On the integrability structure of the deformed rule-54 reversible cellular automaton

• URL: http://arxiv.org/abs/2603.25424v2
• Date: Mon, 30 Mar 2026 20:05:40 GMT
• Title: On the integrability structure of the deformed rule-54 reversible cellular automaton
• Abstract summary: We study quantum and deformations of the cellular automaton RCA54 on a 1+1-dimensional lattice.<n>We show that the shortest-range non conserved charge commuting with the discrete-time evolution operator has a density supported on six consecutive sites.<n>We propose a simple empirical complexity for detecting integrability or exactsolvability in a given model setup.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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 (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.

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