  We present the numerical analysis of effectively interacting Group Field
Theory (GFT) models in the context of the GFT quantum gravity condensate
analogue of the Gross-Pitaevskii equation for real Bose-Einstein condensates
including combinatorially local interaction terms. Thus we go beyond the
usually considered construction for free models.
  More precisely, considering such interactions in a weak regime, we find
solutions for which the expectation value of the number operator N is finite,
as in the free case. When tuning the interaction to the strongly nonlinear
regime, however, we obtain solutions for which N grows and eventually blows up,
which is reminiscent of what one observes for real Bose-Einstein condensates,
where a strong interaction regime can only be realized at high density. This
behaviour suggests the breakdown of the Bogoliubov ansatz for quantum gravity
condensates and the need for non-Fock representations to describe the system
when the condensate constituents are strongly correlated.
  Furthermore, we study the expectation values of certain geometric operators
imported from Loop Quantum Gravity in the free and interacting cases. In
particular, computing solutions around the nontrivial minima of the interaction
potentials, one finds, already in the weakly interacting case, a nonvanishing
condensate population for which the spectra are dominated by the lowest
nontrivial configuration of the quantum geometry. This result indicates that
the condensate may indeed consist of many smallest building blocks giving rise
to an effectively continuous geometry, thus suggesting the interpretation of
the condensate phase to correspond to a geometric phase.
