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:1910.14046 · Physical review. D/Physical review. D.

Decorated Z 2 symmetry defects and their time-reversal anomalies

Clay Córdova, Kantaro Ohmori, Shu-Heng Shao, Fei Yan

Published Aug 27, 2020• 43 citations
arXivHTMLPDF

Abstract

We discuss an isomorphism between the possible anomalies of ($d+1$)-dimensional quantum field theories with ${\mathbb{Z}}_{2}$ unitary global symmetry, and those of $d$-dimensional quantum field theories with time-reversal symmetry $\mathsf{T}$. This correspondence is an instance of symmetry defect decoration. The worldvolume of a ${\mathbb{Z}}_{2}$ symmetry defect is naturally invariant under $\mathsf{T}$, and bulk ${\mathbb{Z}}_{2}$ anomalies descend to $\mathsf{T}$ anomalies on these defects. We illustrate this correspondence in detail for $(1+1)d$ bosonic systems where the bulk ${\mathbb{Z}}_{2}$ anomaly leads to a Kramers degeneracy in the symmetry defect Hilbert space and exhibits examples. We also discuss $(1+1)d$ fermion systems protected by ${\mathbb{Z}}_{2}$ global symmetry where interactions lead to a ${\mathbb{Z}}_{8}$ classification of anomalies. Under the correspondence, this is directly related to the ${\mathbb{Z}}_{8}$ classification of $(0+1)d$ fermions protected by $\mathsf{T}$. Finally, we consider $(3+1)d$ bosonic systems with ${\mathbb{Z}}_{2}$ symmetry where the possible anomalies are classified by ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$. We construct topological field theories realizing these anomalies and show that their associated symmetry defects support anyons that can be either fermions or Kramers doublets.

DOI
10.1103/physrevd.102.045019

Related papers

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.

arXiv:1706.09464•Physical review. B./Physical review. B•Sep 17, 2018•48 citations

Symmetry-protected topological phases and quantum anomalies

Chang-Tse Hsieh

We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.

•Jul 11, 2017•0 citations

Anomaly indicators for topological orders with U(1) and time-reversal symmetry

Matthew F. Lapa, Michael Levin

We study anomalies in time-reversal-(${\mathbb{Z}}_{2}^{T}$) and $U(1)$-symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly $(2+1)$-dimensional system but can be realized on the surface of a $(3+1)$-dimensional [($3+1$)D] symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators, functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e., $U(1)\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{T}$ and $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T}$, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of $(3+1)\mathrm{D}$ SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk $(3+1)\mathrm{D}$ SPT phase on a particular manifold and in the presence of a particular background gauge field for the $U(1)$ symmetry.

arXiv:1905.00435•Physical review. B./Physical review. B•Oct 17, 2019•16 citations

Solvable lattice model for (2+1)D bosonic topological insulator

Yusuke Horinouchi

We construct an exactly sovable commuting projector Hamiltonian for (2+1)D bosonic topological insulator which is one of symmetry-protected topological (SPT) phases protected by U(1) and time-reversal $\mathbb{Z}_2^T$ symmetry, where the symmetry group is U(1)$\rtimes\mathbb{Z}_2^T$. The model construction is based on the decorated domain-wall interpretation of the $E_{\infty}$-page of a spectral sequence of a cobordism group that classifies the SPT phases in question. We demonstrate nontriviality of the model by showing an emergence of a Kramers doublet when the system is put on a semi-infinite cylinder $(-\infty,0]\times S^1$ with an inserted $π$-flux. The surface anomaly manifests itself as a non-onsite representation of the U(1)$\rtimes\mathbb{Z}_2^T$ symmetry. Anomaly matching on a boundary is discussed within a simple boundary theory.

arXiv:2002.01639•Feb 5, 2020•4 citations

Generalization of Anomaly Formula for Time-Reversal Symmetry in (2+1)D Abelian Bosonic Topological Quantum Field Theories

Ippo Orii

Abstract We study time-reversal symmetry in $(2+1)$D abelian bosonic topological phases. Time-reversal anomalies in such systems are classified by $\mathbb {Z}_2 \times \mathbb {Z}_2$ symmetry-protected topological (SPT) phases in $(3+1)$D, and can be diagnosed via partition functions on manifolds such as $\mathbb {RP}^4$ and $\mathbb {CP}^2$. These partition functions are related by the anomaly formula $Z(\mathbb {RP}^4)\, Z(\mathbb {CP}^2) = \theta _{\mathcal {M}}$, where $\theta _\mathcal {M}$ is the Dehn twist phase associated with the crosscap state. Meanwhile, the existence of gapped boundaries is constrained by so-called higher central charges $\xi _n$, which serve as computable invariants encoding obstruction data. Motivated by the known relation $Z(\mathbb {CP}^2) = \xi _1$, we propose a generalization of the anomaly formula that involves both the higher central charges $\xi _n$ and a new time-reversal invariant $\eta _n$. Introducing a distinguished subset $\mathcal {M}^n \subset \mathcal {A}$ of anyons, we establish the relation $\eta _n \cdot \xi _n = \frac{\sum _{a \in \mathcal {M}^n} \theta (a)^n}{\left| \sum _{a \in \mathcal {M}^n} \theta (a)^n \right|},$which generalizes the known anomaly formula. We analyze the algebraic structure of $\mathcal {M}^n$, derive consistency relations it satisfies, and clarify its connection to the original anomaly formula.

arXiv:2508.04990•Progress of Theoretical and Experimental Physics•Oct 30, 2025•0 citations

Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions

Chang-Tse Hsieh, Gil Young Cho, Shinsei Ryu

Quantum anomalies, breakdown of classical symmetries by quantum effects, provide a sharp definition of symmetry protected topological phases. In particular, they can diagnose interaction effects on the noninteracting classification of fermionic symmetry protected topological phases. In this paper, we identify quantum anomalies in two kinds of (3+1)d fermionic symmetry protected topological phases: (i) topological insulators protected by CP (charge conjugation $\ifmmode\times\else\texttimes\fi{}$ reflection) and electromagnetic $\mathrm{U}(1)$ symmetries, and (ii) topological superconductors protected by reflection symmetry. For the first example, which is related to, by CPT-theorem, time-reversal symmetric topological insulators, we show that the CP-projected partition function of the surface theory is not invariant under large $\mathrm{U}(1)$ gauge transformations, but picks up an anomalous sign, signaling a ${\mathbb{Z}}_{2}$ topological classification. Similarly, for the second example, which is related to, by CPT-theorem, class DIII topological superconductors, we discuss the invariance/noninvariance of the partition function of the surface theory, defined on the three-torus and its descendants generated by the orientifold projection, under large diffeomorphisms (coordinate transformations). The connection to the collapse of the noninteracting classification by an integer ($\mathbb{Z}$) to ${\mathbb{Z}}_{16}$, in the presence of interactions, is discussed.

arXiv:1503.01411•Physical review. B./Physical review. B•Feb 18, 2016•74 citations
We discuss an isomorphism between the possible anomalies of (d+1d+1d+1)-dimensional quantum field theories with Z2{\mathbb{Z}}_{2}Z2​ unitary global symmetry, and those of ddd-dimensional quantum field theories with time-reversal symmetry T\mathsf{T}T. This correspondence is an instance of symmetry defect decoration. The worldvolume of a Z2{\mathbb{Z}}_{2}Z2​ symmetry defect is naturally invariant under T\mathsf{T}T, and bulk Z2{\mathbb{Z}}_{2}Z2​ anomalies descend to T\mathsf{T}T anomalies on these defects. We illustrate this correspondence in detail for (1+1)d(1+1)d(1+1)d bosonic systems where the bulk Z2{\mathbb{Z}}_{2}Z2​ anomaly leads to a Kramers degeneracy in the symmetry defect Hilbert space and exhibits examples. We also discuss (1+1)d(1+1)d(1+1)d fermion systems protected by Z2{\mathbb{Z}}_{2}Z2​ global symmetry where interactions lead to a Z8{\mathbb{Z}}_{8}Z8​ classification of anomalies. Under the correspondence, this is directly related to the Z8{\mathbb{Z}}_{8}Z8​ classification of (0+1)d(0+1)d(0+1)d fermions protected by T\mathsf{T}T. Finally, we consider (3+1)d(3+1)d(3+1)d bosonic systems with Z2{\mathbb{Z}}_{2}Z2​ symmetry where the possible anomalies are classified by Z2\ifmmode×\else\texttimes\fiZ2{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}Z2​\ifmmode×\else\texttimes\fiZ2​. We construct topological field theories realizing these anomalies and show that their associated symmetry defects support anyons that can be either fermions or Kramers doublets.
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 Z2{\mathbb{Z}}_{2}Z2​ symmetry which can be associated with either time reversal (denoted Z2T){\mathbb{Z}}_{2}^{\mathbf{T}})Z2T​) 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 Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ symmetry localization anomalies exist in general. We present an infinite series of TQFTs with such anomalies, some examples of which include USp(4)2\mathrm{USp}{(4)}_{2}USp(4)2​ Chern-Simons (CS) theory and SO(4)4\mathrm{SO}{(4)}_{4}SO(4)4​ CS theory. The theories that we find with these Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ anomalies can all be obtained by gauging the unitary Z2{\mathbb{Z}}_{2}Z2​ subgroup of a different TQFT with a Z4T{\mathbb{Z}}_{4}^{\mathbf{T}}Z4T​ symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is Z4T{\mathbb{Z}}_{4}^{\mathbf{T}}Z4T​, 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 Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ localization anomaly can be compatible with Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ 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 Z2{\mathbb{Z}}_{2}Z2​ gauge theory and thus only a subset of the quasiparticles are truly confined to the surface.
We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.
We study anomalies in time-reversal-(Z2T{\mathbb{Z}}_{2}^{T}Z2T​) and U(1)U(1)U(1)-symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly (2+1)(2+1)(2+1)-dimensional system but can be realized on the surface of a (3+1)(3+1)(3+1)-dimensional [(3+13+13+1)D] symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators, functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e., $U(1)\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{T}$ and $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T}$, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of (3+1)D(3+1)\mathrm{D}(3+1)D SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk (3+1)D(3+1)\mathrm{D}(3+1)D SPT phase on a particular manifold and in the presence of a particular background gauge field for the U(1)U(1)U(1) symmetry.
We construct an exactly sovable commuting projector Hamiltonian for (2+1)D bosonic topological insulator which is one of symmetry-protected topological (SPT) phases protected by U(1) and time-reversal Z2T\mathbb{Z}_2^TZ2T​ symmetry, where the symmetry group is U(1)⋊Z2T\rtimes\mathbb{Z}_2^T⋊Z2T​. The model construction is based on the decorated domain-wall interpretation of the E∞E_{\infty}E∞​-page of a spectral sequence of a cobordism group that classifies the SPT phases in question. We demonstrate nontriviality of the model by showing an emergence of a Kramers doublet when the system is put on a semi-infinite cylinder (−∞,0]×S1(-\infty,0]\times S^1(−∞,0]×S1 with an inserted πππ-flux. The surface anomaly manifests itself as a non-onsite representation of the U(1)⋊Z2T\rtimes\mathbb{Z}_2^T⋊Z2T​ symmetry. Anomaly matching on a boundary is discussed within a simple boundary theory.
Abstract We study time-reversal symmetry in (2+1)(2+1)(2+1)D abelian bosonic topological phases. Time-reversal anomalies in such systems are classified by Z2×Z2\mathbb {Z}_2 \times \mathbb {Z}_2Z2​×Z2​ symmetry-protected topological (SPT) phases in (3+1)(3+1)(3+1)D, and can be diagnosed via partition functions on manifolds such as RP4\mathbb {RP}^4RP4 and CP2\mathbb {CP}^2CP2. These partition functions are related by the anomaly formula Z(RP4) Z(CP2)=θMZ(\mathbb {RP}^4)\, Z(\mathbb {CP}^2) = \theta _{\mathcal {M}}Z(RP4)Z(CP2)=θM​, where θM\theta _\mathcal {M}θM​ is the Dehn twist phase associated with the crosscap state. Meanwhile, the existence of gapped boundaries is constrained by so-called higher central charges ξn\xi _nξn​, which serve as computable invariants encoding obstruction data. Motivated by the known relation Z(CP2)=ξ1Z(\mathbb {CP}^2) = \xi _1Z(CP2)=ξ1​, we propose a generalization of the anomaly formula that involves both the higher central charges ξn\xi _nξn​ and a new time-reversal invariant ηn\eta _nηn​. Introducing a distinguished subset Mn⊂A\mathcal {M}^n \subset \mathcal {A}Mn⊂A of anyons, we establish the relation ηn⋅ξn=∑a∈Mnθ(a)n∣∑a∈Mnθ(a)n∣,\eta _n \cdot \xi _n = \frac{\sum _{a \in \mathcal {M}^n} \theta (a)^n}{\left| \sum _{a \in \mathcal {M}^n} \theta (a)^n \right|},ηn​⋅ξn​=∣∑a∈Mn​θ(a)n∣∑a∈Mn​θ(a)n​,which generalizes the known anomaly formula. We analyze the algebraic structure of Mn\mathcal {M}^nMn, derive consistency relations it satisfies, and clarify its connection to the original anomaly formula.
Quantum anomalies, breakdown of classical symmetries by quantum effects, provide a sharp definition of symmetry protected topological phases. In particular, they can diagnose interaction effects on the noninteracting classification of fermionic symmetry protected topological phases. In this paper, we identify quantum anomalies in two kinds of (3+1)d fermionic symmetry protected topological phases: (i) topological insulators protected by CP (charge conjugation $\ifmmode\times\else\texttimes\fi{}$ reflection) and electromagnetic U(1)\mathrm{U}(1)U(1) symmetries, and (ii) topological superconductors protected by reflection symmetry. For the first example, which is related to, by CPT-theorem, time-reversal symmetric topological insulators, we show that the CP-projected partition function of the surface theory is not invariant under large U(1)\mathrm{U}(1)U(1) gauge transformations, but picks up an anomalous sign, signaling a Z2{\mathbb{Z}}_{2}Z2​ topological classification. Similarly, for the second example, which is related to, by CPT-theorem, class DIII topological superconductors, we discuss the invariance/noninvariance of the partition function of the surface theory, defined on the three-torus and its descendants generated by the orientifold projection, under large diffeomorphisms (coordinate transformations). The connection to the collapse of the noninteracting classification by an integer (Z\mathbb{Z}Z) to Z16{\mathbb{Z}}_{16}Z16​, in the presence of interactions, is discussed.