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arXiv:cond-mat/0612659 · Physica Scripta

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

Magnus Ögren, Y Yu, S Åberg, S. M. Reimann, M. Brack

Published Jun 28, 2006• 1 citation
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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.

DOI
10.1088/0031-8949/2006/t125/008

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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 ∝r4\propto r^4∝r4 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.
We show that a dilute harmonically trapped two-component gas of fermionic atoms with a weak repulsive interaction has a pronounced 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 this beating mode depend on the strength of the interaction. We give a simple interpretation of the beat structure in terms of a semiclassical trace formula for the symmetry breaking $\mathrm{U}(3)\ensuremath{\rightarrow}\mathrm{SO}(3)$.
The shell structures for weakly interacting fermions in harmonic oscillator traps at zero temperature undergo several transitions depending on the number of particles in the trap and their interaction strength. Calculations of the one and two-particle spectra give the pairing gaps, and the many particle state is constructed by the seniority scheme. For sufficiently few particles, N less than or similar to 10(4), the pairing field exceeds the mean-field splitting of single particle levels at the Fermi surface and the greater number of almost degenerate states due to the SU(3) symmetry in the harmonic oscillator shell lead to supergaps. Deformation and rotation are also discussed.
We examine the spin asymmetry of ground states for two-dimensional, harmonically trapped two-component gases of fermionic atoms at zero temperature with weakly repulsive short range interactions. Our main result is that, in contrast to the three-dimensional case, in two dimensions a non-trivial spin-asymmetric phase can only be caused by shell structure. A simple, qualitative description is given in terms of an approximate single particle model, comparing well to the standard results of Hartree-Fock or direct diagonalization methods.
Pairing gaps for fermionic atoms in harmonic-oscillator traps are calculated for a wide range of interaction strengths and particle numbers, and compared to pairing in nuclei. Especially systems where the pairing gap exceeds the single-level spacing but is smaller than the shell splitting $\ensuremath{\Elzxh}\ensuremath{\omega}$ are studied, which applies to most trapped Fermi atomic systems as well as to finite nuclei. When solving the gap equation for a large trap with such multilevel pairing, one finds that the matrix elements between nearby harmonic-oscillator levels and the quasi particle energies lead to a double logarithm of the gap, and a pronounced shell structure at magic numbers. It is argued that neutron and proton pairing in nuclei belongs to the class of multilevel pairing, that their shell structure follows naturally, and that the gaps scale as $\ensuremath{\sim}{A}^{\ensuremath{-}1/3}$---all in qualitative agreement with odd-even staggering of nuclear binding energies. Pairing in large systems is related to that in the bulk limit. For large nuclei the neutron and proton superfluid gaps approach the asymptotic value in infinite nuclear matter: $\ensuremath{\Delta}\ensuremath{\simeq}1.1\mathrm{MeV}.$