Phase transitions in tensorial group field theories: Landau-Ginzburg analysis of models with both local and non-local degrees of freedom

TL;DR

This paper investigates phase transitions in tensorial group field theories for quantum gravity, showing how non-compact groups and local degrees of freedom enable continuum spacetime emergence via critical phenomena.

Contribution

It adapts the Landau-Ginzburg approach to mixed local and non-local degrees of freedom in TGFTs, identifying conditions for phase transitions and critical dimensions.

Findings

Phase transitions are possible with non-compact groups or local degrees of freedom.

Critical dimension beyond which Gaussian fixed points exist is determined.

Mean-field theory describes the continuous phase transition in these models.

Abstract

In the tensorial group field theory approach to quantum gravity, the theory is based on discrete building blocks and continuum spacetime is expected to emerge from their collective dynamics, possibly at criticality, via a phase transition. On a compact group of fixed volume this can be expected to be only possible in a large-volume or thermodynamic limit. Here we show how phase transitions are possible in TGFTs in two cases: a) considering the non-local group degrees of freedom on a non-compact Lie group instead of a compact one (or taking a large-volume limit of a compact group); b) in models including $\mathbb{R}$-valued local degrees of freedom (that can be interpreted as discrete scalar fields, often used in this context to provide a matter reference frame). After adapting the Landau-Ginzburg approach to this setting of mixed local/non-local degrees of freedom, we determine the critical dimension beyond which there is a Gaussian fixed point and a continuous phase transition which can be described by mean-field theory. This is an important step towards the realization of a phase transition to continuum spacetime in realistic TGFT models for quantum gravity.