Quasi-Fractonic Behavior in Quantum Systems

• Quasi-fractonic behavior is an emergent phenomenon in quantum systems where excitations exhibit partially restricted mobility due to multipole and subsystem symmetries.
• It arises from generalized conservation laws that forbid single-site hops while allowing finite-step, collective translations, leading to anomalous transport dynamics.
• Model realizations in Z_N stabilizer codes, clock models, and higher-rank gauge theories illustrate its subdiffusive responses and impacts on topological orders.

Quasi-fractonic behavior refers to a class of emergent phenomena in quantum many-body systems and topological phases where excitations possess restricted—but not strictly immobile—mobility, reflecting a partial or conditional enforcement of the stringent mobility constraints characterizing true fracton phases. Unlike strictly fractonic phases, where certain quasiparticles are completely immobile (fractons) or can move only along fixed lower-dimensional submanifolds (lineons, planons), quasi-fractonic behavior manifests as a pronounced, but not absolute, suppression of mobility. Often, such restricted motion is enforced only over a finite spatial or temporal scale, by subsystem symmetries, at finite energy density, or in the presence of specific translation-enrichment structures, resulting in partial confinement and anomalous transport or dynamical signatures that sharply differentiate quasi-fractonic matter from ordinary topological orders, quantum liquids, or crystalline solids.

1. Foundational Principles and Definitions

The essence of quasi-fractonic behavior is rooted in generalized conservation laws (multipole and subsystem symmetries), resulting in partial or emergent mobility constraints:

• Fractonic excitations: Either strictly immobile in isolation, or mobile only in rigid clusters or along lines/planes.
• Quasi-fractonic: Excitations cannot move by arbitrarily small distances or with arbitrary operators; for example, single-site hops are forbidden, but finite-step collective translations are possible. Mobility may emerge dynamically over exponentially long timescales, at higher energy, or after symmetry breaking (He et al., 6 Nov 2025, Delfino et al., 2022, Kumar et al., 2018).
• Symmetry-enforced fractonicity is a related notion in which mobility constraints are contingent upon the preservation of a global symmetry, e.g., $U(1)$ charge conservation enforcing 1D glide-only motion for dislocations in a 2D crystal (Kumar et al., 2018, Gromov et al., 2022).

Quasi-fractonic order manifests in a variety of contexts:

• 2D and 3D stabilizer codes with nontrivial translation-enrichment (He et al., 6 Nov 2025),
• Exactly solvable lattice models with subsystem or higher-form symmetries (Delfino et al., 2022, You et al., 2019),
• Tensor gauge theories with restricted multipole conservation (Prem et al., 2017, Gromov et al., 2022, Shenoy et al., 2019),
• Holographic models and quantum liquids exhibiting anomalous subdiffusion due to emergent dipole conservation (Xia et al., 20 Oct 2025).

2. Theoretical Mechanisms and Model Realizations

2.1 Algebraic and Lattice Origins

• In $\mathbb{Z}_N$ bivariate-bicycle (BB) stabilizer codes, polynomial representations define allowed logical operators and excitation mobility. An anyon at $(i,j)$ cannot be moved incrementally by local operators; it may require translation by a length $l_x$ or $l_y$ before net mobility is restored, with $l_x, l_y$ set by the stabilizer ideal structure (He et al., 6 Nov 2025).
• The mobility limitation arises from the symmetry-enriched topological action of lattice translations, encoded algebraically by the polynomial shift $\lambda-1 \notin (f,g)$ for a translation $\lambda$ unless $\lambda$ is of periodicity $l_x$ or $l_y$.
• In exactly solvable spin/liquid models with higher-form symmetries, e.g., the $\mathbb Z_N$ clock model, conservation of charge, $x$-dipole, $y$-dipole and off-diagonal quadrupole moment ensures that single monopoles are restricted to move only in $N$-site steps, rendering them effectively immobile up to exponential-in-$N$ times $t_*\sim e^{cN^2}$ (Delfino et al., 2022).

2.2 Field Theoretic Descriptions

• Higher-rank gauge theories, such as the rank-2 $U(1)$ scalar charge theory, impose conservation of charge and dipole moment. Isolated fractons are strictly immobile, but breaking these constraints partially (e.g., allowing only certain moments or directions) yields quasi-fractonic (planeon or lineon) phases (Gromov et al., 2022, Shenoy et al., 2019).
• In $(k,n)$-fractonic Maxwell theory, excitations with tensorial source indices are restricted by multipole conservation up to $n$th order, leading to a generalized continuity equation $$\partial_t \rho + \partial_{i^1} ... \partial_{i^n} J^{i^1 \cdots i^n}=0$$. Only bound states of excitations preserving all multipole charges can move collectively (Shenoy et al., 2019).

2.3 Subsystem Symmetry and Symmetry Enrichment

• In symmetry-enforced fractonicity, mobility constraints are present only in the unbroken symmetry phase (e.g., insulators with $U(1)$ particle number conservation), and are lifted when the symmetry is broken, such as in a supersolid where dislocations condense and climb is permitted (Kumar et al., 2018).
• Quasi-fractonicity can also emerge in symmetry-enriched topological orders—SETs—where translation permutes anyon types, making only translations by multiples of the periodicity vectors compatible with local operator dynamics (He et al., 6 Nov 2025).

3. Dynamical, Transport, and Statistical Properties

3.1 Mobility Constraints and Dynamics

• Quasi-fractonic excitations are generally characterized by:
  • Finite-step mobility (e.g. hopping only after $l$ sites, or moving only collectively) (He et al., 6 Nov 2025, Delfino et al., 2022),
  • Subdimensional movement (restricted to lines or planes),
  • Timescale- or system-size-dependent dynamical arrest (dynamically emergent fractonicity for $t \ll t_*$) (Delfino et al., 2022).
• In random circuits constrained to conserve charge and dipole moment, isolated fractons remain localized indefinitely in $d=1,2$ and only delocalize in higher dimensions, owing to the recurrence properties of random walks (Pai et al., 2018).

3.2 Hydrodynamic and Subdiffusive Response

• Dipole conservation elevates hydrodynamic transport from normal diffusion $\omega \sim -ik^2$ to subdiffusion $\omega \sim -ik^4$. This is observed both in effective field theory approaches and in holographic fractonic solids (Xia et al., 20 Oct 2025).
• Subdiffusive fracton modes are robust to explicit translation-symmetry breaking, being protected by the dipole-conservation Ward identity,

$$\partial_\mu J^{I\mu} = -J^\mu \partial_\mu \phi^I.$$

• The anomalously slow relaxation and transport can signal a breakdown of ergodicity and the emergence of disorder-free many-body localized (MBL) phases, even in the absence of quenched disorder or in translation-invariant systems (Pai et al., 2018).

3.3 Spectral and Topological Features

• Ground-state degeneracy in quasi-fractonic models typically depends on geometric parameters (such as system size or the periodicity of allowed translations), e.g.

$$\dim\mathcal H_0 = N \gcd(N,L_x) \gcd(N,L_y) \gcd(N,L_x,L_y)$$

in the $\mathbb Z_N$ clock model (Delfino et al., 2022).

• Braiding statistics may be irrational or size-dependent in certain infinite-component CS theories, contrasting sharply with the usual rational statistics of topological anyons (Ma et al., 2020).
• The edge theory and symmetry anomaly structure of higher-order fractonic topological phases exhibit boundary/hinge-localized fractionalized modes with mixed anomalies, enforceable only in a bulk with global or subsystem symmetry (May-Mann et al., 2022).

4. Examples and Realizations Across Models

Model/Class	Quasi-Fractonic Feature	Mobility Constraint
$\mathbb{Z}_N$ BB stabilizer codes (He et al., 6 Nov 2025)	Anyons hop only by $l_x$ or $l_y$ steps	Local single-step forbidden
$\mathbb{Z}_N$ clock model (Delfino et al., 2022)	Monopoles mobile only on exponentially long times	Mobility “unlocked” for $t\gg t_*$
Symmetry-enforced fractonicity (Kumar et al., 2018)	Dislocation climb forbidden by $U(1)$ symmetry	Glide-only enforced until broken
Fractonic Chern-Simons (You et al., 2019)	Lineons: 1D motion along lattice directions	Subdimensional mobility
Infinite-component CS stacking (Ma et al., 2020)	Planons move in layers; irrational statistics	Plane-confined
Holographic fractonic solids (Xia et al., 20 Oct 2025)	Subdiffusive mode ($\omega \sim -ik^4$), robust	Subdiffusion via dipole cons.

This table summarizes a non-exhaustive cross-section of models and their quasi-fractonic phenomena.

5. Experimental and Physical Implications

• Materials and platforms: Solid-state crystals (with strong $U(1)$ or dipole conservation), cold atom systems engineered for subsystem symmetries, and driven/dissipative optical lattices.
• Experimental signatures:
  • Crossover from normal to subdiffusive transport in response functions or dynamical correlators at long wavelengths.
  • Pinch-point singularities in $\langle E_{ij}(q)E_{kl}(-q)\rangle$ correlation functions, relating to tensor gauge structure (Gromov et al., 2022).
  • Subextensive ground-state degeneracy or anomalous topological edge/hinge states (May-Mann et al., 2022).
  • Slow saturation or area-law scaling of observable entanglement; semi-Poissonian entanglement spectra indicating many-body localization (Pai et al., 2018).
• Potential applications: Memory-enhancing quantum codes with tunable partial confinement, engineered nonergodic phases (MBL) without disorder, and new avenues for understanding ergodicity breaking in quantum systems.

6. Broader Connections and Theoretical Developments

• Quasi-fractonicity bridges the phenomenological gap between fracton topological order and conventional topological order. The interplay of symmetry (global, subsystem, higher-form), translation enrichment, lattice structure, and quantum statistics gives rise to a diverse landscape of emergent sub-dimensional physics.
• It underpins recent theoretical developments in higher-rank Chern-Simons and BF gauge theories (You et al., 2019), generalized Maxwell theories (Shenoy et al., 2019), and the classification of stabilizer codes beyond TQFT through algebraic geometry (e.g., mixed area via BKK theorem) (He et al., 6 Nov 2025).
• Recent explorations extend to fermionic quasi-fractonic orders, where emergent gauge charges can be fermions due to gauged subsystem parity, yielding fracton phases inequivalent to any bosonic model in the presence of fractal symmetries (Shirley, 2020).

A plausible implication is that quasi-fractonic behavior, by permitting limited mobility and tunable partial confinement, suggests new paradigms for quantum memory, quantum error correction, and the design of systems with robust, controllable nonergodic dynamics. Systematic classification and experimental realization of these phases remain key open directions for the field.