Super-shell structure in harmonically trapped fermionic gases and its semi-classical interpretation

TL;DR

This paper explains the super-shell structure observed in harmonically trapped fermionic gases using a semiclassical trace formula, linking quantum shell effects to classical periodic orbits and symmetry breaking.

Contribution

It provides a semiclassical interpretation of super-shell structures in trapped fermionic gases, connecting quantum shell modulations to classical orbit dynamics and symmetry breaking.

Findings

Super-shell structure is dominated by circular and pendulating orbits at low energies.

Beat mode amplitude depends on interaction strength.

Symmetry breaking from U(3) to SO(3) explains shell modulations.

Abstract

It was recently shown in self-consistent Hartree-Fock calculations that a harmonically trapped dilute gas of fermionic atoms with a repulsive two-body interaction exhibits a pronounced {\it super-shell} structure: the shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the ``magic numbers'' occurring between the beat nodes by half a period. The length and amplitude of the beating mode depends on the strength of the interaction. We give a qualitative interpretation of the beat structure in terms of a semiclassical trace formula that uniformly describes the symmetry breaking U(3) $\to$ SO(3) in a 3D harmonic oscillator potential perturbed by an anharmonic term $\propto r^4$ with arbitrary strength. We show that at low Fermi energies (or particle numbers), the beating gross-shell structure of this system is dominated solely by the two-fold degenerate circular and (diametrically) pendulating orbits.