Correlation-function susceptibility sum rule ($\chi ={\beta }_{\mathrm{th}}{\sum }_{r}G(r)$)

When an energy Hamiltonian contains $-h{\sum }_n{\sigma }_n$, differentiating its partition function gives $\chi =({\beta }_{\mathrm{th}}/N)\mathrm{Var}({\sum }_n{\sigma }_n)$. Translation invariance converts this to the connected-correlation sum. A dimensionless source ${\beta }_{\mathrm{th}}h$ 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.