Symmetry induced pairing in dark excitonic condensate at finite temperature
Bose Einstein condensate of dark intervalley excitons must be inherently multi-component because of crystalline symmetries. Since valleys hosting such excitons are separated by large quasi-momenta, a minimal inter-component Josephson-type coupling can only be established between pairs of excitons from the time-reversed valleys. As a result, a paired condensate can emerge at finite temperature, that is, the off-diagonal order exists for the pairs from the time-reversed valleys, while the individual valleys are disordered. This prediction follows from the elementary mean field analysis regardless of the dimensionality. However, as Monte Carlo simulations show, no such a phase exists in 3D crystals. Instead, the multi-component condensation proceeds as the Ist order transition from the normal state. The paired phase does exist in 2D for the number of the components $N_v\geq 6$. It forms by Berezinskii-Kosterlitz-Thouless transition from the high temperature (normal) phase. The multi-component condensate appears upon further lowering of temperature.