Fermions on the surface of a sphere

It is now possible to trap ultracold atomic gases in very peculiar configurations, such as thin spherical shells. After reviewing the main characteristics of these remarkable experimental achievements, we analyze the equilibrium thermodynamics of an ideal gas of fermions constrained to the surface of a sphere by solving the non-interacting one-particle Schrödinger problem and applying the tools of quantum statistical mechanics. We then tackle the more challenging problem of a two-component, weakly interacting gas of fermions confined on the surface of a sphere. Making use of the formalism of quantum field theory within the framework of functional integration, through a dimensional reduction procedure we adapt the standard three-dimensional formalism to the case of a thin spherical surface. This procedure is physically sound and the resulting formalism allows us to retrieve the non-interacting thermodynamics results. In this framework, building on results obtained in the previous sections, we investigate the Stoner itinerant ferromagnetic instability of the gas in the case of a repulsive contact interaction between the fermions. We also develop a BCS mean-field theory for an attractive contact interaction, both at zero and finite temperatures. Throughout the discussion, we highlight the role of the non-trivial topological and geometric features of the sphere - such as its curvature - in the system's physics, comparing our findings to the standard results of its flat two-dimensional counterpart.