Phase structure of lattice Yang-Mills theory on ${\mathbb T}^2 \times {\mathbb R}^2$

We study properties of SU(2) Yang-Mills theory on a four-dimensional Euclidean spacetime in which two directions are compactified into a finite two-dimensional torus ${\mathbb T}^2$ while two others constitute a large ${\mathbb R}^2$ subspace. This Euclidean ${\mathbb T}^2 \times {\mathbb R}^2$ manifold corresponds simultaneously to two systems in a (3+1) dimensional Minkowski spacetime: a zero-temperature theory with two compactified spatial dimensions and a finite-temperature theory with one compactified spatial dimension. Using numerical lattice simulations we show that the model exhibits two phase transitions related to the breaking of center symmetries along the compactified directions. We find that at zero temperature the transition lines cross each other and form the Greek letter $\gamma$ in the phase space parametrized by the lengths of two compactified spatial dimensions. There are four different phases. We also demonstrate that the compactification of only one spatial dimension enhances the confinement property and, consequently, increases the critical deconfinement temperature.