Connected correlation function

The connected correlation of two observables is $⟨AB{⟩}_c=⟨AB⟩-⟨A⟩⟨B⟩$. It vanishes in a product state when $A$ and $B$ act on disjoint factors.

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.

In the scalar-field source Legendre transform, differentiating $h=\delta \Gamma /\delta m$ with respect to the source and using $G=\delta m/\delta h$ yields this inverse-kernel relation. For a scalar quartic Landau-Ginzburg theory, ${\Gamma }_{\mathrm{L}}^{(2)}=-{\nabla }^2+r_0+u_0{m}^2/2$, so the connected response is the Green kernel of that operator. A one-momentum representation requires a translationally invariant background.

An amputated connected correlation function is obtained by removing external propagators. Amputation using full propagators eliminates external self-energy decorations; internal exchange diagrams remain. It is distinct from a one-particle-irreducible correlation function. In a scalar theory with vanishing one-point function, at zero momentum the four-point quantity satisfies ${\mathcal{A}}_4=-{\Gamma }^{(4)}+3[{\Gamma }^{(3)}{]}^2/{\Gamma }^{(2)}$ with a convention in which connected diagram vertices are minus effective-action derivatives. For a constant scalar source, the Legendre relation $J={\Gamma }^{\prime }(\varphi )$ gives $G^{\prime \prime }=1/{\Gamma }^{\prime \prime }$ and $G^{\prime \prime \prime \prime }=-{\Gamma }^{\prime \prime \prime \prime }/({\Gamma }^{\prime \prime }{)}^4+3({\Gamma }^{\prime \prime \prime }{)}^2/({\Gamma }^{\prime \prime }{)}^5$. Dividing by the four full external propagators proves the formula.