We present a brief review on integrability of multispecies zero range process in one dimension introduced recently.The topics range over stochastic R matrices of quantum affine algebra U q A (1) n , matrix product construction of stationary states for periodic systems, q-boson representation of Zamolodchikov-Faddeev algebra, etc.We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of R matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter.
We present a brief review on integrability of multispecies zero range process in one dimension introduced recently.The topics range over stochastic R matrices of quantum affine algebra U q A (1) n , matrix product construction of stationary states for periodic systems, q-boson representation of Zamolodchikov-Faddeev algebra, etc.We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of R matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter.
We consider a three-dimensional (3D) lattice model associated with the intertwiner of the quantized coordinate ring Aq(sl3), and introduce a family of layer to layer transfer matrices on m×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.
We introduce an |n|-species totally asymmetric zero range process (|n|-TAZRP) on one-dimensional periodic lattice with |L| sites. It is a continuous time Markov process in which |n| species of particles hop to the adjacent site only in one direction under the condition that smaller species ones have the priority to do so. Also introduced is an |n|-line process, a companion stochastic system having the uniform steady-state from which the |n|-TAZRP is derived as the image by a certain projection |π|. We construct the |π| by a combinatorial |R| of the quantum affine algebra |Uq(slL)| and establish a matrix product formula of the steady state probability of the |n|-TAZRP in terms of corner transfer matrices of a |q=0|-oscillator-valued vertex model. These results parallel the recent reformulation of the |n|-species totally asymmetric simple exclusion process (|n|-TASEP) by the authors, demonstrating that |n|-TAZRP and |n|-TASEP are the canonical sister models associated with the symmetric and the antisymmetric tensor representations of |Uq(slL)| at |q=0|, respectively. Communicated by: Masatoshi Noumi
Conditions of integrability of general zero range chipping models with factorized steady state, which were proposed in [Evans, Majumdar, Zia 2004 J. Phys. A 37 L275], are examined. We find a three-parametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle models as limiting cases. The solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations, which is proved as a byproduct. We use the Bethe ansatz to solve an eigenproblem for the transition matrix of the Markov process. On its basis we conjecture an integral formula for the Green function of evolution operator for the model on an infinite lattice and derive the Bethe equations for the spectrum of the model on a ring.
Proposed is a method to construct a stationary state of one-dimensional integrable systems in the form of products of matrices. This is a generalization of the so-called “matrix product ansatz (MPA)”. The key idea is that the matrices are chosen to constitute the Zamolodchikov-Faddeev algebra (ZF-algebra) for the R-matrix and the K-matrix, which are the solutions of the Yang-Baxter equation (YBE) and the reflection equation (RE). It is shown that a matrix product state gives a simultaneous stationary state of commuting operators which are expressed in terms of the R-matrices and the K-matrices. As an example, a solution for the isotropic Heisenberg spin chain with non-diagonal boundary fields is given. The connection to the conventional MPA is clarified. Applications to other models and the relationship to the algebraic Bethe ansatz (ABA) are also discussed.
Abstract We introduce a family of layer to layer transfer matrices in a three-dimensional (3D) lattice model which can be viewed as partition functions of the q -oscillator valued six-vertex model on an m × n square lattice. By invoking the tetrahedron equation we establish their commutativity and bilinear relations mixing various boundary conditions. At q = 0 and m = n , they ultimately yield a new proof of the steady state formula for the n -species totally asymmetric simple exclusion process obtained recently by the authors, revealing the 3D integrability in the matrix product construction.