On a renormalized cluster expansion

MANFRED L. RISTIG, J. W. Clark, Francesco Calogero

MANFRED L. RISTIG, J. W. Clark

William J. Shugard, John D. Weeks

We give a simple derivation of a finite cluster expansion for the free energy of a general inhomogeneous lattice system which uses only connected clusters. Two possible ways of defining these clusters are discussed in detail, one most useful for obtaining high-temperature series expansions and another low-temperature expansions. A renormalized finite cluster expansion which uses only multiply connected or star clusters follows from a Legendre transformation on the free energy in which single-site potentials are replaced as independent variables by single-site probability distribution functions. Certain special cases of these results have been known before and successfully used to derive high-temperature series expansions. The relationship of the present approach to the renormalized linked-cluster perturbation expansion used by Wortis and co-workers is discussed, and comparison is also made with the work of Rushbrook and Scoins. Truncations of the renormalized cluster expansion yield closed-form approximations which, except in the critical region, offer a systematic way of improving mean-field theory.

Mark Jarrell

Sherwin J. Singer, John D. Weeks

There are many statistical systems in which degrees of freedom can be grouped according to their degree of correlation. We develop a cluster-expansion method for these systems based on selecting finite clusters of the most strongly interacting degrees of freedom. The starting point is Kubo's generalized cumulant formalism. We show how this expansion can be renormalized; that is, how the contributions of infinite classes of articulated finite clusters can be resummed. The elimination of articulated clusters from the renormalized expansion generates modified interactions in the finite clusters remaining in the expansion. When truncated at different orders, the final result is a hierarchy of mean-field approximations, which generally reproduce an increasing number of terms in exact series expansions for the system. High-order series-expansion coefficients can be obtained by perturbative renormalization of very large clusters, a convenient alternative to the complicated combinatorial analysis associated with standard expansion techniques for lattice models. We compare our work to previous finite-cluster techniques, such as the cluster-variation method of Kikuchi.

M. Bartkowiak, K. A. Chao

We have shown that mean-field-type symmetry-breaking approximations for correlated fermions can be constructed by performing vertex renormalization (VR) in the linked-cluster expansion (LCE), just as for the case of lattice spin models. Using the diagrammatic technique for Hubbard operators, we have constructed explicitly a partial VR in the second order of the LCE for the half-filled Hubbard model around the atomic limit. We obtain a crossover from large-U Heisenberg mean-field behavior to zero Ne\ifmmode\acute\else\textasciiacute\fi{}el temperature at U=0.