Exact Finite-Size-Scaling Corrections to the Critical Two-Dimensional Ising Model on a Torus

We analyze the finite-size corrections to the energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a torus. We extend the analysis of Ferdinand and Fisher to compute the correction of order L−3 to the energy and the corrections of order L−2 and L−3 to the specific heat. We also obtain general results on the form of the finite-size corrections to these quantities: only integer powers of L−1 occur, unmodified by logarithms; and the energy expansion contains only odd powers of L−1. In the specific-heat expansion any power of L−1 can appear, but the coefficients of the odd powers are proportional to the corresponding coefficients of the energy expansion.