Momentum-Conserving Parity Check Cellular Automaton

• MCPCA is a deterministic classical many-body system on a square lattice that evolves binary edge bits using parity-check and momentum conservation rules.
• The system exhibits a second-order phase transition characterized by ballistic information spreading, controlled by initial conserved plaquette-charge densities.
• It leverages spatial diagnostics like the Hamming distance and decorrelators to differentiate localized from chaotic regimes with multifractal signatures.

Searching arXiv for papers on the Momentum-Conserving Parity Check Cellular Automaton and closely related work. Momentum-Conserving Parity Check Cellular Automaton (MCPCA) is a deterministic interacting classical many-body system defined on the two-dimensional square lattice in which binary degrees of freedom live on edges and evolve by sublattice-parallel local maps constrained by parity-check preservation and momentum conservation. In the formulation analyzed in "Phase transition from localization to chaos in classical many-body system" (Kasim et al., 26 Sep 2025), the MCPCA exhibits an extensive family of local conserved loop-charges, a dynamical phase transition in information spreading from a localized regime to a chaotic regime with ballistic spreading, and a multifractal dynamical structure factor whose origin is traced to effective local periodicities rather than to the transition itself.

1. Definition on the square lattice

Let $G=(V,E)$ be the bipartite square-lattice graph with sublattices $A$ and $B$. Each edge $e\in E$ carries a bit $s_e\in\{0,1\}$, and a global configuration is $\underline s\in\{0,1\}^{|E|}$. The dynamics proceeds in two half-steps, first updating all vertices in one sublattice and then all vertices in the other: $$\underline s(t+1)\;=\;\Phi\bigl(\underline s(t)\bigr)\;=\;\Bigl(\!\prod_{v\in B}\Phi_v\Bigr) \Bigl(\!\prod_{v\in A}\Phi_v\Bigr)\,\underline s(t)\,.$$

Each local gate $\Phi_v$ acts on the four incident edge bits at a square-lattice vertex and is required to satisfy two constraints. The first is a parity-check condition: for any pair of incident edges $i,j$, the bit-wise modulo-$2$ sums are preserved,

$A$0

The second is momentum conservation: if an occupied edge $A$1 is interpreted as a unit-mass particle carrying the edge-direction vector $A$2, then the incoming and outgoing momenta satisfy

$A$3

On the square lattice, where $A$4, the most symmetric nontrivial solution is obtained by numbering the four incident edges in cyclic order and applying the rule

$A$5

Thus, if the four bits are not all equal, all four are flipped; if they are all equal, the local configuration is left unchanged. This rule manifestly respects both the parity-check and momentum-conservation constraints (Kasim et al., 26 Sep 2025).

The resulting automaton is deterministic, local, and many-body in the strict sense that its evolution couples edge variables through vertex updates over the entire square lattice. The dynamics is not stochastic; all nontrivial behavior arises from the interplay between the local rule, the lattice geometry, and the conserved quantities.

2. Conserved loop-charges and 1-form symmetry

A central structural feature of the MCPCA is an extensive set of local conserved charges. Define on each edge the staggered-magnetization observable

$A$6

Because each local gate $A$7 flips the sign of any difference $A$8 among the four incident edges, one obtains strict invariants associated with every closed even loop $A$9: $B$0

On an $B$1 square lattice with periodic boundaries, a convenient basis is given by the $B$2 elementary plaquette loops $B$3 of length $B$4. Each plaquette charge takes one of the five values

$B$5

The corresponding plaquette-charge densities are

$B$6

These densities remain constant under time evolution.

The paper characterizes this structure as an extensive family of local “1-form” conserved loop-charges (Kasim et al., 26 Sep 2025). In this setting, the conserved quantities do not attach to vertices or sites in the usual zero-form sense; instead, they are naturally supported on closed loops. This loop-based conservation law strongly constrains the dynamics and is central to the emergence of information localization in part of the phase diagram.

3. Initialization protocols and control of conserved sectors

The conserved plaquette-charge densities $B$7 partition the configuration space into dynamically disconnected sectors. Two initialization schemes are described for sampling these sectors.

In the Bernoulli-product ensemble, each bit is drawn independently with

$B$8

Under this ensemble, the charge densities $B$9 become deterministic functions of the parameter $e\in E$0.

A second procedure uses a Markov-chain (Metropolis) construction at fixed chemical potentials $e\in E$1 for the counts $e\in E$2, thereby targeting arbitrary plaquette-charge densities $e\in E$3. This provides direct control over the conserved-charge content of the initial state.

The localization-chaos transition reported for the MCPCA is not described as a modification of the local update rule; instead, it is induced by selecting initial ensembles with specific charge values. This is a defining point of the model’s phenomenology: the same deterministic dynamics can realize distinct information-spreading regimes depending on the conserved sector in which the system is initialized (Kasim et al., 26 Sep 2025).

4. Diagnostics of information spreading

Two information-theoretic observables are used to diagnose how perturbations propagate: the Hamming distance and the classical decorrelator.

The Hamming distance between two configurations $e\in E$4 and $e\in E$5 is

$e\in E$6

To probe sensitivity to local perturbations, two copies of the system are evolved from identical initial data except for a single flipped bit, so that $e\in E$7. Their time evolution is

$e\in E$8

and the time-dependent Hamming distance is

$e\in E$9

Its long-time saturation defines

$s_e\in\{0,1\}$0

The distinction between phases is then formulated directly in terms of $s_e\in\{0,1\}$1. A delocalized phase is defined by $s_e\in\{0,1\}$2, meaning that the initial error spreads to an extensive fraction of edges. A localized phase is defined by $s_e\in\{0,1\}$3, meaning that the error remains confined.

The classical decorrelator resolves this spreading spatially: $s_e\in\{0,1\}$4 The perturbation is inserted at $s_e\in\{0,1\}$5, so that $s_e\in\{0,1\}$6. Its time average is

$s_e\in\{0,1\}$7

In the localized phase,

$s_e\in\{0,1\}$8

which defines a finite correlation length $s_e\in\{0,1\}$9. In the delocalized phase, by contrast,

$\underline s\in\{0,1\}^{|E|}$0

These diagnostics separate confinement of perturbations from extensive spreading without introducing stochasticity or external disorder. In the MCPCA, the dynamical distinction is encoded entirely in the deterministic evolution and the conserved-charge sector (Kasim et al., 26 Sep 2025).

5. Second-order transition from localization to chaos

For Bernoulli-product initial states, the control parameter is the occupation probability $\underline s\in\{0,1\}^{|E|}$1. Numerical data on lattice sizes

$\underline s\in\{0,1\}^{|E|}$2

show a clean second-order transition at

$\underline s\in\{0,1\}^{|E|}$3

The order parameter is the Hamming-distance density $\underline s\in\{0,1\}^{|E|}$4. In the delocalized regime $\underline s\in\{0,1\}^{|E|}$5, it scales as

$\underline s\in\{0,1\}^{|E|}$6

On the localized side $\underline s\in\{0,1\}^{|E|}$7, the correlation length diverges as

$\underline s\in\{0,1\}^{|E|}$8

At criticality $\underline s\in\{0,1\}^{|E|}$9, the time-averaged decorrelator decays algebraically,

$$\underline s(t+1)\;=\;\Phi\bigl(\underline s(t)\bigr)\;=\;\Bigl(\!\prod_{v\in B}\Phi_v\Bigr) \Bigl(\!\prod_{v\in A}\Phi_v\Bigr)\,\underline s(t)\,.$$0

The three regimes can be summarized concisely as follows.

Regime	$$\underline s(t+1)\;=\;\Phi\bigl(\underline s(t)\bigr)\;=\;\Bigl(\!\prod_{v\in B}\Phi_v\Bigr) \Bigl(\!\prod_{v\in A}\Phi_v\Bigr)\,\underline s(t)\,.$$1	Spatial behavior of $$\underline s(t+1)\;=\;\Phi\bigl(\underline s(t)\bigr)\;=\;\Bigl(\!\prod_{v\in B}\Phi_v\Bigr) \Bigl(\!\prod_{v\in A}\Phi_v\Bigr)\,\underline s(t)\,.$$2
$$\underline s(t+1)\;=\;\Phi\bigl(\underline s(t)\bigr)\;=\;\Bigl(\!\prod_{v\in B}\Phi_v\Bigr) \Bigl(\!\prod_{v\in A}\Phi_v\Bigr)\,\underline s(t)\,.$$3	$$\underline s(t+1)\;=\;\Phi\bigl(\underline s(t)\bigr)\;=\;\Bigl(\!\prod_{v\in B}\Phi_v\Bigr) \Bigl(\!\prod_{v\in A}\Phi_v\Bigr)\,\underline s(t)\,.$$4	$$\underline s(t+1)\;=\;\Phi\bigl(\underline s(t)\bigr)\;=\;\Bigl(\!\prod_{v\in B}\Phi_v\Bigr) \Bigl(\!\prod_{v\in A}\Phi_v\Bigr)\,\underline s(t)\,.$$5 const
$$\underline s(t+1)\;=\;\Phi\bigl(\underline s(t)\bigr)\;=\;\Bigl(\!\prod_{v\in B}\Phi_v\Bigr) \Bigl(\!\prod_{v\in A}\Phi_v\Bigr)\,\underline s(t)\,.$$6	$$\underline s(t+1)\;=\;\Phi\bigl(\underline s(t)\bigr)\;=\;\Bigl(\!\prod_{v\in B}\Phi_v\Bigr) \Bigl(\!\prod_{v\in A}\Phi_v\Bigr)\,\underline s(t)\,.$$7	$$\underline s(t+1)\;=\;\Phi\bigl(\underline s(t)\bigr)\;=\;\Bigl(\!\prod_{v\in B}\Phi_v\Bigr) \Bigl(\!\prod_{v\in A}\Phi_v\Bigr)\,\underline s(t)\,.$$8
$$\underline s(t+1)\;=\;\Phi\bigl(\underline s(t)\bigr)\;=\;\Bigl(\!\prod_{v\in B}\Phi_v\Bigr) \Bigl(\!\prod_{v\in A}\Phi_v\Bigr)\,\underline s(t)\,.$$9	$\Phi_v$0	$\Phi_v$1

The corresponding correlation length is infinite for $\Phi_v$2 and scales as $\Phi_v$3 for $\Phi_v$4 (Kasim et al., 26 Sep 2025).

The transition is described as a phase transition in information spreading within a classical 2D deterministic interacting many-body system. The delocalized phase is also described as a chaotic regime with ballistic information spreading, whereas the localized phase confines perturbations. Because the local update rule is unchanged across the transition, the sharp change in spreading behavior is attributed to the conserved-charge structure of the initial ensemble rather than to a change in microscopic dynamics.

6. Multifractal dynamical structure factor and local periodicities

The same work revisits the multifractal behavior of the dynamical structure factor. Define the vertex-density

$\Phi_v$5

and the two-point function

$\Phi_v$6

The local power spectrum at $\Phi_v$7 is

$\Phi_v$8

Numerically, $\Phi_v$9 displays a dense hierarchy of sharp peaks at rational frequencies $i,j$0. If $i,j$1 denotes the weight of a peak with denominator $i,j$2, the total weight at fixed $i,j$3 is

$i,j$4

and it decays as

$i,j$5

The interpretation advanced in the paper is based on local periodicities. In the localized regime, constraints from plaquette-charges can isolate small regions into near-periodic subsystems of period $i,j$6. Extracting a distribution of local periods from the Fourier peak at $i,j$7 yields

$i,j$8

Averaging uniform spectral combs over periods drawn from $i,j$9 reproduces the observed multifractal structure of $2$0.

A critical conclusion is that the multifractality is present in both phases and is not an order parameter (Kasim et al., 26 Sep 2025). The quantities $2$1 and the multifractal peak-weights vary smoothly across $2$2. The multifractal spectrum therefore does not diagnose the localization-delocalization transition itself; rather, it reflects broad distributions of effective local periods enforced by symmetry constraints.

7. Conceptual significance and common interpretive boundaries

The MCPCA provides an example of a classical deterministic many-body system in which information spreading undergoes a genuine second-order transition while the microscopic update rule remains fixed. The phase structure is organized by conserved loop-charges, and the transition can be accessed by changing the initial charge-sector statistics through $2$3 or, more generally, through targeted $2$4.

Several interpretive boundaries follow directly from the reported results. First, localization in this setting refers to information localization as diagnosed by the Hamming distance and decorrelator, not to the disappearance of nontrivial local dynamics. Second, the multifractal structure factor should not be conflated with the phase transition: it persists on both sides of $2$5 and originates from effective local periodicities. Third, the delocalized regime is identified through extensive perturbation growth and ballistic information spreading, whereas the localized regime is characterized by confinement and an exponentially decaying time-averaged decorrelator with finite $2$6.

The overall picture is that of a constrained classical cellular automaton with local 1-form conserved charges, a sharply defined critical point at $2$7, critical exponents $2$8, $2$9, and $A$00, and a multifractal power spectrum whose mechanism is distinct from the transition in information spreading (Kasim et al., 26 Sep 2025).