Intrinsic Fractonic Matter

• Intrinsic fractonic matter is a class of quantum systems where excitations, known as fractons, exhibit constrained mobility due to the conservation of higher multipole moments.
• The covariant field-theoretic framework employs a symmetric rank-2 tensor gauge field combined with a Chern–Simons-like term to generate a topological mass for one fractonic mode while leaving another gapless.
• The theory enforces generalized conservation laws, including dipole and quadrupole constraints, leading to the emergence of coexisting intrinsic and extrinsic fractonic sectors with distinct mobility properties.

Intrinsic fractonic matter refers to a class of quantum many-body systems or field theories in which certain excitations—fractons—emerge as degrees of freedom with strongly restricted mobility, enforced by fundamental conservation laws or topological terms intrinsic to the low-energy effective action. In contrast to conventional quasiparticles, fractons cannot propagate freely in space due to the conservation not only of total charge but also of higher multipole moments, such as dipole or quadrupole moments. Recent developments have established that these mobility constraints can arise as an intrinsic property of the field theory, linked to internal gauge structure and topological mass-generating mechanisms, rather than solely from extrinsic lattice or symmetry-enrichment effects. Intrinsic fractonic matter thus realizes a novel spectrum of excitations—including both propagating and strictly localized modes—subject to nontrivial conservation laws and robust against various deformations or explicit symmetry breakings.

1. Covariant Field-Theoretic Framework for Intrinsic Fractons

Intrinsic fractonic matter in (2+1)- or (3+1)-dimensional spacetime can be captured within a covariant, gauge-invariant field theory based on a symmetric rank-2 tensor gauge field $h_{\mu\nu}(x)$. The essential gauge invariance is under so-called longitudinal diffeomorphisms: $$\delta h_{\mu\nu} = \partial_\mu \partial_\nu \phi,$$ which preserve the form of the generalized Gauss law and encode the higher-moment conservation central to fractonic behavior.

The full action consists of two key terms:

• A Maxwell-like "fractonic" kinetic term $S_{fr}$, which is a covariant uplift of the quadratic (in derivatives of $h_{\mu\nu}$) action used in fracton gauge theory literature;
• A Chern–Simons-like term $S_m$, given by

$$S_m = \int_{\mathbb{R}^{1,2}} \!\! d^3x\,\epsilon^{\mu\nu\rho} h_\mu{}^\lambda\,\partial_\nu h_{\rho\lambda},$$

where $m$ is a topological mass parameter.

The invariant action is

$S_{inv} = S_{fr} + m\, S_m,$

where $S_{fr}$ is quadratic in $h_{\mu\nu}$ and its derivatives, and $S_m$ serves both to produce mass via a topological mechanism and to act as an intrinsic source of fractonic matter. This construction is a higher-spin analogue of the Maxwell–Chern–Simons action (Bertolini et al., 21 Oct 2025).

2. Spectrum and Propagating Degrees of Freedom

Decomposing $h_{\mu\nu}$ into scalar, longitudinal, and solenoidal (traceless transverse) components, the theory supports two propagating dynamical degrees of freedom. In the massless limit ($m\rightarrow0$), both modes are gapless and resemble standard fracton field-theoretic excitations. When the CS-like term is included, one degree of freedom becomes massive while the other remains massless: $$(\Box - m^2)\,\varphi = 0 \qquad (\Box - m^2)\,\rho_{cs} = 0,$$ where $\varphi$ is a dynamical scalar extracted from $h_{ab}$ and $\rho_{cs}$ is an effective intrinsic fractonic charge density related to the solenoidal part of the spatial tensor. This mirrors the structure of Maxwell–Chern–Simons theory, in which a CS term gives a topological mass to the photon without increasing the field's number of propagating degrees of freedom (cf. the Deser-Jackiw-Templeton mechanism).

The CS-like term is nontrivial: it both endows one fractonic mode with a topological mass and, by encoding a "self-source" for the gauge field, generates intrinsic fractonic charge and current densities in the absence of external matter.

3. Fractonic Conservation Laws

The theory's gauge structure and topological term enforce a hierarchy of conservation laws characteristic of intrinsic fractonic matter:

• Generalized Electric and Magnetic Fields

$$F_{\mu\nu\rho} = \partial_\mu h_{\nu\rho} + \partial_\nu h_{\rho\mu} - 2\partial_\rho h_{\mu\nu}$$

$$E^{ij} = 2(\partial_0 h^{ij} - \partial_i \partial_j\psi), \qquad B^i = -2\epsilon^{0jk}\partial_j h^{ik}$$

with $\psi$ a Lagrange multiplier for the double divergence constraint.

• Gauss-like Law and Intrinsic Charge

$\partial_j E^{ij} = -m\, B^i$

Defining

$\rho_{cs} = \epsilon^{jk} \partial_j B^k,$

which obeys a continuity equation of the form:

$$\partial_0 \rho_{cs} + \partial_i \partial_j J_{cs}^{ij} = 0,$$

where $J_{cs}^{ij}$ is the intrinsic current.

• Multipole Conservation Integrating over space, the theory enforces

$$\partial_0 D_{cs}^i = 0, \qquad D_{cs}^i = \int d^2x\, x^i\rho_{cs},$$

as well as trace quadrupole conservation:

$\partial_0 \!\int d^2x\, x^2\rho_{cs} = 0,$

by virtue of the tracelessness of $J_{cs}^{ij}$.

Thus, isolated fractonic charge densities generated by the CS-like term cannot move, since any motion would violate dipole conservation, and even collective dipolar bound states are constrained by quadrupole conservation, enforcing deeper immobility constraints compared to models with only dipole conservation.

4. Coupling to External Matter and Sectoral Structure

Introducing an external matter current $J^{\mu\nu}$ via the coupling

$S_J = -\int d^3x\, J^{\mu\nu}h_{\mu\nu}$

leads to equations of motion in which both intrinsic ($J^{\mu\nu}_{cs}$) and extrinsic ($J^{\mu\nu}$) fractonic currents coexist: $$-\partial_\mu F^{\alpha\beta\mu} = J_{cs}^{\alpha\beta} + J^{\alpha\beta}.$$ For external matter, the conservation law is less restrictive: $\partial_0\rho + \partial_i\partial_j J^{ij} = 0,$ enforcing charge and dipole conservation, but not generally the trace quadrupole constraint. The net effect is coexistence of two fractonic sectors:

• An intrinsic sector (from the CS-like term), with massive, immobile charge densities and conservation up to quadrupole-trace;
• An extrinsic sector (from external matter), in which isolated charges remain immobile, but dipolar objects may have enhanced mobility due to a relaxed quadrupole constraint.

This formalism permits modeling of systems in which bound states with different mobility or conservation properties interact.

5. Mathematical Structure and Key Equations

Table: Central Equations of Intrinsic Fractonic Matter Theory (Bertolini et al., 21 Oct 2025)

Equation Type	Mathematical Formulation	Physical Content
Gauge transformation	$\delta h_{\mu\nu} = \partial_\mu\partial_\nu \phi$	Longitudinal diffeomorphism invariance
Action (schematic)	$S_{inv} = S_{fr} + m S_m$	Maxwell-like kinetic, Chern–Simons-like mass/intr. source
Gauss-type law	$\partial_j E^{ij} = -m B^i$	Links intrinsic magnetic field to electric divergence
Continuity (intrinsic)	$$\partial_0 \rho_{cs} + \partial_i\partial_j J_{cs}^{ij}=0$$	Ensures immobility and multipole conservation
Dipole conservation	$\partial_0 D_{cs}^i = 0$	$D_{cs}^i = \int d^2x\, x^i \rho_{cs}$
Quadrupole-trace cons.	$\partial_0 \int d^2x\, x^2 \rho_{cs} = 0$	Traceless current further restricts mobility
Massive Klein–Gordon	$(\Box-m^2)\varphi = 0$	Scalar fractonic field with topological mass

This structure tightly constrains the allowed excitations and possible dynamics of the system, underlying the emergent fractonic phenomenology.

6. Physical Implications and Extensions

The dual role of the Chern–Simons-like term is central: it generates a topological mass for tensor gauge modes (realizing massive, propagating fracton excitations) and acts as an intrinsic, self-generated source of fractonic charge and current. The massless-massive crossover is continuous in the number of degrees of freedom, unifying gapless and massive intrinsic fracton phases.

The presence of distinct intrinsic and extrinsic fractonic sectors opens the possibility of hybrid phases, where different types of excitations with different mobility and conservation laws interact, relevant for engineered systems or materials with coexisting strongly and weakly localized modes.

Potential extensions include:

• Coupling to curved spacetime or gravitational backgrounds to paper fractonic analogues of gravitational responses.
• Exploration of dualities between this covariant field-theoretic model and microscopic lattice or defect-based fracton phases.
• Analysis of hydrodynamic limits, where the interplay between intrinsic and extrinsic fractonic sectors manifests in collective transport coefficients and subdiffusive dynamical signatures.

This framework aligns with broader trends in fracton research, including the covariant extension of gauge-theoretic mechanisms, the importance of higher conservation laws, and the search for robust, symmetry-protected topological excitations in both condensed matter and high-energy contexts.

7. Conclusion

Intrinsic fractonic matter, as formulated in covariant gauge theories with Chern–Simons–like terms (Bertolini et al., 21 Oct 2025), provides a unified conceptual and mathematical infrastructure for realizing immobile, massive fractonic excitations obeying generalized (charge, dipole, quadrupole) conservation laws. The coexistence of intrinsic and extrinsic fractonic sectors accommodates a wide spectrum of mobility and conservation hierarchies. This theoretical architecture is poised for further extension into gravitational, topological, and strongly correlated quantum systems, offering avenues for realizing and probing novel phases characterized by robust localization mechanisms and rich conservation law structures.