Graphical Method for Effective Interaction with a New Vertex Function

TL;DR

This paper introduces a new graphical method using a vertex function Z(E) to efficiently compute effective interactions in quantum systems, improving convergence and stability over traditional methods.

Contribution

The paper develops a novel vertex function Z(E) and a graphical approach that enhances the accuracy and convergence speed of effective interaction calculations.

Findings

Z(E) is well-behaved and non-singular.

The method achieves fast convergence compared to KK.

It reliably reproduces true eigenvalues of H.

Abstract

Introducing a new vertex function, Z(E), of an energy variable E, we derive a new equation for the effective interaction. The equation is obtained by replacing the Q-box in the Krenciglowa-Kuo (KK) method by Z(E). This new approach can be viewed as an extension of the KK method. We show that this equation can be solved both in iterative and non-iterative ways. We observe that the iteration procedure with Z(E) brings about fast convergence compared to the usual KK method. It is shown that, as in the KK approach, the procedure of calculating the effective interaction can be reduced to determining the true eigenvalues of the original Hamiltonian H and they can be obtained as the positions of intersections of graphs generated from Z(E). We find that this graphical method yields always precise results and reproduces any of the true eigenvalues of H. The calculation in the present approach can be made regardless of overlaps with the model space and energy differences between unperturbed energies and the eigenvalues of H. We find also that Z(E) is a well-behaved function of E and has no singularity. These characteristics of the present approach ensure stability in actual calculations and would be helpful to resolve some difficulties due to the presence of poles in the Q-box. Performing test calculations we verify numerically theoretical predictions made in the present approach.