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

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 $\chi =N^{-1}{\partial }_h⟨M⟩$ 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.

In a symmetric equal mixture of two ordered pure thermodynamic phases with magnetizations $\pm M_0$, the mean spin vanishes but the large-distance two-spin expectation tends to $M_0^2$. Thus the connected correlation function of the mixture has a nondecaying part. The correlation-function susceptibility sum rule gives an extensive contribution ${\beta }_{\mathrm{th}}N{M}_0^2$ to susceptibility per spin. This is phase-switching response, not the connected susceptibility within one pure phase. The latter uses the thermodynamic limit followed by a one-sided zero-field limit, and subtracts the nonzero pure-phase mean before defining the correlation length.