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In the tensorial group field theory approach to quantum gravity, the theoryis based on discrete building blocks and continuum spacetime is expected toemerge from their collective dynamics, possibly at criticality, via a phasetransition. On a compact group of fixed volume this can be expected to be onlypossible in a large-volume or thermodynamic limit. Here we show how phasetransitions are possible in TGFTs in two cases: a) considering the non-localgroup 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 asdiscrete scalar fields, often used in this context to provide a matterreference frame). After adapting the Landau-Ginzburg approach to this settingof mixed local/non-local degrees of freedom, we determine the criticaldimension beyond which there is a Gaussian fixed point and a continuous phasetransition which can be described by mean-field theory. This is an importantstep towards the realization of a phase transition to continuum spacetime inrealistic TGFT models for quantum gravity.

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