Critical curve of two-matrix models $ABBA$, $A\{B,A\}B$ and $ABAB$, Part I: Monte Carlo

TL;DR

This paper uses Monte Carlo simulations to estimate the critical curves of specific two-matrix models, comparing results with exact solutions and phase diagrams from the functional renormalization group.

Contribution

It provides the first Monte Carlo estimates of the convergence boundaries for three two-matrix models, validating results against known solutions and phase diagrams.

Findings

Monte Carlo estimates of critical curves agree with exact solutions

Phase diagrams from functional renormalization group are consistent with Monte Carlo results

Identifies convergence domains in the (h,g) parameter space for the models

Abstract

For a family of two-matrix models \[ \frac{1}{2} \operatorname{Tr}(A^2+B^2) - \frac{g}{4} \operatorname{Tr}(A^4+B^4) - \begin{cases} \frac{h}{2} \operatorname{Tr}( A BA B) \\ \frac{h}{4} \operatorname{Tr}( A BA B+ ABBA ) \\ \frac{h}{2} \operatorname{Tr}( A B BA ) \end{cases} \] with hermitian $A$ and $B$, we provide, in each case, a Monte Carlo estimate of the boundary of the maximal convergence domain in the $(h,g)$-plane. The results are discussed comparing with exact solutions (in agreement with the only analytically solved case) and phase diagrams obtained by means of the functional renormalization group.