Perpendicular Susceptibility of the Ising Model

A high‐temperature expansion of the partition function for a lattice of N spins with Hamiltonian $$\mathcal{H}\mathrm{=-J}{\displaystyle \sum _{\mathrm{(ij)}}}{\sigma }_i^z{\sigma }_j^z{\mathrm{-mH}}_z{\displaystyle \sum _i}{\sigma }_i^z{\mathrm{-mH}}_x{\displaystyle \sum _i}{\sigma }_i^x$$ is derived and thence an expansion for the zero‐field perpendicular susceptibility is found. By perturbation theory, χ⊥(T) is also expanded at low temperatures and seen, in general, to increase with T from the value χ⊥(O) = Nm²/q | J |, where q is the coordination number of the lattice. The perpendicular susceptibility is re‐expressed in terms of near neighbor pair and higher‐order spin correlation functions in zero field. This yields exact closed formulas for the linear chain, the Bethe pseudolattices, and for the plane square and honeycomb lattices. The behavior of χ⊥(T) in the critical region is investigated for these lattices and for the plane superexchange lattice.