Anisotropic pressure and novel first-order phase transition in $SU(3)$ Yang-Mills theory on $\mathbb{T}^2\times\mathbb{R}^2$

We investigate the thermodynamic behavior and phase diagram of $SU(3)$ Yang-Mills theory on $\mathbb{T}^2 \times \mathbb{R}^2$ in Euclidean spacetime using an effective model. In our approach, the Polyakov loops along the compactified directions are treated as dynamic variables, and the model is calibrated to match lattice simulation results for thermodynamic observables on $\mathbb{T}^2 \times \mathbb{R}^2$. Our analysis reveals a novel first-order phase transition in the deconfined phase that ends at critical points, which appear to belong to the two-dimensional $Z_2$ universality class. This transition is driven by the interplay between the two Polyakov loops, introduced via a cross-term in the Polyakov-loop potential.