(No) phase transition in tensorial group field theory

TL;DR

This paper investigates the phase transition behavior in tensorial group field theories using a local-potential approximation, finding no evidence of phase transition due to zero modes, implying non-compact degrees of freedom are needed for continuum spacetime emergence.

Contribution

It introduces a local-potential approximation for tensorial group field theories, providing a comprehensive RG flow analysis and clarifying the absence of phase transitions in compact models.

Findings

No phase transition observed in tensorial group field theories on compact domains.

Equivalence with scalar field theories at different effective dimensions depending on scale.

Zero modes prevent symmetry breaking, implying the need for non-compact models.

Abstract

Continuum spacetime is expected to emerge via phase transition in discrete approaches to quantum gravity. A promising example is tensorial group field theory but its phase diagram remains an open issue. The results of recent attempts in terms of the functional renormalization group method remain inconclusive since they are restricted to truncations of low order. We overcome this barrier with a local-potential approximation for $\textrm{U}(1)$ tensor fields at arbitrary rank $r$ focusing on a specific class of so-called cyclic-melonic interactions. Projecting onto constant field configurations, we obtain the full set of renormalization-group flow equations. At large cut-offs we find equivalence with $r-1$ dimensional $\textrm{O}(N)$ scalar field theory in the large-$N$ limit, modified by a tensor-specific, relatively large anomalous dimension. However, on small length scales there is equivalence with the corresponding scalar field theory with vanishing dimension and, thus, no phase transition. This is confirmed by numerical analysis of the full non-autonomous equations where we always find symmetry restoration. The essential reason for this effect are isolated zero modes. This result should therefore be true for tensor field theories on any compact domain and including any tensor-invariant interactions. Thus, group field theories with non-compact degrees of freedom will be necessary to describe a phase transition to continuum spacetime.