Exact solution of the Ising model on checkerboard fractals in an external magnetic field

In this paper we use a diagrammatic technique to determine the exact recursion relations for the partition function of the Ising model in an external magnetic field, situated on the first two members of the checkerboard family of fractal lattices embedded in two dimensions. This represents the first exact general solution of this model for the case of nonzero field. The closed-form expression for the partition function is obtained for the first member in the zero-field limit and the nonzero-field recursion relations prove sufficient for exact evaluation of the response functions. We also calculate the temperature dependence of the specific heat and susceptibility