The phase diagram of the multi-matrix model with ABAB interaction from functional renormalization

Abstract

At criticality, discrete quantum-gravity models are expected to give rise to continuum spacetime. Recent progress has established the functional renormalization group method in the context of such models as a practical tool to study their critical properties and to chart their phase diagrams. Here, we apply these techniques to the multi-matrix model with ABAB interaction potentially relevant for Lorentzian quantum gravity in 3 dimensions. We characterize the fixed-point structure and phase diagram of this model, paving the way for functional RG studies of more general multi-matrix or tensor models encoding causality and subjecting the technique to another strong test of its performance in discrete quantum gravity by comparing to known results.

Comment on “The phase diagram of the multi-matrix model with ABAB-interaction from functional renormalization”

“…A consistent phase diagram with trapezoidal form (i.e. predicting « 1{10 for both critical exact a ‹ " b ‹ " 1{4π values) is one of the main results presented in [1], who addressed the model (2.1) using Functional Renormalization. The phase diagram [1, figure 1] obtained from pβ a , β b q follows from a correct expression for β a but also from β b "`η A`ηB`1˘b´2 5 rp5´η A q`p5´η B qs b 2 , which is [1, eq.…”

“…The two-matrix model in question has the following partition function 2 4] a phase diagram of right-angled trapezoidal form for the couplings (a, b), called there (α, β) as well as in [1]. A consistent phase diagram with trapezoidal form (i.e.…”

“…This makes some of the graphs listed above uninteresting from the viewpoint of renormalization, as they do not have a oneloop structure. Those which do, have also some drawbacks: (2.9) has a one-loop structure, and is a graph generated by the RG-flow in the ABAB-model, but that sextic vertex is set to zero in the truncation that [1] considers. On the other hand, the graph (2.12) is a Feynman graph containing only vertices of the ABAB-model, but it leads to a double-trace, and so on.…”

“…We showed that the connection that [1] established between the Functional Renormalization Group (FRG) and a phase diagram -identified there with [2, figure 4]-relies on a β-function that does not have the one-loop structure of the Functional Renormalization equation. In section 3 above, we proposed to extend the minimalist truncation of [1] in order to find a FRG-compatible set of β-functions. Accomplishing this proposal would provide a sound 13 bridge, in the intention of [1], between the FRG and Causal Dynamical Triangulations [23] through the ABAB-model [24,25].…”

“…In section 3 above, we proposed to extend the minimalist truncation of [1] in order to find a FRG-compatible set of β-functions. Accomplishing this proposal would provide a sound 13 bridge, in the intention of [1], between the FRG and Causal Dynamical Triangulations [23] through the ABAB-model [24,25]. Finally, we provided in Remark 3.1 the condition one would need to add in order for the β b -function given by [1, eq.…”