Subvolume method for SU(2) Yang-Mills theory at finite temperature: topological charge distributions

We apply the previously-developed sub-volume method to study the $\theta$-dependence of the four-dimensional SU(2) Yang-Mills theory at finite temperature. We calculate the first two coefficients, the topological susceptibility $\chi$ and the fourth cumulant $b_2$, in the $\theta$-expansion of the free energy density around the critical temperature ($T_c$) for the confinement-deconfinement transition. Lattice calculations are performed with three different spatial sizes ${24}^3,{32}^3,{48}^3$ to monitor finite size effects, while the temporal size is fixed to be $8$. The systematic uncertainty associated with the sub-volume extrapolation is studied with special care. The sub-volume method allows us to determine the values of $b_2$ much more accurately than the standard full-volume method, and we successfully identify the temperature dependence of $b_2$ around $T_c$. Our numerical results suggest that the $\theta$-dependence of the free energy density near $\theta =0$ changes from $4\chi (1-\cos (\theta \mathrm{/}2))$ to $\chi (1-\cos \theta )$ as the temperature crosses $T_c$.