Validity of the scattering length approximation in strongly interacting Fermi systems

TL;DR

This paper examines the limitations of the scattering length approximation in strongly interacting Fermi systems by comparing exact calculations with the approximation, revealing significant discrepancies in critical transition points.

Contribution

It provides a systematic analysis of few-fermion systems, highlighting the inaccuracies of the scattering length approximation in predicting ferromagnetic transition points.

Findings

The critical interaction strength is much higher than predicted by the scattering length approximation.

Discrepancies are approximately a factor of 2 in the 4-particle system.

Exact calculations align with experimental observations.

Abstract

We investigate the energy spectrum of systems of two, three and four spin-1/2 fermions with short range attractive interactions both exactly, and within the scattering length approximation. The formation of molecular bound states and the ferromagnetic transition of the excited scattering state are examined systematically as a function of the 2-body scattering length. Identification of the upper branch (scattering states) is discussed and a general approach valid for systems with many particles is given. We show that an adiabatic ferromagnetic transition occurs, but at a critical transition point kF a much higher than predicted from previous calculations, almost all of which use the scattering length approximation. In the 4-particle system the discrepancy is a factor of 2. The exact critical interaction strength calculated in the 4-particle system is consistent with that reported by experiment. To make comparisons with the adiabatic transition, we study the quench dynamics of the pairing instability using the eigenstate wavefunctions.