Interaction for the trapped fermi gas from a unitary transformation of the exact two-body spectrum

TL;DR

This paper develops a method to analyze few-fermion systems in a harmonic trap using a unitary transformation of the two-body spectrum, achieving accurate energy predictions at unitarity and for various scattering lengths.

Contribution

It introduces a novel approach combining a unitary transformation with the No-Core Shell Model to study few-fermion systems in a trap, extending to multiple particles and interaction regimes.

Findings

Accurate energy calculations for three-fermion systems at unitarity.

Good agreement with exact solutions for few-fermion systems.

Method applicable to systems with up to five fermions.

Abstract

We study systems of few two-component fermions interacting in a Harmonic Oscillator trap. The fermion-fermion interaction is generated in a finite basis with a unitary transformation of the exact two-body spectrum given by the Busch formula. The few-body Schr\"odinger equation is solved with the formalism of the No-Core Shell Model. We present results for a system of three fermions interacting at unitarity as well as for finite values of the S-wave scattering length $a_2$ and effective range $r_2$. Unitary systems with four and five fermions are also considered. We show that the many-body energies obtained in this approach are in excellent agreement with exact solutions for the three-body problem, and results obtained by other methods in the other cases.