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:1604.00407 · Physical review. B./Physical review. B

Modular anomalies in (2+1) - and (3+1) -dimensional edge theories

Moon Jip Park, Chen Fang, Bogdan Andrei Bernevig, Matthew J. Gilbert

Published Jun 16, 2017• 1 citation
arXivHTMLPDF

Abstract

The classification of topological phases of matter in the presence of interactions is an area of intense interest. One possible means of classification is via studying the partition function under modular transforms, as the presence of an anomalous phase arising in the edge theory of a $D$-dimensional system under modular transformation, or modular anomaly, signals the presence of a $(D+1)$-D nontrivial bulk. In this work, we discuss the modular transformations of conformal field theories along a $(2+1)$-D and a $(3+1)$-D edge. Using both analytical and numerical methods, we show that chiral complex free fermions in $(2+1)$-D and $(3+1)$-D are modular invariant. However, we show in $(3+1)$-D that when the edge theory is coupled to a background $U(1)$ gauge field, this results in the presence of a modular anomaly that is the manifestation of a quantum Hall effect in a $(4+1)$-D bulk. Using the modular anomaly, we find that the edge theory of a $(4+1)$-D insulator with space-time inversion symmetry $(PT)$ and fermion number parity symmetry for each spin becomes modular invariant when eight copies of the edges exist.

DOI
10.1103/physrevb.95.235130

Related papers

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

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

A Lattice Chiral Boson Theory in $1+1$d

DeMarco, Michael, Ethan Lake, Xiao-Gang Wen

Chiral field theories describe large classes of matter, from the edges of Quantum Hall systems to the electroweak sector of the Standard Model, but defining them on the lattice has been an ongoing challenge due to a no-go theorem precluding free local models, the potential of symmetry anomalies, and sign problems. Some approaches define a $1+1$d chiral field theory as the edge of a $2+1$d system and argue that the edge decouples from the bulk, but this can be difficult to verify due to finite size effects and strong interactions. On the other hand, recent work has shown how to define the $2+1$d bulk theory as an exactly solvable model with zero correlation length, in which case the edge theory may be extracted exactly. We use these techniques to derive a lattice field theory on a $1+1$d spacetime lattice which carries an anomalous chiral $U(1)$ symmetry with zero chiral central charge. The lattice theory with anomalous chiral $U(1)$ symmetry is always gapless, regardless of lattice interactions. We demonstrate the chiral anomaly by coupling to a background gauge field, develop a field theory which demonstrates the chiral behavior, and show how to assemble a chiral, anomaly-free theory where the gauge field may be taken to be dynamical.

arXiv:2305.03024•May 4, 2023•0 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

( 2 + 1 )-dimensional topological phases with R T symmetry: Many-body invariant, classification, and higher-order edge modes

Ryohei Kobayashi, Yuxuan Zhang, Yanqi Wang +1 more

It is common in condensed matter systems for reflection ($R$) and time-reversal ($T$) symmetry to both be broken while the combination $RT$ is preserved. In this paper, we study invariants that arise due to $RT$ symmetry. We consider many-body systems of interacting fermions with fermionic symmetry groups ${G}_{f}={\mathbb{Z}}_{2}^{f}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{RT}$, $\mathrm{U}{(1)}^{f}\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{RT}$, and $\mathrm{U}{(1)}^{f}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{RT}$. We show that (2+1)D invertible fermionic topological phases with these symmetries have a $\mathbb{Z}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{8}$, ${\mathbb{Z}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$, and ${\mathbb{Z}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}$ classification, respectively, which we compute using the framework of $G$-crossed braided tensor categories. We provide a many-body $RT$ invariant in terms of a tripartite entanglement measure, and which we show can be understood using an edge conformal field theory computation in terms of vertex states. For ${G}_{f}=\mathrm{U}{(1)}^{f}\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{RT}$, which applies to charged fermions in a magnetic field, the nontrivial value of the ${\mathbb{Z}}_{2}$ invariant requires strong interactions. For symmetry-preserving boundaries, the phases are distinguished by zero modes at the intersection of the reflection axis and the boundary. Additional invariants arise in the presence of translation or rotation symmetry.

arXiv:2403.18887•Physical review. B./Physical review. B•Nov 17, 2025•1 citation

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
The classification of topological phases of matter in the presence of interactions is an area of intense interest. One possible means of classification is via studying the partition function under modular transforms, as the presence of an anomalous phase arising in the edge theory of a DDD-dimensional system under modular transformation, or modular anomaly, signals the presence of a (D+1)(D+1)(D+1)-D nontrivial bulk. In this work, we discuss the modular transformations of conformal field theories along a (2+1)(2+1)(2+1)-D and a (3+1)(3+1)(3+1)-D edge. Using both analytical and numerical methods, we show that chiral complex free fermions in (2+1)(2+1)(2+1)-D and (3+1)(3+1)(3+1)-D are modular invariant. However, we show in (3+1)(3+1)(3+1)-D that when the edge theory is coupled to a background U(1)U(1)U(1) gauge field, this results in the presence of a modular anomaly that is the manifestation of a quantum Hall effect in a (4+1)(4+1)(4+1)-D bulk. Using the modular anomaly, we find that the edge theory of a (4+1)(4+1)(4+1)-D insulator with space-time inversion symmetry (PT)(PT)(PT) and fermion number parity symmetry for each spin becomes modular invariant when eight copies of the edges exist.
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.
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.
A Lattice Chiral Boson Theory in 1+11+11+1d
Chiral field theories describe large classes of matter, from the edges of Quantum Hall systems to the electroweak sector of the Standard Model, but defining them on the lattice has been an ongoing challenge due to a no-go theorem precluding free local models, the potential of symmetry anomalies, and sign problems. Some approaches define a 1+11+11+1d chiral field theory as the edge of a 2+12+12+1d system and argue that the edge decouples from the bulk, but this can be difficult to verify due to finite size effects and strong interactions. On the other hand, recent work has shown how to define the 2+12+12+1d bulk theory as an exactly solvable model with zero correlation length, in which case the edge theory may be extracted exactly. We use these techniques to derive a lattice field theory on a 1+11+11+1d spacetime lattice which carries an anomalous chiral U(1)U(1)U(1) symmetry with zero chiral central charge. The lattice theory with anomalous chiral U(1)U(1)U(1) symmetry is always gapless, regardless of lattice interactions. We demonstrate the chiral anomaly by coupling to a background gauge field, develop a field theory which demonstrates the chiral behavior, and show how to assemble a chiral, anomaly-free theory where the gauge field may be taken to be dynamical.
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.
It is common in condensed matter systems for reflection (RRR) and time-reversal (TTT) symmetry to both be broken while the combination RTRTRT is preserved. In this paper, we study invariants that arise due to RTRTRT symmetry. We consider many-body systems of interacting fermions with fermionic symmetry groups ${G}_{f}={\mathbb{Z}}_{2}^{f}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{RT}$, $\mathrm{U}{(1)}^{f}\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{RT}$, and $\mathrm{U}{(1)}^{f}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{RT}$. We show that (2+1)D invertible fermionic topological phases with these symmetries have a $\mathbb{Z}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{8}$, ${\mathbb{Z}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$, and ${\mathbb{Z}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}$ classification, respectively, which we compute using the framework of GGG-crossed braided tensor categories. We provide a many-body RTRTRT invariant in terms of a tripartite entanglement measure, and which we show can be understood using an edge conformal field theory computation in terms of vertex states. For ${G}_{f}=\mathrm{U}{(1)}^{f}\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{RT}$, which applies to charged fermions in a magnetic field, the nontrivial value of the Z2{\mathbb{Z}}_{2}Z2​ invariant requires strong interactions. For symmetry-preserving boundaries, the phases are distinguished by zero modes at the intersection of the reflection axis and the boundary. Additional invariants arise in the presence of translation or rotation symmetry.
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.