# EA order parameter $q(T)$ does not have limits paper says it has

This is related to the paper "Theory of Spin Glasses" by Edwards and Anderson (EA). It can be found at http://iopscience.iop.org/article/10.1088/0305-4608/5/5/017/pdf for those who may want to read the paper.

In explaining equation (2.23) (shown), $$q^2 = 5 \left[ 1 - \left(\frac{T}{T_c}\right)^2 \right] \left( \frac{T}{T_c} \right)^4$$ the authors say

The structure is similar to the standard Curie-Weiss theory, with the proviso that, at $T\rightarrow 0$, $q \rightarrow +1$ and not $-1$, whereas either root is permitted in ferromagnetism.

However, in (2.23) $q \rightarrow 0$ as $T \rightarrow 0$.

On top of this $q = 1$ is not a solution, and not even the sign of $q$ is determined (though with a "physical" argument one might rule out the negative sign.)

I was unable to reproduce (2.23) exactly, but my slightly different equation has the same general behavior.

I am trying to figure out what's going on here. I'd be grateful for any advice or resources.

• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. Jun 7 '17 at 12:56

From the paper, the full equation (2.19) for $q$ is $$q = \coth(\rho q)-\frac{1}{\rho q} \ .$$
In the low temperature limit, $\rho = 2J_0^2/3T^2$ diverges, so eq. (2.19) simply reduces to $q=1$. This gives the correct low-temperature limit solution, and fixes the sign as well.
The paper then states clearly that equations (2.20-23) are obtained by expanding the full equation for $q$ around $q=0$. Equation (2.23) can only be used to study how $q$ varies with $T$ at the transition, i.e. where $q$ "starts" being nonzero, and $T\approx T_c$. You are correct that it does not have $q=1$ as a low-temperature solution, but that's well out of its regime of applicability.