When does the correlation length increase and diverge in an Ising model?

Question

I was reading chapter 5 of Dynamical Processes on Complex Networks and encountered the following paragraph:

The importance of critical phase transitions lies in the emergence at the critical point of cooperative phenomena and critical behavior. Indeed, close to the transition point, for $T$ close to $T_c$, the thermodynamic functions have a singular behavior that can be understood by considering what is happening at the microscopic scale. By definition, the correlation between two spins is $\langle \sigma_i \sigma_j \rangle - \langle \sigma_i \rangle \langle \sigma_j \rangle$, and the correlation function $G(r)$ is the average of such correlations for pairs of spins situated at distance $r$. The correlation function measures the fluctuations of the spins away from their mean values, and vanishes for $r \rightarrow \infty$ at both low and high temperature where spins are either all aligned or fluctuate independently, respectively. In particular, at high temperature the correlation function decays with a correlation length $\xi$ that can be considered as an estimate of the typical size of domains of parallel spins. As $T \rightarrow T_c$, long-range order develops in the system and the correlation length diverges: $\xi \rightarrow \infty$. More precisely, $\xi$ increases and diverges as $|(T - T_c)/T_C|^{-v}$ and exactly at $T_c$ no characteristic length is preferred: domains of all sizes can be found, corresponding to the phenomenon of scale invariance at criticality. Owing to the scale invariance at $T_c$, the ratio $G(r_1)/G(r_2)$ is necessarily a function only of $r_1/r_2$, say $\phi (r_2 / r_1)$. Such identity, which can be rewritten as $G(r/s) = \phi (s)G(r)$, has for consequences $G(r/s_1 s_2)) = \phi (s_1 s_2)G(r) = \phi ( s_1) \phi (s_2) G(r)$, which implies that $\phi$ is a power law and we have therefore at critical point $$G(r) ~ r^{-\lambda}$$ where $\lambda$ is an exponent to be determined.

I'm confused in particular about what $|(T - T_c)/T_C|^{-v}$ means. I couldn't find anywhere in the text where $v$ was defined. Also, what does the text mean by "as $|(T - T_c)/T_C|^{-v}$?

Answer 1

$T_{0} = \lvert(T-T_{c})/T_{c} \rvert$ is used as a reference value and is a ratio of how far above critical you are, $(T-T_{c})$, normalised by the critical value itself, $T_{c}$ (I assume the absolute values are there for the cases where $T_{c} > 0 > T$). For example, in convection in fluid mechanics, the Rayleigh number $\text{Ra}$ is a positive number related to buoyancy and essentially determines the onset of convection (when the system moves from its conduction to its convection state). The critical Rayleigh number at which onset occurs is often denoted by $\text{Ra}_{c}$. The quantity $(\text{Ra}-\text{Ra}_{c})$ then is a measure of how far above the onset of convection you are for a given $\text{Ra}$ and so $\text{R}_{0} = (\text{Ra}-\text{Ra}_{c})/\text{Ra}_{c}$ is a normalised Rayleigh number such that $R_{0} = 0$ at onset (when $\text{Ra} = \text{Ra}_{c}$) and $\text{R}_{0} > 0$ when $\text{Ra} > \text{Ra}_{c}$.

In your text, $\nu$ is just a parameter that tells you how fast the quantity $T_{0}$ grows (decays) i.e does it grow linearly ($\nu = 1$), quadratically ($\nu = 2$), exponentially ($\nu \propto T$) etc. So when the text writes

$$\xi \text{ increases and diverges as } \lvert(T-T_{c})/T_{c} \rvert^{-\nu}$$

they are just saying that the quantity $\xi$ grows at the rate $\lvert(T-T_{c})/T_{c} \rvert^{-\nu}$ for some value of $\nu$. Different systems will have different values of $\nu$ (for differing reasons) and in turn can then have different power law scalings.