Ising-Model Reformulation. II. Relation to the Random-External-Field Method

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Abstract
Starting with the previously derived vacuum-state expectation representation of the Ising partition function, an integral transform is applied to part of the Bose field operator arising from pair interactions, generating thereby the effect of random external fields. The remaining pair interaction is treated as a perturbation, whose effect is small in both the high- and the low-temperature extremes. The transformed part of the potential is selected first to generate a Markoff process, and the perturbation equations may then be regarded as conditions that the perturbation have no effect on either the partition function or the long-range order below the transition. In two dimensions, it is shown how the characteristic integral equation of the Markoff process reduces to the matrix eigenvalue problem solved by Onsager, in the limit of just nearest-neighbor interactions.