Determine the optimal torus modular parameter for extracting γ3

Determine the optimal modular parameter τ = α + i u (i.e., the aspect ratio u and twist α of the toroidal worldsheet) that maximizes the sensitivity to the leading non-universal contribution controlled by the Wilson coefficient γ3 in the effective string theory free energy of the 3d Ising domain wall, taking into account the trade-off with simulation cost and finite-volume constraints.

Background

In this work the free energy of a toroidal 2d domain wall in the 3d Ising model is computed using effective string theory (EST), separating a universal contribution from a leading non-universal correction proportional to the Wilson coefficient γ3. The authors analyze how the modular parameter τ = α + i u of the worldsheet torus influences the relative size of these contributions and thus the practicality of extracting γ3 from Monte Carlo data.

Exploration of different aspect ratios and twists indicates that while changing τ can enhance the non-universal signal relative to the universal baseline, it also increases computational cost because larger physical volumes are required to maintain control of systematics. As a result, despite this exploratory analysis, the value of τ that optimally balances signal enhancement and feasibility remains unresolved in the present paper.