Ground states of lattice gases with "almost" convex repulsive interactions

To the best of our knowledge there is only one example of a lattice system with long-range two-body interactions whose ground states have been determi ned exactly: the one-dimensional lattice gas with purely repulsive and stri ctly convex interactions, Its ground-state particle configurations do not d epend on any other details of the interactions and are known as the general ized Wigner lattices or the most homogeneous particle configurations. The q uestion of the stability of this beautiful and universal result against cer tain perturbations of the repulsive and convex interactions is interesting in itself Additional motivations for studying such perturbations come from surface physics (adsorption on crystal surfaces) and theories of correlated fermion systems (recent results on ground-state particle configurations of the one-dimensional spinless Falicov-Kimball model). As a first step, we s tudied a one-dimensional lattice pas whose two-body interactions are repuls ive and strictly convex only From distance 2 on, while its value at distanc e 1 can be positive or negative, but close to zero. We showed that such a m odification makes the ground-state particle configurations sensitive to the tail of the interactions; if the sum of the strengths of the interactions fi om the distance 3 on is small with respect to the strength of the intera ction at distance 2. then particles form two-particle lattice-connected agg regates that ale distributed in the most homogeneous way. Consequently, des pite breaking of the convexity property. the ground state exhibits the feat ure known as the complete devil's staircase.