= Correlation-function susceptibility sum rule
{title2=$\chi=\beta_{\rm th}\sum_rG(r)$}
When an energy Hamiltonian contains $-h\sum_n\sigma_n$, differentiating its <partition function> gives $\chi=(\beta_{\rm th}/N)\operatorname{Var}(\sum_n\sigma_n)$. Translation invariance converts this to the connected-correlation sum. A dimensionless source $\beta_{\rm th}h$ removes the explicit inverse-temperature factor. In continuum physical coordinates the lattice sum includes the site-density factor $a^{-D}$. A selected <pure thermodynamic phase> is needed below an ordered transition to avoid a macroscopic mixture contribution.
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