The Irreducible String and an Infinity of Additional Constants of Motion in a Deposition-Evaporation Model on a Line

TL;DR

This paper analyzes a generalized deposition-evaporation model with three species on a line, revealing an infinite number of conserved quantities, exact sector sizes, and nontrivial dynamical exponents through analytical and numerical methods.

Contribution

It introduces a generalized model with additional constants of motion and provides exact calculations of sector sizes and eigenvectors, advancing understanding of its dynamical properties.

Findings

The model has an infinite number of constants of motion.

Exact sector sizes are determined.

The dynamical exponent is approximately 2.39.

Abstract

We study a model of stochastic deposition-evaporation with recombination, of three species of dimers on a line. This model is a generalization of the model recently introduced by Barma {\it et. al.} (1993 {\it Phys. Rev. Lett.} {\bf 70} 1033) to $q\ge 3$ states per site. It has an infinite number of constants of motion, in addition to the infinity of conservation laws of the original model which are encoded as the conservation of the irreducible string. We determine the number of dynamically disconnected sectors and their sizes in this model exactly. Using the additional symmetry we construct a class of exact eigenvectors of the stochastic matrix. The autocorrelation function decays with different powers of $t$ in different sectors. We find that the spatial correlation function has an algebraic decay with exponent 3/2, in the sector corresponding to the initial state in which all sites are in the same state. The dynamical exponent is nontrivial in this sector, and we estimate it numerically by exact diagonalization of the stochastic matrix for small sizes. We find that in this case $z=2.39\pm0.05$.