Conformal universality of bulk spin and energy correlations for weakly interacting square-lattice Ising models

For sufficiently small square-symmetric finite-range even multispin perturbations of either sign, with one critical-temperature branch and two field normalizations, mixed bulk spin and energy correlations converge to the nearest-neighbor conformal limits in every bounded simply connected $C^2$ Jordan domain with free, plus or minus boundary conditions, and in the periodic plane state; energies are centered in their actual states. Companion: weakly perturbed contour energies have a domain-independent temperature at which the Dobrushin interface converges to chordal $\mathrm{SLE}_3$. Not shown: perturbations without square symmetry, multiply connected domains, boundary spin observables, or strong perturbations.

For sufficiently small square-symmetric finite-range even multispin perturbations of either sign, with one critical-temperature branch and two field normalizations, mixed bulk spin and energy correlations converge to the nearest-neighbor conformal limits in every bounded simply connected $C^2$ Jordan domain with free, plus or minus boundary conditions, and in the periodic plane state; energies are centered in their actual states. Companion: weakly perturbed contour energies have a domain-independent temperature at which the Dobrushin interface converges to chordal $\mathrm{SLE}_3$. Not shown: perturbations without square symmetry, multiply connected domains, boundary spin observables, or strong perturbations.