On the Convergence of Cluster Expansions in Finite Nuclei

E. Krotscheck

T. Papenbrock, David J. Dean, J. R. Gour +4 more

I. Moliner, Niels R. Walet, R. F. Bishop

We study a formulation of the translationally invariant coupled cluster method in coordinate space for finite nuclei. The new formulation remedies convergence problems that plagued previous calculations in configuration space. The method is applied to light nuclei using semi-realistic central interactions.

David Brydges, Paul G. Federbush

Lothar Schäfer

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.

Deepak Iyer, Yuyi Wan

The linked cluster expansion has been shown to be highly efficient in calculating equilibrium and nonequilibrium properties of a variety of 1D and 2D classical and quantum lattice models. In this article, we extend the linked cluster method to the Cayley tree and its boundaryless cousin the Bethe lattice. We aim to (a) develop the linked cluster expansion for these lattices, a novel application, and (b) to further understand the surprising convergence efficiency of the linked cluster method, as well as its limitations. We obtain several key results. First, we show that for nearest-neighbor Hamiltonians of a specific form, all finite treelike clusters can be mapped to one dimensional finite chains. We then show that the qualitative distinction between the Cayley tree and Bethe lattice appears due to differing lattice constants that is a result of the Bethe lattice being boundaryless. We use these results to obtain the explicit closed-form formula for the zero-field susceptibility for the entire disordered phase up to the critical point for Bethe lattices of arbitrary degree; remarkably, only 1D chainlike clusters contribute. We also obtain the exact zero field partition function for the Ising model on both trees with only the two smallest clusters, similar to the 1D chain. Finally, these results achieve a direct comparison between an infinite lattice with a nonnegligible boundary and one without any boundary, allowing us to show that the linked cluster expansion eliminates boundary terms at each order of the expansion, answering the question about its surprising convergence efficiency. We conclude with some ramifications of these results, and possible generalizations and applications.