True, quasi and unstable Nambu-Goldstone modes of the two-dimensional Bose-Einstein condensed magnetoexcitons

The collective elementary excitations of two-dimensional magnetoexcitons in a Bose-Einstein condensate (BEC) with wave vector k→=0 were investigated in the framework of the Bogoliubov theory of quasi-averages. The Hamiltonian of the electrons and holes lying in the lowest Landau levels (LLLs) contains supplementary interactions due to virtual quantum transitions of the particles to the excited Landau levels (ELLs) and back. As a result, the interaction between the magnetoexcitons with k→=0 does not vanish and their BEC becomes stable. The energy spectrum contains only one gapless, true Nambu-Goldstone (NG) mode of the second kind with dependence ω(k)≈k² at small values k describing the optical-plasmon-type oscillations. There are two exciton-type branches corresponding to normal and abnormal Green's functions. Both modes are gapped with roton-type segments at intermediary values of the wave vectors and can be named as quasi-NG modes. The fourth branch is the acoustical plasmon-type mode with absolute instability in the region of small and intermediary values of the wave vectors. All branches have a saturation-type dependencies at great values of the wave vectors. The number and the kind of the true NG modes are in accordance with the number of the broken symmetry operators.