  A two-dimensional (2D) harmonically trapped interacting Fermi gas is
anticipated to exhibit a quantum anomaly and possesses a breathing mode at
frequencies different from a classical scale invariant value
$\omega_{B}=2\omega_{\perp}$, where $\omega_{\perp}$ is the trapping frequency.
The predicted maximum quantum anomaly ($\sim10\%$) has not been confirmed in
experiments. Here, we theoretically investigate the zero-temperature density
equation of state and the breathing mode frequency of an interacting Fermi
superfluid at the dimensional crossover from three to two dimensions. We find
that the simple model of a 2D Fermi gas with a single $s$-wave scattering
length is not adequate to describe the experiments in the 2D limit, as commonly
believed. A more complete description of quasi-2D leads to a much weaker
quantum anomaly, consistent with the experimental observations. We clarify that
the reduced quantum anomaly is due to the significant confinement-induced
effective range of interactions, which is overlooked in previous theoretical
and experimental studies.
