A sufficient criterion for integrability of stochastic many-body dynamics and quantum spin chains

Abstract

We propose a dynamical matrix product ansatz describing the stochastic dynamics of two species of particles with excluded-volume interaction and the quantum mechanics of the associated quantum spin chains, respectively. The time-dependent algebra which is obtained from the action of the Markov generator of the exclusion process (or quantum Hamiltonian of the spin chain, respectively) is given in terms of a set of quadratic relations. By analysing the permutation consistency of the induced cubic relations, we obtain sufficient conditions on the hopping rates (i.e. the quantum mechanical interaction constants) which allow us to identify integrable models. From the dynamical algebra we construct the quadratic algebra of Zamolodchikov type, the associativity of which is a Yang–Baxter equation. We also obtain directly from the dynamical matrix product ansatz the Bethe ansatz equations for the spectra of these models.