Relativistic Fracton Fluid Phase: Theory & Implications

• Relativistic fracton fluid phases are states of matter defined by constraints on charge and dipole transport due to generalized symmetry conservation laws.
• They exhibit hybrid dispersion relations with both linear sound modes and quadratic magnon-like modes that modify standard hydrodynamic behavior.
• Theoretical models incorporate comoving dipole symmetry and non-Newtonian stress responses, providing insights for condensed matter and cosmological applications.

A relativistic fracton fluid phase constitutes a distinct state of matter characterized by constraints on charge and dipole transport that arise from enlarged conservation laws in relativistic settings. Such phases interpolate between conventional relativistic fluids, in which charges are freely advected by the fluid flow, and superfluid phases, in which the condensate breaks phase rotation symmetry and supports frictionless transport (Głódkowski, 12 Sep 2025). In the fracton fluid phase, symmetry principles enforce conservation not only of total charge but also of dipole moment (or higher multipole moments) on the comoving hypersurface, restricting the mobility of constituent quasiparticles in subdimensional fashion, even in the relativistic regime. This results in anomalous hydrodynamic behaviors, non-Newtonian transport, and novel dispersion relations for collective modes, with implications for both condensed matter systems and early-universe cosmology.

1. Symmetry Principles and Conservation Laws

Relativistic fracton fluid phases are founded on generalized symmetry algebras which extend beyond the usual U(1) charge conservation to include emergent dipole and higher-moment conservation. The field-theoretic realization employs scalar fields $\Phi$ defined on a spacelike hypersurface comoving with the fluid. The key symmetry—termed "comoving dipole symmetry"—is implemented via phase rotations linear in the comoving coordinates: $\Phi \to e^{i(\Lambda_0 + \Lambda_I \phi^I)} \Phi$ where $\Lambda_I$ are parameters for dipole shifts in comoving space (Głódkowski, 12 Sep 2025). This truncation of the full chemical shift symmetry from arbitrary polynomial functions to linear forms enforces conservation of both the total charge and the dipole moment on the comoving manifold, thereby pinning charges to move only in ways compatible with dipole preservation.

This symmetry structure effectively reduces the invariance group from the area-preserving diffeomorphisms $\text{SDiff}(\mathbb{R}^2)$ to its affine subgroup $SL(2, \mathbb{R}) \ltimes \mathbb{R}^2$, coupled to the truncated chemical shift group. The consequences are:

• Charge conservation (monopole moment)
• Dipole moment conservation in comoving coordinates
• Constraints on allowed field configurations and collective motions

These principles have profound implications for the form of hydrodynamic equations and collective excitations.

2. Field-Theoretic Realization

The UV-complete field-theoretic model for relativistic fracton fluid hydrodynamics is encoded in an action constructed from a complex scalar field $\Phi$ with comoving coordinates $\phi^I$: $$S = \int d^3x\, \sqrt{-g}\Big[ \frac{i}{2}(\Phi^\dagger D_0 \Phi - \Phi D_0 \Phi^\dagger) - \frac{\eta}{2} B^{IJ} B^{KL} D_{IK}(\Phi, \Phi) D_{JL}(\Phi^\dagger, \Phi^\dagger) - V(|\Phi|) + F(b) \Big]$$ where

• $D_0 \Phi = u^\mu (\partial_\mu - iA_\mu)\Phi$, with $u^\mu$ the fluid four-velocity
• $D_{IJ}$ is a covariant operator involving two derivatives along comoving directions, potentially coupled to a dipole gauge field via $A_{|\mu|J}$
• $\eta$ is a dipole interaction coefficient
• $B^{IJ}$ is the internal comoving metric

This model retains a single time derivative in its kinetic term, consistent with relativistic hydrodynamics, while spatial derivatives appear only in higher-order operators reflecting dipole conservation.

Upon linearization,

$$\Phi = \sqrt{\rho_0 + \delta\rho} \, e^{i\varphi} \qquad \phi^I = \sqrt{s_0}(x^I + \pi^I)$$

the effective Lagrangian for Goldstone fields $\varphi$ (phase mode) and $\pi^I$ (fluid momenta/displacements) becomes: $$\mathcal{L}_{\text{eff}} = \frac{1}{2\lambda} (\partial_t \varphi)^2 + \rho_0 \partial_t \pi^i \partial_i \varphi - \frac{\eta}{2} \rho_0^2 (\partial_i \partial_j \varphi)^2 + \frac{w_0}{2} (\partial_t \pi^i)^2 + \frac{F_{bb} s_0^2}{2} (\partial_i \pi^i)^2$$ The absence of linear spatial gradients for $\varphi$ is a direct consequence of dipole symmetry.

3. Hydrodynamic Modes and Dispersion Relations

The partial constraint on charge mobility modifies the mode structure of the fluid:

• Linear (sound) mode: $\omega \sim k$ arises from standard collective motion of charge densities.
• Quadratic ("magnon-like") mode: $\omega \sim k^2$ emerges from the higher-derivative terms $(\partial^2 \varphi)^2$ mandated by dipole conservation.

This hybrid spectrum is the haLLMark of a relativistic fracton fluid (Głódkowski, 12 Sep 2025):

• In the normal fluid (full chemical shift symmetry), only sound modes are present, with charges strictly advected by the flow.
• In the superfluid phase (ordinary U(1) symmetry), second sound appears due to free charge motion.
• In the fracton fluid phase (linear chemical shifts), both sound and magnon-like modes coexist, indicating constrained but nonzero charge transport.

Such mode structure has important thermodynamic and transport implications, including modified attenuation rates, anisotropic relaxation, and nonstandard scaling exponents.

4. Connections with Non-Newtonian Relativistic Hydrodynamics

The phenomenology of the fracton fluid phase aligns closely with the broader non-Newtonian nature of relativistic fluids (Koide, 2010). Here, modification of constitutive relations to include "memory effects" and finite relaxation times (via Maxwell–Cattaneo–Vernotte equations) yields causal, stable fluid dynamics. In fracton fluids, the additional symmetry constraints and the associated nonlocal terms naturally suppress instantaneous response, enforcing finite propagation velocities and further stabilizing hydrodynamic evolution.

Modified transport coefficients derived from projection-operator methods or kinetic theory, such as

$$\frac{\eta}{[\beta(\epsilon + P)]} = \eta_{GKN} \beta^2 \int d^3x\, \langle \hat{T}^{0x}(x), \hat{T}^{0x}(0) \rangle$$

(with analogous formulas for relaxation times) exhibit agreement with the predictions of generalized Boltzmann approaches that accommodate higher-moment conservation (Jaiswal et al., 2012). These theoretical advances have direct application to fracton fluid modeling, where nonlocality and anisotropic transport must be included.

5. Physical Implications and Experimental Signatures

Relativistic fracton fluid phases are characterized by:

• Restricted charge/dipole mobility: Individual quasiparticles cannot move freely; only dipole-preserving motions are allowed. This results in subdimensional transport and robust ground state degeneracies.
• Nonstandard hydrodynamic response: Attenuation of sound and thermal modes is altered due to effective viscosity and relaxation time modifications, as well as higher-order gradient corrections.
• Hybrid excitation spectrum: Coexistence of linearly and quadratically dispersing modes due to symmetry constraints.
• Enhanced stability and causal propagation: Theoretical frameworks ensure that signal speeds are finite and instabilities are suppressed, even for relativistically boosted systems.

Experimental detection could rely on characteristic responses in conductivity, sound attenuation, and collective mode spectra, as well as emergent pinch-point structures in stress/strain correlation functions, as predicted in elasticity–fracton dualities (Pretko et al., 2017, Pretko et al., 2019).

6. Interpolating Between Fluid and Superfluid Phases

The realization of fracton fluid phases as an interpolation between normal fluids and superfluids is enabled by symmetry engineering. By adjusting the allowed chemical shift symmetries (full polynomial, constant only, or constant plus linear), one can tune the extent to which charge carriers are locked to the fluid's comoving plane, thus controlling the transition between strictly immobile (fractonic), freely mobile (superfluid), and intermediate (fracton fluid) regimes (Głódkowski, 12 Sep 2025).

This interpolation framework provides a flexible approach for modeling diverse physical systems, ranging from high-energy relativistic plasmas to topological matter and quantum information platforms, where multipole conservation and subdimensional transport are essential.

7. Outlook and Theoretical Extensions

Ongoing research explores the ramifications of fracton symmetries and their augmentation to higher-form, subsystem, or multipole conservation laws—including the impact of exotic gauge structures, coupling to curvature or gravity, and generalized hydrodynamic EFTs (Qi et al., 2020, Ahmadi-Jahmani et al., 27 Mar 2025). A plausible implication is the development of fracton gravity frameworks and connections to Carrollian/Aristotelian geometries.

Another direction involves generalizations of nonlinear fluctuating hydrodynamics, where the coexistence of conservation laws (charge, dipole, momentum) yields instability below four spatial dimensions and flow governed by fractonic generalizations of the KPZ universality class (Glorioso et al., 2021). The mathematical formalism underpinning these behaviors continues to develop, with implications for cosmology, quantum criticality, and emergent collective phenomena.