  In the tensorial group field theory (TGFT) approach to quantum gravity, the
basic quanta of the theory correspond to discrete building blocks of geometry.
It is expected that their collective dynamics gives rise to continuum spacetime
at a coarse grained level, via a process involving a phase transition. In this
work we show for the first time how phase transitions for realistic TGFT models
can be realized using Landau-Ginzburg mean-field theory. More precisely, we
consider models generating 4-dimensional Lorentzian triangulations formed by
spacelike tetrahedra whose quantum geometry is encoded in non-local degrees of
freedom on the non-compact group $\mathrm{SL}(2,\mathbb{C})$ and subject to
gauge and simplicity constraints. Further we include $\mathbb{R}$-valued
variables which may be interpreted as discretized scalar fields typically
employed as a matter reference frame. We apply the Ginzburg criterion finding
that fluctuations around the non-vanishing mean-field vacuum remain small at
large correlation lengths regardless of the combinatorics of the non-local
interaction validating the mean-field theory description of the phase
transition. This work represents a first crucial step to understand phase
transitions in compelling TGFT models for quantum gravity and paves the way for
a more complete analysis via functional renormalization group techniques.
Moreover, it supports the recent extraction of effective cosmological dynamics
from TGFTs in the context of a mean-field approximation.
