Collective Excitations in Spin-2 Bose-Einstein Condensates

The Green’s functions and the correlation functions in spin-2 Bose Einstein condensates at finite temper-ature are defined and the generalized Dyson-Beliaev equations are introduced. We discuss the spin conservation in zdirection and decouple the Green’s functions and the generalized Dyson-Beliaev equations according to different spinconservations in z direction. The anomalous vertex functions are introduced and the self-energies are separated into theproper self-energies and the improper self-energies. The generalized Dyson-Beliaev equations are decoupled accordingto separation of the self-energies. We calculate the Green’s functions step by step in the Bogoliubov approximationand discuss the collective excitations in spin-2 Bose-Einstein condensates in the polar, ferromagnetic, and cyclic cases,respectively.