### Correlation length

In statistical mechanics, the correlation function is a measure of the order in a system, as characterized by a mathematical correlation function, and describes how microscopic variables at different positions are correlated.

In a spin system, it is the thermal average of the scalar product of the spins at two lattice points over all possible orderings. The correlation function is hence,

$G \left(r\right) = \langle \mathbf\left\{s\right\}\left(R\right) \cdot \mathbf\left\{s\right\}\left(R+r\right)\rangle\ - \langle \mathbf\left\{s\right\} \rangle^2\,.$

Here the brackets mean the above-mentioned thermal average.

Even in a disordered phase, spins at different positions are correlated, i.e., if the distance r is very small (compared to some length scale $\xi$), the interaction between the spins will cause them to be correlated. The alignment that would naturally arise as a result of the interaction between spins is destroyed by thermal effects. At high temperatures one sees an exponential correlation with the correlation function being given asymptotically by

$G \left(r\right) \approx \frac\left\{1\right\}\left\{r^\left\{d-2+\eta\right\}\right\}\exp\left\{\left\left(\frac\left\{-r\right\}\left\{\xi\right\}\right\right)\right\}\,,$

where r is the distance between spins, d is the dimension of the system. The correlation decays to zero exponentially with the distance between the spins.

Note that this is true not only above, but also below the critical temperature, although here the mean value of the spin is not 0.

Furthermore, $\eta$ is a critical exponent.

As the temperature is lowered, thermal disordering is lowered, and in a continuous phase transition the correlation length diverges, namely

$\xi\propto |T-T_c|^\left\{-\nu\right\}\,,$

with another exponent $\nu$.

This power law correlation is responsible for the scaling, seen in these transitions. All exponents mentioned are independent of temperature. They are in fact universal, i.e. found to be the same in a wide variety of systems.

One very important correlation function is the radial distribution function which is seen often in statistical mechanics. The correlation function can be calculated in exactly solvable models (one dimensional Bose gas, spin chains, Hubbard model) by means of Quantum inverse scattering method and Bethe ansatz. In an isotropic XY model, time and temperature correlations were evaluated by Its, Korepin, Izergin & Slavnov (see external link).

## References

• C. Domb, M.S. Green, J.L. Lebowitz editors, Phase Transitions and Critical Phenomena, vol. 1-20 (1972–2001), Academic Press.
• A.R. Its, V.e. Korepin, A.G. Izergin & N.A. Slavnov (2009) arxiv.org.
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