Repulsive fermions and shell effects on the surface of a sphere

TL;DR

This work analyzes a two-component Fermi gas constrained to the surface of a sphere, showing that curvature-induced finite-size effects yield a distinctive shell structure that reshapes low-temperature thermodynamics compared to flat 2D systems. It combines an exact non-interacting treatment of fermions on the sphere with a path-integral Hartree-Fock approach to include repulsive interactions, deriving the grand potential and self-consistent density equations, and then performing a finite-temperature Stoner analysis to determine the stability of the spin-balanced state. A key finding is that shell effects modulate the critical interaction strength $g_{2D,c}$ for spin polarization, producing sharp peaks at magic numbers and vanishing thresholds away from closed shells as $T\to0$, while high temperatures recover the semiclassical flat-$2D$ behavior. The results illuminate how geometry and finite size alter many-body fermionic physics on curved manifolds and propose experimental pathways using spherical bubble traps, though observing shell effects demands ultra-low temperatures or smaller radii for practicality.

Abstract

In recent years, ultracold atomic gases confined in curved geometries have obtained considerable theoretical interest. This is motivated by recent realizations of bubble traps in microgravity conditions, which open the possibility of investigating quantum many-body physics beyond the conventional flat-space paradigm. The theoretical interest up to now was mainly focused on Bose gases and their phenomenology, and had left the study of Fermi gases behind. In this paper, we investigate a two-component repulsive Fermi gas constrained to the surface of a sphere at finite temperature. We first analyze the non-interacting case, showing how the intrinsic geometrical features of the spherical surface give rise to a shell structures and modify the low-temperature thermodynamics compared to the flat two-dimensional gas. Repulsive interactions are then considered through an effective path-integral approach within a Hartree-Fock mean-field approximation, enabling us to derive the grand canonical potential and to regularize the associated Matsubara summation. We then investigate the stability of the spin-balanced state and obtain the finite-temperature Stoner criterion for fermions on a sphere, highlighting the interplay between the repulsive interactions and shell effects.

Figures (2)

• Figure 1: (a) Dimensionless chemical potential $\mu_\parallel/\zeta$ as a function of the number of fermions $N$ for different temperatures, namely $k_BT/\zeta=10$ (blue bold line), $k_BT/\zeta=1.0$ (orange bold line) and $k_BT/\zeta=0.1$ (green bold line). The gray vertical dashed lines mark the magic numbers, corresponding to completely filled shells. (b) Chemical potential per particle $\mu_\parallel(T)/(N \zeta)$ as a function of the dimensionless temperature for two different $N$ values, i.e. $N=10^3$ (in blue) and $N=10^{4}$ (in light blue) in the semiclassical limit (dashed lines) and the exact result (solid lines).
• Figure 2: (a) Dimensionless critical interaction strength $g_{2D,c} m/\hbar^2$ as a function of the number of fermions on the sphere $N$ for different temperatures: $k_BT/\zeta=1.0$ (blue solid line), $k_BT/\zeta=0.2$ (orange solid line) and $k_BT/\zeta=0.1$ (green solid line). The gray dashed vertical lines indicates the $N$ magic numbers, i.e the shells closure. (b) Dimensionless critical interaction strength $g_{2D,c} m/\hbar^2$ as a function of the dimensionless temperature $k_BT/\zeta$ for different number of fermions $N$, namely $N=9.0\times10^3$ (blue solid line), $N=9.0\times10^3$ (green solid line) and $N=1.1\times10^4$ (gold solid line). The colored plane portions correspond to the $g_{2D}$ values for which the spin-balanced solution is stable, namely for which Eq. (\ref{['STONER CRITERION']}) is not satisfied. The solid lines represent the critical coupling strength $g_{2D,c}$ above which Eq. (\ref{['STONER CRITERION']}) is satisfied. The semiclassical curves (provided by Eq. (\ref{['semiclassical critical']}), dashed lines) are also plotted as a reference. In the inset we highlight the convergence to zero of $g_{2D,c}$ as $T\to 0$.