Hard-Wall and Non-Uniform Lattice Monte Carlo Approaches to One-Dimensional Fermi Gases in a Harmonic Trap

TL;DR

This paper introduces two lattice Monte Carlo methods for simulating one-dimensional trapped Fermi gases, demonstrating their effectiveness in calculating energies and densities, and paving the way for more complex future studies.

Contribution

The paper develops and compares a hard-wall and a non-uniform Gauss-Hermite lattice Monte Carlo approach for one-dimensional Fermi gases in traps, improving computational efficiency and accuracy.

Findings

Both methods accurately compute ground-state energies and density profiles.

Hard-wall basis effectively resolves the trapping potential.

Fast Fourier transforms can accelerate calculations with the hard-wall basis.

Abstract

We present in detail two variants of the lattice Monte Carlo method aimed at tackling systems in external trapping potentials: a uniform-lattice approach with hard-wall boundary conditions, and a non-uniform Gauss-Hermite lattice approach. Using those two methods, we compute the ground-state energy and spatial density profile for systems of N=4 - 8 harmonically trapped fermions in one dimension. From the favorable comparison of both energies and density profiles (particularly in regions of low density), we conclude that the trapping potential is properly resolved by the hard-wall basis. Our work paves the way to higher dimensions and finite temperature analyses, as calculations with the hard-wall basis can be accelerated via fast Fourier transforms, the cost of unaccelerated methods is otherwise prohibitive due to the unfavorable scaling with system size.