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arXiv:1608.02779 · Journal of Physics A Mathematical and Theoretical

Matrix product formula for U q ( A 2 ( 1 ) ) -zero range process

Atsuo Kuniba, Masato Okado

Published Dec 23, 2016• 7 citations
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Abstract

Abstract The U q ( A n ( 1 ) ) -zero range processes introduced recently by Mangazeev, Maruyama and the authors are integrable discrete and continuous time Markov processes associated with the stochastic R matrix derived from the well-known U q ( A n ( 1 ) ) quantum R matrix. By constructing a representation of the relevant Zamolodchikov–Faddeev algebra, we present, for n = 2, a matrix product formula for the steady state probabilities in terms of q -boson operators.

DOI
10.1088/1751-8121/50/4/044001

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Abstract The U q ( A n ( 1 ) ) -zero range processes introduced recently by Mangazeev, Maruyama and the authors are integrable discrete and continuous time Markov processes associated with the stochastic R matrix derived from the well-known U q ( A n ( 1 ) ) quantum R matrix. By constructing a representation of the relevant Zamolodchikov–Faddeev algebra, we present, for n = 2, a matrix product formula for the steady state probabilities in terms of q -boson operators.
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.
The stochastic R matrix for U q ( A n ( 1 ) ) introduced recently gives rise to an integrable zero range process of n classes of particles in one dimension. For n = 2 we investigate how finitely many first class particles fixed as defects influence the grand canonical ensemble of the second class particles. By using the matrix product stationary probabilities involving infinite products of q -bosons, exact formulas are derived for the local density and current of the second class particles in the large volume limit.
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