  Two-dimensional Fermi gases with universal short-range interactions are known
to exhibit a quantum anomaly, where a classical scale and conformal invariance
is broken by quantum effects at strong coupling. We argue that in a quasi
two-dimensional geometry, a conformal window remains at weak interactions.
Using degenerate perturbation theory, we verify the conformal symmetry by
computing the energy spectrum of mesoscopic particle ensembles in a harmonic
trap, which separates into conformal towers formed by so-called primary states
and their center-of-mass and breathing-mode excitations, the latter having
excitation energies at precisely twice the harmonic oscillator energy. In
addition, using Metropolis importance sampling, we compute the hyperradial
distribution function of the many-body wave functions, which are predicted by
the conformal symmetry in closed analytical form. The weakly interacting Fermi
gas constitutes a system where the nonrelativistic conformal symmetry can be
revealed using elementary methods, and our results are testable in current
experiments on mesoscopic Fermi gases.
