Title: Direct Monte Carlo Computation of the 't~Hooft Partition Function Show Chinese title

Title: 直接蒙特卡罗计算't~霍夫特划分函数

Abstract: The 't~Hooft partition function~$\mathcal{Z}_{\text{tH}}[E;B]$ of an $SU(N)$ gauge theory with the $\mathbb{Z}_N$ 1-form symmetry is defined as the Fourier transform of the partition function~$\mathcal{Z}[B]$ with respect to the spatial-temporal components of the 't~Hooft flux~$B$. Its large volume behavior detects the quantum phase of the system. When the integrand of the functional integral is real-positive, the latter partition function~$\mathcal{Z}[B]$ can be numerically computed by a Monte Carlo simulation of the $SU(N)/\mathbb{Z}_N$ gauge theory, just by counting the number of configurations of a specific 't~Hooft flux~$B$. We carry out this program for the $SU(2)$ pure Yang--Mills theory with the vanishing $\theta$-angle by employing a newly-developed hybrid Monte Carlo (HMC) algorithm (the halfway HMC) for the $SU(N)/\mathbb{Z}_N$ gauge theory. The numerical result clearly shows that all non-electric fluxes are ``light'' as expected in the ordinary confining phase with the monopole condensate. Invoking the Witten effect on~$\mathcal{Z}_{\text{tH}}[E;B]$, this also indicates the oblique confinement at~$\theta=2\pi$ with the dyon condensate. Show Chinese abstract

Abstract: 具有$\mathbb{Z}_N$1-形式对称性的$SU(N)$规范理论的't Hooft 分区函数$\mathcal{Z}_{\text{tH}}[E;B]$被定义为关于't Hooft 通量$B$的时空分量的傅里叶变换分区函数$\mathcal{Z}[B]$。 其大体积行为检测系统的量子相位。 当泛函积分的被积函数为实正值时，后者（即分区函数~$\mathcal{Z}[B]$）可以通过对$SU(N)/\mathbb{Z}_N$规范理论的蒙特卡洛模拟来数值计算，只需统计特定't~Hooft通量~$B$的构型数量即可。 我们通过采用一种新开发的混合蒙特卡洛（HMC）算法（半程HMC）对$SU(N)/\mathbb{Z}_N$规范理论执行了这个程序，研究了具有零$\theta$-角项的$SU(2)$纯杨-米尔斯理论。数值结果显示，正如预期，在单极凝聚的普通 confinement 相中，所有非电通量都是“轻”的。 根据Witten效应在~$\mathcal{Z}_{\text{tH}}[E;B]$的体现，这也表明了在~$\theta=2\pi$存在磁单极子凝聚下的斜 confinement。