A novel approach to the exact calculation of correlation functions of a one-dimensional random Ising chain

An exact result for the partition function for a general one‐dimensional Ising chain of N spin‐1/2 particles described by the Hamiltonian $H_N{\mathrm{=-\sum }}_{\text{i=1}}^N{J}_i{\sigma }_i{\sigma }_{\text{i+1}}{\mathrm{-\sum }}_{\text{i=1}}^N{H}_i{\sigma }_i$ is given. For an open strand, J_N = 0; for a closed chain, σ_N+1 = σ₁, J_N ≠ 0. The novelty of the trick used enables one to obtain the partition function and all the spin correlation functions for open and closed chains with equal ease. Special cases of this model have been discussed before to elucidate certain features of some biological systems. New expressions for parallel and perpendicular susceptibilities for this model are also derived. When J_i and H_i are treated as random variables, the above Hamiltonian describes a one‐dimensional Ising spin glass. In this case some simple models and formal averaging procedures are discussed.