Why the Ising and continuous-space models of ionic systems exhibit essentially different critical behavior

A Landau–Ginzburg–Wilson functional of two order-parameters—the density of charge φ and the density of mass ρ—is constructed for ionic systems in which the positions of ions are either in the Euclidean space or are restricted to the lattice sites. We find two phase-transitions: (i) a line of continuous transitions to the charge-ordered phase, induced by the fluctuations φ(r)∝cos(r⋅k) with 2π/k of molecular size, which terminates at a tricritical point and (ii) a transition between two uniform, ion-poor and ion-rich phases, induced by the whole spectrum of the charge fluctuations. Due to the dominant role of the short-wavelength charge fluctuations, the positions of the transitions depend significantly on the short-distance properties of the system. In different systems (continuous or on different lattices) one or the other transition may be preempted by the occurrence of the other, by which qualitatively different phase diagrams are obtained.