Numerical determination of the width and shape of the effective string using Stochastic Normalizing Flows

TL;DR

This paper develops and validates a physics-informed Stochastic Normalizing Flow (PI-SNF) framework to numerically study Effective String Theory on the lattice. By using a free-boson prior and a linear $1/\sigma$-driven protocol through the Nambu-Gotò action, the method accurately reproduces the NG partition function and enables precise measurement of the string width, as well as systematic exploration of higher-order terms $\mathcal{K}^2$ and $\mathcal{K}^4$ and their impact on flux-tube profiles. The results confirm a conjectured NG-based width scaling in the high-temperature regime, quantify the strong influence of rigidity terms on width, and demonstrate how the Binder cumulant responds to different action terms, with boundary effects playing a significant role. Overall, the study establishes SNFs as a reliable and scalable tool for lattice EST simulations, capable of addressing observables beyond analytical reach and guiding future investigations into more complex EST constructions and confinement mechanisms.

Abstract

Flow-based architectures have recently proved to be an efficient tool for numerical simulations of Effective String Theories regularized on the lattice that otherwise cannot be efficiently sampled by standard Monte Carlo methods. In this work we use Stochastic Normalizing Flows, a state-of-the-art deep learning architecture based on non-equilibrium Monte Carlo simulations, to study different effective string models. After testing the reliability of this approach through a comparison with exact results for the Nambu-Gotō model, we discuss results on observables that are challenging to study analytically, such as the width of the string and the shape of the flux density. Furthermore, we perform a novel numerical study of Effective String Theories with terms beyond the Nambu-Gotō action, including a broader discussion on their significance for lattice gauge theories. The combination of these findings enables a quantitative description of the fine details of the confinement mechanism in different lattice gauge theories. The results presented in this work establish the reliability and feasibility of flow-based samplers for Effective String Theories and pave the way for future applications on more complex models.

Figures (11)

• Figure 1: Schematic representation of a lattice configuration: cyan and magenta sites represent the active volume of the lattice, while yellow sites represent the Dirichlet boundaries where the field is fixed to $0$; the boundary in $\epsilon=R$ is considered as a part of the lattice. The width $\sigma w^2$ of eq. \ref{['eq:LatticeWidth']} is computed averaging only on the magenta sites.
• Figure 2: Results for the coefficient $a(L)$ of the partition function of the lattice Nambu-Gotō model, for various values of $L$ and for $\sigma=1/30$ (left panel) and $\sigma=1/50$ (right panel). The expected behaviour for the Free Boson and the Nambu-Gotō actions are also shown.
• Figure 3: Results for the numerical prediction of the term $\sigma/\sigma(L)$, which corresponds to the coefficient $f(L)$ of the string width for $\sigma = 1/10$, after removing the divergent term $f_1$ and multiplying by $4L$. The solid curve corresponds to the true value from eq. \ref{['eq:SigmaL']}.
• Figure 4: Results for the width $\sigma w^2(R,L,\gamma_2)$ at high temperature ($R \gg L=20$, left panel) and at low temperature ($R \ll L=80$, right panel). The $\gamma_2=0$ case is equivalent to the Nambu-Gotō action.
• Figure 5: Results for $a^{(\mathrm{HT})}(\gamma_2)$ and $a^{(\mathrm{LT})}(\gamma_2)$ (left panel), and for $R_c(\gamma_2)$ (right panel).