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Physical review. B, Condensed matter

Renormalized finite-cluster expansions

Sherwin J. Singer, John D. Weeks

Published Aug 1, 1987• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.36.2228

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Physical review. B, Condensed matter•Feb 15, 1993•8 citations
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.
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.
The linked-cluster series expansion is developed fur use on models of quantum spin systems with single-ion potentials. The linked-cluster series is defined as a statistical mechanical perturbation series in which the mean-field Hamiltonian and the single-ion potential are treated exactly with the perturbation representing the effects of fluctuation correlations. All terms of the free energy up to a given finite order are calculated, and estimates of thermodynamic functions are made based on extrapolation methods including the ratio method and Pade approximants.
An approximation method developed previously to deal with many particles in strong interaction is examined in further detail. It is shown that the series giving the interaction energy is a development in a sequence of linked or irreducible cluster terms each of which gives a contribution to the energy proportional to the total number of particles. Consequently the convergence of the expansion is independent of the total number of particles. The origin of this simple feature is illustrated by showing that a similar situation exists in the expansion of standard perturbation theory. The numerical convergence of the expansion is quantitatively discussed for the nuclear problem where it is shown that the correction arising from the first cluster term involving three particles is less than the leading term by a factor of about ${10}^{\ensuremath{-}4}$. The smallness of the correction is largely a result of the action of the exclusion principle.
Perturbation series expansion methods are sophisticated numerical tools used to provide quantitative calculations in many areas of theoretical physics. This book gives a comprehensive guide to the use of series expansion methods for investigating phase transitions and critical phenomena, and lattice models of quantum magnetism, strongly correlated electron systems and elementary particles. Early chapters cover the classical treatment of critical phenomena through high-temperature expansions, and introduce graph theoretical and combinatorial algorithms. The book then discusses high-order linked-cluster perturbation expansions for quantum lattice models, finite temperature expansions, and lattice gauge models. Also included are numerous detailed examples and case studies, and an accompanying resources website, www.cambridge.org/9780521842426, contains programs for implementing these powerful numerical techniques. A valuable resource for graduate students and postdoctoral researchers working in condensed matter and particle physics, this book will also be useful as a reference for specialized graduate courses on series expansion methods.
Under quite general conditions critical phenomena can be described with high order linked cluster expansions. The coefficients of the series admit a graphical expansion that is generated with the aid of computers. Our generalization of linked cluster expansions from an infinite to a finite volume allows to perform a finite size scaling analysis. We also indicate a generalization to Dynamical Linked Cluster Expansions with possible applications to spin glasses and neural networks with coupled spin and interaction dynamics.
We show that mean-field-type symmetry-breaking approximations for correlated fermions can be constructed by performing vertex renormalization (VR) in the linked-cluster expansion, just as for the case of lattice spin models. In the limit of infinite spatial dimensions we sum up exactly the renormalized diagrams for the grand-canonical potential of the Hubbard model and, using the diagrammatic technique for Hubbard operators, we construct explicitly an approximate VR. The procedure leads to a self-consistent symmetry breaking analogous to that of the mean-field theory for spin systems. We present solutions for the case of half-filled energy band. They are valid for arbitrary temperatures and become exact for both large-U limit and for U=0. The antiferromagnetic order parameter is found to have the conventional mean-field form and, in the regime of strong correlations, the obtained critical temperature coincides with the mean-field-theory result for the N\'eel temperature of the effective Heisenberg Hamiltonian.