Exact solution of the Rule 150 reversible cellular automaton

TL;DR

This paper provides an exact analytical solution for the dynamics of the Rule 150 reversible cellular automaton, revealing its integrability, quasiparticle behavior, and detailed stationary states, relaxation dynamics, and large deviation statistics.

Contribution

It presents the first exact solution of Rule 150 RCA, including stationary states, spectrum, and dynamical statistics, highlighting its noninteracting, integrable nature.

Findings

Exact equilibrium and nonequilibrium stationary states derived.

Full spectrum of the time evolution operator obtained.

Large deviation statistics of dynamical observables characterized.

Abstract

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 reversible discrete version of the kinetically constrained XOR-Fredrickson-Andersen model, whereby the local dynamics is restricted: a site flips if and only if the states of its neighbouring sites are different from each other. Like other RCA which have been studied recently, such as Rule 54 and Rule 201, Rule 150 is integrable, however, in contrast is noninteracting. In particular, the emergent quasiparticles - the domain walls - behave as free fermions. This then allows us to solve the model by means of matrix product ans\"{a}tze. We find the exact equilibrium and nonequilibrium stationary states for systems with closed (periodic) and open (stochastic) boundaries, respectively, resolve the full spectrum of the time evolution operator and, therefore, gain access to the relaxation dynamics, and obtain the exact large deviation statistics of dynamical observables in the long time limit.