  For the quantum well in an optical microcavity, the interplay of the Coulomb
interaction and the electron-photon (e-ph) coupling can lead to the
hybridizations of the exciton and the cavity photon known as polaritons, which
can form the Bose-Einstein condensate above a threshold density. Additional
physics due to the nontrivial Berry phase comes into play when the quantum well
consists of the gapped two-dimensional (2D) Dirac material such as the
transition metal dichalcogenide (TMDC) MoS$_2$ or WSe$_2$. Specifically, in
forming the polariton, the e-ph coupling from the optical selection rule due to
the Berry phase can compete against the Coulomb electron-electron (e-e)
interaction. We find that this competition gives rise to a rich phase diagram
for the polariton condensate involving both topological and symmetry breaking
phase transitions, with the former giving rise to the quantum anomalous Hall
and the quantum spin Hall phases.
