Phase transitions in optical turbulence

Abstract: We consider turbulence in the Gross-Pitaevsky model and study the creation of a coherent condensate via an inverse cascade originated at small scales. The growth of the condensate leads to a spontaneous breakdown of symmetries of small-scale over-condensate fluctuations: first, statistical isotropy is broken, then series of phase transitions mark the change of symmetry from the two-fold to three-fold to four-fold. At the highest condensate level reached, we observe a short-range positional and long-range orientational order (similar to a hexatic phase in the condensed matter physics). In other words, the longer one pumps the system the more ordered it becomes. We show that these phase transitions happen when the driving term corresponds to an instability (i.e. it is multiplicative in the k-space) but not when the system is pumped by a random force. Thus we demonstrate for the first time non-universality of the inverse-cascade turbulence. We also describe anisotropic spectral flux flows in k-space, anomalous correlations of fluctuations and collective oscillations of turbulence-condensate system.