Breathing mode of two-dimensional atomic Fermi gases in harmonic traps

For two-dimensional (2D) atomic Fermi gases in harmonic traps, the SO(2,1) symmetry is broken by the interatomic interaction explicitly via the contact correlation operator. Consequently the frequency of the breathing mode $\omega_B$ of the 2D Fermi gas can be different from $2\omega_0$, with $\omega_0$ the trapping frequency of harmonic potentials. 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 $2\omega_0$ which can be as large as about 8%.