Periodic ordering of clusters in a one-dimensional lattice model

A generic lattice model for systems containing particles interacting with short-range attraction long-range repulsion (SALR) potential that can be solved exactly in one dimension is introduced. We assume attraction J1 between the first neighbors and repulsion J2 between the third neighbors. The ground state of the model shows existence of two homogeneous phases (gas and liquid) for J2/J1 <1/3. In addition to the homogeneous phases, the third phase with periodically distributed clusters appears for J2/J1 > 1/3. Phase diagrams obtained in the self-consistent mean-field approximation for a range of values of J2/J1 show very rich behavior, including reentrant melting, and coexistence of two periodic phases (one with strong and the other one with weak order) terminated at a critical point. We present exact solutions for the equation of state as well as for the correlation function for characteristic values of J2/J1. Based on the exact results, for J2/J1 > 1/3 we predict pseudo-phase transitions to the ordered cluster phase indicated by a rapid change of density for a very narrow range of pressure, and by a very large correlation length for thermodynamic states where the periodic phase is stable in mean field. For 1/9 < J2/J1 < 1/3 the correlation function decays monotonically below certain temperature, whereas above this temperature exponentially damped oscillatory behavior is obtained. Thus, even though macroscopic phase separation is energetically favored and appears for weak repulsion at T = 0, local spatial inhomogeneities appear for finite T. Monte Carlo simulations in canonical ensemble show that specific heat has a maximum for low density ρ that we associate with formation of living clusters, and if the repulsion is strong, another maximum for ρ = 1/2.