Reaction-Diffusion Processes Described by Three-State Quantum Chains and Integrability

TL;DR

This paper maps one-dimensional three-species reaction-diffusion processes onto integrable quantum chains, identifying specific models and transformations that relate these processes to known integrable systems.

Contribution

It identifies all three-state integrable quantum chains with known spectra related to reaction-diffusion systems and describes transformations to standard Hamiltonian forms.

Findings

Two integrable models identified: $U_q\widehat{SU(2)}$-invariant and Perk-Schultz models.

A nonlocal similarity transformation relates chemical process Hamiltonians to standard forms.

Generalization of Dzialoshinsky-Moriya interaction for periodic boundary conditions.

Abstract

The master equation of one-dimensional three-species reaction-diffusion processes is mapped onto an imaginary-time Schr\"odinger equation. In many cases the Hamiltonian obtained is that of an integrable quantum chain. Within this approach we search for all $3$-state integrable quantum chains whose spectra are known and which are related to diffusive-reactive systems.Two integrable models are found to appear naturally in this context: the $U_{q}\widehat{SU(2)}$-invariant model with external fields and the $3$-state $U_q Su(P/M)$-invariant Perk-Schultz models with external fields. A nonlocal similarity transformation which brings the Hamiltonian governing the chemical processes to the known standard forms is described, leading in the case of periodic boundary conditions to a generalization of the Dzialoshinsky-Moriya interaction.