  In the group field theory approach to quantum gravity, continuous spacetime
geometry is expected to emerge via phase transition. However, understanding the
phase diagram and finding fixed points under the renormalization group flow
remains a major challenge. In this work we tackle the issue for a tensorial
group field theory using the functional renormalization group method. We derive
the flow equation for the effective potential at any order restricting to a
subclass of tensorial interactions called cyclic melonic and projecting to a
constant field in group space. For a tensor field of rank $r$ on U$(1)$ we
explicitly calculate beta functions and find equivalence with those of O$(N)$
models but with an effective dimension flowing from $r-1$ to zero. In the $r-1$
dimensional regime, the equivalence to O$(N)$ models is modified by a tensor
specific flow of the anomalous dimension with the consequence that the
Wilson-Fisher type fixed point solution has two branches. However, due to the
flow to dimension zero, fixed points describing a transition between a broken
and unbroken phase do not persist and we find universal symmetry restoration.
To overcome this limitation, it is necessary to go beyond compact configuration
space.
