Mixed exciton–plasmon collective elementary excitations of the Bose–Einstein condensed two-dimensional magnetoexcitons with motional dipole moments

Abstract

The collective elementary excitations of two-dimensional magnetoexcitons in aBose-Einstein condensate (BEC) with wave vector k = 0 wereinvestigated in the framework of the Bogoliubov theory of quasiaverages. The Hamiltonianof the electrons and holes lying in the lowest Landau levels (LLLs) contains supplementaryinteractions due to virtual quantum transitions of the particles to the excited Landaulevels (ELLs) and back. As a result, the interaction between the magnetoexcitons withk = 0 does not vanish and their BEC becomes stable. Theequations of motion for the exciton operators d(P) andd†(P) are interconnected with equations ofmotion for the density operators ρ(P) andD(P). Instead of a set of two equations of motion, asin the case of usual Bose gas, corresponding to normal and abnormal Green’s functions, wehave a set of four equations of motion. This means we have to deal simultaneously withfour branches of the energy spectrum, the two supplementary branches being the opticalplasmon branch represented by the operator ρ(P) and theacoustical plasmon branch represented by the operatorD(P). The perturbation theory on the small parameterv2(1 − v2), wherev2 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 thesecond kind with dependenceω(k) ≈ k2 at small valuesk describing the optical-plasmon-type oscillations. There are twoexciton-type branches corresponding to normal and abnormal Green’s functions. Both modesare gapped with roton-type segments at intermediary values of the wave vectors and can benamed as quasi-NG modes. The fourth branch is the acoustical plasmon-type mode withabsolute 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. Thenumber and the kind of the true NG modes is in accordance with the number of the brokensymmetry operators. The comparison of the results concerning two Bose-Einstein condensatesnamely of the coplanar magnetoexcitons and of the quantum Hall excitons in the bilayerelectron systems reveals their similarity.