Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation

TL;DR

This paper explores a 3D lattice model linked to quantum algebra, deriving transfer matrix relations and providing a new proof for the steady state of a multi-species zero range process, highlighting 3D integrability.

Contribution

It introduces a novel 3D lattice model and uses the tetrahedron equation to establish transfer matrix relations, offering a new proof of the zero range process steady state.

Findings

Derived commutativity and bilinear relations of transfer matrices

Established 3D integrability in the matrix product framework

Provided a new proof for the steady state probability of the multi-species zero range process

Abstract

We consider a three-dimensional (3D) lattice model associated with the intertwiner of the quantized coordinate ring $A_q(sl_3)$, and introduce a family of layer to layer transfer matrices on $m\times n$ square lattice. By using the tetrahedron equation we derive their commutativity and bilinear relations mixing various boundary conditions. At $q=0$ and $m=n$, they lead to a new proof of the steady state probability of the $n$-species totally asymmetric zero range process obtained recently by the authors, revealing the 3D integrability in the matrix product construction.