Phase structure of the generalized two dimensional Yang-Mills theories on sphere

TL;DR

This paper analyzes the phase structure of generalized two-dimensional Yang-Mills theories on a sphere, deriving free energy expressions, identifying phase transitions, and exploring models with polynomial potentials at large N.

Contribution

It provides a general expression for free energy in $G()=^{2k}$ models and investigates their phase transitions and density functions at large N.

Findings

Identified a third order phase transition in the $^6$ model.

Described the density function as a three-cut problem.

Analyzed phase structure of combined $^2 + g ^4$ models.

Abstract

We find a general expression for the free energy of $G(\phi)=\phi^{2k}$ generalized 2D Yang-Mills theories in the strong ($A>A_c$) region at large $N$. We also show that in this region, the density function of Young tableau of these models is a three-cut problem. In the specific $\phi^6$ model, we show that the theory has a third order phase transition, like $\phi^2$ (YM_2) and $\phi^4$ models. We have some comments for $k \geq 4$ cases. At the end, we study the phase structure of $\phi^2 + g \phi^4$ model for $g \leq A/4$ region.