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We discuss the collective modes in an alkaline-earth Fermi gas close to anorbital Feshbach resonance. Unlike the usual Feshbach resonance, the orbitalFeshbach resonance in alkaline-earth atoms realizes a two-band superfluidsystem where the fermionic nature of both the open and the closed channel hasto be taken into account. We show that apart from the usual Anderson-Bogoliubovmode which corresponds to the oscillation of total density, there also appearsthe long-sought Leggett mode corresponding to the oscillation of relativedensity between the two channels. The existence of the phonon and the Leggettmodes and their evolution are discussed in detail. We show how these collectivemodes are reflected in the density response of the system.

Collective Modes in a Two-band Superfluid of Ultracold Alkaline-earth Atoms Close to an Orbital Feshbach Resonance