Hydrodynamic collective modes in bilayer graphene

We investigate the hydrodynamic collective mode dispersions of bilayer graphene in the long wavelength and low-frequency limit and in the absence and presence of a metallic gate, following D. Svintsov [Phys Rev B 97, 121405(R) (2018)] who extended the classical  Bhatnagar-Gross-Krook formalism to two-component quantum gases and used it to derive hydrodynamic collective mode dispersions in monolayer graphene.  At high density, we recover well-known expressions for degenerate Fermi gases. At charge neutrality, two collective modes are obtained. The first mode is an in-phase electron-hole oscillation with a sound-like dispersion (real frequency component varies linearly with wave vector) that is unaffected by electron-hole collisions and the presence of the gate to leading order. The second mode is an out-of-phase electron-hole oscillation that is purely damped at low wave vectors but develops a real frequency component above a critical wave vector that then varies as the square root (first power) of the wave vector in the absence (presence) of an external gate. We compare and contrast these modes with known results from monolayer graphene.