Chern-Simons-like theories for fracton phases

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.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:2010.08917Physical review. B./Physical review. BMay 18, 202225 citations

Higher-form symmetry perspective on fracton phases

Marvin Qi, Leo Radzihovsky, Michael Hermele

Bulletin of the American Physical SocietyMar 5, 20200 citations

A Magnetic Model with a Possible Chern-Simons Phase

Michael Hartley Freedman

arXiv:quant-ph/0110060Communications in Mathematical PhysicsMar 1, 200380 citations

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.comFINDING: 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, 20260 citations

Fracton phases via exotic higher-form symmetry-breaking

Marvin Qi, Leo Radzihovsky, Michael Hermele

arXiv:2010.02254Annals of PhysicsNov 25, 202085 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.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:1904.11530Physical Review ResearchMay 29, 202095 citations

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