Superfluidity of dipolar excitons in a double layer of $\alpha -T_3$ with a mass term

We predict Bose-Einstein condensation and superfluidity of dipolar excitons, formed by electron-hole pairs in spatially separated gapped hexagonal $\alpha -T_3$ (GHAT3) layers. In the $\alpha -T_3$ model, the AB-honeycomb lattice structure is supplemented with C atoms located at the centers of the hexagons in the lattice. We considered the $\alpha -T_3$ model in the presence of a mass term which opens a gap in the energy dispersive spectrum. The gap opening mass term, caused by a weak magnetic field, plays the role of Zeeman splitting at low magnetic fields for this pseudospin-1 system. The band structure of GHAT3 monolayers leads to the formation of two distinct types of excitons in the GHAT3 double layer. We consider two types of dipolar excitons in double-layer GHAT3: (a) A excitons'', which are bound states of electrons in the conduction band (CB) and holes in the intermediate band (IB) and (b) B excitons’', which are bound states of electrons in the CB and holes in the valence band (VB). The binding energy of A and B dipolar excitons is calculated. For a two-component weakly interacting Bose gas of dipolar excitons in a GHAT3 double layer, we obtain the energy dispersion of collective excitations, the sound velocity, the superfluid density, and the mean-field critical temperature $T_c$ for superfluidity.