Compressible Ising double chains

We obtain exact expressions for the free energy and the magnetic susceptibility in zero field of a compressible double Ising chain with first and second neighbour interactions. The chain is supposed to be made of rigid rods which move like dumbbells in an elastic harmonic potential. The exchange interactions along the direction of the chain are linear functions of the spacing between rods. The effective spin hamiltonian of the double chain involves short-range two and four-spin interactions. Due to the existence of compensation points, we obtain regions of peculiar thermodynamic properties in the pressure-temperature phase diagram.