Reaction-diffusion processes and their connection with integrable quantum spin chains

TL;DR

This paper explores the connection between reaction-diffusion systems and integrable quantum spin chains, highlighting their mathematical relationships and implications for understanding non-equilibrium phenomena.

Contribution

It provides a pedagogical review of how reaction-diffusion models relate to integrable quantum chains and discusses methods to identify and solve these models using algebraic techniques.

Findings

Identification of integrable reaction-diffusion models via Hecke algebra

Application of Baxterization and Yang-Baxter equations to these models

Discussion of local scale invariance in non-equilibrium ageing phenomena

Abstract

This is a pedagogical account on reaction-diffusion systems and their relationship with integrable quantum spin chains. Reaction-diffusion systems are paradigmatic examples of non-equilibrium systems. Their long-time behaviour is strongly influenced through fluctuation effects in low dimensions which renders the habitual mean-field cinetic equations inapplicable. Starting from the master equation rewritten as a Schr\"odinger equation with imaginary time, the associated quantum hamiltonian of certain one-dimensional reaction-diffusion models is closely related to integrable magnetic chains. The relationship with the Hecke algebra and its quotients allows to identify integrable reaction-diffusion models and, through the Baxterization procedure, relate them to the solutions of Yang-Baxter equations which can be solved via the Bethe ansatz. Methods such as spectral and partial integrability, free fermions, similarity transformations or diffusion algebras are reviewed, with several concrete examples treated explicitly. An outlook on how the recently-introduced concept of local scale invariance might become useful in the description of non-equilibrium ageing phenomena is presented, with particular emphasis on the kinetic Ising model with Glauber dynamics.