Dynamic Matrix Ansatz for Integrable Reaction-Diffusion Processes

TL;DR

This paper introduces a novel algebraic approach to analyze integrable reaction-diffusion systems, enabling exact solutions for their dynamics and spectral properties, with applications to exclusion processes and spin relaxation models.

Contribution

It develops a time-dependent operator algebra framework that generalizes previous methods, providing new insights and exact results for specific reaction-diffusion models.

Findings

Constructed a 2D time-dependent algebra representation for the symmetric exclusion process.

Obtained new results on eigenvectors and eigenvalues of the related quantum spin chain.

Showed that coupling to an infinite-temperature bath does not alter late-time behavior of zero-temperature dynamics.

Abstract

We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Generalizing earlier work \cite{Stin95a,Stin95b} we present an alternative description of these processes in terms of a time-dependent operator algebra with quadratic relations. These relations generate the Bethe ansatz equations for the spectrum and turn the calculation of time-dependent expectation values into the problem of either finding representations of this algebra or of solving functional equations for the initial values of the operators. We use both strategies for the study of two specific models: (i) We construct a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. In this way we obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal, symmetry-breaking boundary fields. (ii) We consider the non-equilibrium spin relaxation of Ising spins with zero-temperature Glauber dynamics and an additional coupling to an infinite-temperature heat bath with Kawasaki dynamics. We solve the functional equations arising from the algebraic description and show non-perturbatively on the level of all finite-order correlation functions that the coupling to the infinite-temperature heat bath does not change the late-time behaviour of the zero-temperature process. The associated quantum chain is a non-hermitian anisotropic Heisenberg chain related to the seven-vertex model.