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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 $\chi =N^{-1}{\partial }_h⟨M⟩$ at fixed temperature. Differentiating its partition function givesFor a translation-invariant array of spins, write $G(r)=⟨{\sigma }_0{\sigma }_r⟩-⟨{\sigma }_0⟩⟨{\sigma }_r⟩$. Expanding the variance of the sum then yields $\chi ={\beta }_{\mathrm{th}}{\sum }_{r}G(r)$. The derivative with respect to the dimensionless source ${\beta }_{\mathrm{th}}h$ 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 givesThis is an integrated imaginary-time-ordered correlation function, not generally ${\beta }_{\mathrm{th}}$ times the equal-time variance. It follows by insertinginto 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 $\chi =\tanh ({\beta }_{\mathrm{th}}\Delta )/\Delta$ at $h=0$, whereas ${\beta }_{\mathrm{th}}\mathrm{Var}({\sigma }_x)={\beta }_{\mathrm{th}}$. 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.

An imaginary-time-ordered correlation function orders operators along the Euclidean thermal circle. Bosonic correlators are periodic with period $\beta$.

A path integral is obtained by slicing an evolution operator into short time intervals and inserting complete sets of field eigenstates between successive factors. An operator inserted at a later slice therefore appears to the left of one inserted at an earlier slice. Summing over the intermediate fields preserves this ordering, so real-time path integrals generate time-ordered correlators and Euclidean thermal path integrals generate imaginary-time-ordered correlation functions.

A thermal fluctuation is equilibrium variation of a random variable representing an observable. In a classical canonical ensemble with $H_h=H_0-hA$, assume $A$ and the state space measure are independent of $h$, and fix temperature $T$. Differentiating the normalized Boltzmann distribution givesThus the variance measures response to a conjugate field. For a two-state observable $A=\pm a$ with energies $-hA$, $⟨A⟩=a\tanh (\beta ah)$ and $\mathrm{Var}(A)=a^2{\mathrm{sech}}^2(\beta ah)$, verifying the identity directly.

The same simple formula holds in a finite quantum system when the commutator of $A$ and $H_0$ vanishes. For noncommuting quantum observables, differentiation of the operator exponential produces an imaginary-time-ordered correlation function response rather than automatically $\beta \mathrm{Var}(A)$. In a classical field theory, the spatial two-point analogue uses the connected correlation function.