Bose-Einstein condensate of dark intervalley excitons must be inherently multicomponent because of crystalline symmetries. Since valleys hosting such excitons are separated by large quasimomenta, a minimal intercomponent 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 three-dimensional (3D) crystals. Instead, the multicomponent condensation proceeds as the first-order transition from the normal state. The paired phase does exist in two dimensions (2D) for the number of the components, $N_v\ge 6$. It forms by Berezinskii-Kosterlitz-Thouless transition from the high-temperature (normal) phase. The multicomponent condensate appears upon further lowering of the temperature.

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• Classical statistical mechanics
• Phase transitions
• Monte Carlo methods