Static and dynamic properties of a few spin $1/2$ interacting fermions trapped in an harmonic potential

TL;DR

This paper investigates the static and dynamic properties of a few spin-1/2 fermions in a one-dimensional harmonic trap, combining numerical diagonalization with analytical insights across different interaction regimes.

Contribution

It provides a comprehensive analysis of few-fermion systems in a harmonic trap, including numerical methods, static properties, and dynamical responses, with benchmarking against exact solutions.

Findings

Numerical methods accurately reproduce known solutions for two particles.

Static properties like energy spectra and densities are characterized for up to five particles.

Dynamical responses such as breathing modes and quench dynamics are analyzed.

Abstract

We provide a detailed study of the properties of a few interacting spin $1/2$ fermions trapped in a one-dimensional harmonic oscillator potential. The interaction is assumed to be well represented by a contact delta potential. Numerical results obtained by means of direct diagonalization techniques are combined with analytical expressions for both the non-interacting and strongly interacting regime. The $N=2$ case is used to benchmark our numerical techniques with the known exact solution of the problem. After a detailed description of the numerical methods, in a tutorial-like manner, we present the static properties of the system for $N=2, 3, 4$ and 5 particles, e.g. low-energy spectrum, one-body density matrix, ground-state densities. Then, we consider dynamical properties of the system exploring first the excitation of the breathing mode, using the dynamical structure function and corresponding sum-rules, and then a sudden quench of the interaction strength.