Integrable Floquet dynamics, generalized exclusion processes and "fused" matrix ansatz

TL;DR

This paper introduces a general framework for constructing integrable stochastic processes with discrete-time Floquet dynamics, including new generalized exclusion processes, and develops a fused matrix ansatz for explicit stationary distributions and physical observables.

Contribution

It presents a novel method to build integrable stochastic models with two-step Floquet dynamics using transfer matrix formalism and introduces the fused matrix ansatz for stationary distributions.

Findings

Constructed integrable models for SSEP and ASEP with explicit stationary states.

Developed a fused matrix ansatz for generalized exclusion processes.

Computed correlation functions and particle currents using the new algebraic framework.

Abstract

We present a general method for constructing integrable stochastic processes, with two-step discrete time Floquet dynamics, from the transfer matrix formalism. The models can be interpreted as a discrete time parallel update. The method can be applied for both periodic and open boundary conditions. We also show how the stationary distribution can be built as a matrix product state. As an illustration we construct a parallel discrete time dynamics associated with the R-matrix of the SSEP and of the ASEP, and provide the associated stationary distributions in a matrix product form. We use this general framework to introduce new integrable generalized exclusion processes, where a fixed number of particles is allowed on each lattice site in opposition to the (single particle) exclusion process models. They are constructed using the fusion procedure of R-matrices (and K-matrices for open boundary conditions) for the SSEP and ASEP. We develop a new method, that we named "fused" matrix ansatz, to build explicitly the stationary distribution in a matrix product form. We use this algebraic structure to compute physical observables such as the correlation functions and the mean particle current.