Correlation function

A correlation function measures the joint fluctuation of observables at separated points. For a scalar field, the connected two-point function is $G(x-y)=⟨\varphi (x)\varphi (y)⟩-⟨\varphi (x)⟩⟨\varphi (y)⟩$.

The four-point correlation function measures a time-ordered product of four field insertions. For a centered scalar field its disconnected terms are the three products of two-point functions, and subtracting those products leaves the connected correlation function. In a free Gaussian scalar vacuum the connected four-point function vanishes, by Wick's theorem, so only the three paired products remain. The connected four-point function of phi-fourth theory starts with one quartic vertex; amputation removes its external propagators. The full correlator, its connected part and the four-point one-particle-irreducible correlation function should therefore be distinguished.

The correlation function with a single field insertion is its expectation value. In a translation-invariant vacuum it is independent of position. A tadpole subtraction chooses a linear counterterm, or shifts the expansion point of the field, so that this expectation is zero. This removes attached tadpole diagrams from higher connected correlation functions without discarding the physical need to renormalize the chosen vacuum.

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.

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.

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.

A two-point correlation function is the expectation of a product of fields or observables at two points. Its connected version subtracts the product of the one-point expectations.

The dynamic structure factor is the space-time Fourier transform of an equilibrium two-point correlation function. It resolves fluctuations by angular frequency and wavevector and is related to the dissipative part of a retarded Green function by the Fluctuation-dissipation theorem.

The static structure factor is the equal-time spatial Fourier transform of a two-point correlation function. Depending on convention it is obtained either by integrating the dynamic structure factor over frequency or by taking an appropriate zero-frequency limit.

For $N$ equal scattering sites on one polymer, define $g(\mathbf{k})=N^{-1}{\sum }_{m,n}⟨e^{i\mathbf{k}\cdot ({\mathbf{R}}_n-{\mathbf{R}}_m)}⟩$. It has $g(0)=N$; its normalized form factor is $g/N$. Under a rotationally invariant joint distribution, each separation vector has a uniform direction at fixed magnitude $r$. The angular average is $\frac{1}{2}{\int }_{-1}^1{e}^{ikr\mu }d\mu =\sin (kr)/(kr)$, with value one at $r=0$. Thus $g$ depends only on $k=|\mathbf{k}|$, and $g(k)=N^{-1}{\sum }_{m,n}⟨\sin (k|{\mathbf{R}}_n-{\mathbf{R}}_m|)/(k|{\mathbf{R}}_n-{\mathbf{R}}_m|)⟩$. The diagonal self terms are essential in a finite chain and give the large-$k$ limit one for a discrete Gaussian chain.

Expand the isotropic angular factor as $\sin (kr)/(kr)=1-k^2{r}^2/6+O(k^4{r}^4)$. For a polymer scattering function and finite fourth pair moments, the constant double sum gives $g(0)=N$, and the quadratic sum is determined by the radius of gyration:In the range $k{R}_g\ll 1$, this is also approximated by $e^{-k^2{R}_g^2/3}$. The coefficient follows from isotropy and the definition of $R_g$ and does not require a Gaussian chain.

The correlation length is the characteristic distance over which a connected correlation function decays. It diverges at an ordinary continuous critical point as $\xi \sim |t{|}^{-\nu }$.

A volume over which order-parameter fluctuations are correlated. For one isotropic correlation length $\xi$, its scale is ${\xi }^D$; with anisotropic lengths it is the product of the appropriate directional lengths. A Ginzburg criterion compares fluctuations averaged over this volume with the squared mean order parameter.

For a stable homogeneous scalar quartic saddle with gradient coefficient one, the Ornstein--Zernike correlation function has denominator $q^2+r_0+u_0{m}_0^2/2$. At zero source and $u_0>0$, the correlation length is $r_0^{-1/2}$ in the disordered phase and $(-2r_0{)}^{-1/2}$ on either selected ordered branch. Linear thermal tuning of $r_0$ gives correlation-length critical exponent $\nu =1/2$ with different amplitudes on the two sides.