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Theoretical and Mathematical Physics

Bogolyubov's statistical variational principle and the Green's function method applied to the Heisenberg-Ising model

Yu. G. Rudoi

Published Jan 1, 1970• 2 citations
Publisher
DOI
10.1007/bf01028861

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The Ising model has been employed in a study of the variational principles which are associated with renormalized diagram expansions in statistical mechanics. For this model the variational functional is expressed in terms of the renormalized semi-invariants, which play the role of the one-particle density or the one-particle Green's function in quantum statistics. We review the derivation of this functional and discuss some of its properties. In order to examine the mathematical content of the variational principle in more detail, we specialize to the exactly soluble infinite-dimensional (infinite-range) model. We find that, by virtue of the Dyson relation, the variational theorem is not valid over all possible values of the renormalized semi-invariants but applies only within a restricted domain. Within this domain the variational functional is multiple valued. The renormalized expansion is asymptotically convergent to the branch of this functional which describes a single phase with uniform magnetization.
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