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

TL;DR

This paper uses the functional renormalization group to analyze phase transitions in tensorial group field theories, revealing an equivalence to O(N) models with a flowing effective dimension and universal symmetry restoration.

Contribution

It derives the flow equations for cyclic melonic tensorial interactions and demonstrates their equivalence to O(N) models with a dynamic effective dimension.

Findings

Beta functions match those of O(N) models with flowing dimension.

Fixed points are absent in the zero-dimensional limit, indicating symmetry restoration.

The effective dimension flows from r-1 to zero, affecting the phase structure.

Abstract

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.