Breathing mode of a harmonically trapped two-dimensional Fermi gas

For two-dimensional (2D) atomic Fermi gases in harmonic traps, the $SO(2,1)$ symmetry is broken by the inter-atomic interaction explicitly via the contact correlation operator. Consequently, the frequency of the breathing mode $\omega_B$ of a 2D Fermi gases in a harmonic trap with frequency $\omega_0$ can be different from $2\omega_0$, which is implied by symmetry. At zero temperature, we use the sum rules of density correlation functions to yield upper bounds for $\omega_B$.We further calculate $\omega_B$ through the Euler equations in the hydrodynamic regime. The obtained value of $\omega_B$ satisfies the upper bounds and shows deviation from $2\omega_0$.