  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.
