Magnetoplasmons and magnetoexcitons in doped double-layered $\alpha$-T$_3$ lattice in a strong magnetic field

We investigate the conditions for the occurrence of Bose-Einstein condensation and superfluidity of indirect magnetoexcitons for a pair of quasi-two-dimensional spatially separated $\alpha$-T$_3$ layers. The energy dispersion of collective excitations, the spectrum of sound velocity, and the effective magnetic mass of magnetoexcitons are obtained in the integer quantum-Hall regime for strong magnetic fields. The superfluid density and the temperature of the Kosterlitz-Thouless phase transition are probed as functions of the excitonic density, magnetic field and the interlayer separation.