Monte Carlo reconstruction of symmetry-twisted partition function ratios: the critical 3D Ising

We introduce a Monte Carlo strategy for directly estimating partition function ratios between distinct global sectors of a lattice theory. It enlarges the configuration space to sample an interpolating family whose endpoints are the desired sectors, and uses flat histogram methods to reconstruct the corresponding free energy difference. Although the construction is more general, we focus here on the three-dimensional Ising model on the slab $\mathbb{R}^{2}\times S^{1}_{L_{z}}$ at the bulk critical point, comparing the untwisted periodic sector with the $\mathbb{Z}_{2}$-twisted antiperiodic sector. A large-volume and aspect ratio extrapolation gives the symmetry-twisted thermodynamic Casimir difference $\Delta_{\mathbb{Z}_{2}}=0.327(2)$ directly, without lattice derivatives or bulk subtractions. This provides an independent twisted sector probe of tensions observed in periodic sector thermodynamic Casimir observables. More generally, the method gives direct but selective numerical access to CFT compactification data, including estimates of the effective thermal screening scale and the $\mathbb{Z}_{2}$-odd sector energy gap on $T^{2}$.