Analytical and Machine Learning study of one-dimensional non-interacting spinless trapped fermionic systems

Abstract

In this work we study the ground-state properties of three different one-dimensional systems of N identical, non-interacting, spinless fermions trapped in a potential well. We consider a harmonic trap, an infinite potential well and a Morse potential, and for all of them we prove that the ground-state wave-function can be written in terms of a Vandermonde determinant. We compute and plot the one-body density matrix and the pair correlation function for systems of 2 to 5 particles using two different analytical methods to check that both provide the same results. Moreover, we derive closed expressions for these functions in terms of polynomials for a general number of particles. These polynomials, in turn, can be expressed using Vandermonde vectors and square matrices. To complement the mathematical study of the systems, we reproduce and validate the analytical results using a Machine Learning approach. We use a Neural Quantum State as an ansatz for the ground-state wavefunction and a Variational Monte-Carlo method to find the best neural network parameters. Both the energies and the density matrices are correctly reproduced using Machine Learning.