# Exact matrix product decay modes of a boundary driven cellular automaton

## TL;DR

This paper investigates the integrability and spectral properties of a boundary-driven cellular automaton, providing explicit decay modes, Bethe-like equations, and insights into the spectrum's structure and thermodynamic behavior.

## Contribution

It introduces an exact matrix product ansatz for decay modes and conjectures Bethe-like equations for all spectral orbitals, advancing understanding of boundary-driven automaton dynamics.

## Key findings

- Explicit decay modes constructed for the automaton
- Spectrum separates into orbitals with Bethe-like equations
- Thermodynamic properties and spectral gap scaling analyzed

## Abstract

We study integrability properties of a reversible deterministic cellular automaton (the rule 54 of [Bobenko et al., Commun. Math. Phys. 158, 127 (1993)]) and present a bulk algebraic relation and its inhomogeneous extension which allow for an explicit construction of Liouvillian decay modes for two distinct families of stochastic boundary driving. The spectrum of the many-body stochastic matrix defining the time propagation is found to separate into sets, which we call orbitals, and the eigenvalues in each orbital are found to obey a distinct set of Bethe-like equations. We construct the decay modes in the first orbital (containing the leading decay mode) in terms of an exact inhomogeneous matrix product ansatz, study the thermodynamic properties of the spectrum and the scaling of its gap, and provide a conjecture for the Bethe-like equations for all the orbitals and their degeneracy.