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:2305.02492 · SciPost Physics

A Chern-Simons theory for dipole symmetry

Xiaoyang Huang

Published Oct 11, 2023• 24 citations
arXivHTMLPDF

Abstract

We present effective field theories for dipole symmetric topological matters that can be described by the Chern-Simons theory. Unlike most studies using higher-rank gauge theory, we develop a framework with both U(1) U ( 1 ) and dipole gauge fields. As a result, only the highest multipole symmetry can support the ’t Hooft anomaly. We show that with appropriate point group symmetries, the dipolar Chern-Simons theory can exist in any dimension and, moreover, the bulk-edge correspondence can depend on the boundary. As two applications, we draw an analogy between the dipole anomaly and the torsional anomaly and generalize particle-vortex duality to dipole phase transitions. All of the above are in the flat spacetime limit, but our framework is able to systematically couple dipole symmetry to curved spacetime. Based on that, we give a proposal about anomalous dipole hydrodynamics. Moreover, we show that the fracton-elasticity duality arises naturally from a non-abelian Chern-Simons theory in 3D.

DOI
10.21468/scipostphys.15.4.153

Related papers

Fractonic Chern-Simons Theory: Higher-Rank Gauge Fields and Subdimensional Topological Order — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Fractonic Chern-Simons theory extends topological gauge theory to higher-rank symmetric tensor gauge fields, enabling fracton phases with restricted mobility and subdimensional topological order. | MATH: Higher-rank gauge fields: \( A_{ij} \) (symmetric tensor) with gauge transformation \( \delta A_{ij} = \partial_i \partial_j \alpha \); fractonic Chern-Simons action: \( S = \frac{k}{4\pi} \int \epsilon^{\mu\nu\rho} A_{\mu} \partial_\nu A_{\rho} \) generalized to higher-rank; fracton charge conservation: \( \partial_i \partial_j J_{ij} = 0 \); dipole moment conservation: \( \int x^i \rho \) constant. | CONNECTION: Crystallographic group symmetries (e.g., cubic, hexagonal) constrain allowed higher-rank tensor gauge fields; lattice structures (e.g., square, triangular) yield subdimensional particle mobility; no direct golden ratio or base-60 link found. | DEPTH: 8 — Fractonic Chern-Simons theory unifies higher-rank gauge theory with topological order, revealing new quantum phase Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Jul 20, 2026•0 citations

Fractonic Chern-Simons and BF theories

Yizhi You, Trithep Devakul, S. L. Sondhi +1 more

Fracton order is an intriguing new type of order which shares many common features with topological order, such as topology-dependent ground-state degeneracies, and excitations with mutual statistics.However, it also has several distinctive geometrical aspects, such as excitations with restricted mobility, which naturally lead to effective descriptions in terms of higher-rank gauge fields.In this paper, we investigate possible effective field theories for three-dimensional fracton order, by presenting a general philosophy whereby topological-like actions for such higher-rank gauge fields can be constructed.Our approach draws inspiration from Chern-Simons and BF theories in 2+1 dimensions, and imposes constraints binding higher-rank gauge charge to higher-rank gauge flux.We show that the resulting fractonic Chern-Simons and BF theories reproduce many of the interesting features of their familiar two-dimensional cousins.We analyze one example of the resulting fractonic Chern-Simons theory in detail, and show that upon quantization it realizes a gapped fracton order with quasiparticle excitations that are mobile only along a subset of one-dimensional lines, and display a form of fractional self-statistics.The ground-state degeneracy of this theory is both topology and geometry dependent, scaling exponentially with the linear system size when the model is placed on a three-dimensional torus.By studying the resulting quantum theory on the lattice, we show that it describes a Z s generalization of the Chamon code.

arXiv:1904.11530•Physical Review Research•May 29, 2020•95 citations

Fractonic order in infinite-component Chern-Simons gauge theories

Xiuqi Ma, Wilbur E. Shirley, Meng Cheng +3 more

The Chern-Simons theory is a powerful tool in the study of 2+1D topological phases, especially quantum Hall systems. Here, the authors extend the framework by allowing an infinite number of gauge fields and use it to study the 3+1D fracton order. Fracton order has been studied mostly using lattice models and higher-rank gauge theories. The introduction of the Chern-Simons formulation allows the user to take advantage of theoretical tools developed over the past decades for quantum Hall systems. In particular, the authors are able to study foliated fracton order more systematically and identify a new type of fracton order, not possible in previous formulations.

arXiv:2010.08917•Physical review. B./Physical review. B•May 18, 2022•25 citations

Effective hydrodynamic field theory and condensation picture of topological insulators

AtMa P. O. Chan, Thomas Klein Kvorning, Shinsei Ryu +1 more

While many features of topological band insulators are commonly discussed at the level of single-particle electron wave functions, such as the gapless Dirac boundary spectrum, it remains elusive to develop a hydrodynamic or collective description of fermionic topological band insulators in 3+1 dimensions. As the Chern-Simons theory for the 2+1-dimensional quantum Hall effect, such a hydrodynamic effective field theory provides a universal description of topological band insulators, even in the presence of interactions, and that of putative fractional topological insulators. In this paper, we undertake this task by using the functional bosonization. The effective field theory in the functional bosonization is written in terms of a two-form gauge field, which couples to a $U(1)$ gauge field that arises by gauging the continuous symmetry of the target system [the $U(1)$ particle number conservation]. Integrating over the $U(1)$ gauge field by using the electromagnetic duality, the resulting theory describes topological band insulators as a condensation phase of the $U(1)$ gauge theory (or as a monopole condensation phase of the dual gauge field). The hydrodynamic description of the surface of topological insulators and the implication of its duality are also discussed. We also touch upon the hydrodynamic theory of fractional topological insulators by using the parton construction.

arXiv:1510.08975•Physical review. B./Physical review. B•Apr 14, 2016•27 citations

Higher SPT's and a generalization of anomaly in-flow

Ryan Thorngren, Curt W. von Keyserlingk

Symmetry protected topological (SPT) phases of bosons in $d$ spatial dimensions have been characterized by the action of the protecting global symmetry $G$ on their boundary. The symmetry acts on the boundary in a way that would be impossible to realize in a purely $d-1$ dimensional system i.e., without the bulk. This is often formalized by saying the $G$ symmetry is anomalous when the boundary theory is gauged. Simultaneously gauging the symmetry on the boundary and in the bulk yields a gauge-invariant composite system. One says there is an anomaly in-flow from the boundary to the bulk. Recently it has been appreciated that some anomalies are too severe to be regulated by an SPT bulk. These do not satisfy anomaly in-flow in the traditional sense. However, we show that these anomalous systems can be regulated as the symmetric boundaries of the newly discovered higher SPT phases. These higher SPT's are protected by a symmetry $d$-group, a higher categorical version of a group which can charge not just particles but also strings, volumes, etc. This structure emerges naturally from computation of the anomaly. One can interpret these severe anomalies as producing an emergent higher-form gauge field when $G$ is gauged, and this gauge field participates in the bulk topological action.

arXiv:1511.02929•Nov 9, 2015•39 citations

Chern-Simons-Matter Conformal Field Theory on the Fuzzy Sphere: Higgs Transition of the Kalmeyer-Laughlin Chiral Spin Liquid

Zheng Zhou, Chong Wang, Yin-Chen He

Gauge theories form a large class of interacting conformal field theories in 3D, among which critical Chern-Simons-matter theories play a prominent role. Here, we study a simple instance: a gapless complex scalar coupled to a U(1)_{2} Chern-Simons gauge field. This theory is conjectured to be dual to three other Chern-Simons-matter theories and to describe the transition between a Kalmeyer-Laughlin chiral spin liquid (a ν=1/2 bosonic Laughlin state) and a trivially gapped phase. Using the fuzzy-sphere regularization, we realize the corresponding quantum phase transition between a ν_{f}=2 fermionic integer quantum Hall state and a ν_{b}=1/2 bosonic fractional quantum Hall state. We show that the transition is continuous and governed by a conformal field theory with SO(3) global symmetry. The operator spectrum contains only one relevant Lorentz scalar, the SO(3) singlet with scaling dimension Δ_{S}=1.52(18), which tunes the transition. Our nonperturbative results identify the universality class naturally associated with the U(1)_{2} Chern-Simons-Higgs theory and show that the conjectured dualities remain self-consistent.

Physical Review Letters•Jul 21, 2026•1 citation
We present effective field theories for dipole symmetric topological matters that can be described by the Chern-Simons theory. Unlike most studies using higher-rank gauge theory, we develop a framework with both U(1) U ( 1 ) and dipole gauge fields. As a result, only the highest multipole symmetry can support the ’t Hooft anomaly. We show that with appropriate point group symmetries, the dipolar Chern-Simons theory can exist in any dimension and, moreover, the bulk-edge correspondence can depend on the boundary. As two applications, we draw an analogy between the dipole anomaly and the torsional anomaly and generalize particle-vortex duality to dipole phase transitions. All of the above are in the flat spacetime limit, but our framework is able to systematically couple dipole symmetry to curved spacetime. Based on that, we give a proposal about anomalous dipole hydrodynamics. Moreover, we show that the fracton-elasticity duality arises naturally from a non-abelian Chern-Simons theory in 3D.
FINDING: Fractonic Chern-Simons theory extends topological gauge theory to higher-rank symmetric tensor gauge fields, enabling fracton phases with restricted mobility and subdimensional topological order. | MATH: Higher-rank gauge fields: Aij A_{ij} Aij​ (symmetric tensor) with gauge transformation δAij=∂i∂jα \delta A_{ij} = \partial_i \partial_j \alpha δAij​=∂i​∂j​α; fractonic Chern-Simons action: S=k4π∫ϵμνρAμ∂νAρ S = \frac{k}{4\pi} \int \epsilon^{\mu\nu\rho} A_{\mu} \partial_\nu A_{\rho} S=4πk​∫ϵμνρAμ​∂ν​Aρ​ generalized to higher-rank; fracton charge conservation: ∂i∂jJij=0 \partial_i \partial_j J_{ij} = 0 ∂i​∂j​Jij​=0; dipole moment conservation: ∫xiρ \int x^i \rho ∫xiρ constant. | CONNECTION: Crystallographic group symmetries (e.g., cubic, hexagonal) constrain allowed higher-rank tensor gauge fields; lattice structures (e.g., square, triangular) yield subdimensional particle mobility; no direct golden ratio or base-60 link found. | DEPTH: 8 — Fractonic Chern-Simons theory unifies higher-rank gauge theory with topological order, revealing new quantum phase Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Fracton order is an intriguing new type of order which shares many common features with topological order, such as topology-dependent ground-state degeneracies, and excitations with mutual statistics.However, it also has several distinctive geometrical aspects, such as excitations with restricted mobility, which naturally lead to effective descriptions in terms of higher-rank gauge fields.In this paper, we investigate possible effective field theories for three-dimensional fracton order, by presenting a general philosophy whereby topological-like actions for such higher-rank gauge fields can be constructed.Our approach draws inspiration from Chern-Simons and BF theories in 2+1 dimensions, and imposes constraints binding higher-rank gauge charge to higher-rank gauge flux.We show that the resulting fractonic Chern-Simons and BF theories reproduce many of the interesting features of their familiar two-dimensional cousins.We analyze one example of the resulting fractonic Chern-Simons theory in detail, and show that upon quantization it realizes a gapped fracton order with quasiparticle excitations that are mobile only along a subset of one-dimensional lines, and display a form of fractional self-statistics.The ground-state degeneracy of this theory is both topology and geometry dependent, scaling exponentially with the linear system size when the model is placed on a three-dimensional torus.By studying the resulting quantum theory on the lattice, we show that it describes a Z s generalization of the Chamon code.
The Chern-Simons theory is a powerful tool in the study of 2+1D topological phases, especially quantum Hall systems. Here, the authors extend the framework by allowing an infinite number of gauge fields and use it to study the 3+1D fracton order. Fracton order has been studied mostly using lattice models and higher-rank gauge theories. The introduction of the Chern-Simons formulation allows the user to take advantage of theoretical tools developed over the past decades for quantum Hall systems. In particular, the authors are able to study foliated fracton order more systematically and identify a new type of fracton order, not possible in previous formulations.
While many features of topological band insulators are commonly discussed at the level of single-particle electron wave functions, such as the gapless Dirac boundary spectrum, it remains elusive to develop a hydrodynamic or collective description of fermionic topological band insulators in 3+1 dimensions. As the Chern-Simons theory for the 2+1-dimensional quantum Hall effect, such a hydrodynamic effective field theory provides a universal description of topological band insulators, even in the presence of interactions, and that of putative fractional topological insulators. In this paper, we undertake this task by using the functional bosonization. The effective field theory in the functional bosonization is written in terms of a two-form gauge field, which couples to a U(1)U(1)U(1) gauge field that arises by gauging the continuous symmetry of the target system [the U(1)U(1)U(1) particle number conservation]. Integrating over the U(1)U(1)U(1) gauge field by using the electromagnetic duality, the resulting theory describes topological band insulators as a condensation phase of the U(1)U(1)U(1) gauge theory (or as a monopole condensation phase of the dual gauge field). The hydrodynamic description of the surface of topological insulators and the implication of its duality are also discussed. We also touch upon the hydrodynamic theory of fractional topological insulators by using the parton construction.
Symmetry protected topological (SPT) phases of bosons in ddd spatial dimensions have been characterized by the action of the protecting global symmetry GGG on their boundary. The symmetry acts on the boundary in a way that would be impossible to realize in a purely d−1d-1d−1 dimensional system i.e., without the bulk. This is often formalized by saying the GGG symmetry is anomalous when the boundary theory is gauged. Simultaneously gauging the symmetry on the boundary and in the bulk yields a gauge-invariant composite system. One says there is an anomaly in-flow from the boundary to the bulk. Recently it has been appreciated that some anomalies are too severe to be regulated by an SPT bulk. These do not satisfy anomaly in-flow in the traditional sense. However, we show that these anomalous systems can be regulated as the symmetric boundaries of the newly discovered higher SPT phases. These higher SPT's are protected by a symmetry ddd-group, a higher categorical version of a group which can charge not just particles but also strings, volumes, etc. This structure emerges naturally from computation of the anomaly. One can interpret these severe anomalies as producing an emergent higher-form gauge field when GGG is gauged, and this gauge field participates in the bulk topological action.
Gauge theories form a large class of interacting conformal field theories in 3D, among which critical Chern-Simons-matter theories play a prominent role. Here, we study a simple instance: a gapless complex scalar coupled to a U(1)_{2} Chern-Simons gauge field. This theory is conjectured to be dual to three other Chern-Simons-matter theories and to describe the transition between a Kalmeyer-Laughlin chiral spin liquid (a ν=1/2 bosonic Laughlin state) and a trivially gapped phase. Using the fuzzy-sphere regularization, we realize the corresponding quantum phase transition between a ν_{f}=2 fermionic integer quantum Hall state and a ν_{b}=1/2 bosonic fractional quantum Hall state. We show that the transition is continuous and governed by a conformal field theory with SO(3) global symmetry. The operator spectrum contains only one relevant Lorentz scalar, the SO(3) singlet with scaling dimension Δ_{S}=1.52(18), which tunes the transition. Our nonperturbative results identify the universality class naturally associated with the U(1)_{2} Chern-Simons-Higgs theory and show that the conjectured dualities remain self-consistent.