Correlation Function

Let the order parameter $M$ be written as a volume integral over an order-parameter density $m(\vec{ r})$. Then

$$M = <\int d^3 r m(\vec{r})>$$ (4.32)

where $<>$ represents the statistical average by which one obtains thermodynamic functions. A useful quantity that can now be defined is the correlation function

$$\Gamma(\vec{r}) = < m(\vec{r}) m(0) > - <m(\vec{r})> <m(0)>.$$ (4.33)

It measures how the value of the order parameter at one point is correlated to its value at some other point. If $\Gamma$ decreases very fast with distance, then far away points are relatively uncorrelated and the system is dominated by its microscopic structure and short-ranged forces. On the other hand, a slow decrease of $\Gamma$ would imply that faraway points have a large degree of correlation or influence on each other. The system thus becomes organsied at a macroscopic level with the possibility of new structure beyond the obvious one dictated by the short-ranged microscopic forces. As we shall see below, this possibility does actually occur. Usually, near the critical point ($T \to T_c$), the correlation function can be written in the form

$$\Gamma (r) \to r^{-p} \exp({-r/\xi})$$ (4.34)

where $\xi$ is the correlation length. The correlation length is a measure of the range over which fluctuations in one region of space are correlated with (influence) those in another region. Two points which are separated by a distance larger than the correlation length will each have fluctuations which are relatively independent, that is, uncorrelated. Experimentally, the correlation length is found to diverge at the critical point. Thus near the critical point, the correlation length may be written as

$$\xi \sim \vert t\vert^{-\nu} ,$$ (4.35)

where $t= {T-T_c \over T}$. The divergence of the correlation length at the critical point means that very far points become correlated. In other words, the long-wavelength fluctuations dominate. Thus the system near a second-order phase transition ‘loses memory’ of its microscopic structure and begins to display new long-range macroscopic correlations. Exactly at the critical point, the correlation function (4.34) therefore displays a power law behaviour $\sim r^{-p}$ with

$$p \equiv d-2 +\eta$$ (4.36)

Here $d$ represents the effective space dimensionality of the system. The quantities $\nu$ and $\eta$ are examples of what is known as critical exponents. Experiments, supported by renormalization group theory, have shown that systems undergoing second-order phase transitions can be grouped into universality classes. Within each universality class, very different systems with widely different critical temperatures, have approximately the same critical exponents. The reason for this is precisely the ‘loss of memory’ mentioned above, so that systems with different microscopic structures can give rise to the same long-range behaviour. In addition to the above two critical exponents, there are four more, $\alpha, \beta, \gamma, \delta$ defined by Heat Capacity: $C \sim \vert t\vert^{-\alpha}$
Order Parameter: ${ M \over V} \sim \vert t\vert^{\beta}$
Susceptibility: $\chi \sim \vert t\vert^{-\gamma}$
Equation of state (at t=0): ${M \over V} \sim \vert{\cal{H}}\vert^{1/ \delta}$ .
Here $M/V$ refers to the magnetisation density. The first three exponents refer to the case ${\cal{H}} =0$. Note that in general the thermodynamic quantities near the critical temperature may contain both finite and ‘singular’ parts (or a part with singular derivatives). It is the singular dominant parts to which one refers to above with the $\sim$ symbol above (the symbol also implies a proportionality constant).