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Phases and phase transitions of Bose condensed light

Bose-Einstein condensation of light in 2D is characterized by two classical fields corresponding to two polarizations of light as well as by the distribution of dye molecules inducing light thermalization through dipolar transition. In the case when this transition is triple-degenerate the resulting field theory for the condensate of light is O(4) symmetric, which precludes algebraic long range order in 2D at any finite temperature $T$. If the dipolar degeneracy is removed, then, equilibrium phases with lower symmetries -- O(2)$\times$Z$_2$ and O(2) can emerge. Accordingly, algebraic off diagonal order of light condensate becomes possible. An orientationsl disorder introduced by local dipolar anisotropy can destroy algebraic order in one-photon density matrix while preserving it in the two-photon one. This represents formation of the condensate of photon pairs.