Matrix product states of three families of one-dimensional interacting particle systems

The steady states of three families of one-dimensional non-equilibrium models with open boundaries, first proposed in N. J. Phys. 5 (2003) 145.1, are studied using a matrix product formalism. It is shown that their associated quadratic algebras have two-dimensional representations, provided that the transition rates lie on specific manifolds of parameters. Exact expressions for the correlation functions of each model have also been obtained. We have also studied the steady-state properties of one of these models, first introduced in J. Phys. A: Math. Gen. A 36 (2003) 74, with more details. By introducing a canonical ensemble we calculate the canonical partition function of this model exactly. Using the Yang–Lee theory of phase transitions we spot a second-order phase transition from a power law to a jammed phase. The density profile of particles in each phase has also been studied. A simple generalization of this model in which both the left and the right boundaries are open has also been introduced. It is shown that double shock structures may evolve in the system under certain conditions