arXiv:2109.10922 · Physical review. B./Physical review. B

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

Daniel Bulmash, Maissam Barkeshli

Published Apr 15, 2022• 19 citations

Abstract

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.

DOI: 10.1103/physrevb.105.155126

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Fermionic symmetry fractionalization in ( 2 + 1 ) dimensions

Daniel Bulmash, Maissam Barkeshli

arXiv:2109.10913Physical review. B./Physical review. BMar 11, 202244 citations

Characterization and Classification of Fermionic Symmetry Enriched Topological Phases

David Aasen, Parsa Bonderson, Christina Knapp

arXiv:2109.10911Sep 22, 202127 citations

Bosonic topological phases of matter: Bulk-boundary correspondence, symmetry protected topological invariants, and gauging

Apoorv Tiwari, Xiao Chen, Ken Shiozaki +1 more

arXiv:1710.04730Physical review. B./Physical review. BJun 21, 201869 citations

Unwinding fermionic symmetry-protected topological phases: Supersymmetry extension

Abhishodh Prakash, Juven Wang

arXiv:2011.13921Physical review. B./Physical review. BFeb 22, 202118 citations

Time-reversal and spatial-reflection symmetry localization anomalies in (2+1)-dimensional topological phases of matter

Maissam Barkeshli, Meng Cheng

arXiv:1706.09464Physical review. B./Physical review. BSep 17, 201848 citations

Sixteen-Fold Way for Fermionic Topological Orders

Ryohei Kobayashi, Abhinav Prem, Matthew Yu

cond-mat.str-elCross-listedarXiv:2606.28682Jun 30, 20260 citations

arXiv:2109.10922 · Physical review. B./Physical review. B

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

Daniel Bulmash, Maissam Barkeshli

Published Apr 15, 2022• 19 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.105.155126

Related papers

Fermionic symmetry fractionalization in ( 2 + 1 ) dimensions

Daniel Bulmash, Maissam Barkeshli

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 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.

arXiv:2109.10913Physical review. B./Physical review. BMar 11, 202244 citations

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.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:2109.10911Sep 22, 202127 citations

Bosonic topological phases of matter: Bulk-boundary correspondence, symmetry protected topological invariants, and gauging

Apoorv Tiwari, Xiao Chen, Ken Shiozaki +1 more

We analyze $2+1d$ and $3+1d$ bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group $G$ by using dual bulk and boundary approaches. In the bulk, we study an effective field theory, which upon coupling to a background flat $G$ gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat $G$ gauge fields to obtain Dijkgraaf-Witten topological $G$ gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in $2+1d$ and condensing bosonic gauge charges in $3+1d$ . In the dual boundary approach, we study $1+1d$ and $2+1d$ quantum field theories that have $G$ 't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct $G$ characters that generate topological data for the bulk topological gauge theory. Finally, we study a $2+1d$ quantum field theory with a mixed ${\mathbb{Z}}_{2}^{T/R}\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ anomaly where ${\mathbb{Z}}_{2}^{T/R}$ is time-reversal/reflection symmetry, and the $\text{U}(1)$ could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in $3+1d$ that cancels this anomaly. This signals the existence of SPTs in $3+1d$ protected by 0,1-form $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.We analyze $2+1d$ and $3+1d$ bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group $G$ by using dual bulk and boundary approaches. In the bulk, we study an effective field theory, which upon coupling to a background flat $G$ gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat $G$ gauge fields to obtain Dijkgraaf-Witten topological $G$ gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in $2+1d$ and condensing bosonic gauge charges in $3+1d$. In the dual boundary approach, we study $1+1d$ and $2+1d$ quantum field theories that have $G$ 't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct $G$ characters that generate topological data for the bulk topological gauge theory. Finally, we study a $2+1d$ quantum field theory with a mixed ${\mathbb{Z}}_{2}^{T/R}\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ anomaly where ${\mathbb{Z}}_{2}^{T/R}$ is time-reversal/reflection symmetry, and the $\text{U}(1)$ could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in $3+1d$ that cancels this anomaly. This signals the existence of SPTs in $3+1d$ protected by 0,1-form $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.

arXiv:1710.04730Physical review. B./Physical review. BJun 21, 201869 citations

Unwinding fermionic symmetry-protected topological phases: Supersymmetry extension

Abhishodh Prakash, Juven Wang

We show how $1+1$ -dimensional fermionic symmetry-protected topological states (SPTs, i.e., nontrivial short-range entangled gapped phases of quantum matter whose boundary exhibits 't Hooft anomaly and whose bulk cannot be deformed into a trivial tensor product state under finite-depth local unitary transformations only in the presence of global symmetries), indeed can be unwound to a trivial state by enlarging the Hilbert space via adding extra degrees of freedom and suitably extending the global symmetries. The extended projective global symmetry on the boundary can become supersymmetric in a specific sense, i.e., it contains group elements that do not commute with the fermion number parity ${(\ensuremath{-}1)}^{F}$, while the antiunitary time-reversal symmetry becomes fractionalized. This also means we can uplift and remove certain exotic fermionic anomalies (e.g., ``parity'' anomaly in time-reversal or reflection symmetry) via appropriate supersymmetry extensions in terms of group extensions. We work out explicit examples for multilayers of $1+1\mathrm{d}$ Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK) interactions, intrinsic fermionic gapless SPTs protected by supersymmetry, and generalizations to higher space-time dimensions via a cobordism theory.We show how $1+1$-dimensional fermionic symmetry-protected topological states (SPTs, i.e., nontrivial short-range entangled gapped phases of quantum matter whose boundary exhibits 't Hooft anomaly and whose bulk cannot be deformed into a trivial tensor product state under finite-depth local unitary transformations only in the presence of global symmetries), indeed can be unwound to a trivial state by enlarging the Hilbert space via adding extra degrees of freedom and suitably extending the global symmetries. The extended projective global symmetry on the boundary can become supersymmetric in a specific sense, i.e., it contains group elements that do not commute with the fermion number parity ${(\ensuremath{-}1)}^{F}$, while the antiunitary time-reversal symmetry becomes fractionalized. This also means we can uplift and remove certain exotic fermionic anomalies (e.g., ``parity'' anomaly in time-reversal or reflection symmetry) via appropriate supersymmetry extensions in terms of group extensions. We work out explicit examples for multilayers of $1+1\mathrm{d}$ Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK) interactions, intrinsic fermionic gapless SPTs protected by supersymmetry, and generalizations to higher space-time dimensions via a cobordism theory.

arXiv:2011.13921Physical review. B./Physical review. BFeb 22, 202118 citations

Time-reversal and spatial-reflection symmetry localization anomalies in (2+1)-dimensional topological phases of matter

Maissam Barkeshli, Meng Cheng

We study a class of anomalies associated with time-reversal and spatial-reflection symmetry in (2+1)-dimensional bosonic topological phases of matter. In these systems, the topological quantum numbers of the quasiparticles, such as the fusion rules and braiding statistics, possess a ${\mathbb{Z}}_{2}$ symmetry which can be associated with either time reversal (denoted ${\mathbb{Z}}_{2}^{\mathbf{T}})$ or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low-energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a consistent notion of symmetry fractionalization in such systems. We present simple sufficient conditions which determine when ${\mathbb{Z}}_{2}^{\mathbf{T}}$ symmetry localization anomalies exist in general. We present an infinite series of TQFTs with such anomalies, some examples of which include $\mathrm{USp}{(4)}_{2}$ Chern-Simons (CS) theory and $\mathrm{SO}{(4)}_{4}$ CS theory. The theories that we find with these ${\mathbb{Z}}_{2}^{\mathbf{T}}$ anomalies can all be obtained by gauging the unitary ${\mathbb{Z}}_{2}$ subgroup of a different TQFT with a ${\mathbb{Z}}_{4}^{\mathbf{T}}$ symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is ${\mathbb{Z}}_{4}^{\mathbf{T}}$ , or (2) the theory can be considered to be a theory of fermions, with ${\mathbf{T}}^{2}={(\ensuremath{-}1)}^{{N}_{f}}$ corresponding to fermion parity. Finally, we demonstrate that theories with the ${\mathbb{Z}}_{2}^{\mathbf{T}}$ localization anomaly can be compatible with ${\mathbb{Z}}_{2}^{\mathbf{T}}$ if they are ``pseudorealized'' at the surface of a (3+1)D symmetry-enriched topological phase. The ``pseudorealization'' refers to the fact that the bulk (3+1)D system is described by a dynamical ${\mathbb{Z}}_{2}$ gauge theory and thus only a subset of the quasiparticles are truly confined to the surface.We study a class of anomalies associated with time-reversal and spatial-reflection symmetry in (2+1)-dimensional bosonic topological phases of matter. In these systems, the topological quantum numbers of the quasiparticles, such as the fusion rules and braiding statistics, possess a ${\mathbb{Z}}_{2}$ symmetry which can be associated with either time reversal (denoted ${\mathbb{Z}}_{2}^{\mathbf{T}})$ or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low-energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a consistent notion of symmetry fractionalization in such systems. We present simple sufficient conditions which determine when ${\mathbb{Z}}_{2}^{\mathbf{T}}$ symmetry localization anomalies exist in general. We present an infinite series of TQFTs with such anomalies, some examples of which include $\mathrm{USp}{(4)}_{2}$ Chern-Simons (CS) theory and $\mathrm{SO}{(4)}_{4}$ CS theory. The theories that we find with these ${\mathbb{Z}}_{2}^{\mathbf{T}}$ anomalies can all be obtained by gauging the unitary ${\mathbb{Z}}_{2}$ subgroup of a different TQFT with a ${\mathbb{Z}}_{4}^{\mathbf{T}}$ symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is ${\mathbb{Z}}_{4}^{\mathbf{T}}$, or (2) the theory can be considered to be a theory of fermions, with ${\mathbf{T}}^{2}={(\ensuremath{-}1)}^{{N}_{f}}$ corresponding to fermion parity. Finally, we demonstrate that theories with the ${\mathbb{Z}}_{2}^{\mathbf{T}}$ localization anomaly can be compatible with ${\mathbb{Z}}_{2}^{\mathbf{T}}$ if they are ``pseudorealized'' at the surface of a (3+1)D symmetry-enriched topological phase. The ``pseudorealization'' refers to the fact that the bulk (3+1)D system is described by a dynamical ${\mathbb{Z}}_{2}$ gauge theory and thus only a subset of the quasiparticles are truly confined to the surface.

arXiv:1706.09464Physical review. B./Physical review. BSep 17, 201848 citations

Sixteen-Fold Way for Fermionic Topological Orders

Ryohei Kobayashi, Abhinav Prem, Matthew Yu

Fermionic topological orders can host 't Hooft anomalies with no bosonic counterpart. We identify a new sixteen-fold family of (2+1)D fermionic topological orders, forming a fermionic analogue of Kitaev's sixteen-fold way. This family is distinguished by the mod 16 't Hooft anomaly of a $\mathbb{Z}_2$ one-form symmetry, generated in each theory by a single nontrivial $\mathbb{Z}_2$ anyon. This intrinsically fermionic anomaly permits anyon spins that are forbidden in bosonic phases; the simplest new example is an Abelian fermionic topological order containing a single $\mathbb{Z}_2$ Abelian anyon of spin 1/8. Each theory can be realized as the gapped boundary of a (3+1)D fermionic symmetry-protected topological (SPT) phase protected by the $\mathbb{Z}_2$ one-form symmetry, which acquires a $\mathbb{Z}_{16}$ classification once the spacetime spin structure is twisted by the one-form symmetry. We realize these phases microscopically via lattice models built from Walker-Wang models coupled to local fermions.Fermionic topological orders can host 't Hooft anomalies with no bosonic counterpart. We identify a new sixteen-fold family of (2+1)D fermionic topological orders, forming a fermionic analogue of Kitaev's sixteen-fold way. This family is distinguished by the mod 16 't Hooft anomaly of a $\mathbb{Z}_2$ one-form symmetry, generated in each theory by a single nontrivial $\mathbb{Z}_2$ anyon. This intrinsically fermionic anomaly permits anyon spins that are forbidden in bosonic phases; the simplest new example is an Abelian fermionic topological order containing a single $\mathbb{Z}_2$ Abelian anyon of spin 1/8. Each theory can be realized as the gapped boundary of a (3+1)D fermionic symmetry-protected topological (SPT) phase protected by the $\mathbb{Z}_2$ one-form symmetry, which acquires a $\mathbb{Z}_{16}$ classification once the spacetime spin structure is twisted by the one-form symmetry. We realize these phases microscopically via lattice models built from Walker-Wang models coupled to local fermions.

cond-mat.str-elCross-listedarXiv:2606.28682Jun 30, 20260 citations

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

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.

We analyze

2+1d

and

3+1d

bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group

G

by using dual bulk and boundary approaches. In the bulk, we study an effective field theory, which upon coupling to a background flat

G

gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat

G

gauge fields to obtain Dijkgraaf-Witten topological

G

gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in

2+1d

and condensing bosonic gauge charges in

3+1d

. In the dual boundary approach, we study

1+1d

and

2+1d

quantum field theories that have

G

't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct

G

characters that generate topological data for the bulk topological gauge theory. Finally, we study a

2+1d

quantum field theory with a mixed ${\mathbb{Z}}_{2}^{T/R}\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ anomaly where

{\mathbb{Z}}_{2}^{T/R}

is time-reversal/reflection symmetry, and the

\text{U}(1)

could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in

3+1d

that cancels this anomaly. This signals the existence of SPTs in

3+1d

protected by 0,1-form $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.

We show how

1+1

-dimensional fermionic symmetry-protected topological states (SPTs, i.e., nontrivial short-range entangled gapped phases of quantum matter whose boundary exhibits 't Hooft anomaly and whose bulk cannot be deformed into a trivial tensor product state under finite-depth local unitary transformations only in the presence of global symmetries), indeed can be unwound to a trivial state by enlarging the Hilbert space via adding extra degrees of freedom and suitably extending the global symmetries. The extended projective global symmetry on the boundary can become supersymmetric in a specific sense, i.e., it contains group elements that do not commute with the fermion number parity ${(\ensuremath{-}1)}^{F}$, while the antiunitary time-reversal symmetry becomes fractionalized. This also means we can uplift and remove certain exotic fermionic anomalies (e.g., ``parity'' anomaly in time-reversal or reflection symmetry) via appropriate supersymmetry extensions in terms of group extensions. We work out explicit examples for multilayers of

1+1\mathrm{d}

Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK) interactions, intrinsic fermionic gapless SPTs protected by supersymmetry, and generalizations to higher space-time dimensions via a cobordism theory.

{\mathbb{Z}}_{2}

symmetry which can be associated with either time reversal (denoted

{\mathbb{Z}}_{2}^{\mathbf{T}})

or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low-energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a consistent notion of symmetry fractionalization in such systems. We present simple sufficient conditions which determine when

{\mathbb{Z}}_{2}^{\mathbf{T}}

symmetry localization anomalies exist in general. We present an infinite series of TQFTs with such anomalies, some examples of which include

\mathrm{USp}{(4)}_{2}

Chern-Simons (CS) theory and

\mathrm{SO}{(4)}_{4}

CS theory. The theories that we find with these

{\mathbb{Z}}_{2}^{\mathbf{T}}

anomalies can all be obtained by gauging the unitary

{\mathbb{Z}}_{2}

subgroup of a different TQFT with a

{\mathbb{Z}}_{4}^{\mathbf{T}}

symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is

{\mathbb{Z}}_{4}^{\mathbf{T}}

, or (2) the theory can be considered to be a theory of fermions, with ${\mathbf{T}}^{2}={(\ensuremath{-}1)}^{{N}_{f}}$ corresponding to fermion parity. Finally, we demonstrate that theories with the

{\mathbb{Z}}_{2}^{\mathbf{T}}

localization anomaly can be compatible with

{\mathbb{Z}}_{2}^{\mathbf{T}}

if they are ``pseudorealized'' at the surface of a (3+1)D symmetry-enriched topological phase. The ``pseudorealization'' refers to the fact that the bulk (3+1)D system is described by a dynamical

{\mathbb{Z}}_{2}

gauge theory and thus only a subset of the quasiparticles are truly confined to the surface.

\mathbb{Z}_2

one-form symmetry, generated in each theory by a single nontrivial

\mathbb{Z}_2

anyon. This intrinsically fermionic anomaly permits anyon spins that are forbidden in bosonic phases; the simplest new example is an Abelian fermionic topological order containing a single

\mathbb{Z}_2

Abelian anyon of spin 1/8. Each theory can be realized as the gapped boundary of a (3+1)D fermionic symmetry-protected topological (SPT) phase protected by the

\mathbb{Z}_2

one-form symmetry, which acquires a

\mathbb{Z}_{16}

classification once the spacetime spin structure is twisted by the one-form symmetry. We realize these phases microscopically via lattice models built from Walker-Wang models coupled to local fermions.