Effective string theory on a torus: the 3d Ising domain wall

TL;DR

The paper develops and tests an effective string theory description of a toroidal domain wall in a 3D torus, deriving the universal GGRT contribution and the leading non-universal correction controlled by the Wilson coefficient $\gamma_3$. Using high-precision flat-histogram Monte Carlo simulations of the 3D Ising model with anti-periodic boundary conditions, the authors fit the EST predictions to obtain a precise value for $\gamma_3$, finding $\gamma_3/|\gamma_3^{\text{min}}| = -0.82(15)$ (i.e., $\gamma_3 = -0.00106(18)$ in their normalization). They also validate the area-independent normalization, explore the modular dependence of the non-universal terms, and demonstrate that finite-transverse-volume effects reveal couplings between the domain wall and bulk excitations. Overall, the work extends EST to toroidal domain walls, provides a robust numerical determination of $\gamma_3$, and highlights finite-volume effects that motivate further study of wall–bulk interactions in the 3D Ising model.

Abstract

We use effective string theory (EST) to describe a toroidal 2d domain wall embedded in a 3d torus. In particular, we compute the free energy of the domain wall in an expansion in inverse powers of the area, up to the second non-universal order that involves the Wilson coefficient $γ_3$. In order to test our predictions, we simulate the 3d Ising model with anti-periodic boundary conditions, using a two-step flat-histogram Monte Carlo method in an ensemble over the boundary coupling $J$ that delivers high-precision free energy data. The predictions from EST reproduce the lattice results with only two adjustable parameters: the string tension, $1/\ell_s^2$, and $γ_3$. We find $γ_3 /|γ_3^{\text{min}}|= -0.82(15)$, which is compatible with previous estimates.

Figures (12)

• Figure 1: Schematic representation of the domain wall/worldsheet on a three-dimensional torus. For 3d Ising, the boundary conditions are periodic along $\xi_1$ and $\xi_2$ and anti-periodic along $z$.
• Figure 2: Comparison between the numerical evaluation of eq. \ref{['eq:partition function rectangular']}, dashed line, and the expansion in eq. \ref{['eq:free energy']}, truncated at order $1/\mathcal{A}^{n}$, full lines. $Z_0^\prime \equiv e^{-\mathcal{A}} \left(\frac{\sigma L_z^2}{2\pi u} \right)^{\frac{1}{2}}Z_0$. The shaded region denotes the parameter range used to extract the value of $\gamma_3$.
• Figure 3: Comparison of the exact $F_{\gamma_3}$, eq. \ref{['def:Fg3']}, with its asymptotic expansion, eq. \ref{['eq:free energy perturbatively in gamma3']}. The gray region marks the small $\mathcal{A}$ range relevant to our numerics, where the two differ by as much as a factor of two. Left of the zero, the 1st-order prediction has the wrong sign.
• Figure 4: Dependence of the ratio $F_{\gamma_3}/F_\text{U}$ between the leading non-universal correction proportional to $\gamma_3$ and the universal free-energy, as a function of the modular parameter of the torus for 3 representative areas. The red region for ${\cal A}=2$ is beyond the Hagedorn phase transition due to the smallest size when we change the aspect ratio at fixed area. The ratio is normalized by the value at $\tau=i$.
• Figure 5: Discrete (left) and continuous (right) contribution to $\omega(J)$ for square domain walls of size $64^2$ and $\beta=0.23$, with 512 bins.