We propose a subvolume method to study the θ dependence of the free energy density of the four-dimensional SU(N) Yang-Mills theory on the lattice. As an attempt, the method is first applied to SU(2) Yang-Mills theory at $T=1.2T_c$ to understand the systematics of the method. We then proceed to the calculation of the vacuum energy density and obtain the θ dependence qualitatively different from the high temperature case. The numerical results combined with the theoretical requirements provide the evidence for the spontaneous CP violation at $\theta =\pi$, which is in accordance with the large N prediction and indicates that the similarity between 4d SU(N) and 2d CP${}^{N-1}$ theories does not hold for $N=2$.

我们提出了一种子体积方法来研究四维 SU( N ) Yang-Mills 理论的自由能密度对晶格的θ依赖性。作为一种尝试，该方法首先应用于 SU(2) Yang-Mills 理论$吨=1.2{吨}_C$了解该方法的系统性。然后，我们继续计算真空能量密度，并获得与高温情况下性质不同的θ依赖性。数值结果结合理论要求为自发CP违反提供了证据$\theta =\pi$，这与大N预测一致，表明 4d SU( N ) 和 2d CP之间的相似性${}^{N-1}$ 理论不成立 $N=2$.