Correlation length ($\xi$)

The correlation length is a characteristic distance over which connected spatial fluctuations remain correlated. Away from a critical point, a common large-distance form is $G(r)\sim r^{-p}{e}^{-r/\xi }$; the exponential decay scale $\xi$ is one standard definition. Other definitions, such as a second-moment length, can differ by finite factors.

Near an ordinary continuous phase transition, $\xi$ grows as $|t{|}^{-\nu }$. In scattering, a larger correlation length produces a narrower feature around the ordering wavevector: a Gaussian kernel $A+\kappa q^2$ gives a width of order $1/\xi$ with $\xi =\sqrt{\kappa /A}$. Finite system size cuts off this growth. A correlation length measures collective organization rather than the size of an individual atom, molecule or magnetic moment.

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.