Correlations:
Quantifying "Clumpiness"

Contents

Nature Of The Problem

The pictures generated using the Ising Sampler page clearly show a transition from "disorder" to "order" as temperature is decreased. For example

System States At Two Temperatures

J/kT = 0.40
("Hot")		 J/kT = 0.47
("Cold")

At lower temperatures (the right hand picture) there are large regions which are almost entirely one color - meaning large areas where most of the spin vectors point in a common direction. As the temperature increases, this large-scale order disappears.

We need a way to measure the tendency towards order as the temperature decreases (after all, this is supposed to be a physics unit, and physicists make careers of measuring things.) The "obvious" quantity would be the total magnetization of the system:

Total Magnetization Of The System

The system is supposed to "become magnetic" as the temperature decreases, suggesting a noticeable increase in M. However, from the results seen in the pictures above, this is clearly not always the case. Both the "Hot" and "Cold" results are about half blue and half red - meaning comparable total magnetization.

The important difference between the "Hot" and "Cold" results is not just the relative proportions of Up and Down spins. Rather, it is the segregation of the different spins into largely localized regions. That is,

As the temperature of the system decreases, the probability increases that "reasonably close" sites have the same spin value

The phase transition in the Ising Model (and in real magnetic materials, for that matter) is characterized by an increase in localized spin alignments. This is the effect that needs to be made more quantitative.

("Confession": The phase transition is, in fact, also characterized by a definite jump in mean magnetization per site. However, for very large systems, this is somewhat harder to measure due to cancellations among the large "up" and "down" domains.)

Measuring Spatial Correlations

Let "s" be some fixed separation with the grid (e.g., "two steps up") and consider the function

A Generic Correlation Function

That is:
  1. Pick a site j within the grid.
  2. Multiply the value of the spin at that site by the value of the spin which is s steps away.
  3. Repeat steps 1 and 2 over all possible initial positions j and average the results.
The function C(s) thus measures the likelihood that sites separated by s steps have "related" spin states. The important limiting values of the correlation function are:

C(s) = 1 Complete Correlation: The spins at two sites s-steps apart are always the same.
C(s) = -1 Complete "Anti-Correlation": The spins at two sites s-steps apart are always the opposite.
C(s) = 0 Complete Randomness: The spins at two sites s-steps apart are completely independent.

Studies of magnetization in the Ising Model are naturally done by measuring values of the correlation C(s). As the temperature decreases, we expect that

  1. Values of C(s) for fixed separation s will become larger as the temperature decreases.
  2. As temperature decreases, C(s) will be "significantly non-zero" for increasingly larger values of s.
These are the most easily measured "signatures" for the phase transition in the Ising Model system.

Digression: Some Correlation Conventions

The discussions above are not particularly precise as to what "Separation s" really means - largely because there is no obvious, best definition. The correlation evaluations in this unit are done by averaging over the four possible displacements (up,down,left,right) as shown below.

Practical Implementation Of C(S)

That is, for fixed sampling site (x,y) (the yellow dot in the picture) multiply the spin at that site by the average of the spins ate the (up,down,left,right)-displaced sites (the green dots).

Some Sample Results

The computational pages within this unit calculate and list correlations for several separation values. The figure below plots results from runs on the 120x160 grid at two different temperatures.

Some Simulation Results

At lower temperatures (larger coupling J/kT), the correlations larger at all values of the separation.

At fixed J/kT, the correlations fall off quickly for increasing separations "s". It is customary to parameterize this behavior by fitting the correlations to an exponential form:

Assumed Exponential Form For The Correlation Function

The creation of magnetization domains with decreasing temperature is seen through values for A and x_C which increase with decreasing temperature. The parameter x_C is often called the "Correlation Length".

The results in the previous figure are for a single Monte Carlo sample (a "single realization"), and one would expect that both the data points in the plot and the corresponding parameter values from the fit would "wiggle" a bit for a different random realization. This is indeed the case - as can be seen , for example, by generating several configurations at the same value of J/kT using the Monte Carlo Engine

Typical Monte Carlo Results

The plot above shows typical results form the Monte Carlo engine. The Correlation Length and Mean Magnetization values at each temperature represent averages over several realizations. The "error bars" show the typical fluctuations in these values from run to run. The estimated errors in the correlation lengths are too small to be seen on the plot, but these "measurement errors" in the Mean Magnetization are comparable to the jump in the magnetization at the phase transition.