A CLT for the total energy of the two-dimensional critical Ising model

Consider the Ising model on ([1, 2N] x [1, 2M])boolean AND Z(2) at critical temperature with periodic boundary condition in the horizontal direction and free boundary condition in the vertical direction. Let E-M,E- (N) be its total energy (or Hamiltonian). Suppose M is a function of N satisfying M >= N=(ln N)(alpha) for some alpha is an element of [0, 1). In particular, one may take M = N. We prove that E-M,E- (N) + 4 root 2M N - (4/pi)N ln N/root(32/pi)M N ln N converges weakly to a standard Gaussian distribution as N -> infinity.