Direct Monte Carlo Computation of the 't~Hooft Partition Function

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.

• The paper introduces a novel halfway hybrid Monte Carlo algorithm that directly computes partition function ratios for 't Hooft fluxes.
• It distinguishes 'light' and 'heavy' fluxes in SU(2) Yang-Mills theory, confirming theoretical predictions in the confining phase.
• The study validates duality and reflection positivity, paving the way for probing extended gauge theories and quantum phase transitions.

Direct Monte Carlo Computation of the 't~Hooft Partition Function

This paper introduces and executes a numerical approach to calculating the 't~Hooft partition function, $\mathcal{Z}_{\text{tH}[E;B]}$, within $SU(N)$ gauge theories, emphasizing the case for $SU(2)$. The 't~Hooft partition function is pivotal for exploring quantum phases, being defined as the Fourier transform of the partition function $\mathcal{Z}[B]$ concerning 't~Hooft flux $B$.

Key Methodologies and Findings

The core methodology revolves around Monte Carlo simulations, crucially employing a novel hybrid Monte Carlo (HMC) algorithm called the "halfway HMC." This technique facilitates explicit treatment of the 't~Hooft flux as a dynamic variable, enabling direct computation of the partition function ratios $\mathcal{Z}[B]/\mathcal{Z}[0]$ by counting configurations of specific 't~Hooft fluxes. The authors demonstrate this through $SU(2)$ pure Yang-Mills theory simulations, revealing numerical results that classify 't~Hooft fluxes into "light" or "heavy" categories, consistent with theoretical expectations in the confinement phase.

Numerical Insights and Theoretical Implications

The numerical results, obtained via detailed Monte Carlo simulations, show a distinct classification of fluxes. Specifically, all non-electric fluxes are characterized as "light," indicative of the ordinary confining phase associated with monopole condensation. This aligns with the theoretical framework that anticipates light fluxes yielding $$\mathcal{Z}_{\text{tH}[E;B]}/\mathcal{Z}_{\text{tH}[E=0;B=0]} \sim 1$$, whereas heavy fluxes lean towards zero. Additionally, the satisfactory compliance with the duality equation and the reflection positivity are significant, reflecting the consistency and reliability of the results.

Explorations were further extended by incorporating $\theta$-terms to study $\theta = 2\pi$, leveraging the Witten effect. Fluxes conforming to relations including $E_i$ and $B_{ij}$ manifest signs of oblique confinement at $\theta = 2\pi$, attributed to dyons' condensation.

Prospects and Future Directions

While the study commendably implements a direct numerical strategy for the 't~Hooft partition function, it chiefly provides a foundational basis for further exploration across diverse $SU(N)$ gauge theories incorporating $\mathbb{Z}_N$ symmetries. One prospect is the examination of finite temperature scenarios that could divulge insights into phase transitions between confining and deconfining states. Furthermore, extensions to matter-coupled theories, including adjoint scalars or fermions, promise richer phase structures ripe for examination through this computational lens.

Finally, the study's techniques possess potential for broader applicability, including systems where symmetry operators act on diverse topological sectors, providing valuable insights into the interplay between gauge symmetries and quantum phases.

Conclusion

The paper succeeds in showcasing a direct approach to computing $\mathcal{Z}_{\text{tH}[E;B]}$, corroborated by numerical results consistent with theoretical predictions for pure Yang-Mills theories. As computational methods evolve, integrating these insights with ongoing theoretical advancements promises breakthroughs in understanding gauge theories' quantum characteristics. The implications of such studies extend beyond theoretical physics, offering potential insights for fields engaging with complex quantum system simulations.