Exact finite-size-scaling corrections to the critical two-dimensional Ising model on a torus

We analyse 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 (except of course for the leading log L term in the specific heat); 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.