}Gg​) and Abelian symmetry groups (Gs{G}_{s}Gs​), we successfully establish equivalence classes that lead to a finite number of patterns of symmetry fractionalization, although there are notoriously infinite number of Lagrangian-descriptions of the system. Based on this, we compute topologically distinct types of fractional symmetry charges carried by particles. Then, for each type, we compute topologically distinct statistical phases of braiding processes among loop excitations and external symmetry fluxes. As a result, we are able to unambiguously list all physical observables for each pattern of symmetry fractionalization. We present detailed calculations on many concrete examples. As an example, we find that the symmetry fractionalization in an untwisted ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological order with Z2{\mathbb{Z}}_{2}Z2​ symmetry is classified by ${({\mathbb{Z}}_{2})}^{6}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}$. If the topological order is twisted, the classification reduces to (Z2)6{({\mathbb{Z}}_{2})}^{6}(Z2​)6 in which particle excitations always carry integer charge. Inspired by the field-theoretical analysis, we find that our classification of symmetry fractionalization with twist $\ensuremath{\omega}$ can be formally organized into an algebraic formalism: ${⨁}_{{\ensuremath{\nu}}_{i}}{\mathcal{H}}^{4}({G}_{g}{\ensuremath{\leftthreetimes}}_{{\ensuremath{\nu}}_{i}}{G}_{s},U(1))/{\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i}),$ where anomaly-free symmetry fractionalization on particles ${\ensuremath{\nu}}_{i}\ensuremath{\in}{\mathcal{H}}_{non}^{2}({G}_{s},{G}_{g})$. Despite the lack of detailed dependence of ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$ on $\ensuremath{\omega}$ and ${\ensuremath{\nu}}_{i}$, the present field-theoretical approach allows us to efficiently calculate and understand ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$. Several future directions are presented at the end of this paper.