Gapless Fracton Phases

• Gapless fracton phases are defined by the coexistence of gapless gauge modes and subdimensional excitations with restricted mobility.
• They extend familiar quantum spin liquids through higher-rank U(1) and multipole gauge theories, resulting in unconventional dispersion relations and robust ground-state degeneracies.
• Lattice models and elasticity dualities reveal actionable experimental signatures, such as characteristic pinch points and unique low-temperature scaling laws.

Gapless fracton phases are quantum phases of matter characterized by the coexistence of gapless gauge modes with exotic "fractonic" excitations exhibiting subdimensional or purely immobile behavior. These phases generalize conventional quantum spin liquids and topological ordered phases beyond the Landau and TQFT paradigms, uniting features such as restricted-mobility quasiparticles, higher-rank gauge structures, subsystem symmetries, and UV–IR entanglement. The theoretical framework underlying gapless fracton phases spans higher-rank U(1) gauge theories, multipole gauge theories, symmetry-enriched constructions based on higher-form symmetries, and exact lattice models—providing a rich taxonomy of both Abelian and non-Abelian variants in two and three dimensions.

1. Field-Theoretic Framework and Defining Properties

Gapless fracton phases are most fundamentally understood as phases governed by generalizations of gauge theory, specifically higher-rank (tensor) U(1) or multipole gauge theories, in which the gauge constraints enforce conservation of higher moments (e.g., dipole or quadrupole) of the charge distribution (Pretko, 2017, Pretko et al., 2017). The archetypal example is the symmetric rank-2 scalar-charge theory, with dynamical field $A_{ij}$ (symmetric tensor), canonical conjugate $E^{ij}$, and Gauss law $\partial_i \partial_j E^{ij} = 0$. Such constraints enforce fractonic behavior, as single charge motion would violate dipole conservation. The theory admits gapless gauge modes ("higher-spin photons"), whose nature and dispersion are controlled by the underlying gauge and symmetry structure. Various models realize additional subdimensional excitation types (planons, lineons, etc.), with mobility encoded in the rank and symmetry of the Gauss law and in possible nonuniform higher-form symmetries (Hirono et al., 2022).

Distinguishing features of gapless fracton phases include:

• Restricted mobility of isolated excitations: fractons (immobile), dipoles (1D motion), planons (plane-confined motion).
• Existence of gapless higher-rank photon modes, typically with non-Lorentz-invariant dispersion (e.g., quadratic, quartic).
• Planar or subdimensional conservation laws for U(1) charge and/or higher moments, often resulting in extensive or subsystem algebraic structure (Radicevic, 2019).
• Robust, often exponentially large, ground-state degeneracy on toroidal manifolds.
• Universal, singular signatures in both correlation and response functions, e.g., pinch points in the structure factor (with characteristic angular or parabolic contours) (Hart et al., 2021, Niggemann et al., 8 Aug 2025).

These properties sharply distinguish gapless fracton phases both from conventional U(1) gauge spin liquids (with fully mobile charges and linear photon dispersion) and from gapped fracton codes.

2. Archetypal Models and Lattice Realizations

Microscopic realizations of gapless fracton phases encompass both exactly solvable models and numerically tractable quantum Hamiltonians. A landmark result is the identification of a spin-1 "spiderweb" model on the square lattice, which provides the first quantum spin realization of a gapless scalar-charge fracton quantum spin liquid in $d = 2$ (Niggemann et al., 8 Aug 2025). The model enforces an eight-site Gauss law, $C_X = 0$, combined with ring-exchange dynamics and an RK-type chemical potential. The low-energy sector is mapped to an emergent trace-free rank-2 U(1) gauge theory, with $S^z_i$ as the electric tensor $E^{\mu\nu}$ and $S^\pm_i$ generating the conjugate vector potential $A^{\mu\nu}$. The theory yields a quadratic photon dispersion, $\omega(\mathbf{q}) \sim |\mathbf{q}|^2$, and immobile, gapped fractonic charge excitations associated with local Gauss-law violations.

Significantly, the gapless fracton phase observed in this model is robust not only in the ground state but across an exponentially fragmented manifold of excited sectors, establishing that the phenomenon is not an artifact of fine-tuning but a generic feature in this constrained setting (Niggemann et al., 8 Aug 2025). This provides a concrete platform for probing gapless fracton spin liquids and for potential experimental engineering (e.g., in multi-level Rydberg or trapped-ion arrays).

Other classes of models, such as type-II U(1) Haah codes and lattice implementations of $\mathfrak{F}_2$ gauge theories, realize multipole-constrained gapless fracton phases in three dimensions (Hart et al., 2021, Radicevic, 2019). The multipole/U(1) Haah code exhibits UV–IR-mixed photon dispersion, eccentric "needle-like" pinch points in the structure factor, and a $T^2$-law in low-temperature heat capacity.

3. Symmetry Principles: Nonuniform Higher-Form Symmetries

A unifying organizing principle for gapless fracton phases emerges from the engineering and spontaneous breaking of nonuniform higher-form symmetries—continuous symmetries whose conserved charges do not commute with spatial translations (Hirono et al., 2022). In such systems, the nontrivial algebra between symmetry charges and translations encodes a "kinematic" origin of subdimensional mobility, resulting in fractons, lineons, and planons as worldlines of topological operators charged under these symmetries.

Upon gauging such a symmetry, the gapless excitations become higher-rank Nambu–Goldstone modes, described by effective field theories of (generalized) symmetric tensor gauge fields. The possible mobility constraints and their bulk-boundary correspondence are determined algebraically by the commutation relations $[iP_i, Q_\alpha]$, and the inverse Higgs mechanism selects which modes are gapped or gapless. Magnetic (dual) nonuniform symmetries further augment the theory, yielding fractonic monopoles and dual tensor SPT bulk actions. All known gapless fracton phases with symmetric-tensor gauge theory descriptions can be viewed as instances of spontaneously broken, nonuniform higher-form symmetry phases (Hirono et al., 2022).

4. Dualities with Elasticity and Coset Constructions

A fruitful duality relates certain gapless fracton gauge theories to elasticity theories of quantum crystals. The rank-2 symmetric tensor gauge theory is dual to 2D quantum elasticity, with disclinations (fractons) and dislocations (dipoles/planons) mapping directly to fracton and dipole excitations (Pretko et al., 2017, Hirono et al., 2021). The longitudinal and transverse phonons become the gapless gauge modes of the fracton theory, carrying linear dispersion and matching precisely the symmetry-based coset-construction of elasticity.

The coset construction, starting from the Galilean or Poincaré group and breaking down to discrete translations and rotations, yields the effective Cosserat elasticity theory, where topological defects (encoded as gauge fields) inherit fractonic immobility from group-theoretic semidirect product/Bianchi identities. At long wavelengths, the theory reduces to ordinary symmetric elasticity, with two gapless modes but with fractonic defect structure (Hirono et al., 2021).

Wess–Zumino terms in the supersolid case encode universal defect–quasiparticle couplings, and the approach systematically generalizes to higher dimensions and symmetry-breaking patterns.

5. Experimental Signatures and Classification

Key spectroscopic and thermodynamic signatures of gapless fracton phases include:

• Distinctive "pinch-point" or "needle-like" singularities in static and dynamical structure factors. In rank-2 scalar-charge theories, these appear as fourfold pinch points, which are suppressed in the presence of a quadratic photon (Niggemann et al., 8 Aug 2025). In type-II multipole gauge phases (e.g., U(1) Haah code), parabolic pinch-point loci emerge due to UV–IR mixing, with strong directional anisotropy in reciprocal space (Hart et al., 2021).
• Low-temperature specific heat exhibiting power laws differing from standard U(1) spin liquids or phonon systems (e.g., $C(T) \sim T^2$ for mixed-dispersion photons).
• Persistence of gapless behavior and restricted mobility in a broad range of Hilbert-space sectors, characteristic of highly fragmented ground-state manifolds (Niggemann et al., 8 Aug 2025).
• Robust exponential (system-size dependent) ground-state degeneracies and subsystem symmetries (Radicevic, 2019, Katsura et al., 2022).

These features provide a route for experimental diagnostics in neutron scattering and thermodynamic measurements.

6. Generalizations and Unconventional Examples

Beyond the standard bosonic models, there exist gapless fracton phases with additional structure, such as:

• Type-II fracton gauge theories, including models with UV–IR-mixed photon dispersions, unconventional pinch-point structure, and anomalous thermal scaling (Hart et al., 2021).
• Non-Abelian higher-rank $\mathfrak{F}_2$ gauge theories, realized via systematic lattice constructions, possessing plane-by-plane symmetries and unconventional gapless excitations not mappable to ordinary tensor gauge frameworks (Radicevic, 2019).
• Fermionic and supersymmetric gapless fracton models, where subsystem fermionic symmetries, spontaneous breaking of supersymmetry, and 't Hooft anomalies produce protected subdimensional dispersions and area- to volume-law ground-state degeneracies (Katsura et al., 2022).

These generalizations deepen the landscape of possible fracton quantum matter, with implications for the topological classification of quantum phases.

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Key References:

• Higher-spin U(1) tensor gauge theory and $\theta$-term boundary structure (Pretko, 2017)
• Fracton–elasticity duality (Pretko et al., 2017)
• Two-dimensional spin-1 gapless fracton quantum spin liquid (Niggemann et al., 8 Aug 2025)
• Multipole gauge theory and gapless type-II fracton phase signatures (Hart et al., 2021)
• Symmetry principle for gapless fracton gauge theories (Hirono et al., 2022)
• Coset construction for elasticity and fractonic defects (Hirono et al., 2021)
• $\mathfrak{F}_2$ gauge theory constructions and gapless fracton regime (Radicevic, 2019)
• Fermionic supersymmetric fracton phases (Katsura et al., 2022)