A variational approach to repulsively interacting three-fermion systems in a one-dimensional harmonic trap

TL;DR

This paper introduces a variational method for accurately approximating the energy and density profiles of a three-fermion system in a one-dimensional harmonic trap with zero-range interactions, including mass imbalance effects.

Contribution

It proposes an interpolation ansatz for the wave function of a 2+1 fermionic system with zero-range interactions, validated against numerical solutions, and explores mass imbalance effects on the spectrum.

Findings

Interpolation ansatz matches numerical energies and densities closely.

Strong and weak interaction limits are well described by hyperspherical coordinates.

Mass imbalance lifts degeneracy in the spectrum at strong coupling.

Abstract

We study a three-body system with zero-range interactions in a one-dimensional harmonic trap. The system consists of two spin-polarized fermions and a third particle which is distinct from two others (2+1 system). First we assume that the particles have equal masses. For this case the system in the strongly and weakly interacting limits can be accurately described using wave function factorized in hyperspherical coordinates. Inspired by this result we propose an interpolation ansatz for the wave function for arbitrary repulsive zero-range interactions. By comparison to numerical calculations, we show that this interpolation scheme yields an extremely good approximation to the numerically exact solution both in terms of the energies and also in the spin-resolved densities. As an outlook, we discuss the case of mass imbalanced systems in the strongly interacting limit. Here we find spectra that demonstrate that the triply degenerate spectrum at infinite coupling strength of the equal mass case is in some sense a singular case as this degeneracy will be broken down to a doubly degenerate or non-degenerate ground state by any small mass imbalance.