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arXiv:cond-mat/0403235 · Journal of Physics A Mathematical and General

Construction of a matrix product stationary state from solutions of a finite-size system

Yasuhiro Hieida, Tomohiro Sasamoto

Published Oct 7, 2004• 12 citations

Abstract

Stationary states of stochastic models, which have $N$ states per site, in matrix product form are considered. First we give a necessary condition for the existence of a finite $M$-dimensional matrix product state for any ${N,M}$. Second, we give a method to construct the matrices from the stationary states of small size system when the above condition and $N\le M$ are satisfied. Third, the method by which one can check that the obtained matrices are valid for any system size is presented for the case where $M=N$ is satisfied. The application of our methods is explained using three examples: the asymmetric exclusion process, a model studied in [F. H. Jafarpour: J. Phys. A: Math. Gen. 36 (2003) 7497] and a hybrid of both of the models.

DOI: 10.1088/0305-4470/37/42/003

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Stationary states of stochastic models, which have

N

states per site, in matrix product form are considered. First we give a necessary condition for the existence of a finite

M

-dimensional matrix product state for any

{N,M}

. Second, we give a method to construct the matrices from the stationary states of small size system when the above condition and

N\le M

are satisfied. Third, the method by which one can check that the obtained matrices are valid for any system size is presented for the case where

M=N

is satisfied. The application of our methods is explained using three examples: the asymmetric exclusion process, a model studied in [F. H. Jafarpour: J. Phys. A: Math. Gen. 36 (2003) 7497] and a hybrid of both of the models.

We find the exact solution for the stationary state measure of the partially asymmetric exclusion process on a ring with multiple species of particles. The solution is in the form of a matrix product representation where the matrices for a system of N species are defined recursively in terms of the matrices for a system of N − 1 species. A complete proof is given, based on the quadratic relations verified by these matrices. This matrix product construction is interpreted in terms of the action of a transfer matrix.

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 construct matrix product steady states for a class of interacting particle systems where particles do not obey hardcore exclusion, meaning each site can occupy any number of particles subjected to the global conservation of the total number of particles in the system. To represent the arbitrary occupancy of the sites, the matrix product ansatz here requires an infinite set of matrices which in turn leads to an algebra involving an infinite number of matrix equations. We show that these matrix equations, in fact, can be reduced to a single functional relation when the matrices are parametric functions of the representative occupation number. We demonstrate this matrix formulation in a class of stochastic particle hopping processes on a one dimensional periodic lattice where hop rates depend on the occupation numbers of the departure site and its neighbors within a finite range; this includes some well known stochastic processes like, the totally or partially asymmetric zero range process, misanthrope process and finite range process.

Recently, it has been shown that the zero-energy eigenstate – corresponding to the stationary state – of a stochastic Hamiltonian with nearest-neighbour interaction in the bulk and single-site boundary terms, can generically be written in the form of a so-called matrix-product state. We generalize this result to stochastic Hamiltonians with arbitrary, but finite, interaction range. As an application two different particle-hopping models with three-site bulk interaction are studied. For these models which can be interpreted as cellular automata for traffic flow, we present exact solutions for periodic boundary conditions and some suitably chosen boundary interactions.