The dual model for an Ising model with nearest and next-nearest   neighbors
  We construct and analyse a dual model to the Ising model with the nearest and next-nearest neighbors on the rectangular lattice (NNNI model). The Hamiltonian of the dual model turns out to contain two- and four-spin interactions. 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. 
Adam Strycharski and Zbigniew Koza
cond-mat.stat-mech