Correlation length ($\xi$)

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.