The dual model for an Ising model with nearest and next-nearest neighbors (Q2843741)

The authors construct and analyze a dual model to the next-nearest-neighbor Ising model on the rectangular lattice, which is described by the Hamiltonian NEWLINE\[NEWLINEH=-\sum_{<i,j>}J_{ij}\sigma_i \sigma_j -J_4\sum_{<i,j,k,l>}\sigma_i \sigma_j\sigma_k \sigma_lNEWLINE\]NEWLINE defined on the brick-wall lattice. Here \(J_4\) is the four-spin interaction constant for the four spins occupying the right-hand side of a brick, and the values of the nearest-neighbor interaction constants \(J_{ij}\) that are non-zero only if \(i\) and \(j\) are on the same side of a brick, in which case they are equal to \(J_1\) and \(J_2\) for interactions along the vertical and horizontal direction, respectively. The free fermion approximation suggests that an increase in the critical temperature of the dual model caused by the four-spin interactions is limited to a finite range.