Representations of the quadratic Algebra and Partially Asymmetric Diffusion with Open Boundaries

TL;DR

This paper analyzes a partially asymmetric exclusion process with open boundaries, deriving exact stationary measures and correlation functions using quadratic algebra representations, and connects these results to quantum spin chains.

Contribution

It provides explicit algebraic representations and exact solutions for the model's density profiles and correlations, extending to quantum spin chain correspondence.

Findings

Exact density profiles and correlation functions derived

Representations of the quadratic algebra obtained

Connection established with Heisenberg XXZ spin chain

Abstract

We consider the one-dimensional partially asymmetric exclusion model with open boundaries. The model describes a system of hard-core particles that hop stochastically in both directions with different rates. At both boundaries particles are injected and extracted. By means of the method of Derrida, Evans, Hakim and Pasquier the stationary probability measure can be expressed as a matrix-product state involving two matrices subject to a quadratic algebra. We obtain the representations of this algebra and use the two-dimensional one to derive exact expressions for the density profile and correlation functions. Using the correspondence between the stochastic model and a quantum spin chain, we obtain exact correlation functions for a spin-$\frac{1}{2}$ Heisenberg XXZ chain with non-diagonal boundary terms. Generalizations to other reaction-diffusion models are discussed.