A Fortran 90 program to solve the Hartree-Fock equations for interacting spin-1/2 Fermions confined in Harmonic potentials

TL;DR

This paper presents a Fortran 90 program that solves Hartree-Fock equations for interacting spin-1/2 Fermions in harmonic traps, enabling studies of their shell structure and magnetic properties.

Contribution

The paper introduces a novel Fortran 90 code capable of solving both restricted and unrestricted Hartree-Fock equations for spin-1/2 Fermions in harmonic and polynomial potentials.

Findings

Demonstrates shell structure in confined Fermionic systems

Shows the code can handle both repulsive and attractive interactions

Allows investigation of magnetic properties in imbalanced Fermi gases

Abstract

A set of weakly interacting spin-1/2 Fermions, confined by a harmonic oscillator potential, and interacting with each other via a contact potential, is a model system which closely represents the physics of a dilute gas of two-component Fermionic atoms confined in a magneto-optic trap. In the present work, our aim is to present a Fortran 90 computer program which, using a basis set expansion technique, solves the Hartree-Fock (HF) equations for spin-1/2$ Fermions confined by a three-dimensional harmonic oscillator potential, and interacting with each other via pair-wise delta-function potentials. Additionally, the program can also account for those anharmonic potentials which can be expressed as a polynomial in the position operators $x,$ $y$, and $z$. Both the restricted-HF (RHF), and the unrestricted-HF (UHF) equations can be solved for a given number of Fermions, with either repulsive or attractive interactions among them. The option of UHF solutions for such systems also allows us to study possible magnetic properties of the physics of two-component confined atomic Fermi gases, with imbalanced populations. Using our code we also demonstrate that such a system exhibits shell structure, and follows Hund's rule.