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 quasiaverages. 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 equations of motion for the exciton operators d(P) and d†(P) are interconnected with equations of motion for the density operators ρ(P) and D(P). Instead of a set of two equations of motion, as in the case of usual Bose gas, corresponding to normal and abnormal Green’s functions, we have a set of four equations of motion. This means we have to deal simultaneously with four branches of the energy spectrum, the two supplementary branches being the optical plasmon branch represented by the operator ρ(P) and the acoustical plasmon branch represented by the operator D(P). The perturbation theory on the small parameter v2(1 − v2), where v2 is the filling factor and (1 − v2) is the phase space filling factor was developed. 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 plasmontype 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 is in accordance with the number of the broken symmetry operators. The comparison of the results concerning two Bose-Einstein condensates namely of the coplanar magnetoexcitons and of the quantum Hall excitons in the bilayer electron systems reveals their similarity.
Eur. Phys. J. B (2012) 85: 359