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Suppose we now consider a lattice of spin, say Ising model, and the phase transition at the critical temperature $T_c$. There are few scaling laws describe the regime around the critical temperature such as correlation length $\xi$:

$\xi\sim|T-T_{c}|{}^{-\nu}$

where $\nu$ is the critical exponent. This equation describes the divergence of the correlation length as the temperature approaching $T_c$.

However, it gives no direct physical picture of the spins near the $T_c$ because we need to compute the summation over each pair of spins to find the correlation. So the question is: If we really look at the lattice, what is the most obvious feature and scaling that we can obverse?

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Fractals. –  Marek Nov 28 '10 at 16:02
    
Any elaborate? The figure is around the critical temperature, but is it above or below $T_c$? I find it difficult to get information from those random fractal. –  hwlau Nov 28 '10 at 16:48
    
I just answered the question "what is the most obvious feature?". I am not really sure what else are you asking for. In particular, do you want to compute some quantity or do you just want some qualitative discussion? For the first case you can compute everything because free energy is known (due to Onsager). For the second case I am not sure what else can one say beyond pointing out the appearance of fractals near $T_c$. –  Marek Nov 28 '10 at 17:51
    
@Marek: Yes, the fractal is the obvious pattern because of the scaling form here, but whether there is a way to determine that the fractal pattern of spin is above or below the $T_c$. i.e. is there any different in the fractal pattern. –  hwlau Nov 28 '10 at 18:21
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The renormalization group approach suggests one way of distinguishing between temperatures above and below $T_{c}$. If you are above $T_{c}$ and you 'coarse-grain' the lattice by averaging the spins in blocks, you will get another lattice which looks like it came from a simulation at even higher temperature. After enough coarse-graining you will get a completely randomized lattice (as in $T=\infty$). If instead you start below $T_{c}$ you will eventually get the opposite, a lattice where all the spins are the same (as they are at $T=0$). –  Greg P Nov 28 '10 at 19:24
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2 Answers

up vote 9 down vote accepted

Generalities on Conformal Invariance

In two dimensions, a lot is known / conjectured about statistical models at criticality. For instance, at $T_c$, the spin configuration that you see will not only be self-similar (what others here have been calling "fractal") but actually fully conformally invariant (in the continum limit); that is, the probability distributions ("ensembles") are invariant not only under scale transformations, but indeed, under any transformation which preserves angles. (I think conformal invariance is expected in models with "short-range" interactions at criticality in any dimension, but it's only well understood in 2 dimensions).

Let me go a bit more into what "conformal invariance" means. Consider the following "thought experiment", which you might try with a computer simulation. First, intersect a disk with a square lattice with side length $\epsilon$ with $\epsilon$ quite small. On this finite graph, draw a random spin configuration for the Ising model at criticality. Call this ensemble "A".

Now, perform a conformal mapping of a disk onto whatever (simply connected) shape you like, say a triangle or a square (this is guaranteed by the Riemann mapping theorem). Approximate this shape with a square lattice with side $\epsilon$ and draw a random critical Ising spin configuration for this. Call this ensemble "B".

The claim is that if you perform a conformal transformation of ensemble "A" to get A', the typical picture from A' will be identical to the typical picture from B (this is a statement that the distribution of spin configurations will be nearly identical, if $\epsilon$ is small).

Here's some illustrations from slides of Stanislav Smirnov (who recently won a Fields medal for related work!).

Ising model random walk

On the left is a critical Ising model configuration under a conformal transformation; on the right is a random walk under a conformal transformation. You have to kind of use your imagination to see that (other than finite lattice effects) the pictures would be similar to those if you just ran a Ising model or random walk in the funny shapes below.

This has recently become a hot topic in mathematics, due to the breakthroughs in Schramm-Loewner evolution, though it has been studied a bit longer in physics using the less rigorous techniques of conformal field theory.

A specific "geometrical observable"

I've been kind of vague above about what conformal invariance means for individual "pictures" of spins (well, most of the claims are about probability distributions). You seem to be dissatisfied with various claims about correlation functions; I also feel that these are often a bit hard to grasp geometrically. There is one solidly geometric thing that one can say about certain lattice models at criticality, and this is about the boundaries between spin clusters.

The boundaries between up and down spins in the Ising model at criticality form an ensemble of closed loops in the plane. This loop ensemble is believed to be related to SLE $\kappa=3$ curves; this is a very particular (conformally invariant) ensemble of random planar curves. SLE curves are parametrized by a real number $\kappa\geq0$, when $\kappa=0$ one just has straight lines, as $\kappa$ gets larger, the curve gets more "wriggly", the fractal dimension of these curves is equal to $1+\kappa/8$ for $\kappa<8$ (thus in the continuum limit, boundaries between Ising spin clusters are expected to be 11/8 = 1.375 dimensional!). For $\kappa\geq8$, the curves actually become plane-filling!

All of the things I talked about above do not hold except when you are exactly at $T=T_c$. For the Ising model, it is sometimes hard to generate spin configurations at $T=T_c$ due to critical slowing down of Markov chains which are used, but there are many other systems such as critical percolation which are easier to study.

References

One good place to start reading about conformal invariance in lattice models for a theoretical physicist is Malte Henkel's book "Conformal Invariance and Critical Phenomena". Of course if you're serious, you'll eventually turn to the big yellow book by DiFrancesco et al on conformal field theory...

A good intro to SLE for physicists is this one by Ilya Gruzberg. Probably it won't be long before some books combining the SLE approach with the CFT approach start coming out.

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Well, nice answer but I wasn't aware that OP was asking about continuum limit. Actually, I am still not sure what precisely he is asking for. –  Marek Nov 28 '10 at 23:12
    
Phase transitions only exist in the thermodynamic limit, which is basically equivalent to the continuum limit in these lattice models. –  j.c. Nov 29 '10 at 0:20
    
Good answer. I have not read about the boundary between cluster befoer. @Marek: I want a geometrical explanation other than the correlation function. –  hwlau Nov 30 '10 at 1:52
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wait, what? How is continuum limit equivalent to thermodynamic limit? It's true that the ratio of lattice scale and model scale goes to zero in both limits but in the usual thermodynamic limit the model remains discrete and infinite whereas in the continuum limit it is continuous and finite. If an equivalence exists it's certainly non-trivial and I never heard about it. Could you provide some links? –  Marek Dec 2 '10 at 13:32
    
OK, good point Marek. I was sweeping too much under the rug with "basically equivalent". –  j.c. Dec 2 '10 at 16:36
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The correlation length is something that you can see from the spins (looking at the lattice). You can determine it from a sequence of snapshots of an Ising model simulation. Visually, as the critical point is reached, you will see that the typical size of clusters having the same magnetization (up or down) becomes larger and larger, and this size is none other than the correlation length. This seems to me to be a very "direct physical picture."

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Then is there any different for $ T > T_c $ and $ T < T_c $ –  hwlau Nov 28 '10 at 16:53
    
There is of course a difference in the average magnetization, which is zero above $T_{c}$ and nonzero below. But as one approaches $T_{c}$ either from above or below, the correlation length diverges, and in fact the critical exponent $\nu$ is the same for $T > T_{c}$ and $T < T_{c}$. This can be shown using the renormalization group analysis of the Ising model. –  Greg P Nov 28 '10 at 17:34
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