Large-$N$ Dynamics of a QCD-Inspired Unitary Matrix Model

TL;DR

This paper analyzes the large-N behavior of QCD-inspired unitary matrix models, revealing phase transitions and spectral properties in real and complex action regimes.

Contribution

It provides analytic and numerical solutions for spectral density, Wilson loops, and free energy in different phases, highlighting phase transition orders.

Findings

Spectral density and free energy match low-temperature QCD behaviour.

Identifies a third-order phase transition at zero chemical potential.

Finds a continuous phase transition at finite chemical potential.

Abstract

We study the large-$N$ limit of $U(N)$ and $SU(N)$ unitary matrix models inspired by QCD. The model is analyzed in two cases: $\mu = 0$, where the potential is real, and finite $\mu$, where it becomes complex. The complex action drives the eigenvalues into the complex plane, leading to $\langle U \rangle \neq \langle U^{-1} \rangle$. In the ungapped phase, we obtain analytic expressions for the spectral density, Wilson loops, and free energy, which reproduce the low-temperature behaviour of QCD. In contrast, the gapped phase involves a nontrivial resolvent and is solved partially analytically and numerically. At $\mu=0$, the model exhibits a $3^{rd}$ order phase transition, while at finite $\mu$, it shows a continuous phase transition of at least second order.