Numerical determination of the internal energies of the multiple-spin-interaction Ising model

We present a method of evaluating the internal energies in coexisting phases at a thermally driven phase transition by extrapolating finite-lattice Monte Carlo results for cumulants of the energy distributions. The method is demonstrated by determining accurately the internal energies of a two-dimensional Ising model with multiple-spin interactions in the horizontal direction and two-body interactions in the vertical direction.