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arXiv:2608.13080

Local and quasilocal conservation laws of three-state IRF cellular automata and their quantum deformations

Tomaž Prosen

cond-mat.stat-mechmath-phnlin.CGquant-phSubmitted Aug 13, 2026• Announced Aug 14, 2026• 0 citations
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Abstract

Using patch-matrix-product methods, we study two reversible three-state interaction-round-a-face cellular automata introduced by Klobas and Prosen [J. Phys. A 55, 094003 (2022)] - the species-preserving and species-flipping rules - and a coherent quantum deformation interpolating between them. Within an explicit translationally invariant ansatz, the two classical rules share a one-parameter family of quasi-local conservation laws with an auxiliary-dimension-three realization. Every regular member is also the first centered logarithmic derivative of an analytic auxiliary-dimension-two family passing through the identity observable. On the closed span of this family and the three elementary local charges, the only nonzero Euler velocities are $\pm\sqrt{3/23}$. Consequently, $\sqrt{3/23}$ is a rigorous lower bound on the maximal Euler speed in any larger conserved sector. For the species-flipping rule, this value agrees with the reported extrapolated value $0.361$ of Klobas and Prosen, strongly supporting completeness of the known sound-active charges. The species-preserving rule admits, in addition, a staggered generating family whose logarithmic derivatives form an infinite tower of strictly local charges. In the quantum deformation, the same algebraic structures yield exact low-bond-dimension invariant states (scar candidates), exponentially long-lived quasi-local quasimodes, and diagonal quasi-local charges that produce a nonzero Mazur bound after projection away from the three elementary local charges.

Comments
41 pages, 4 figures; Research in this work has been assisted by generative AI

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Using patch-matrix-product methods, we study two reversible three-state interaction-round-a-face cellular automata introduced by Klobas and Prosen [J. Phys. A 55, 094003 (2022)] - the species-preserving and species-flipping rules - and a coherent quantum deformation interpolating between them. Within an explicit translationally invariant ansatz, the two classical rules share a one-parameter family of quasi-local conservation laws with an auxiliary-dimension-three realization. Every regular member is also the first centered logarithmic derivative of an analytic auxiliary-dimension-two family passing through the identity observable. On the closed span of this family and the three elementary local charges, the only nonzero Euler velocities are ±3/23\pm\sqrt{3/23}±3/23​. Consequently, 3/23\sqrt{3/23}3/23​ is a rigorous lower bound on the maximal Euler speed in any larger conserved sector. For the species-flipping rule, this value agrees with the reported extrapolated value 0.3610.3610.361 of Klobas and Prosen, strongly supporting completeness of the known sound-active charges. The species-preserving rule admits, in addition, a staggered generating family whose logarithmic derivatives form an infinite tower of strictly local charges. In the quantum deformation, the same algebraic structures yield exact low-bond-dimension invariant states (scar candidates), exponentially long-lived quasi-local quasimodes, and diagonal quasi-local charges that produce a nonzero Mazur bound after projection away from the three elementary local charges.
Abstract We introduce a pair of time-reversible models defined on the discrete space–time lattice with three states per site, specifically, a vacancy and a particle of two flavours (species). The local update rules reproduce the rule 54 reversible cellular automaton when only a single species of particles is present, and satisfy the requirements of flavour exchange ( C ), space-reversal ( P ), and time-reversal ( T ) symmetries. We find closed-form expressions for three local conserved charges and provide an explicit matrix product form of the grand canonical Gibbs states, which are identical for both models. For one of the models this family of Gibbs states seems to be a complete characterisation of equilibrium (i.e. space and time translation invariant) states, while for the other model we empirically find a sequence of local conserved charges, one for each support size larger than 2, hinting to its algebraic integrability. Finally, we numerically investigate the behaviour of spatio-temporal correlation functions of charge densities, and test the hydrodynamic prediction for the model with exactly three local charges. Surprisingly, the numerically observed ‘sound velocity’ does not match the hydrodynamic value. The deviations are either significant, or they decay extremely slowly with the simulation time, which leaves us with an open question for the mechanism of such a glassy behaviour in a deterministic locally interacting system.
A family of reversible deterministic cellular automata, including the rules 54 and 201 of [Bobenko et al., Commun. Math. Phys. 158, 127 (1993)] as well as their kinetically constrained quantum (unitary) or stochastic deformations, is shown to correspond to integrable Floquet circuit models with local interactions round-a-face. Using inhomogeneous solutions of the star-triangle relation with a one or two dimensional spectral parameter, changing their functional form depending on the orientation, we provide an explicit construction of the transfer matrix and establish its conservation law and involutivity properties. Integrability is independently demonstrated by numerically exploring the spectral statistics via the Berry-Tabor conjecture. Curiously, we find that the deformed rule 54 model generically possesses no other local conserved quantities besides the net soliton current.
We study quantum and stochastic deformations of the rule-54 reversible cellular automaton (RCA54) on a 1+1-dimensional spatiotemporal lattice, focusing on their integrability structures in two distinct settings. First, for the quantum deformation, which turns the model into an interaction-round-a-face brickwork quantum circuit (either on an infinite lattice or with periodic boundary conditions), we show that the shortest-range nontrivial conserved charge commuting with the discrete-time evolution operator has a density supported on six consecutive sites. By constructing the corresponding range-6 Lax operator, we prove that this charge belongs to an infinite tower of mutually commuting conserved charges generated by higher-order logarithmic derivatives of the transfer matrix. With the aid of an intertwining operator, we further prove that the transfer matrix commutes with the discrete-time evolution operator. Second, for the stochastic deformation, which renders the model into a Markov-chain circuit, we investigate open boundary conditions that couple the system at its edges to stochastic reservoirs. In this setting, we explicitly construct the non-equilibrium steady state (NESS) by means of a staggered patch matrix ansatz, a hybrid construction combining the previously used commutative patch-state ansatz for the undeformed RCA54 with the matrix-product ansatz. Finally, we propose a simple empirical criterion for detecting integrability or exact solvability in a given model setup, introducing the notion of digit complexity.
Abstract We study quantum and stochastic deformations of the rule-54 reversible cellular automaton (RCA54) on a 1+1–dimensional spatiotemporal lattice, focusing on their integrability structures in two distinct settings. First, for the quantum deformation, which turns the model into an interaction-round-a-face brickwork quantum circuit (either on an infinite lattice or with periodic boundary conditions), we show that the shortest-range nontrivial conserved charge commuting with the discrete-time evolution operator has a density supported on six consecutive sites. By constructing the corresponding range-6 Lax operator, we prove that this charge belongs to an infinite tower of mutually commuting conserved charges generated by higher-order logarithmic derivatives of the transfer matrix. With the aid of an intertwining operator, we further prove that the transfer matrix commutes with the discrete-time evolution operator. Second, for the stochastic deformation, which renders the model into a Markov-chain circuit, we investigate open boundary conditions that couple the system at its edges to stochastic reservoirs. In this setting, we explicitly construct the non-equilibrium steady state by means of a staggered patch matrix ansatz, a hybrid construction combining the previously used commutative patch-state ansatz for the undeformed RCA54 with the matrix-product ansatz. Finally, we propose a simple empirical criterion for detecting integrability or exact solvability in a given model setup, introducing the notion of digit complexity.
Quantum cellular automata (QCAs) are automorphisms of tensor product algebras that preserve locality, with local quantum circuits as a simple example. We study approximate QCAs, where the locality condition is only satisfied up to a small error, as occurs for local quantum dynamics on the lattice. A priori, approximate QCAs could exhibit genuinely new behavior, failing to be well-approximated by any exact QCA. We show this does not occur in one dimension: every approximate QCA on a finite circle can be rounded to a strict QCA with approximately the same action on local operators, so these systems are classified by the same index as in the exact case. Previous work considered the case of the infinite line, by using global methods not amenable to finite systems. Our new approach proceeds locally and now applies to finite systems, including circles or homomorphisms from sub-intervals. We extract exact local boundary algebras from the approximate QCA restricted to local patches, then glue these to form a strict QCA. The key technical ingredient is a robust notion of the intersection of two subalgebras: when the projections onto two subalgebras approximately commute, we construct an exact subalgebra that serves as a stable proxy for their intersection. This construction uses a recent theorem of Kitaev on the rigidity of approximate C∗C^*C∗-algebras.
We propose and discuss two variants of kinetic particle models—cellular automata in 1 + 1 dimensions—that have some appeal due to their simplicity and intriguing properties, which could warrant further research and applications. The first model is a deterministic and reversible automaton describing two species of quasiparticles: stable massless matter particles moving with velocity ±1 and unstable standing (zero velocity) field particles. We discuss two distinct continuity equations for three conserved charges of the model. While the first two charges and the corresponding currents have support of three lattice sites and represent a lattice analogue of the conserved energy–momentum tensor, we find an additional conserved charge and current with support of nine sites, implying non-ergodic behaviour and potentially signalling integrability of the model with a highly nested R-matrix structure. The second model represents a quantum (or stochastic) deformation of a recently introduced and studied charged hardpoint lattice gas, where particles of different binary charge (±1) and binary velocity (±1) can nontrivially mix upon elastic collisional scattering. We show that while the unitary evolution rule of this model does not satisfy the full Yang–Baxter equation, it still satisfies an intriguing related identity which gives birth to an infinite set of local conserved operators, the so-called glider operators.