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.

Figure 1

Expectations values of Polyakov loops (12) in various directions at different lattice geometries (the detailed description is given in the text).

Figure 2

Examples of the susceptibilities of the Polyakov lines (13) defined along the compactified directions $\mu =1$ and $\mu =2$ of unequal length ($N_1=4$ and $N_2=6$, respectively).

Figure 3

The critical lattice couplings associated with $\mu =1$ (top) and $\mu =2$ (bottom) compactified dimensions on $N_1\times N_2\times 32^2$ lattice as a function of the lattice extension $N_2$ at the fixed sets of the lattice extension $N_1$ in the other direction. The dashed lines of are shown to guide the eye.

Figure 4

Lattice spacing $a$ expressed via the physical string tension $\sigma$ and the lattice coupling $\beta$. The data points are taken from Ref. [25]. The dashed line corresponds to a spline interpolation of the data, while the solid line is given by the two-loop renormalization group relation (14) and Eq. (4).

Figure 5

The phase diagram of the zero-temperature $SU(2)$ Yang-Mills theory with two compactified spatial dimensions $L_1$ and $L_2$. The colored arrows point to the asymptotic critical lengths (15) of compactification transition related to a finite-temperature phase transition (6). The colored points correspond to the numerical data, the dashed blue line (the dot-dashed red line) is the $L_1$ ($L_2$) compactification transition. A detailed description is given in the text.

Figure 6

The natural logarithm of the gluon anisotropy (16) vs the length of the compactified direction $L_1$ for fixed ratios between the compactification lengths $L_1/L_2=0.8$. All three representative phases are shown. The notation for the labels “$\pm \mu \nu$” is as follows: “$\pm$” implies a positive or negative value of the anisotropy $\delta F_{\mu \nu }^2$ and $\mu =\perp$ denotes the long direction, either $\mu =3$ or $\mu =4$. The lower figure is a zoom in on the I–II transition of the whole plot (the upper figure). The vertical dot-dashed red and dashed blue lines mark the positions of the phase transition lines of Fig. 5.

Figure 7

The phase diagram of finite-temperature $SU(2)$ Yang-Mills theory with one compactified spatial dimension of the length $L$. The thick red line is the deconfinement phase transition and the thin blue line represents the critical line of the breaking of the global ${\mathbb{Z}}_2$ symmetry associated with the compactified spatial dimension. The red- and blue-shaded areas are the confinement and $\mathrm{spatial}\text{−}{\mathbb{Z}}_2$ restored phases, respectively. The critical temperature (6) and the critical compactified length (15) are shown by the arrows.