Ising model on the square M x N lattice : exact finite - size calculations

Let f , U and C represent, respectively, the free energy, the internal energy and the specific heat of the critical Ising model on the square M×N lattice with periodic boundary conditions, and f∞ represents f as M,N → ∞. We find that f , U and C can be written as: N(f −f∞) = ∑ ∞ i=1 fi/N 2i−1, U = − √ 2+ ∑ ∞ i=1 ui/N 2i−1 and C = 8 π lnN+ ∑ ∞ i=0 ci/N i andNf and U are well-defined odd function of N−1. We also find that the ratio u2i+1/c2i+1 of subdominant (N −2i−1) finite-size correction amplitudes for the internal energy and the specific heat are constant, namely, u2i+1/c2i+1 = 1/ √ 2 and u2i/c2i = 0. We obtain analytic equations for fi, ui and ci up to i = 3.