= Correlation-function susceptibility sum rule
{title2=$\chi=\beta_{\rm th}\sum_rG(r)$}
For a classical <spin model> with <Hamiltonian> $H_h=H_0-hM$, $M=\sum_n\sigma_n$, define the static <magnetic susceptibility> per site by $\chi=N^{-1}\partial_h\langle M\rangle$ at fixed <temperature>. Differentiating its <partition function> gives
$$
\chi=\frac{\beta_{\rm th}}N\bigl(\langle M^2\rangle-\langle M\rangle^2\bigr),\qquad \beta_{\rm th}=\frac1{k_BT}.
$$
For a translation-invariant array of <spins>, write $G(r)=\langle\sigma_0\sigma_r\rangle-\langle\sigma_0\rangle\langle\sigma_r\rangle$. Expanding the <variance> of the sum then yields $\chi=\beta_{\rm th}\sum_rG(r)$. The derivative with respect to the dimensionless source $\beta_{\rm th}h$ removes the explicit inverse-<temperature> factor. Replacing the sum by an integral over physical coordinates requires the site-density factor $a^{-D}$.
The equal-time <variance> formula also holds in a quantum system when $[H_0,M]=0$. Without that commutation assumption, differentiating a <matrix exponential> instead gives
$$
\chi=\frac1N\int_0^{\beta_{\rm th}}\!d\tau\,
\langle\delta M(\tau)\delta M(0)\rangle,
\quad M(\tau)=e^{\tau H_h}Me^{-\tau H_h},\quad
\delta M=M-\langle M\rangle.
$$
This is an integrated <imaginary-time-ordered correlation function>, not generally $\beta_{\rm th}$ times the equal-time <variance>. It follows by inserting
$$
\partial_h e^{-\beta_{\rm th}H_h}
=\int_0^{\beta_{\rm th}}e^{-(\beta_{\rm th}-\tau)H_h}M e^{-\tau H_h}\,d\tau
$$
into the derivative of the normalized <expectation value> and using the cyclic <matrix trace>.
For example, a single <qubit> with $H_h=\Delta\sigma_z-h\sigma_x$, $\Delta>0$, has $\chi=\tanh(\beta_{\rm th}\Delta)/\Delta$ at $h=0$, whereas $\beta_{\rm th}\operatorname{Var}(\sigma_x)=\beta_{\rm th}$. Their difference directly exhibits the quantum noncommutation correction. Below an ordered transition, selecting a <pure thermodynamic phase> also avoids a macroscopic phase-mixture contribution to the classical sum.
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