Vortices as Fractons

Fracton phases of matter are characterized by the presence of immobile or partially mobile local excitations, with mobility constraints stemming from conservation laws of multipole moments of charge density. Such phases were first discovered in exactly solvable quantum lattice models and have been systematically studied using tensor and multipole gauge theories. Despite extensive theoretical advances and proposed experimental realizations, conclusive experimental evidence remains lacking. Crystalline defects in quantum crystals and liquid crystals provide a prominent example of restricted-mobility excitations, where dislocations obey a glide constraint and disclinations are immobile. In this work, the authors point out that superfluid vortices—long accessible experimentally—exhibit fracton physics. They show that vortices in two spatial dimensions share mobility constraints with the traceless scalar charge theory (TSCT). Using a Hamiltonian formulation of vortex dynamics, they demonstrate conservation of dipole and trace of quadrupole moments of vorticity, identifying quantized vorticity with scalar charge. These laws imply isolated vortices are immobile while vortex dipoles move perpendicular to their dipole moment. The study then considers a hydrodynamic limit with many vortices, finding an emergent hydrodynamics with conserved moments and a Hamiltonian structure (classical w-algebra). Finally, it discusses vortices and fractons on curved manifolds, relevant to curved He films, and notes the formal equivalence of results to charged particles in a strong magnetic field (lowest Landau level).

The paper situates its contribution within extensive prior work on fracton phases, including tensor and multipole gauge theories and exactly solvable lattice models that first revealed fracton excitations. It references reviews summarizing classifications and potential applications (quantum memory, elasticity, gravity), and notes proposed but as yet inconclusive experimental realizations. It also connects to elasticity dualities where crystalline defects manifest restricted mobility (glide constraints), and to developments in fracton hydrodynamics, higher-rank gauge theories, and behavior on curved spaces. Prior results on vortex dynamics in fluids and superfluids, including Hamiltonian point-vortex models, vortex hydrodynamics (Wiegmann–Abanov), emergent non-Eulerian hydrodynamics for mixed-sign vortices, and vortices on curved surfaces, provide the theoretical groundwork for the present mapping to TSCT.

- Model system: Two-dimensional incompressible ideal fluid governed by Euler equations with incompressibility ∇·u=0, yielding the Helmholtz vorticity equation (∂t + u·∇) ω = 0. Consider solutions with a finite number of point vortices, with complex velocity field u(z) = −i Σα γα/(z − zα(t)), where 2πγα is circulation and γ is the quantized vortex strength. - Hamiltonian point-vortex dynamics: Use the Hamiltonian H = − Σα≠β γαγβ ln|xα − xβ| with nontrivial Poisson brackets {zα, zβ} = i(πγ)−1 δαβ, leading to equations of motion ∂t zα = i Σβ≠α γβ/(zα − zβ). This framework also describes charged particles in the lowest Landau level (infinite cyclotron frequency limit). - Conservation laws and multipole algebra: From translation and rotation invariance, define impulse p = −2π ε Σαγα xα and angular impulse L = 2π Σαγα |xα|^2, identifying p with negative dipole moment of vorticity D and L with the trace of the quadrupole Q. Show these generate a multipole algebra and imply conservation of total charge Γ=Σαγα, dipole, and quadrupole trace, matching TSCT constraints. - Traceless Scalar Charge Theory (TSCT): Summarize conservation laws via ∂t ρ + ∂i j^i = 0 with j^i = ∂j J^{ij} and Tr(J)=0. Provide a symmetry-based Lagrangian for a complex scalar Φ with invariance under phase transformations with linear, quadratic, and isotropic quadratic spatial polynomials, whose Noether currents enforce the TSCT conservation laws. - Mobility constraints analysis: Use conserved multipole moments to deduce kinematics: isolated vortices are immobile (fracton-like), neutral dipoles move perpendicular to their dipole moment (lineon-like), equal-sign pairs orbit their center of vorticity with fixed separation, and dipole–dipole scattering leads to characteristic π/2 deflections. - Statistical mechanics context: Discuss emergence of vortex crystals (equilibria/relative equilibria) in specific configurations and implications such as negative temperature states and ergodicity issues. - Vortex hydrodynamics (chiral case): In the many-vortex limit with same-sign vorticity, adopt Wiegmann–Abanov hydrodynamics with Hamiltonian H_WA and incompressible vortex flow v, related to density ρ by εij ∂j v_i = 2πγ ρ + η Δ ln ρ (η=γ/4). Employ Poisson brackets forming the classical w_∞ algebra to derive the continuity equation ∂t ρ + ∂i(ρ v_i)=0. Identify tensor currents to recast the continuity equation in TSCT form with ρ = ω/(2πγ). - Mixed-sign hydrodynamics: Reference emergent non-Eulerian hydrodynamics for positive and negative vortices, noting conservation of impulse and angular impulse and separating number and charge densities. Derive tensor current in Supplementary Discussion. - Collective field theory derivation: Define charge density and current from discrete vortices ρ(z)=Σa γa δ(z−za), j_i=Σa γa v_a δ(z−za). Using complex analysis identities and equations of motion, derive a continuity equation of TSCT form with a symmetric traceless tensor current I_{ij} expressed microscopically; show cancellation of second-order poles ensuring well-defined current. - Curved-space generalization: Extend hydrodynamics to curved manifolds, deriving the modified Helmholtz/continuity relations with curvature R and geometric spin s, and express curvature-corrected current. Analyze mobility on constant- vs variable-curvature surfaces.

- Equivalence to TSCT: Two-dimensional superfluid vortex dynamics constitutes a Hamiltonian realization of a traceless scalar charge theory with conservation of total vorticity (charge), dipole moment, and trace of the quadrupole moment of vorticity. Consequently, isolated vortices are immobile and vortex dipoles move perpendicular to their dipole moment. - Multipole conservation from spatial symmetries: The conservation of dipole and quadrupole trace arises from spatial symmetries and non-commutativity of vortex coordinates, not from an internal symmetry. - Hydrodynamic limit: In the many-vortex limit (chiral flows), emergent hydrodynamics preserves the same conservation laws. With ρ = ω/(2πγ), the continuity equation takes TSCT form ∂t ρ + ∂i j^i=0 with j^i=∂j J^{ij}, Tr(J)=0, and Poisson brackets realizing the classical w_∞ algebra. The vortex velocity is determined by density via εij ∂j v_i = 2πγ ρ + (γ/4) Δ ln ρ. - Microscopic tensor current: A collective-field derivation yields an explicit symmetric traceless tensor current I_{ij} in terms of vortex positions and densities, demonstrating that the discrete vortex dynamics obeys the TSCT conservation structure and that singularities cancel. - Mobility phenomenology: Single vortices are stationary (fractons) with effectively divergent mass; neutral dipoles propagate perpendicular to their dipole moments; equal-sign pairs orbit their center of vorticity; dipole–dipole scattering induces π/2 turns, consistent with TSCT expectations. - Statistical mechanics parallels: Vortex crystals and negative-temperature clustering of like-signed vortices correspond to fracton-like gravitational attraction; mobility constraints and reduced phase space lead to nontrivial ergodicity questions. - Curved spaces: On constant-curvature surfaces, isolated vortices remain immobile and dipoles move along geodesics perpendicular to their moment, matching fracton predictions. On variable curvature, dipole conservation is broken and isolated vortices experience curvature-induced forces, losing fractonic immobility. - Broader applicability: The formalism and conclusions apply directly to charged particles confined to the lowest Landau level (strong magnetic field).

By mapping the Hamiltonian point-vortex system to a traceless scalar charge theory, the work demonstrates that familiar, experimentally accessible superfluid vortices naturally realize fracton-like mobility constraints arising from conserved multipole moments. The identification of vorticity as a conserved scalar charge and the derivation of conserved dipole and quadrupole-trace moments explain the immobility of isolated vortices and constrained motion of vortex dipoles. Extending to the hydrodynamic regime shows that these conservation principles survive coarse-graining, with a Hamiltonian vortex hydrodynamics possessing w_∞ Poisson structure and a continuity equation in TSCT form. The explicit construction of a symmetric traceless tensor current at the collective-field level provides a microscopic underpinning of the hydrodynamic conservation laws. On curved manifolds, the analysis elucidates when fracton-like constraints persist (constant curvature) and when they are modified or broken (variable curvature), aligning with expectations for symmetric tensor gauge theories. These insights suggest concrete experimental routes in superfluid helium, BECs, polariton superfluids, nonlinear media, and chiral active fluids to probe fracton physics and its interplay with geometry. The formal equivalence to lowest-Landau-level charged particle dynamics opens connections to fractional quantum Hall phenomena.

The paper establishes a concrete equivalence between two-dimensional superfluid vortex dynamics and the traceless scalar charge theory of fractons. It shows that both finite-vortex systems and the emergent vortex hydrodynamics conserve total charge (vorticity), dipole moment, and quadrupole trace, leading to fracton-like mobility constraints (immobile isolated vortices and perpendicular motion of dipoles). A microscopic collective-field expression for the symmetric rank-2 current is provided, and the curved-space analysis clarifies how geometry preserves or alters these constraints. These results identify superfluid vortices as a readily accessible platform for experimental studies of fracton physics and suggest broader implications for systems such as electrons in the lowest Landau level. Future directions include extending to three-dimensional vortex lines, studying traps and finite lifetimes in superfluids and BECs, exploring chiral superfluids like 3He, refining probes of fracton dynamics, addressing ergodicity and statistical mechanics questions in vortex turbulence, and applying fracton-inspired frameworks to quantum Hall systems.

- The work is theoretical and does not present experimental data; proposed realizations rely on existing superfluid and BEC platforms. - The mapping emphasizes chiral vortex flows, whereas traditional TSCT is non-chiral; while correspondences are established, chirality introduces subtle differences (e.g., right-handed relation between dipole and motion). - On curved spaces with inhomogeneous curvature, dipole conservation is broken and fracton-like immobility is lost, limiting generality beyond constant-curvature manifolds. - Ergodicity and validity of statistical mechanics in many-vortex dynamics remain open problems; certain results depend on idealizations (incompressible, inviscid fluids, point vortices). - Three-dimensional generalizations (vortex lines) and effects of trapping potentials, dissipation, and finite lifetimes are left for future work.