Comparison of renormalized interactions using one-dimensional few-body systems as a testbed

TL;DR

This paper compares different renormalization methods for one-dimensional few-body systems, showing that effective interactions and running coupling constants improve convergence and accuracy in numerical calculations.

Contribution

It provides a detailed comparison of renormalization techniques, including the running coupling constant and effective interactions, for improving convergence in few-body quantum systems.

Findings

Running coupling constant improves convergence significantly.

Effective interactions enhance accuracy when including excited states.

Transforming observables alongside potential renormalization is crucial.

Abstract

Even though the one-dimensional contact interaction requires no regularization, renormalization methods have been shown to improve the convergence of numerical ab initio calculations considerably. In this work, we compare and contrast these methods: `the running coupling constant' where the two-body ground state energy is used as a renormalization condition, and two effective interaction approaches that include information about the ground as well as excited states. In particular, we calculate the energies and densities of few-fermion systems in a harmonic oscillator with the configuration interaction method, and compare the results based upon renormalized and bare interactions. We find that the use of the running coupling constant instead of the bare interaction improves convergence significantly. A comparison with an effective interaction, which is designed to reproduce the relative part of the energy spectrum of two particles, showed a similar improvement. The effective interaction provides an additional improvement if the center-of-mass excitations are included in the construction. Finally, we discuss the transformation of observables alongside the renormalization of the potential, and demonstrate that this might be an essential ingredient for accurate numerical calculations.