Dipole conservation alone does not imply fracton order or k⁴ relaxation

The claim under pressure

Conserving total dipole moment neither proves fracton topological order nor fixes the slowest density decay rate to $\mathrm{\Gamma }(k)\propto k^4$.

Restricted motion appears in several settings:

• a microscopic Hamiltonian whose local operators preserve charge and dipole moment;
• a scalar-charge tensor gauge theory with a double-divergence Gauss law;
• a thermalizing fluid with a rank-2 continuity equation;
• a fragmented Hilbert space that does not reach local equilibrium;
• a gapped lattice phase such as the X-cube model.

These systems can share kinematics while differing in dynamics, excitations, and phase structure.

The task is to identify which statement has actually been established.

1. Three meanings of “dipole conservation”

Let

Three claims are often compressed into one phrase.

Global conservation

One may find

for a chosen geometry and boundary condition.

This statement concerns integrated quantities. It need not reveal a microscopic selection rule or a local current algebra.

Exact microscopic conservation

A Hamiltonian may obey

with each allowed local term respecting the same charges.

This is stronger. Local operator support and the algebra of conserved charges can then restrict which excitations move without creating additional defects.

Local higher-moment conservation

At long wavelengths, one may have

This Ward identity identifies a rank-2 current. It still does not select a constitutive relation, guarantee local equilibration, or classify a zero-temperature phase.

The hierarchy is therefore

The symbols summarize logical strength, not a theorem that every microscopic model flows through every intermediate description.

2. A rank-2 Gauss law immobilizes an isolated gauge charge

Consider scalar-charge rank-2 $U(1)$ gauge theory:

Assume fields decay fast enough on ${\mathbb{R}}^d$, or impose boundary conditions that remove the relevant fluxes.

The total charge becomes a boundary quantity:

After one further weighted integration,

is also determined by boundary data.

For a point charge $q$ at $\mathbf{r}$,

Translating it by $\delta r^a$ changes the dipole moment by

An operator supported in the bulk cannot perform this translation while preserving the Gauss constraint and leaving all other excitations unchanged.

A neutral dipole can move without changing the total dipole moment. Additional microscopic constraints may restrict its direction, but charge and dipole conservation alone do not.

This is a kinematic statement about gauge charge. It does not yet tell us whether the gauge theory is gapless, whether a lattice regularization is deconfined, or whether the phase has X-cube-type order.

Pretko, Chen, and You review this route from multipole conservation to restricted mobility and tensor gauge structure.

3. Boundaries and periodic space are part of the statement

The formula

uses a globally defined coordinate. On a torus, $x^a$ jumps across the periodic cut.

Lattice dipole moment is then commonly defined modulo a system-size-dependent quantity, through exponentiated polarization, or after choosing a cut and tracking boundary transport.

A particle crossing the cut can change the naive $P^a$ by $q{L}_a$ without a local violation of periodic dynamics.

Open systems also exchange charge and dipole moment through their boundaries. The integrated Ward identity contains boundary currents that must be measured or set to zero.

Statements about immobility should therefore name:

• the geometry;
• the allowed boundary operators;
• whether dipole moment is exact or defined modulo a lattice period;
• the time window over which symmetry-breaking processes are negligible.

Without these details, “dipole is conserved” is incomplete.

4. The rank-2 Ward identity does not yet give $z=4$

Suppose a thermalizing system admits

Near homogeneous equilibrium, define

so $\mu =\delta \rho /\chi$ at linear order.

Exact conservation of $Q$ and $P^a$ permits a family of generalized equilibrium profiles

Their Hessian vanishes:

A dissipative constitutive relation must therefore respond to the Hessian rather than to $\mu$ or its first derivative.

For an isotropic, parity-even fluid, take

More generally,

where the longitudinal contraction of $B^{ij,kl}$ must be nonnegative for stable dissipation.

Indeed,

for a positive dissipative tensor.

In the isotropic scalar sector,

A Fourier mode then obeys

The relaxation time at length scale $\ell$ grows as

This is the Gaussian $z=4$ hydrodynamic result derived in the framework of Gromov, Lucas, and Nandkishore.

Its assumptions are visible:

1. locality at the scale of the derivative expansion;
2. local equilibration;
3. exact charge and dipole symmetry in the hydrodynamic window;
4. a stable dissipative constitutive tensor;
5. no additional slow mode that changes the leading pole;
6. fluctuations and nonlinearities that do not drive another infrared fixed point.

The Ward identity supplies the allowed current structure. Hydrodynamics supplies the constitutive law.

5. Ordinary diffusion is the decisive counterexample

Ordinary diffusion satisfies

On ${\mathbb{R}}^d$, with sufficiently rapid decay,

The global dipole moment is conserved even though

The counterexample is stronger than it first appears. Diffusion can be written formally as

So the bare double-divergence equation, viewed only as a differential identity, also fails to force $k^4$.

What excludes ${\tilde{J}}^{ij}$ in an exact dipole-symmetric thermal fluid?

The microscopic symmetry and local-equilibrium construction require every profile

to carry no dissipative charge current. The ordinary diffusive current

is nonzero for a linear chemical potential.

Ordinary diffusion preserves $P^a$ only as an integrated boundary identity. Its local dynamics does not implement the equilibrium family associated with an exact dipole charge.

This distinction is more precise than saying “rank-2 current instead of rank-1 current.” Current definitions admit improvements; the source symmetry, equilibrium states, and constitutive response carry the physical content.

6. Weak dipole breaking restores diffusion at the longest scale

Real platforms rarely conserve dipole moment forever. Landau-level mixing, off-resonant hopping, boundary loss, or higher-order processes can open an ordinary current channel.

Write

with

Linear modes decay at

The crossover wave number is

and the corresponding length is

For $k\gg k_{\times }$, a preasymptotic $k^4$ regime can be visible. For $k\ll k_{\times }$, the ordinary $k^2$ term wins.

Observation of $z=4$ over a finite window therefore establishes an emergent constraint on that window. It does not prove exact microscopic symmetry.

7. Local equilibration can fail

Hydrodynamics assumes that all nonconserved local information decays before the slow density mode.

Kinetic constraints can instead split the Hilbert space into many disconnected Krylov sectors. Two states with the same $Q$, $P^a$, and energy can then retain different memories.

This is Hilbert-space fragmentation, not hydrodynamic local equilibrium.

A 2024 quantum-gas-microscope experiment observed such fragmentation and subdimensional defect motion in a tilted two-dimensional Bose-Hubbard system (Adler et al., Nature 636, 80–85).

The experiment establishes constrained dynamics and fractonic excitations in that effective model. It does not establish a gapped fracton topological phase.

The reverse situation is also possible. A chaotic system can thermalize and display fracton hydrodynamics without possessing immobile deconfined quasiparticles in its ground state.

8. Even the Gaussian $k^4$ fixed point has limits

The linear equation

is a starting point, not the endpoint of every dipole-conserving theory.

If momentum is also conserved, nonlinear fluctuating hydrodynamics introduces additional couplings. A 2022 analysis found the naive hydrodynamic theory unstable below four spatial dimensions for a class of fluids conserving charge, dipole moment, and momentum.

The flow reaches different dynamical behavior rather than retaining the Gaussian result unchanged (Glorioso et al., Nature Physics 18, 912–917).

This does not invalidate $k^4$ hydrodynamics in lattice systems where momentum relaxes. It limits the claim to the conserved quantities and nonlinearities of the model being studied.

Long-range interactions, subsystem conservation laws, broken time-reversal symmetry, and coupling to phonons can also change the constitutive expansion or add modes.

The safe inference is conditional:

9. Hydrodynamics does not classify a phase of matter

Compare four cases.

System	Established structure	Missing before calling it the same phase
Scalar-charge $U(1)$ gauge theory	Gauss constraint, restricted gauge-charge mobility	compactness, deconfinement, photon spectrum, operator content
Dipole-conserving thermal fluid	Ward identity and hydrodynamic poles	ground-state superselection and long-range entanglement
Fragmented tilted-lattice model	disconnected dynamical sectors and constrained defects	local equilibration and topological phase invariants
X-cube-type gapped model	fracton, lineon, and planon sectors with nonlocal order	equivalence still requires a chosen fracton-phase relation

A gapped fracton order is characterized using data absent from a density relaxation law:

• superselection sectors and their mobility;
• fusion rules;
• remote detection or braiding processes;
• geometry-dependent ground-state structure;
• entanglement renormalization, often allowing two-dimensional topological layers as resources.

The 2025 planon-modular framework defines a class in which every nontrivial point excitation can be detected by braiding with a planon. It contains several type-I models, including X-cube variants.

It does not contain Haah’s cubic code or provide a classification of all fracton orders. See Wickenden et al., Physical Review B 112, 115129.

A measured $k^4$ pole contains none of this fusion or detection data.

10. Elasticity gives the same algebra a different origin

In two spatial dimensions, fracton-elasticity duality maps

The dipole orientation is related to the Burgers vector by a convention-dependent ${90}^{\circ }$ rotation.

Elastic stress variables map to the tensor gauge fields, while crystal phonons map to gapless gauge modes. The correspondence was developed in Pretko and Radzihovsky and reviewed in the 2024 Colloquium on fracton matter.

The duality also exposes a physical caveat. Vacancies and interstitials affect dislocation climb and can relax the constraint that appears exact in a defect-only continuum theory.

Melting, lattice-scale cores, and dynamical geometry can change the effective conservation law. Mobility restriction belongs to a specified elastic regime.

11. The geometry problem has moved forward

The Cartesian charge

ties dipole symmetry to affine coordinates. General curved spaces need not admit the required global functions.

Modern formulations distinguish gauge invariance of higher-rank sources from the existence of a global dipole symmetry on a chosen geometry.

A 2026 construction defines dipole moment relative to dynamical crystal coordinates. Fractons remain fixed relative to the material while moving in absolute space as the solid deforms.

This “fractonic solid” is compatible with boost symmetry and gravitational coupling in a way that an absolute Cartesian dipole can obstruct. See Jain, Physical Review D 113, 105015.

The proposal changes the symmetry principle. It should not be presented as a derivation from the original scalar-charge model.

12. A current frontier map

Landau-level projection

Projection to a partially filled lowest Landau level can generate an effective one-dimensional lattice dynamics that conserves charge and dipole moment.

Numerical work finds late-time subdiffusive relaxation under the stated hydrodynamic conditions (Zerba et al., PRX Quantum 6, 020321).

In thin-torus regimes, strong fragmentation and long prethermal windows can appear before hydrodynamics. The same model can therefore test the boundary between fragmentation and equilibration.

Continuum few-body fractons

A 2026 quantization of continuum dipole-conserving mechanics found that locality and symmetry produce nontrivial few-body spectral transitions and long-lived nonergodic structures.

Those results concern a specified continuum Hamiltonian and its regularization, not a gapped many-body topological order. See Sadki, Prakash, and Sondhi, Physical Review B 114, 045102.

Algebraic classification

Planon-modular invariants organize a broad type-I family. Detector-based approaches are beginning to state which excitation theories qualify as physically nondegenerate.

Type-II models remain outside this particular framework. Hydrodynamic multipole algebra is therefore much coarser than the emerging algebra of gapped fracton phases.

The deliberately false inference

If a translationally invariant system conserves total charge and total dipole moment, its longest-lived density mode must have $\mathrm{\Gamma }(k)\propto k^4$.

Ordinary diffusion on ${\mathbb{R}}^d$ preserves both integrated moments under decaying boundary conditions and still has $\mathrm{\Gamma }=D_2{k}^2$.

To obtain the $k^4$ pole, specify the exact local symmetry, the generalized equilibrium family, the rank-2 constitutive law, local equilibration, and the rest of the slow sector.

Collision: four levels

Kinematics

A double-divergence Gauss law can forbid translation of an isolated gauge charge by a local operator.

Conservation theory

A rank-2 Ward identity encodes a local current structure stronger than an observed constant value of $\int x^a\rho$.

Dynamics

$k^4$ relaxation follows from a constitutive and equilibration problem. Weak symmetry breaking, nonlinear fluctuations, or extra conserved quantities can change the infrared law.

Phase structure

Restricted motion and subdiffusion do not supply fusion, braiding, entanglement, or ground-state data. They cannot identify an X-cube or Haah-type phase.

Do this now

Consider

on ${\mathbb{R}}^d$, with fields decaying fast enough to remove all boundary terms.

Take

Target A: integrated charges

Show that

are conserved.

Target B: the hydrodynamic pole

Derive

and determine $D_4$.

State the positivity condition needed for stability.

Target C: defeat the false inference

For

show that global $P^a$ is also conserved on ${\mathbb{R}}^d$.

Rewrite ordinary diffusion in double-divergence form. Explain which equilibrium or constitutive condition it fails for an exact dipole-symmetric fluid.

Target D: weak symmetry breaking

Suppose

Find $k_{\times }$, ${\ell }_{\times }$, and the regime in which an experiment would see apparent $z=4$ scaling.

Oral check 1

Would $\tau (L)\propto L^4$ establish fracton topological order?

Oral check 2

Why is a linear chemical-potential profile an equilibrium diagnostic for exact dipole conservation?

Hints

• Integrate twice when differentiating $P^a$.
• In Fourier space, ${\mathrm{\nabla }}^4\mapsto k^4$.
• Ordinary diffusion has ${\tilde{J}}^{ij}=-D_2{\delta }^{ij}\rho$.
• Equate the $k^2$ and $k^4$ contributions at the crossover.

Solution outline

For charge,

$$\dot{Q}=-\int d^{d}x{\partial }_i{\partial }_j{J}^{ij}=0.$$Copy LaTeX

For dipole moment,

$${\dot{P}}^a=-\int d^{d}x{x}^a{\partial }_i{\partial }_j{J}^{ij}=\int d^{d}x{\partial }_j{J}^{aj}=0.$$Copy LaTeX

The constitutive relation gives

$${\partial }_i{\partial }_j{J}^{ij}=\frac{B_1+B_2}{\chi }{\mathrm{\nabla }}^4\rho .$$Copy LaTeX

Thus

$${\partial }_t\rho =-D_4{\mathrm{\nabla }}^4\rho ,D_4=\frac{B_1+B_2}{\chi }.$$Copy LaTeX

For a plane wave,

$$\rho \propto e^{\mathrm{i}\mathbf{k}\cdot \mathbf{x}-\mathrm{\Gamma }t},$$Copy LaTeX

so

$$\mathrm{\Gamma }=D_4{k}^4.$$Copy LaTeX

Stability requires the longitudinal dissipative contraction to be nonnegative. In this scalar isotropic channel, $D_4\ge 0$.

Ordinary diffusion gives

$${\dot{P}}^a=D_2\int d^{d}x{x}^a{\mathrm{\nabla }}^2\rho =-D_2\int d^{d}x{\partial }_a\rho =0.$$Copy LaTeX

It can also be written as

$${\partial }_t\rho +{\partial }_i{\partial }_j(-D_2{\delta }^{ij}\rho )=0.$$Copy LaTeX

This tensor rewrite does not turn ordinary diffusion into exact dipole hydrodynamics. A linear chemical potential represents a generalized equilibrium state when $P^a$ is an exact conserved charge.

Ordinary diffusion assigns it a nonzero vector current:

$$j^i=-D_2\chi {\partial }_i\mu .$$Copy LaTeX

The dipole-symmetric dissipative current instead begins with ${\partial }_i{\partial }_j\mu$, which vanishes for constant and linear $\mu$.

For weak symmetry breaking, equate

$$D_2{k}_{\times }^2=D_4{k}_{\times }^4.$$Copy LaTeX

Therefore

$$k_{\times }=\sqrt{\frac{D_2}{D_4}},{\ell }_{\times }=\frac{1}{k_{\times }}=\sqrt{\frac{D_4}{D_2}}.$$Copy LaTeX

The $k^4$ term dominates for $k\gg k_{\times }$, provided the wavelength remains long compared with microscopic scales. The strict infrared, $k\ll k_{\times }$, is diffusive.

Exit ticket

An X-cube sample and a Landau-level-projected atomic fluid both show restricted density relaxation.

What additional data would you require before claiming that they realize the same phase rather than related conservation kinematics?

A strong answer should separate:

1. the local conservation algebra;
2. thermalization or fragmentation;
3. the spectrum of mobile and immobile excitations;
4. fusion, detection, and entanglement data;
5. stability under symmetry-preserving local perturbations.

Research checks worth doing next

1. Current reconstruction. Measure or compute the rank-2 current rather than inferring it only from a fitted density exponent.
2. Crossover collapse. Fit $\mathrm{\Gamma }(k)$ to $D_2{k}^2+D_4{k}^4$ over system size and symmetry-breaking strength.
3. Equilibration test. Compare late-time states from different initial conditions within the same $Q$, $P^a$, and energy sector.
4. Boundary audit. Track dipole flux through open edges and winding events on periodic geometry.
5. Nonlinear scaling. Test whether momentum conservation and fluctuations renormalize the Gaussian exponent.
6. Phase diagnosis. Calculate superselection, fusion, planon-detection, and entanglement-RG data independently of transport.
7. Geometry response. Compare an absolute-space dipole symmetry with one defined relative to dynamical crystal coordinates.

Further reading

• Fracton Phases of Matter — broad review of tensor gauge theories, lattice phases, localization, elasticity, and higher-moment conservation.
• Fracton hydrodynamics — local multipole conservation and the hierarchy of subdiffusive hydrodynamic classes.
• Colloquium: Fracton matter — 2024 review centered on tensor-gauge and elasticity dualities.
• Breakdown of hydrodynamics below four dimensions in a fracton fluid — nonlinear fluctuation instability when momentum is also conserved.
• Observation of Hilbert space fragmentation and fractonic excitations in 2D — constrained dynamics in a tilted two-dimensional Bose-Hubbard experiment.
• Emergent Fracton Hydrodynamics of Ultracold Atoms in Partially Filled Landau Levels — projection-induced dipole conservation, fragmentation regimes, and late-time subdiffusion.
• Planon-modular fracton orders — a 2025 algebraic class of type-I orders and associated phase invariants.
• Fractonic solids — 2026 formulation of multipole symmetry relative to dynamical crystal coordinates.
• Continuum fractons: Quantization and the few-body problem — 2026 few-body quantum mechanics with dipole conservation and locality.

What to retain

• Global dipole conservation is weaker than an exact microscopic dipole symmetry.
• A Gauss law can impose immobility without determining finite-temperature hydrodynamics.
• The rank-2 Ward identity needs a constitutive and equilibration problem before it yields $k^4$.
• Ordinary diffusion preserves global dipole moment on infinite space.
• Weak dipole breaking restores $k^2$ behavior at the longest wavelength.
• Hilbert-space fragmentation can prevent hydrodynamics.
• Momentum conservation and nonlinear fluctuations can change the Gaussian universality class.
• Restricted mobility does not classify gapped fracton order.
• Periodic boundaries and curved geometry change how dipole symmetry is defined.
• Transport exponents and phase invariants answer different questions.

Next: derive fracton-elasticity duality from the stress Hubbard-Stratonovich field and track how vacancies relax the dislocation glide constraint.