True, quasi and unstable Nambu–Goldstone modes of the 2D Bose–Einstein condensed magnetoexcitons with different wave vectors

Abstract

The collective elementary excitations of the two-dimensional magnetoexcitons in the state of their Bose-Einstein condensation with any wave vector k and in-plane parallel oriented motional dipole moments are investigated in the Hartree-Fock-Bogoliubov approximation. The breaking of the gauge symmetry is achieved using the Bogoliubov theory of quasiaverages and the Keldysh-Kozlov-Kopaev method. The starting Hamiltonian and the Green's functions are determined using the integral two-particle operators instead of the single-particle Fermi operators. The infinite chains of equations of motion for the multioperator four and six-particle Green's functions are truncated following the Zubarev method. The energy spectrum at k  0 contains only one gapless optical-plasmon-type oscillations. There are two excitontype branches corresponding to normal and abnormal Green’s functions. Both modes are gapped with rotontype segments at intermediary values of the wave. The fourth branch is the acoustical plasmon-type mode with absolute instability in the region of small and intermediary values of the wave vectors. The energy spectrum at k  3.6 consists of the mixed exciton-plasmon energy braches, mixed exciton-plasmon quasienergy branches as well as the optical and acoustical plasmon energy branches. The exciton branches of the spectrum have the gaps related with the negative values of the chemical potential and attractive interaction between the two-dimensional megnetoexcitons with in-plane, parallel oriented motional dipole moments. The acoustical and optical plasmon energy branches are gapless. Their dependence on the small wave vectors accounted from the condensate wave vector is linear and quadratic correspondingly, with saturation in the range of high values of the wave vectors.