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In the group field theory approach to quantum gravity, continuous spacetimegeometry is expected to emerge via phase transition. However, understanding thephase diagram and finding fixed points under the renormalization group flowremains a major challenge. In this work we tackle the issue for a tensorialgroup field theory using the functional renormalization group method. We derivethe flow equation for the effective potential at any order restricting to asubclass of tensorial interactions called cyclic melonic and projecting to aconstant field in group space. For a tensor field of rank $r$ on U$(1)$ weexplicitly 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 tensorspecific flow of the anomalous dimension with the consequence that theWilson-Fisher type fixed point solution has two branches. However, due to theflow to dimension zero, fixed points describing a transition between a brokenand unbroken phase do not persist and we find universal symmetry restoration.To overcome this limitation, it is necessary to go beyond compact configurationspace.

Phase transitions in TGFT: Functional renormalization group in the cyclic-melonic potential approximation and equivalence to O$(N)$ models