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We discuss a new mechanism leading to a matrix product form for the stationary state of one-dimensional stochastic models. The corresponding algebra is quadratic and involves four different matrices. For the example of a coagulation-decoagulation model explicit four-dimensional representations are given and exact expressions for various physical quantities are recovered. We also find the general structure of n-point correlation functions at the phase transition.cond-mat/9510104

We show that all zero energy eigenstates of an arbitrary m-state quantum spin chain Hamiltonian with nearest neighbor interaction in the bulk and single site boundary terms, which can also describe the dynamics of stochastic models, can be written as matrix product states. This means that the weights in these states can be expressed as expectation values in a Fock representation of an algebra generated by 2m operators fulfilling m 2 quadratic relations which are defined by the Hamiltonian.

We study the one dimensional partially asymmetric simple exclusion process (ASEP) with open boundaries, that describes a system of hard-core particles hopping stochastically on a chain coupled to reservoirs at both ends. Derrida, Evans, Hakim and Pasquier [J. Phys. A 26, 1493 (1993)] have shown that the stationary probability distribution of this model can be represented as a trace on a quadratic algebra, closely related to the deformed oscillatoralgebra. We construct all finite dimensional irreducible representations of this algebra. This enables us to compute the stationary bulk density as well as all correlation lengths for the ASEP on a set of special curves of the phase diagram.