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arXiv:2609.18386

Geometry-Driven Suppression of Fermionic Pairing on a Spherical Surface

Lorenzo Frigato, Andrea Tononi, Luca Salasnich

Published Sep 16, 2026• 0 citations
arXivHTMLPDF

Abstract

We investigate the ground-state properties of a two-component attractive Fermi gas confined to a spherical surface, demonstrating how curvature and finite-size effects lead to qualitatively different behavior compared to the planar limit. In the strongly attractive regime, the system forms a Bose-Einstein condensate (BEC) of tightly bound dimers, yielding a gap and chemical potential that approach flat-space results with leading-order geometric corrections. However, as the attraction decreases, the system crosses over into a Bardeen-Cooper-Schrieffer (BCS) regime where the Cooper pair size becomes constrained by the sphere radius, producing pronounced deviations from flat-space behavior. We identify strong shell effects at magic particle numbers set by the single-particle spectrum, revealing the geometry-driven suppression of fermionic pairing at weak interactions. Our results highlight a fundamental interplay among geometry, quantum statistics, and interactions, motivating future experiments with shell-trapped ultracold fermions and inspiring geometry-controlled condensed matter devices.

DOI
10.48550/arxiv.2609.18386

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We investigate the ground-state properties of a two-component attractive Fermi gas confined to a spherical surface, demonstrating how curvature and finite-size effects lead to qualitatively different behavior compared to the planar limit. In the strongly attractive regime, the system forms a Bose-Einstein condensate (BEC) of tightly bound dimers, yielding a gap and chemical potential that approach flat-space results with leading-order geometric corrections. However, as the attraction decreases, the system crosses over into a Bardeen-Cooper-Schrieffer (BCS) regime where the Cooper pair size becomes constrained by the sphere radius, producing pronounced deviations from flat-space behavior. We identify strong shell effects at magic particle numbers set by the single-particle spectrum, revealing the geometry-driven suppression of fermionic pairing at weak interactions. Our results highlight a fundamental interplay among geometry, quantum statistics, and interactions, motivating future experiments with shell-trapped ultracold fermions and inspiring geometry-controlled condensed matter devices.
We present a theory of a two-component atomic Fermi gas with tunable attractive contact interactions on a spherical shell going through the Bardeen-Cooper-Schrieffer (BCS)--Bose-Einstein condensation (BEC) crossover, inspired by the realizations of spherical bubble traps for ultracold atoms in microgravity. The derivation follows the BCS-Leggett theory to obtain the gap and number equations. The BCS-BEC crossover can be induced by tuning the interaction, and the properly normalized gap and chemical potential exhibit universal behavior regardless of the planar or spherical geometry. Nevertheless, the spherical-shell geometry introduces another way of inducing the crossover by the curvature. The curvature-induced BCS-BEC crossover is made possible by fixing the particle number and interaction strength while shrinking the sphere, causing a reduction to the ratio of the pairing and kinetic energies and pushing the system toward the BCS limit. The saturation of the superfluid density further confirms that the ground state is a Fermi superfluid.
Fermion pairing underlies collective quantum phenomena across widely different forms of matter. In extended systems such as ultracold Fermi gases, pairing is commonly understood through the BCS--BEC crossover, where the pair size evolves from large, overlapping Cooper pairs to tightly bound dimers. In finite systems such as atomic nuclei, superconducting grains and quantum dots, however, the same pairing tendency competes with confinement, shell filling and spatial inhomogeneity, making the microscopic structure of pairs much harder to access. Here, we image pair correlations in a finite, tunable system of few fermionic atoms with single-particle resolution and full counting statistics. We observe that confinement and shell structure re-organize pairing in real space: In the weakly interacting, confinement-dominated regime, closed-shell configurations suppress correlations in the high-density trap center. Pairing is mainly observed toward the low-density surface. Open-shell systems, however, support substantially stronger central pairing. Already for surprisingly small systems, increasing either interaction strength or particle number restores a locally bulk-like Cooper-pair profile in the trap center, whereas the edge retains dimer-like correlations. By resolving where pairs form and how their character changes from localized dimers to overlapping Cooper pairs, our measurements provide a microscopic view of pairing in finite fermionic matter and connect the physics of mesoscopic cold atoms to pairing phenomena in nuclei and superconducting nanostructures.
Quantum-size oscillations of the basic physical characteristics of a confined fermionic condensate are a well-known phenomenon. Its conventional understanding is based on the single-particle physics, whereby the oscillations follow variations in the single-particle density of states driven by the size quantization. Here we present a study of a cigar-shaped ultracold superfluid Fermi gas, which demonstrates an important many-body aspect of the quantum-size coherent effects, overlooked previously. The many-body physics is revealed here in the atypical crossover from the Bardeen-Cooper-Schrieffer (BCS) superfluid to the Bose-Einstein condensate (BEC) induced by the size quantization of the particle motion. The single-particle energy spectrum for the transverse dimensions is tightly bound, whereas for the longitudinal direction it resembles a quasi-free dispersion. This results in the formation of a series of single-particle subbands (shells) so that the aggregate fermionic condensate becomes a coherent mixture of subband condensates. Each time when the lower edge of a subband crosses the chemical potential, the BCS-BEC crossover is approached in this subband, and the aggregate condensate contains both BCS and BEC-like components.
In recent years, ultracold atomic gases confined in curved geometries have attracted 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 the study of Fermi gases has been left 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 noninteracting case, showing how the intrinsic geometrical features of the spherical surface give rise to a shell structure 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.
We investigate the Bose-Einstein condensation of fermionic pairs in a two-dimensional uniform two-component Fermi superfluid obtaining an explicit formula for the condensate density as a function of the chemical potential and the energy gap. By using the mean-field extended Bardeen-Cooper-Schrieffer theory, we analyze, as a function of the bound-state energy, the off-diagonal long-range order in the crossover from the Bardeen-Cooper-Schrieffer state of weakly bound Cooper pairs to the Bose-Einstein condensate of strongly-bound molecular dimers.
We explore the evolution of an ultracold quantum gas of interacting fermions crossing from a Bardeen-Cooper-Schrieffer (BCS) superfluidity to a Bose-Einstein condensation (BEC) of molecular bosons in the presence of a tunable-range interaction among the fermions and of an artificial magnetic field, which can be used to simulate a pseudo-spin-orbit coupling (SOC) and to produce topological states. We find that the crossover is affected by a competition between the finite range of the interaction and the SOC and that the threshold for the topological transition is affected by the interactions only in the small pair size, BEC-like, regime. Below , we find persistence of universal behavior in the critical temperature, chemical potential, and condensate fraction, provided that the pair correlation length is used as a driving parameter. Above threshold, universality is lost in the regime of large pair sizes. Here, the limiting ground state departs from a weakly interacting BCS-like one so that a different description is required. Our results can be relevant in view of current experiments with cold atoms in optical cavities, where tunable-range effective atomic interactions can be engineered.