Nonclassical scissors mode of a vortex lattice in a Bose-Einstein condensate

We show that a Bose-Einstein condensate with a vortex lattice in a rotating anisotropic harmonic potential exhibits a very low frequency scissors mode. The appearance of this mode is due to the SO(2) symmetry breaking introduced by the vortex lattice. The corresponding Goldstone mode in an isotropic trap becomes, in an anisotropic trap, a scissors mode, whose frequency is finite and tends to zero with the trap anisotropy $ϵ$, with a generic √ϵ dependence. We present analytical formulas giving the mode frequency in the low $ϵ$ limit and we show that the mode frequency for some class of vortex lattices can tend to zero as $ϵ$ or faster. We demonstrate that the standard classical hydrodynamics approach fails to reproduce this low frequency mode, because it does not contain the discrete structure of the vortex lattice.