Phase transitions in TGFT: a Landau-Ginzburg analysis of Lorentzian quantum geometric models

TL;DR

This paper demonstrates how phase transitions in tensorial group field theories, relevant for quantum gravity, can be analyzed using Landau-Ginzburg mean-field theory, validating the emergence of continuum spacetime from discrete quantum geometric building blocks.

Contribution

First application of Landau-Ginzburg mean-field theory to realistic Lorentzian TGFT models with non-local interactions and scalar fields, advancing understanding of phase transitions in quantum gravity.

Findings

Fluctuations remain small at large correlation lengths, validating mean-field approximation.

Phase transitions can be realized in models with non-local interactions on SL(2,C).

Supports effective cosmological dynamics derived from TGFTs.

Abstract

In the tensorial group field theory (TGFT) approach to quantum gravity, the basic quanta of the theory correspond to discrete building blocks of geometry. It is expected that their collective dynamics gives rise to continuum spacetime at a coarse grained level, via a process involving a phase transition. In this work we show for the first time how phase transitions for realistic TGFT models can be realized using Landau-Ginzburg mean-field theory. More precisely, we consider models generating 4-dimensional Lorentzian triangulations formed by spacelike tetrahedra whose quantum geometry is encoded in non-local degrees of freedom on the non-compact group $\mathrm{SL}(2,\mathbb{C})$ and subject to gauge and simplicity constraints. Further we include $\mathbb{R}$-valued variables which may be interpreted as discretized scalar fields typically employed as a matter reference frame. We apply the Ginzburg criterion finding that fluctuations around the non-vanishing mean-field vacuum remain small at large correlation lengths regardless of the combinatorics of the non-local interaction validating the mean-field theory description of the phase transition. This work represents a first crucial step to understand phase transitions in compelling TGFT models for quantum gravity and paves the way for a more complete analysis via functional renormalization group techniques. Moreover, it supports the recent extraction of effective cosmological dynamics from TGFTs in the context of a mean-field approximation.