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So for a quantum spin chain, one can easily prove via the partition function that you have a fluctuation-dissipation type relation between the magnetic susceptibility and the variance of the magnetization in cases where the Hamiltonian commutes with the external field.

For example, if I have a Heisenberg Hamiltonian in a transverse field, I can show

$$\frac1\beta\frac{d\langle M\rangle}{dh}=\langle M^2\rangle-\langle M\rangle^2$$

Where $M$ is the magnetization operator, and $\beta$ is the inverse temperature.

My question is whether this relationship should be expected to hold in the thermodynamic limit, i.e. if I let $T\to0$ and the length of the chain go to infinity. The usual derivation would seems to break down.

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