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

We use effective string theory (EST) to describe a toroidal 2d domain wall embedded in a 3d torus. In particular, we use TBA to 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 ${\gamma}_3/\left|{\gamma}_3^{\mathrm{min}}\right|$ = −0.82(15), which is compatible with previous estimates.