Charge-and-Dipole Conserving Circuits

• Charge-and-dipole conserving circuits are networks that enforce both charge and dipole moment conservation via capacitor and transformer architectures, leading to fracton-like mobility constraints.
• They utilize transformer-induced current mirroring to convert first-derivative continuity into a second-derivative form, resulting in unique linear equilibrium charge profiles and effective DC filtering.
• Extensions to higher multipole moments and superconducting designs offer broad applications, paving the way for deeper exploration of fracton kinematics and subdiffusive hydrodynamics.

Charge-and-dipole conserving circuits are electrical networks engineered so that the dynamical conservation law for charge is upgraded from ordinary charge continuity to a higher-moment continuity law that preserves not only total charge but also dipole moment. In the canonical construction, capacitors store charge while transformers couple adjacent wires so that current in one branch induces an equal-and-opposite current in a nearby branch. For a suitable lattice geometry, this converts the usual first-derivative continuity equation into a second-derivative form, producing fracton-like mobility restrictions: isolated charge cannot move freely, and relaxation proceeds only through collective rearrangements compatible with dipole conservation. The original proposal introduced these systems as classical “fractolectric” circuits, emphasized their linear-in-position equilibrium charge profiles and DC-filtering properties, and outlined extensions to higher moments, higher dimensions, and superconducting implementations (Pretko, 2019).

1. Transformer-constrained circuit architecture

The basic circuit architecture consists of a lattice of capacitors connected by ideal transformers. The capacitors are the only elements that store charge; the transformers do not store charge but constrain how charge can move between neighboring capacitors. The microscopic mechanism is the transformer relation between two adjacent wires. With self-inductance $L$ and mutual inductance $M$, the coupled voltages are written as

$$V_{1/2} = -L\,\partial_t I_{1/2} - M\,\partial_t I_{2/1}.$$

When the windings are arranged so that the secondary voltage is the negative of the primary voltage,

$V_1=-V_2,$

one obtains

$\frac{dI_1}{dt}=-\frac{dI_2}{dt},$

and hence, if the initial currents are opposite, they remain opposite: $I_1(t)=-I_2(t).$ This makes the transformer a current “mirror” with sign reversal. The same conclusion can be obtained from power balance: neglecting resistance, $I_1V_1=I_2V_2$, and opposite voltages imply opposite currents (Pretko, 2019).

In the one-dimensional realization, each capacitor sits at a lattice site $x_n$, and transformers connect neighboring sites so that current through one link induces opposite current in an adjacent link. Because there are two neighboring paths available in the bulk, a current pulse cannot simply move a single isolated charge to the next site; it necessarily generates compensating motion nearby. This is the circuit implementation of constrained transport usually associated with fracton quasiparticles.

2. Generalized continuity and dipole conservation

The central conservation law of these circuits is a generalized continuity equation. For a transformer carrying current $I(x_n)$, the circuit obeys

$\partial_t Q + \partial_x^2 I = 0,$

where $M$0 is a lattice second difference. This differs from ordinary charge conservation,

$M$1

by the extra spatial derivative. That additional derivative is what enforces conservation of the first moment of charge (Pretko, 2019).

If the dipole moment is defined as

$M$2

then, under the boundary conditions used in the circuit, the time derivatives of both total charge and total dipole moment vanish: $M$3 An isolated charge is therefore effectively immobile in the precise sense that moving it would change the dipole moment unless accompanied by compensating charge motion elsewhere. The mobility restriction is thus not an incidental feature of a particular drive protocol; it is a consequence of the circuit geometry and the higher-derivative continuity law.

This mechanism is the direct circuit analogue of fracton kinematics. Individual charges are constrained, while certain bound combinations or collective patterns can still move. A common misconception is that the relevant conservation law is merely charge conservation with unusual impedances. In fact, the defining property is the replacement of first-derivative transport by second-derivative transport, with the resulting exact conservation of $M$4 in the ideal limit.

3. Energetics, relaxation, and experimental diagnostics

For uniform capacitance $M$5, the electrostatic energy stored on the capacitors is

$M$6

With dissipation present, the circuit relaxes toward the minimum-energy state consistent with the conserved quantities. Minimizing

$M$7

with respect to $M$8 yields

$M$9

so the steady-state charge profile is linear in position: $$V_{1/2} = -L\,\partial_t I_{1/2} - M\,\partial_t I_{2/1}.$$0 This linear equilibrium profile is the clearest diagnostic of dipole conservation. Without dipole conservation, charge would relax to a uniform profile; with dipole conservation, the circuit retains a memory of its initial dipole moment and relaxes only to the lowest-energy configuration compatible with that memory (Pretko, 2019).

The original work verified this diagnostic in CircuitLab. Starting from charge localized on one capacitor, the simulated circuit relaxed through oscillations to a steady state whose voltage and charge profile was linear in position, as predicted. The same study also identified a transport signature: the circuit acts as a perfect DC filter. A constant applied voltage can drive a current through the network, but for general time-dependent driving only the zero-frequency component $$V_{1/2} = -L\,\partial_t I_{1/2} - M\,\partial_t I_{2/1}.$$1 passes in the idealized limit. This filtering behavior follows from the same transformer-induced constraints that enforce dipole conservation.

4. Nonidealities, dissipation, and superconducting realizations

Exact dipole conservation requires ideal transformers. In realistic devices, the principal symmetry-breaking mechanism is flux leakage: not all magnetic flux generated by the primary coil is captured by the secondary. Flux leakage weakens the equal-and-opposite current constraint and allows dipole moment to relax at very long times. The resulting violation of fracton-like behavior is therefore expected first at the longest timescales (Pretko, 2019).

An important technical point is that ordinary internal resistance does not by itself spoil dipole conservation if it is symmetric on the two sides of the transformer. The conservation law is tied to the flux-sharing geometry rather than to the complete absence of dissipation. This distinguishes these circuits from architectures in which constrained transport relies only on fine-tuned lossless dynamics.

The same paper argued that superconducting circuits provide a natural extension. In superconducting wires, current is directly tied to magnetic flux, so the equal-and-opposite current relation can hold even for DC currents. A superconducting flux transformer can transfer flux more faithfully than a classical magnetic-core transformer, thereby reducing flux-leakage violations. Quantum dots can then store quantized charge, suggesting a route from classical charge-and-dipole conserving circuits to quantized fracton realizations in superconducting hardware (Pretko, 2019).

5. Higher moments, higher dimensions, and multipolar response

The transformer-based construction generalizes beyond dipole conservation. By stacking transformers into hierarchical layered structures, one can engineer circuits that conserve quadrupole moment and higher multipole moments. The paper presenting fractolectric circuits gave an explicit example conserving both dipole and quadrupole moment and stated that the method extends to arbitrarily high moments by adding more transformer layers. In effect, increasingly elaborate transformer networks enforce increasingly restrictive generalized continuity equations (Pretko, 2019).

Higher-dimensional generalizations replace the scalar current by a tensorial current $$V_{1/2} = -L\,\partial_t I_{1/2} - M\,\partial_t I_{2/1}.$$2, with continuity equation

$$V_{1/2} = -L\,\partial_t I_{1/2} - M\,\partial_t I_{2/1}.$$3

In the limit $$V_{1/2} = -L\,\partial_t I_{1/2} - M\,\partial_t I_{2/1}.$$4, so that no charge is stored locally, the constraint reduces to

$$V_{1/2} = -L\,\partial_t I_{1/2} - M\,\partial_t I_{2/1}.$$5

and the energy becomes

$$V_{1/2} = -L\,\partial_t I_{1/2} - M\,\partial_t I_{2/1}.$$6

The resulting current correlations are expected to show pinch-point singularities, analogous to spin ice; this regime was termed fracton “current-ice” (Pretko, 2019).

Related interacting lattice models show that charge-and-dipole conservation is also compatible with multipolar topological response. In two dimensions, a ring-exchange model conserving charge and dipole realizes a quadrupole response

$$V_{1/2} = -L\,\partial_t I_{1/2} - M\,\partial_t I_{2/1}.$$7

and at $$V_{1/2} = -L\,\partial_t I_{1/2} - M\,\partial_t I_{2/1}.$$8 symmetry the quadrupole moment is pinned to $$V_{1/2} = -L\,\partial_t I_{1/2} - M\,\partial_t I_{2/1}.$$9 or $V_1=-V_2,$0. The same work connected a periodic adiabatic cycle to quantized pumping with $V_1=-V_2,$1, and in three dimensions related chiral hinge modes to a dipolar Chern-Simons response (May-Mann et al., 2021). This suggests that charge-and-dipole conserving circuits belong to a broader class of multipole-conserving platforms rather than an isolated circuit trick.

6. Hydrodynamics, criticality, and symmetry constraints in related dipole-conserving systems

The dynamical consequences of charge-and-dipole conservation extend well beyond static circuit equilibration. In one-dimensional dipole-conserving systems, hydrodynamics predicts subdiffusion rather than diffusion. For conservation of charge and dipole moment, the long-wavelength equation takes the generalized form

$V_1=-V_2,$2

with $V_1=-V_2,$3, giving dynamical exponent $V_1=-V_2,$4 and return probability

$V_1=-V_2,$5

Automaton simulations found $V_1=-V_2,$6, while quadrupole-conserving systems gave $V_1=-V_2,$7 (Feldmeier et al., 2020). In coupled charge-energy hydrodynamics, charge remains subdiffusive while energy supports an ordinary diffusive mode, and the two sectors are linked by a generalized diffusion matrix (Burchards et al., 2022). For charge-and-dipole conserving circuits, this implies that late-time transport signatures should be interpreted through higher-derivative hydrodynamics rather than ordinary RC diffusion.

Two-dimensional dipole-conserving systems also modify the conventional Kosterlitz-Thouless paradigm. In the dipole-conserving XY model, the ordinary phase correlator decays super-exponentially rather than algebraically, the conventional vortex has finite self-energy, and KT criticality is instead driven by two unconventional vortices in compact dipole fields whose self-energies diverge logarithmically. In the isotropic model they deconfine simultaneously in a single KT transition, while coupling anisotropy can split the transition into two (Wang et al., 24 Jun 2026). This establishes that higher-moment conservation changes not only transport but also the identity of the critical topological defects.

Symmetry constraints also become subtler on periodic lattices. In one-dimensional translation-invariant systems on a ring, dipole symmetry is not continuous $V_1=-V_2,$8 but discrete $V_1=-V_2,$9, generated by

$\frac{dI_1}{dt}=-\frac{dI_2}{dt},$0

A symmetric, gapped, non-degenerate ground state then requires integer charge filling together with a special dipole filling obeying

$\frac{dI_1}{dt}=-\frac{dI_2}{dt},$1

Other fractional dipole fillings enforce either gaplessness or symmetry breaking (Burnell et al., 2023). Although this result concerns lattice systems rather than electrical hardware directly, it clarifies that dipole conservation is sensitive to boundary conditions and global topology.

Monitored quantum circuits with exact conservation of both charge and dipole moment reveal a further layer of structure. In one spatial dimension, charge is always easy to learn from local measurements, while dipole moment can be either easy or hard. In two dimensions, there are three phases: for frequent measurements, both charge and dipole moment are easy to learn; as the measurement rate is decreased, first dipole moment and then charge become hard; and the low-measurement phase is an exotic critical phase with anisotropic spacetime scaling analogous to a smectic liquid crystal (Zerba et al., 16 Dec 2025). This suggests that charge-and-dipole conserving circuit ideas naturally extend into information-theoretic settings where conservation laws govern not only transport but also inferability from measurement records.