Spontaneous breaking of translational invariance and spatial condensation in stationary states on a ring. I. The neutral system

We consider a model in which positive and negative particles with equal densities diffuse in an asymmetric, CP-invariant way on a ring. The positive par tides hop clockwise, the negative counterclockwise, and oppositely charged adjacent particles may swap positions. The model depends on two parameters. Analytic calculations using quadratic algebras, inhomogeneous solutions of the mean-field equations, and Monte Carlo simulations suggest that the model has three phases: (1) a pure phase in which one has three pinned blocks of only positive or negative particles and vacancies and in which translational invariance is broken; (2) a mixed phase in which the current has a linear dependence on one parameter, but is independent of the other one and of the density of the charged particles; in this phase one has a bump and a fluid, the bump (condensate) containing positive and negative particles only, the fluid containing charged particles and vacancies uniformly distributed; and (3) the mixed phase is separated from the disordered phase by a second-order phase transition which has many properties of the Bose-Einstein phase transition observed in equilibrium. Various critical exponents are found.