A set of exactly solvable fermionic spin-S Ising Model lattice with quartic interaction

We present a set of exactly solvable fermionic spin-S Ising model on a square-type lattice including a quartic interaction term in the Hamiltonian, using an auxiliary mixed fermionic spin-(S,1/2) square-type lattice with only first nearest-neighbor interaction. The particular properties of the mixed lattice, associated to the fermionic mixed spin-(S,1/2), allow us to map this system either into a purely spin-1/2 lattice or into a purely spin-S lattice, respectively. By imposing the condition that the mixed fermionic spin-(S,1/2) lattice must have an exact solution, we found a set of exact solutions that satisfy the free fermion condition, and the number of solutions for a general fermionic spin-S is given by S + 1/2. Then we conclude that this transformation is equivalent to a simple spin transformation which is independent of the coordination number. This transformation could be extended to higher fermionic spin-S to yield an exactly solvable lattice. The subject of two-dimensional lattices is one of the most interesting ones, both experimentally[1, 2] and theoretically; several approximation methods such as the mean-field theory[1, 3], the Bethe approximation[4], the correlated effective field theory[5], the renormalization group[6], series expansion methods[7], Monte Carlo methods[8] and the cluster variation methods are used to investigate this interesting lattice. On the other hand, exact solutions were obtained only in very particular cases, mainly the honeycomb lattices[9, 10]. Some exact results have been obtained with parameter restrictions, as investigated by Mi and Yang[11] using a non-one-toone transformation[10]. Some fermionic Ising spin lattices were already discussed in the literature[12]. Using the method proposed by Wu[13], Izmailian [14] obtained an exact solution for a spin-3/2 square lattice with only nearest-neighbor twobody spin interaction. Izmailian and Ananikian[15] have also obtained an exact solution for a honeycomb lattice with spin-3/2. Particular solutions of these models could be obtained by the method proposed by Joseph[16] where any spin-S could be decomposed in terms of spin-1/2. Another interesting method for mapping the spin-S lattice into a spin-1/2 lattice has been proposed by Horiguchi[17]. It is possible to transform a mixed spin lattice into an effective spin-1/2 lattice such as presented in the literature[14] if we consider the spin-S as a decorated Ising model of the lattice[18] Lb. Then the Hamiltonian for a fermionic spin-(1/2,3/2) lattice is given by H1/2,3/2 = ∑ (K r Siσj +K (3) r S 3 i σj) + ∑ i DS i , (1) with < i, j > meaning summation over nearest interacting neighbors on the square lattice, and the last summation is performed over all spin-3/2 sites. The coefficient K (1) r is the first-neighbor bilinear interaction parameter; K (3) r corresponds to the parameter of the non-bilinear interaction, for each coordination number r = 4; D is the single ion-anisotropy parameter acting on spin-3/2; σi represents the spin-1/2 particle, with two possible values ±1; whereas Si represents the fermionic spin-3/2 particle.