Breathing-mode frequency of a strongly interacting Fermi gas across the two- to three-dimensional crossover

We address the interplay between dimension and quantum anomaly on the breathing mode frequency of a strongly interacting Fermi gas harmonically trapped at zero temperature. Using a beyond-mean-field Gaussian pair fluctuation theory, we employ periodic boundary conditions to simulate the dimensionality of the system and impose a local-density approximation, with two different schemes, to model different trapping potentials in the tightly confined axial direction. By using a sum-rule approach, we compute the breathing mode frequency associated with a small variation of the trapping frequency along the weakly confined transverse direction and describe its behavior as a function of the dimensionality, from two to three dimensions, and of the interaction strength. We compare our predictions with previous calculations on the two-dimensional breathing mode anomaly and discuss their possible observation in ultracold Fermi gases of Li-6 and K-40 atoms.