Covariant Fracton-like Effective Field Theory

• Covariant fracton-like effective field theory is a class of quantum field theories characterized by higher-moment symmetries and constrained quasiparticle mobility emerging from gauged higher-rank tensor fields.
• These theories extend conventional gauge symmetries with double-derivative transformations, leading to field equations that enforce both charge and dipole conservation.
• They provide effective continuum descriptions of fracton phases, revealing dualities with topological order, elasticity, and gravitational analogues.

Covariant fracton-like effective field theory encompasses a broad class of quantum field theories in which fractonic quasiparticles—excitations with constrained or subdimensional mobility—emerge as a result of higher-moment symmetries gauged in a manner consistent with Lorentz covariance. These theories generalize the gauge principles of electromagnetism and gravity to higher-rank tensor fields, introduce novel conservation laws (such as dipole moment conservation), and display rich connections to topological order, elasticity, and gravitational analogues. Central constructions utilize rank-2 symmetric tensor gauge fields, multipole algebraic symmetries, and their covariantizations, yielding field equations in which the constrained dynamics and conservation laws of fracton phases arise intrinsically from local symmetries and the structure of the action.

1. Covariant Gauge Symmetries and Higher-Rank Tensor Fields

At the core of covariant fracton-like effective field theories is the generalization of gauge symmetry from the vector (Maxwell) case to higher-rank symmetric tensor fields, typically denoted $A_{\mu\nu}(x)$ or $h_{\mu\nu}(x)$. The defining gauge redundancies are of “double-derivative” type,

$$\delta_\text{fract}A_{\mu\nu}(x) = \partial_\mu\partial_\nu \Lambda(x),$$

where $\Lambda(x)$ is a scalar gauge parameter. This transformation generalizes $U(1)$ gauge symmetry, is a special (longitudinal) limit of infinitesimal diffeomorphism invariance, and enforces the immobility of isolated charges by imposing conservation of both charge and dipole moment. In models such as the covariant extensions of the Haah code or scalar charge theories, this “multipole” gauge invariance is systematically encoded through the multipole algebra, which extends the usual (space-)time symmetry algebra to include polynomial shift symmetries (Gromov, 2018).

The covariant nature of these symmetries is emphasized by constructing Lagrangians that are Poincaré or Lorentz invariant, such as

$$\mathcal{L} = \frac{1}{6}F^{\mu\nu\rho}F_{\mu\nu\rho},$$

where the “field strength” $F_{\mu\nu\rho}$ is constructed to be invariant under double-derivative gauge transformations (Bertolini et al., 2022). The symmetry is commonly realized with symmetric tensor gauge fields, but extensions to mixed, antisymmetric, or multi-potential (electric/magnetic) cases have also been considered (Bertolini et al., 30 Jan 2025).

2. Gauge-Invariant Field Strengths and Analogies to Maxwell Theory and Gravity

An essential ingredient is the construction of covariant, gauge-invariant field strengths that generalize the electromagnetic tensor $F_{\mu\nu}$: $$F_{\mu\nu\rho}(x) = \partial_\mu A_{\nu\rho}(x) + \partial_\nu A_{\rho\mu}(x) - 2\partial_\rho A_{\mu\nu}(x).$$ This structure ensures invariance under $$\delta_\text{fract}A_{\mu\nu} = \partial_\mu\partial_\nu\Lambda$$ and exhibits a cyclic Bianchi identity similar to standard gauge theories: $$F_{\mu\nu\rho} + F_{\nu\rho\mu} + F_{\rho\mu\nu} = 0.$$ The most general quadratic action invariant under this symmetry is a linear combination of two terms: a “fracton” term (Maxwell-like, $F^2$) and the linearized gravity term ($S_\text{LG}$) (Bertolini et al., 2022Bertolini et al., 2023). The field equations,

$\partial^\alpha F_{\alpha\mu\nu} = 0,$

are direct analogs of Maxwell's equations and linearized Einstein equations, with additional constraints imposed by the higher-rank and trace structures.

This duality results in a formal analogy between covariant fracton gauge theory and linearized gravity: both possess a generalized Gauss constraint and, crucially, limited mobility for their respective sources or excitations, despite the seemingly disparate physical origins (Bertolini et al., 2022). Both theories are invariant under a subset of diffeomorphisms, but the fracton case restricts to longitudinal diffeomorphisms of the form $\xi_\mu(x) = \partial_\mu\Lambda(x)$, leading to distinct physical sectors.

3. Conservation Laws and Constrained Dynamics

A defining feature is that the field equations themselves encode conservation of charge {\it and} dipole moment, rendering the limited mobility of fractons a consequence of symmetry rather than an imposed constraint: $$\partial_i\partial_j E_{ij} = \rho, \qquad \Rightarrow \qquad Q = \int d^3x\,\rho = \text{const}, \quad P_i = \int d^3x\,x_i\rho = \text{const}.$$ Here $E_{ij}$ is the symmetric “electric field” derived from the canonical momentum of the fracton action (Bertolini et al., 2022Bertolini et al., 2023). These higher-moment conservation laws mean that isolated charges cannot move without violating gauge invariance; only dipolar (or, in more exotic cases, multipolar or fractal) bound states are mobile, leading to the “fractonic” constraint.

In recent topological and higher-rank Chern-Simons-like generalizations, such as (Bertolini et al., 29 May 2024), there emerges a “Hall-like” response at the dipole level, with generalized electric and magnetic fields: $$\bar{E}_{ab} = \bar{F}_{ab0}, \qquad B^a = \frac{1}{2}\epsilon^{0mn}\bar{F}^a{}_{mn},$$ and a flux-attachment relation for dipoles, $d^i = -2B^i$, implemented via the equations of motion. These relations enforce the immobility of single fracton charges and allow only transverse dipole motion.

4. Gauge Fixing, Propagators, and Degrees of Freedom

Covariant gauge fixing in these theories is nontrivial due to the rank and symmetry of the fields and the unusual scaling of the gauge parameter. Standard scalar gauge fixing may “freeze” the gauge in Landau gauge and preclude the recovery of linearized gravity, as found in (Bertolini et al., 2023). To circumvent this, a vectorial (de Donder/harmonic-type) gauge fixing is adopted: $\partial^\nu h_{\mu\nu} + k\partial_\mu h = 0,$ with a corresponding gauge-fixing action. This choice yields well-defined propagators and allows smooth passage to the linearized gravity sector. The kinetic operators decompose in a basis of tensorial projectors, ensuring that only the physical degrees of freedom remain after gauge fixing. In the traceless sector, two independent degrees of freedom for the physical spin-2 tensor remain, matching expectations from the gravitational analogy (Bertolini et al., 2023Bertolini et al., 29 May 2024).

For non-topological higher-rank Chern-Simons-like theories, peculiarities such as nonlocal poles in the propagator (and vanishing integrated energy on-shell) are observed, reflecting the “almost topological” nature of these models (Bertolini et al., 29 May 2024).

5. Boundary Phenomena, Holography, and Anomalies

Introducing a boundary, as in (Bertolini et al., 2023), results in the emergence of lower-dimensional effective theories that inherit the higher-rank gauge structure. For a 4D bulk covariant fracton theory, the induced 3D boundary theory takes the form of a Maxwell–Chern–Simons-like action involving traceless symmetric tensor fields: $$S_{3D} = \int d^3X\,[-g_{12}(\text{kinetic terms in }\phi_{abc}) + \omega_5(\text{CS-like term})].$$ Boundary theories may be characterized by generalized Kac–Moody-type algebras (without central extension) and exhibit fractonic analogs of electromagnetic phenomena (e.g., “Ampère-like” and “Gauss-like” laws). As in conventional topological phases, anomalies (particularly mixed ’t Hooft anomalies between Wilson lines and fractal membrane operators) encode protected degeneracies, UV/IR mixing, and non-trivial edge physics (Casasola et al., 27 Jun 2024). These anomalies arise from the underlying subsystem symmetries and their fractal or non-on-site nature.

Cohomological methods (BRST) are employed to classify possible anomalies arising in covariant fracton gauge theories, confirming the existence of non-trivial gauge anomalies in specific dimensions (Rovere, 10 Jun 2024).

6. Relation to Lattice Models, Topological Phases, and Dualities

Covariant fracton-like field theories provide effective descriptions for fracton phases originally discovered in lattice models. The mapping from discrete systems such as the Haah code, the X-cube model, and the rank-2 toric code (R2TC) is realized by constructing continuum limits that encode the lattice’s fractal or fracton symmetries via higher-derivative operators, multipole algebras, or polynomial identities over finite fields (Gromov, 2018Fontana et al., 2021Bertolini et al., 31 Jan 2025).

In 3D, higher-rank BF or Chern–Simons–like actions can be mapped onto the low-energy field theories of the rank-2 toric code: $$S_\text{BF} = \int d^3x\,\epsilon^{\mu\nu\rho} B_{\mu\nu}\partial^\mu a^{\nu\rho} + \cdots,$$ and in the case of full tensor symmetry, the action can be decomposed as a difference of two rank-2 Chern–Simons actions (Bertolini et al., 31 Jan 2025). These effective theories capture the constrained mobility of quasiparticles, topological ground state degeneracy, and nontrivial statistical phases, in agreement with the discrete lattice physics.

Subsystem symmetries (including fractal symmetries that act only on fractal subsets of sites) result in behaviors where ground state degeneracy and anomalies display strong UV/IR mixing, with degeneracy depending intricately on system size and geometry (Casasola et al., 27 Jun 2024).

7. Extensions: Supersymmetry, Dualities, and Interacting Generalizations

Supersymmetric extensions of covariant fracton field theories have been constructed using superfield formalisms, yielding fermionic partners exhibiting fractonic statistics, area law entropy, and BPS-protected fractonic excitations (Yamaguchi, 2021). Duality generalizations, such as tensorial self-duality and doubled-potential frameworks, enable symmetric treatment of covariant “electric” and “magnetic” fracton sectors and clarify the nature of electromagnetic duality and the absence of dyonic (Witten effect) fracton analogs (Bertolini et al., 30 Jan 2025). Hall-like responses for dipole currents, “flux attachment” for fractonic dipoles, and notions of fracton–vortex dualities arise in higher-rank Chern-Simons-like theories (Bertolini et al., 29 May 2024).

Interacting (beyond linear) covariant fracton models have been developed via the Frölicher–Nijenhuis bracket formalism, providing a true nonabelianization of the fracton gauge structure—paralleling the passage from Maxwell to Yang–Mills theory—while maintaining covariance and restricted symmetry under longitudinal diffeomorphisms (Bertolini et al., 2 Oct 2024).

Summary Table: Fundamental Structures in Covariant Fracton-like EFT

Structure/Concept	Role/Consequence	Key Reference(s)
Double-derivative (multipole)	Enforces charge & dipole conservation, immobility	(Gromov, 2018, Bertolini et al., 2022)
Symmetric tensor gauge field	Mediator field; encodes fractonic gauge symmetry	(Bertolini et al., 2022, Bertolini et al., 2023)
Gauge-invariant field strength	Covariantization, Bianchi identity, field equations	(Bertolini et al., 2022)
Maxwell-like and gravity terms	Duality, constraint structure, linearized gravity link	(Bertolini et al., 2022, Afxonidis et al., 2023)
Anomaly and boundary algebra	Kač–Moody structure, UV/IR mixing, edge physics	(Bertolini et al., 2023, Rovere, 10 Jun 2024)
Chern–Simons–like action	Hall-like dipole response, topological features	(Bertolini et al., 29 May 2024, Bertolini et al., 31 Jan 2025)
Interacting (nonabelian) theory	Frölicher–Nijenhuis formalism, Yang–Mills analogy	(Bertolini et al., 2 Oct 2024)

Outlook

The covariant fracton-like effective field theory framework greatly enriches the landscape of emergent gauge theories and supports connections between condensed matter, topological phases, and gravitational analogues. Its mathematical foundations provide robust mechanisms by which the characteristic features of fracton order—restricted mobility, higher-moment conservation, unconventional statistics, and topological degeneracy—can be systematically encoded, analyzed, and potentially exploited for further applications, including dualities with elasticity, holography, and supersymmetric extensions.