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. Analyzing 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, 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.

Critical phenomena and universal dynamics in one-dimensional driven diffusive systems with two species of particles

“…For integrable models [128,129] one may use the Bethe-ansatz and symmetry properties [18] for the explicit construction of stationary states which are not simple product measures. Using the Bethe ansatz has not been attempted yet for this class of models.…”

“…The approach can also be extended to describe the full time evolution and hence yield time-dependent probabilities of the system evolving from some nonstationary initial distribution [131,132,133]. Popkov et al have identified the parameter manifold for which the dynamics of the 2-species ASEP can be solved using the dynamical matrix product ansatz [129]. Such models are all integrable in the sense of being associated with an integrable vertex model.…”

Asymmetric exclusion model with several kinds of impurities

“…In this model particles of kind 1 can jump to neighboring sites if they are empty and particles of kind α = 2, 3, ..., N (called impurities) only exchange positions with others particles, satisfying a well defined dynamics. We solve this model with periodic boundary condition through a new matrix product ansatz [24,21] and we analyze the spectral gap for some special cases. Our N-ASEPI model displays the full spectrum of the ASEP [6] plus new levels.…”

Integrable inhomogeneous spin chains in generalized Lunin-Maldacena backgrounds

“…The MPA becomes also a successful tool for the exact calculation of the stationary probability distribution of some stochastic one dimensional systems [15][16][17]. An extension of this last MPA, called dynamical MPA was introduced in [18,19] and extended in [20]. This last ansatz gives the time-dependent probability distribution for some exact integrable systems.…”