The CVM is a mean-field variational method which gives an upper bound to the exact free energy. It has several implementations since the original formulation by Kikuchi in 1951. The basic idea of all mean-field cluster formulations is the same: define a cluster within the real system and then solve the partition function exactly for the cluster and consider interactions with the rest of the system via a mean field. There are several ways of doing this in practice. The one point cluster approximation leads to the usual mean field results, the two-points approx. is equivalent to the so called Bethe-Peierls approx., the four-point was originally solved by Krammers-Wannier for the Ising model, and so on. Larger clusters allow to improve the upper bound on the free energy at the expense of growing computational complexity to solve the variational problem. A nice exposition of the CVM and some of its formulations and technical details of implementation is given in the book by T. Tanaka, "Methods of Statistical Physics", (Cambdrige University Press, 2002).
Daniel, thank you. I know about "Methods of Statistical Physics". Before questioning here I was trying to find an answer in the book. I'll try to explain the reasons of question.
As I understood, the general variational method for the density matrix is starting point of CVM. Then variational potential is expressed in terms of cumulant functions and reduced density matrices. Then approximation is made for cumulant functions.
I don't understand what form has trial density matrix in CVM approach. Is it a product of cluster density matrices? In the simple mean field approach trial density matrix is product of one-site density matrices.
Every density matrix generates set of reduced density matrices and cumulant functions naturally. Is there density matrix for every artificially constructed set of reduced density matrices? There is no upper bound if trial density matrix doesn't exist.
Gennadiy, I think the answer to your question about the nature of the trial density matrix is contained in pages 128-129 of Tanaka's book, where the cumulant expansion is described. Note that the largest cluster function G^N(1,2,...,N) is nothing but the entropy term in the variational potential, which is expressed in terms of the trial density matrix of the whole system (see equation 6.12 in page 129 of the book). This cluster function can, in turn, be expressed as a sum of cumulant functions of 1, 2, 3, ...
N sites. This representation is exact in the sense that minimizing the variational potential with respect to the N-point density matrix one should obtain the exact equilibrium density. In practice one has to truncate the expansion to a low order, equivalent to the size of the maximum cluster that can be solved with the numerical resourses at hand. There is no a priori form for the trial density. More precisely, the form of the trial density is encoded in the cumulant expansion. But, in practice, you don't have to worry about it, just minimize the variational potential with respect to all the reduced densities and then apply the normalization and reducibility conditions, as explained in the book. Hope this helps.
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