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

TL;DR

This paper critiques a recent phase diagram derivation for a two-matrix model, highlighting inconsistencies in the renormalization group approach and proposing a simple truncation to reconcile the results.

Contribution

It provides a consistent set of beta-functions and identifies the specific ensemble where the previous phase diagram is accurate.

Findings

The phase diagram's beta-function lacks one-loop Wetterich-Morris structure.

A simple truncation can reproduce the phase diagram.

The correct ensemble for the phase diagram is identified.

Abstract

Recently, [JHEP 20 131 (2020)] obtained (a similar, scaled version of) the ($a,b$)-phase diagram derived from the Kazakov--Zinn-Justin solution of the Hermitian two-matrix model with interactions \[\mathrm{Tr\,}\Big\{\frac{a}{4} (A^4+B^4)+\frac{b}{2} ABAB\Big\}\,,\] starting from Functional Renormalization. We comment on something unexpected: the phase diagram of [JHEP 20 131 (2020)] is based on a $\beta_b$-function that does not have the one-loop structure of the Wetterich-Morris Equation. This raises the question of how to reproduce the phase diagram from a set of $\beta$-functions that is, in its totality, consistent with Functional Renormalization. A non-minimalist, yet simple truncation that could lead to the phase diagram is provided. Additionally, we identify the ensemble for which the result of op. cit. would be entirely correct.