Toward Universality in Similarity Renormalization Group Evolved Few-body Potential Matrix Elements

TL;DR

This paper investigates the universality of SRG-evolved potential matrix elements driven by T-matrix equivalence, explores local decoupling in two- and three-body sectors, and introduces a 1-D bosonic model with a harmonic oscillator basis to study evolution artifacts and truncation effects.

Contribution

It introduces a simplified 1-D bosonic model and analyzes how basis truncations affect SRG evolution, highlighting challenges in extending universality to three-body forces.

Findings

Universality is observed in local energy regions where T-matrix equivalence holds.

Basis truncation causes evolution artifacts in 3-body potential matrix elements.

Increasing basis size can mitigate artifacts in 2-body evolution, but is challenging for 3-body systems.

Abstract

We first examine how T-matrix equivalence drives the flow of similarity renormalization group (SRG) evolved potential matrix elements to a universal form, with the ultimate goal of gaining insight into universality for three-nucleon forces. In agreement with observations made previously for Lee-Suzuki transformations, regions of universal potential matrix elements are restricted to where half-on-shell T-matrix equivalence holds, but the potentials must also reproduce binding energies. We find universality in local energy regions, reflecting a local decoupling by the SRG. To continue the study in the 3-body sector, we create a simple 1-D spinless boson "theoretical laboratory" for a dramatic improvement in computational efficiency. We introduce a basis-transformation, harmonic oscillator (HO) basis, which is used for current many-body calculations and discuss the imposed truncations. When SRG evolving in a HO-basis, we show that the evolved matrix elements, once transformed back into momentum-representation, differ from those when evolving with momentum representation. This is because the generator in each basis is not exactly the same due to the truncation, which creates evolution artifacts in the 3-body potential matrix elements. In the 2- body sector, this can be avoided by increasing the basis size, but it remains unclear whether this is possible in the 3-body sector, as truncation errors in the 3-body sector are more difficult to avoid, and the computational power required is greatly increased for three-body evolution.