arXiv:1507.04673 · Physical review. B./Physical review. B

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

Tian Lan, Liang Kong, Xiao-Gang Wen

Published Oct 10, 2016• 126 citations

Abstract

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 $d=1+\sqrt{2}$.

DOI: 10.1103/physrevb.94.155113

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A Classification of (2+1)D Topological Phases with Symmetries

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arXiv:1507.04673 · Physical review. B./Physical review. B

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

Tian Lan, Liang Kong, Xiao-Gang Wen

Published Oct 10, 2016• 126 citations

arXiv HTML PDF

Abstract

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}

DOI: 10.1103/physrevb.94.155113

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A Classification of (2+1)D Topological Phases with Symmetries

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This thesis aims at concluding the classification results for topological phases with symmetry in 2+1 dimensions. The main result is that topological phases are classified by a triple of unitary braided fusion categories $\mathcal E\subset\mathcal C\subset\mathcal M$ plus the chiral central charge $c$ . Here $\mathcal E$ is a symmetric fusion category, $\mathcal E=\mathrm{Rep}(G)$ for boson systems or $\mathcal E=\mathrm{sRep}(G^f)$ for fermion systems, consisting of the representations of the symmetry group and describing the local excitations with symmetry; $\mathcal C$ is the category of all the quasiparticle excitations in the bulk, containing $\mathcal E$ as its Müger center; $\mathcal M$ is a minimal modular extension of $\mathcal C$ , that also includes the gauged symmetry defects. We also study the stacking of topological phases with symmetry and two types of anyon condensations based on such classification.This thesis aims at concluding the classification results for topological phases with symmetry in 2+1 dimensions. The main result is that topological phases are classified by a triple of unitary braided fusion categories $\mathcal E\subset\mathcal C\subset\mathcal M$ plus the chiral central charge $c$. Here $\mathcal E$ is a symmetric fusion category, $\mathcal E=\mathrm{Rep}(G)$ for boson systems or $\mathcal E=\mathrm{sRep}(G^f)$ for fermion systems, consisting of the representations of the symmetry group and describing the local excitations with symmetry; $\mathcal C$ is the category of all the quasiparticle excitations in the bulk, containing $\mathcal E$ as its Müger center; $\mathcal M$ is a minimal modular extension of $\mathcal C$, that also includes the gauged symmetry defects. We also study the stacking of topological phases with symmetry and two types of anyon condensations based on such classification.

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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.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.

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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.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.

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Two-dimensional topologically ordered states such as fractional quantum Hall fluids host anyonic excitations, which are relevant for realizing fault-tolerant topological quantum computers. Classification and characterization of topological orders have been intensely pursued in both the condensed matter and mathematics literature. These topological orders can be bosonic or fermionic depending on whether the system hosts fundamental fermionic excitations or not. In particular, emergent topological orders in usual solid state systems are fermionic topological orders because the electron is a fermion. Recently, bosonic topological orders have been extensively completely classified up to rank 6 using representation theory. Inspired by their method, we provide in this paper a systematic method to classify the fermionic topological orders by explicitly building their modular data, which encodes the self and mutual statistics between anyons. Our construction of the modular data relies on the fact that the modular data of a fermionic topological order forms a projective representation of the ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\theta}}$ subgroup of the modular group ${\mathrm{SL}}_{2}(\mathbb{Z})$ . We carry out the classification up to rank 10 and obtain both unitary and nonunitary modular data. This includes all previously known unitary modular data, and also two new classes of modular data of rank 10. We also determine the chiral central charges (mod $\frac{1}{2}$ ) via a novel method, which does not require the explicit computation of modular extensions.Two-dimensional topologically ordered states such as fractional quantum Hall fluids host anyonic excitations, which are relevant for realizing fault-tolerant topological quantum computers. Classification and characterization of topological orders have been intensely pursued in both the condensed matter and mathematics literature. These topological orders can be bosonic or fermionic depending on whether the system hosts fundamental fermionic excitations or not. In particular, emergent topological orders in usual solid state systems are fermionic topological orders because the electron is a fermion. Recently, bosonic topological orders have been extensively completely classified up to rank 6 using representation theory. Inspired by their method, we provide in this paper a systematic method to classify the fermionic topological orders by explicitly building their modular data, which encodes the self and mutual statistics between anyons. Our construction of the modular data relies on the fact that the modular data of a fermionic topological order forms a projective representation of the ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\theta}}$ subgroup of the modular group ${\mathrm{SL}}_{2}(\mathbb{Z})$. We carry out the classification up to rank 10 and obtain both unitary and nonunitary modular data. This includes all previously known unitary modular data, and also two new classes of modular data of rank 10. We also determine the chiral central charges (mod $\frac{1}{2}$) via a novel method, which does not require the explicit computation of modular extensions.

arXiv:2210.03681Physical review. B./Physical review. BSep 1, 20238 citations

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}

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)

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}

\mathcal E\subset\mathcal C\subset\mathcal M

plus the chiral central charge

c

. Here

\mathcal E

is a symmetric fusion category,

\mathcal E=\mathrm{Rep}(G)

for boson systems or

\mathcal E=\mathrm{sRep}(G^f)

for fermion systems, consisting of the representations of the symmetry group and describing the local excitations with symmetry;

\mathcal C

is the category of all the quasiparticle excitations in the bulk, containing

\mathcal E

as its Müger center;

\mathcal M

is a minimal modular extension of

\mathcal C

, that also includes the gauged symmetry defects. We also study the stacking of topological phases with symmetry and two types of anyon condensations based on such classification.

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.

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.

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.

{\mathrm{SL}}_{2}(\mathbb{Z})

. We carry out the classification up to rank 10 and obtain both unitary and nonunitary modular data. This includes all previously known unitary modular data, and also two new classes of modular data of rank 10. We also determine the chiral central charges (mod

\frac{1}{2}

) via a novel method, which does not require the explicit computation of modular extensions.