Matrix Product States for Interacting Particles without Hardcore Constraints

TL;DR

This paper develops a matrix product state approach for interacting particle systems without hardcore constraints, enabling exact steady state solutions for a broad class of stochastic processes with variable site occupancy.

Contribution

It introduces a novel matrix product ansatz with an infinite set of matrices reduced to a single functional relation, applicable to non-hardcore interacting particle systems.

Findings

Derived a matrix product formulation for non-hardcore particles.

Applied the method to various stochastic processes including zero range and misanthrope processes.

Unified treatment of different particle hopping models with finite-range interactions.

Abstract

We construct matrix product steady state for a class of interacting particle systems where particles do not obey hardcore exclusion, meaning each site can occupy any number of particles subjected to the global conservation of total number of particles in the system. To represent the arbitrary occupancy of the sites, the matrix product ansatz here requires an infinite set of matrices which in turn leads to an algebra involving infinite number of matrix equations. We show that these matrix equations, in fact, can be reduced to a single functional relation when the matrices are parametric functions of the representative occupation number. We demonstrate this matrix formulation in a class of stochastic particle hopping processes on a one dimensional periodic lattice where hop rates depend on the occupation numbers of the departure site and its neighbors within a finite range; this includes some well known stochastic processes like, totally asymmetric zero range process, misanthrope process, finite range process and partially asymmetric versions of the same processes but with different rate functions depending on the direction of motion.