Analytic verification of the droplet picture in the two-dimensional Ising model

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It is widely accepted that the free energy F(H) of the two-dimensional Ising model in the ferromagnetic phase T<Tc has an essential branch cut singularity at the origin H=0. The phenomenological droplet theory predicts that near the cut drawn along the negative real axis H<0, the imaginary part of the free energy per lattice site has the form Im F[exp (± iπ|H|)] = ± B |H| exp (-A/|H|) for small |H|. We verify this prediction in analytical perturbative transfer matrix calculations for the square lattice Ising model for all temperatures 0<T<Tc and arbitrary anisotropy ratio J₁/J₂. We obtain an expression for the constant A which coincides exactly with the prediction of the droplet theory. For the amplitude B we obtain B =πM/18, where M is the equilibrium spontaneous magnetization. In addition we find discrete-lattice corrections to the above mentioned phenomenological formula for ImF, which oscillate in H⁻¹.