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

Consider the Ising model on $([1,2N]\times [1,2M])\cap {\mathbb{Z}}^2$ at critical temperature with periodic boundary condition in the horizontal direction and free boundary condition in the vertical direction. Let $E_{M,N}$ be its total energy (or Hamiltonian). Suppose $M$ is a function of $N$ satisfying $M\ge N\mathrm{/}(\ln N{)}^{\alpha }$ for some $\alpha \in [0,1)$. In particular, one may take $M=N$. We prove that \begin{equation*} \frac{E_{M,N}+4\sqrt{2}M N-(4/\pi)N\ln N}{\sqrt{(32/\pi)MN\ln N}} \end{equation*} converges weakly to a standard Gaussian distribution as $N\to \mathrm{\infty }$.