I'm trying to form a better understanding of the 2D Ising Model, in particular the behaviour of the correlation functions between spins of distance $r$.
I've found a number of explanatory texts that seem to indicate that at both above and below the critical temperature $T_c$, the correlation function $C(r)$ decays exponentially at some correlation length $\xi$, and this helps us determine typical domain sizes. At $T_c$, the correlation length diverges to infinity. Here's an image from (http://math.arizona.edu/~tgk/541/chap1.pdf) to illustrate what I mean ($\beta$ of course is inverse temperature).
$T > T_c$: This makes sense to me - adjacent spins are practically independent so domains are tiny, and the correlation length tends to zero as temperature increases.
$T < T_c$: This doesn't make intuitive sense to me - I was under the impression that below $T_c$ the domains were large enough to ensure spontaneous magnetisation was observed. However the above is indicating that the typical correlation length ($\xi$) tends to zero as the temperature rises - and therefore domains shrink? I would have thought:
- The correlation function $C_r$ should not decay exponentially at all, but remain at a constant (as is indicated on Chapter 1, page 6 of Paul Fendley's "Modern Statistical Mechanics" draft)
- If they do decay exponentially, they would decay to a value > 0 (as is indicated on page 216 of Sethna's "Entropy, Order Parameters and Complexity")
- If they do decay exponentially and to 0, then surely as $T \rightarrow 0$, the correlation length $\xi$ should tend to infinity, as at $T = 0$ we know that all spins must perfectly correlate with arbitrarily far spins?
$T = T_c$: If the correlation length is infinity, surely we would see perfect correlation and total magnetisation in the region close to the critical temperature? Instead at the critical temperature references indicate the correlation function takes the form $C_r \sim r^ {- \lambda} $ - the intuition of how this leap is made isn't clear to me.
Clearly, I have fundamentally misunderstood either how the correlation function and critical length work, how it relates to domain sizes, or both. I would really appreciate if someone could indicate where I have misunderstood.
For reference, I have an economics, not physics background, but need to understand the intuition of these models for my doctoral research in opinion dynamics. My primary reference text is "Dynamical Processes on Complex Systems" by Barrat et al.