I want to know why the magnetic susceptibility in zero external field is :
(the fluctuations in magnetization)
$$ \chi = \frac{1}{k_bT} \left( \langle M^2 \rangle - \langle M \rangle^2 \right)$$
can any one give a derivation?
Second question(if any one is familiar): I want to calculate magnetic susceptibility in The Ising Model using Wolff Cluster algorithm, this is the formula I should use?!!
I don't know about Wolf's cluster algorithm. Here is an answer to your first question.
Define partition function in the presence of external magnetic field (source field-$h$) as, $$Z[h]=\sum_{\{\sigma_{i}^{}\}\in\text{Phase space}}^{} e^{-\beta\left[H[\{\sigma_{i}\};\{J_{ij}\}]-h \hat{M}\right]},$$ where $H[\{\sigma_{i}\};\{J_{ij}\}]$ is a generic (classical) Hamiltonian model of a magnet with magnetic moments, $\{\sigma_{i}\}$ on the lattice interacting with each other by generic exchange couplings $\{J_{ij}\}$, and the total magnetic moment (defined as $\hat{M}=\sum_{i\in \text{Lattice site}}^{}\sigma_{i}^{}$) is coupled to an external magnetic field, $h$.
Average of the total magnetic moment (in the presence of external magnetic field) can be calculated using
$$M[h]=\langle\hat{M}\rangle=\frac{1}{\beta}\frac{\partial}{\partial h}\ln[Z[h]],$$
where $\langle \hat{O} \rangle=\frac{}{Z[h]}\sum_{\{\sigma_{i}\}\in \text{Phase space}}^{} \hat{O} e^{-\beta\left[H[\{\sigma_{i}\};\{J_{ij}\}]-h \hat{M}\right]}.$
As the magnetic susceptibility (at the zero external magnetic field) is defined as $$\chi=\frac{\partial M[h]}{\partial h}\Big|_{h \to 0}^{}.$$
Use the above, $$\begin{eqnarray} \chi&=&\frac{\partial M[h]}{\partial h}\Big|_{h \to 0}^{}\\ &=&\beta[\langle\hat{M}^2_{}\rangle-\langle\hat{M}\rangle_{}^2].\end{eqnarray}$$
