The magnetic susceptibility

The magnetic susceptibility $\chi$ is an example of a ``response function '', since it measures the ability of a spin to ``respond'' or flip with a change in the external magnetic field. The zero isothermal magnetic susceptibility is defined by the thermodynamic derivative

$$\chi=\lim _{H \rightarrow 0} \frac{\partial \langle M \rangle}{\partial H}.$$

The zero field susceptibility can be related to the magnetization fluctuations in the system:

$$\chi=\frac{1}{k_BT} \left( \langle M^2 \rangle - \langle M \rangle ^2 \right),$$

where $\langle M^2 \rangle$ and $\langle M \rangle ^2$ are zero field values.