Reversible Cellular Automata as Integrable Interactions Round-a-Face: Deterministic, Stochastic, and Quantized

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.