Correlation length at low temperatures?
Answers
Answered by
2
HEY!!!
UR ANS IS...
assume that the question is not about how one shows that the correlation length diverges, as this is a simple computation when d=1d=1, but rather about intuition behind the result.
First, note that the same also happens when you consider the dd-dimensional Ising model, with d≥2d≥2, as the temperature approaches TcTc from above. (Actually, also from below, but let's stick to the situation you are interested in.)
In both cases, the reason is the same (although the mechanism is more subtle in higher dimensions). The correlation length is the relevant length-scale in the system. So, above TcTc, it measures, for example, the typical size of clusters of spins taking the same values. The size of these clusters diverges as the system gets closer and closer to the critical temperature.
Rather than the kind of dynamical interpretation you seem to be using, you should rather interpret the correlation length in the following statistical manner: suppose you only observe a single spin of your system (say, the spin at 00, σ0σ0) and you discover that it takes the value +1+1. What can you say about the value of the spin at i≠0i≠0? Well, if ii is "close enough" to 00, then knowing that you have a +1+1 at 00 makes it more likely to observe a +1+1 also at ii. The correlation length quantifies what "close enough" means. Namely it is the typical distance up to which the probability of observing a +1+1, given that σ0=+1σ0=+1, differs significantly from 1/21/2.
Now, for the one-dimensional model at low temperature, observing that σ0=+1σ0=+1 makes it very likely that you'll see only +1+1up to very large distances, precisely because the energetic cost is huge to flip spin. This will occur (the cost being finite), but the density of pairs of neighbors with spins taking different values goes to zero as T↓0T↓0. The average distance between two consecutive such pairs will be of the order of the correlation length, so it diverges.
Addendum
Let me stress the difference with what happens as T↓0T↓0 in higher dimensions. In this case, the system is in an ordered phase. For definiteness, let's assume that it is in the ++ state μ+TμT+. In this case, the correlation length measures the typical distance over which
μ+T(σi=s|σ0=s) differs significantly from μ+T(σi=s).μT+(σi=s|σ0=s) differs significantly from μT+(σi=s).
And this distance decays to 00 as T↓0T↓0 , for reasons explained in THIS ANSWER.
HOPE IT HELPS.
UR ANS IS...
assume that the question is not about how one shows that the correlation length diverges, as this is a simple computation when d=1d=1, but rather about intuition behind the result.
First, note that the same also happens when you consider the dd-dimensional Ising model, with d≥2d≥2, as the temperature approaches TcTc from above. (Actually, also from below, but let's stick to the situation you are interested in.)
In both cases, the reason is the same (although the mechanism is more subtle in higher dimensions). The correlation length is the relevant length-scale in the system. So, above TcTc, it measures, for example, the typical size of clusters of spins taking the same values. The size of these clusters diverges as the system gets closer and closer to the critical temperature.
Rather than the kind of dynamical interpretation you seem to be using, you should rather interpret the correlation length in the following statistical manner: suppose you only observe a single spin of your system (say, the spin at 00, σ0σ0) and you discover that it takes the value +1+1. What can you say about the value of the spin at i≠0i≠0? Well, if ii is "close enough" to 00, then knowing that you have a +1+1 at 00 makes it more likely to observe a +1+1 also at ii. The correlation length quantifies what "close enough" means. Namely it is the typical distance up to which the probability of observing a +1+1, given that σ0=+1σ0=+1, differs significantly from 1/21/2.
Now, for the one-dimensional model at low temperature, observing that σ0=+1σ0=+1 makes it very likely that you'll see only +1+1up to very large distances, precisely because the energetic cost is huge to flip spin. This will occur (the cost being finite), but the density of pairs of neighbors with spins taking different values goes to zero as T↓0T↓0. The average distance between two consecutive such pairs will be of the order of the correlation length, so it diverges.
Addendum
Let me stress the difference with what happens as T↓0T↓0 in higher dimensions. In this case, the system is in an ordered phase. For definiteness, let's assume that it is in the ++ state μ+TμT+. In this case, the correlation length measures the typical distance over which
μ+T(σi=s|σ0=s) differs significantly from μ+T(σi=s).μT+(σi=s|σ0=s) differs significantly from μT+(σi=s).
And this distance decays to 00 as T↓0T↓0 , for reasons explained in THIS ANSWER.
HOPE IT HELPS.
Similar questions