Hugenholtz-Pines theorem for Bose–Einstein condensates with internal degrees of freedom

The Hugenholtz-Pines theorem for the spin-f spinor BEC with broken U(1)×SO(3) symmetry is an unsolved problem in the field of ultracold quantum gases since 1998 when the experiment of the spin-1 BEC was first realized, and its Bogoliubov theories were first proposed. In this presentation, we will propose the Hugenholtz-Pines theorem for BECs with internal degrees of freedom. In order to that, we clarified the generalized version of the Hugenholtz-Pines theorem for BECs with broken U(1)×SO(N) or U(1)×SU(N) symmetry in the presence of the SO(N) or SU(N) symmetry breaking external fields. This theorem serves to deductively organize the Hugenhotlz-Pines theorem for spin-f spinor BECs in the presence of an external magnetic field. We will introduce some specific results of the Hugenholtz-Pines theorem in certain phases of the spin-1 or spin-2 BEC.