On some variational approximations in two-dimensional classical lattice systems

A new derivation is presented of some variational approximations for classical lattice systems that belong to the class of cluster-variation methods, among them the well-known Bethe-Peierls and Kramers-Wannier approximations. The limiting behavior of a hierarchical sequence of cluster-variation approximations, the so-calledC hierarchy, is discussed. It is shown that this hierarchy provides a monotonically decreasing sequence of upper boundsf _n on the free energy per lattice sitef and thatf _n f asn . Our results are based on extension theorems for states given on subsets of the lattice, which might be of some independent interest, and on an application of transfer matrix concepts to the variational characterization of translation-invariant equilibrium states.