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The correlation length, $\xi(T)$, defined from the correlation function $$C(r)\sim \frac{1}{r^{d-2+\eta}}e^{-r/\xi(T)},~~T>T_C\tag{1}$$ is exponentially small above the transition temperature ($T>T_C$). The power law behaviour of the correlation function at $T=T_C$: $$C(r)\sim \frac{1}{r^{2-d+\eta}},~~T=T_C\tag{2}$$ tells that $\xi$ diverges at $T=T_C$. When the transition is associated with long-range order for $T<T_C$, the correlation function becomes homogeneous i.e., spatially uniform: $$C(r)\sim {\rm const.},~T<T_C\tag{3}$$

Question What is the correlation length for $T<T_C$ and how does it change as $T$ is gradually decreased from $T=T_C$ to $T\to 0$? How can we see and understand that?

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(I discuss the case of the Ising model, for concreteness, but most of what I say below holds much more generally.)

The correlation length below $T_c$ is defined through the truncated 2-point function, namely $$ \langle \sigma_0\sigma_x\rangle^+ - \langle \sigma_0\rangle^+ \langle\sigma_x\rangle^+ \sim \frac{1}{|x|^{d-2+\eta}}e^{-|x|/\xi}, $$ where $\langle\cdot\rangle^+$ denotes the expectation w.r.t. the $+$ state (one must precise the state below $T_c$ as there are several). I am cheating a bit (as you do), by doing as if $\xi$ was isotropic (it's not for lattice models, except asymptotically close to $T_c$).

Also, in the definition of $\xi$, one has to let $|x|$ go to infinity above. More precisely, one would define $$ \frac{1}{\xi} = - \lim_{|x|\to\infty} \frac1{|x|} \log \bigl(\langle \sigma_0\sigma_x\rangle^+ - \langle \sigma_0\rangle^+ \langle\sigma_x\rangle^+ \bigr). $$ In particular, for $T>T_c$, you really have $$ \langle \sigma_0\sigma_x\rangle \sim \frac{1}{|x|^{(d-1)/2}}e^{-|x|/\xi}, $$ for all $|x|$ large compared to $\xi$. This exponent $(d-1)/2$ is typical of what is known as the Ornstein-Zernike behavior.

Similarly, when $|x|\gg\xi$, one has, for $T<T_c$, $$ \langle \sigma_0\sigma_x\rangle^+ - \langle \sigma_0\rangle^+ \langle\sigma_x\rangle^+ \sim \frac{1}{|x|^{(d-1)/2}}e^{-|x|/\xi}, $$ when $d\geq 3$, while $$ \langle \sigma_0\sigma_x\rangle^+ - \langle \sigma_0\rangle^+ \langle\sigma_x\rangle^+ \sim \frac{1}{|x|^{2}}e^{-|x|/\xi}, $$ when $d=2$ (this anomalous behavior when $d=2$ is the only claim that it really specific to the nearest-neighbor Ising model, and some relatives; for generic models, the behavior is also of Ornstein-Zernike type when $d=2$, that is, the exponent is $1/2$, not $2$).

As to the interpretation of the correlation length below $T_c$ and its behavior as $T\downarrow 0$, I already answered that elsewhere: see this answer (and also this one). Here, I'll only provide a graph of $\xi$ (along the horizontal direction) as a function of the temperature for the two-dimensional Ising model, where it can be computed explicitly (note that $\xi$ tends to $0$ both as $T\downarrow 0$ and as $T\uparrow\infty$, although the latter isn't visible in the picture below). enter image description here

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  • $\begingroup$ Is it a typo in the 3rd equation? Shouldn't it be $\frac{1}{|x|}e^{-x/\xi}$ instead of $\frac{1}{|x|^2}e^{-x/\xi}$? $\endgroup$
    – SRS
    May 8 '20 at 5:59
  • $\begingroup$ No, it's indeed $1/|x|^2$. This is a (well understood now) "pathology" of the planar Ising model. In the two-dimensional Ising model with non-nearest-neighbor interactions (thus non-planar), it is expected that the prefactor is again the usual $1/|x|^{(d-1)/2} = 1/\sqrt{|x|}$. $\endgroup$ May 8 '20 at 8:10
  • $\begingroup$ I thought, for $T<T_C$, when a long-range order is established, the two-point correlation function should be nonzero no matter how large $x$ is. Is that true? If so, I can't see that from your expressions. @YvanVelenik $\endgroup$
    – SRS
    May 15 '20 at 1:59
  • $\begingroup$ The non-truncated 2-point function is bounded below. This follows from the FKG inequality: $\langle\sigma_0\sigma_x\rangle^+ \geq \langle\sigma_0\rangle^+\langle\sigma_x\rangle^+$, since the right-hand side is just the square of the magnetization density, which $>0$ in the $+$ state below $T_c$. The truncated 2-point function $\langle\sigma_0\sigma_x\rangle^+ - \langle\sigma_0\rangle^+\langle\sigma_x\rangle^+$ tends to $0$ (as $|x|\to\infty$) at all temperatures. It does so exponentially fast when $T<T_c$. $\endgroup$ May 15 '20 at 7:40
  • $\begingroup$ As a complement to the previous comment: the fact that the truncated 2-point function tends to $0$ as $|x|\to\infty$ is true in any pure state, but can fail otherwise: for instance, if you consider the state obtained using free or periodic boundary conditions, then $\langle\sigma_0\sigma_x\rangle - \langle\sigma_0\rangle\langle\sigma_x\rangle = \langle\sigma_0\sigma_x\rangle$, which is still larger than the square of the magnetization density and thus $>0$ uniformly in $x$. $\endgroup$ May 15 '20 at 7:55

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