Collective modes for a Fermi Liquid under Harmonic Trap

Based on the Boltzmann transport equation of quasi-particles, we examine the low energy excitations of a normal Fermi liquid conned in a two dimensional harmonic trap. In the non-interacting limit, the excitation spectrum is discrete and the highly degenerate eigenfrequencies locate exactly at integer multiples of the corresponding trap frequency. When the interaction is switched on, the degeneracy will be broken and every degenerate frequency will extend to be a continuous band but with some discrete modes out of the band. The continuum and discrete modes in the spectrum can be interpreted as particle-hole excitations and collective modes (“zero sound”) respectively. For a repulsive interaction, the lowest collective mode frequencies are higher than those of the corresponding particle-hole excitations and can be detected by the experiments on cold atom systems.