Direct Monte Carlo Computation of the 't~Hooft Partition Function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908402933825536 |
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| author | Morikawa, Okuto Suzuki, Hiroshi |
| author_facet | Morikawa, Okuto Suzuki, Hiroshi |
| contents | 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 $θ$-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~$θ=2π$ with the dyon condensate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07042 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Direct Monte Carlo Computation of the 't~Hooft Partition Function Morikawa, Okuto Suzuki, Hiroshi High Energy Physics - Lattice 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 $θ$-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~$θ=2π$ with the dyon condensate. |
| title | Direct Monte Carlo Computation of the 't~Hooft Partition Function |
| topic | High Energy Physics - Lattice |
| url | https://arxiv.org/abs/2501.07042 |