  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 $M\geq N/(\ln N)^{\alpha}$ 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 $N\rightarrow\infty$.
