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We study the dynamics of the Rule 150 reversible cellular automaton (RCA).This is a one-dimensional lattice system of binary variables with synchronous(Floquet) dynamics, corresponding to a bulk deterministic and reversiblediscrete version of the kinetically constrained XOR-Fredrickson-Andersen model,whereby the local dynamics is restricted: a site flips if and only if thestates of its neighbouring sites are different from each other. Like other RCAwhich have been studied recently, such as Rule 54 and Rule 201, Rule 150 isintegrable, however, in contrast is noninteracting. In particular, the emergentquasiparticles - the domain walls - behave as free fermions. This then allowsus to solve the model by means of matrix product ans\"{a}tze. We find the exactequilibrium and nonequilibrium stationary states for systems with closed(periodic) and open (stochastic) boundaries, respectively, resolve the fullspectrum of the time evolution operator and, therefore, gain access to therelaxation dynamics, and obtain the exact large deviation statistics ofdynamical observables in the long time limit.

Exact solution of the Rule 150 reversible cellular automaton