generalized Hamiltonian where µ$_{i}$ is the chemical potential of the species i . We shall require charge neutrality for ionized species 3 and 4 in the bulk, and assume µ$_{3}$ = µ$_{4}$ . For the short-range interaction energy we make the standard nearest-neighbor (NN) approximation for all the species where ˆ e$_{n}$ is the unit lattice vector in the n -th direction and summation convention is assumed for i, j = 1 , ..., 4. The coupling constants are symmetric, J$_{ij}$ = J$_{ji}$ , and represent the sum of all short-range interactions. The Hamiltonian of the lattice model with the electrostatic interactions disregarded takes the form where ∑ 〈 x , x ′ 〉 is the summation over all pairs of nearest-neighbors belonging to the system. The system volume is denoted by V , ∂V denotes the boundary layer, and the new operators are From close packing (1) it follows that ˆ o$_{1}$ + ˆ o$_{2}$ = ˆ s 2 = 1 − ˆ φ $^{2}$. The new coupling constants in (5) are linear combinations of the coupling constants J$_{ij}$ , and Δ µ and Δ µ$_{c}$ are linear combinations of µ$_{i}$ . We further assume that the species 3 and 4 are similar, i.e., interact in a similar way with 6 H $^{SR}$[ { ˆ o$_{i}$ ( x ) } ] = E $^{SR}$[ { ˆ o$_{i}$ ( x ) } ] − µ$_{1}$ ∑ x ˆ o$_{1}$ ( x ) − µ$_{2}$ ∑ x ˆ o$_{2}$ ( x ) − µ$_{3}$ ∑ x (ˆ o$_{3}$ ( x ) + ˆ o$_{4}$ ( x )) , (3) E $^{SR}$[ { ˆ o$_{i}$ ( x ) } ] = − ∑ x 3 ∑ n =1 ˆ o$_{i}$ ( x ) J$_{ij}$ ˆ o$_{j}$ ( x + ˆ e$_{n}$ ) , (4) H $^{SR}$[ˆ s, ˆ φ ] = − ∑ 〈 x , x ′ $_{〉}$[ 1 2 ( ˆ s ( x ) J$_{ss}$ ˆ s ( x $^{′}$) + ˆ φ $^{2}$( x ) J$_{ρρ}$ ˆ φ $^{2}$( x $^{′}$) + ˆ φ ( x ) J$_{φφ}$ ˆ φ ( x $^{′}$) ) (5) +ˆ s ( x ) J$_{sρ}$ ˆ φ $^{2}$( x $^{′}$) + ˆ φ ( x ) J$_{φs}$ ˆ s ( x $^{′}$) + ˆ φ ( x ) J$_{φρ}$ ˆ φ $^{2}$( x $^{′}$) ] − ∑ x ∈ V ( Δ µ ˆ s ( x ) + Δ µ$_{c}$ ˆ φ $^{2}$( x ) ) − ∑ x ∈ ∂V ( h$_{s}$ ˆ s ( x ) + h$_{ρ}$ ˆ φ $^{2}$( x ) ) + const, ˆ s = ˆ o$_{1}$ − ˆ o$_{2}$ = − 1 , 0 , 1 (6) ˆ φ = ˆ o$_{3}$ − ˆ o$_{4}$ = − 1 , 0 , 1 ˆ φ 2 = ˆ o$_{3}$ + ˆ o$_{4}$ = 0 , 1 .