Matrix Product Steady States as Superposition of Product Shock Measures in 1D Driven Systems

TL;DR

This paper demonstrates that steady states in certain 1D driven lattice systems can be represented as superpositions of shock measures, with their algebraic structure always admitting a two-dimensional matrix representation.

Contribution

It establishes that the quadratic algebra for these models always has a two-dimensional representation, linking shock superpositions to matrix product states.

Findings

Steady states can be expressed as superpositions of product shocks.

Quadratic algebra associated with these models has a universal two-dimensional representation.

Examples for systems with n=1 and n=2 are provided.

Abstract

It is known that exact traveling wave solutions exist for families of (n+1)-states stochastic one-dimensional non-equilibrium lattice models with open boundaries provided that some constraints on the reaction rates are fulfilled. These solutions describe the diffusive motion of a product shock or a domain wall with the dynamics of a simple biased random walker. The steady state of these systems can be written in terms of linear superposition of such shocks or domain walls. These steady states can also be expressed in a matrix product form. We show that in this case the associated quadratic algebra of the system has always a two-dimensional representation with a generic structure. A couple of examples for n=1 and n=2 cases are presented.