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Journal of Mathematical Physics

Diagrammatic methods and renormalization in the Iwamoto-Yamada cluster expansion

Lothar Schäfer

Published Oct 1, 1973• 5 citations
Publisher

Abstract

We develop diagrammatic methods which give the explicit expression for the n th order term of the linked Iwamoto-Yamada cluster expansion in terms of the cluster integrals. The method applies independently of the number A of particles of the system. We carry through partial summations which for finite A result in a representation of the expansion by skeleton diagrams, whereas in the limit A → ∞ the expansion is reduced to two analytical equations. In addition, for finite A the renormalization establishes a connection between the expectation value of an arbitrary operator and the probability to find n hole states, 1 ≤ n ≤ A, occupied. Simple considerations raise some doubts concerning the convergence of the unrenormalized expansion.

DOI
10.1063/1.1666180

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We develop diagrammatic methods which give the explicit expression for the n th order term of the linked Iwamoto-Yamada cluster expansion in terms of the cluster integrals. The method applies independently of the number A of particles of the system. We carry through partial summations which for finite A result in a representation of the expansion by skeleton diagrams, whereas in the limit A → ∞ the expansion is reduced to two analytical equations. In addition, for finite A the renormalization establishes a connection between the expectation value of an arbitrary operator and the probability to find n hole states, 1 ≤ n ≤ A, occupied. Simple considerations raise some doubts concerning the convergence of the unrenormalized expansion.
A simple alternative derivation of conventional cluster expansion formulas of Iwamoto and Yamada and others for the expectation value ≪H > of the energy of a many-Fermion system is offered. The assumed trial function is of the general form FΦp/(FΦp, FΦp)1/2, where Φp is a Slater determinant of appropriate single-particle functions and the factor F is attached to account for the possibility of strong interparticle correlations. The proposed scheme is applicable to finite as well as infinite systems and is easily generalized to the calculation of non-diagonal as well as diagonal matrix elements, with respect to the members of the set {FΦp/(FΦp, FΦp)1/2}. Explicit results for ≪H > are displayed to second cluster order, contributions being classified according to their assigned order of magnitude in a correlation parameter ξ=|Nω/Ω| of the sort introduced by Iwamoto and Yamada and Clark and Westhaus.
The ``factor-cluster'' formalisms introduced in the preceding paper are used as a tool for a further development of the cluster theories proposed by Iwamoto and Yamada and by Aviles, Hartogh, and Tolhoek. The primitive decomposition characterizing each of the older formalisms is rearranged into the exponential of a series with a uniform N dependence in the limit of large N. In addition a ``linked cluster'' theorem is proven for the series comprising the exponent in the Iwamoto-Yamada formalism. Our derivations, unlike those of earlier authors, are valid for all N.
We develop a perturbation expansion in the intersite hopping around the atomic limit of the Hubbard model. It is valid for arbitrary finite temperatures and interaction strengths. Diagrammatic rules that determine the grand-canonical potential and the Green's functions are derived. They reduce the calculation of any finite-order contribution to simple algebra. This opens the way for series extrapolations from computer-aided high-finite-order evaluations. Discrepancies in earlier expansions around the atomic limit are clarified. The present expansion scheme involves only connected diagrams with unrestricted lattice sums. This allows one to perform a vertex renormalization as for the linked-cluster expansion of the Ising model. The renormalized perturbation expansion can be used to construct self-consistent approximations which are automatically exact in the atomic limit. In the limit of high lattice dimensions, only fully two-particle reducible embeddings of diagrams on the lattice contribute. The single-particle properties of the infinite-dimensional Hubbard model reduce to those of independent tight-binding fermions hopping between dressed sites.
The ground-state energy of nuclear matter is expanded in such a way that each term of the expansion is characterized by the number of hole lines. This procedure reverses the usual sequence of steps, which consists in the use of the linked-cluster theorem, the introduction of GGG matrices, and the partial summation of GGG-matrix diagrams leading to the hole-line expansion. In this way, we find an algebraic method which yields automatically all diagrams with a fixed number of independent hole lines. As a by-product, a new proof for the linked-cluster theorem is obtained. We have applied the method to diagrams with four independent hole lines and find complete agreement with results published previously by B. D. Day.
Cluster development may furnish a powerful device for the calculation of the expectation values of the observables of a many-fermion system with respect to dynamically correlated state vectors. The generalized normalization integral, a generating function for the required expectation values, is defined, and four of the many possible decompositions of this function into cluster integrals are explored. Two of these decompositions are slight extensions of the conventional ones of Iwamoto and Yamada and Aviles, Hartogh, and Tolhoek. The other two are product decompositions, leading to new ``factor-cluster'' formalisms. A factor-cluster expansion is applied to the evaluation of the n-particle spatial distribution function.