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Questions on the paper "Theory of Spin Glasses" by Edwards and Anderson

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I have two questions related to the paper "Theory of Spin Glasses" by Edwards and Anderson. 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.

  1. In explaining equation (2.23), \[q^2 = 5 \left[ 1 - \left(\frac{T}{T_c}\right)^2 \right] \left( \frac{T}{T_c} \right)^4\] 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 even a solution, and the case that $q = -1$ is not entirely ruled out. I was unable to reproduce their equation given their variables -- I think there may be a typo in their definition of $\rho$ -- but even with the fixed-up expression, I get roughly the same dependence on $T$ as they do.

  2.  Edwards and Anderson define an order parameter $q = \langle \mathbf{s}_i^{(1)} \cdot \mathbf{s}_i^{(2)} \rangle$. For those unfamiliar with the idea, this is an average of the $i$th spin $\mathbf{s}_i^{(1)} \equiv \mathbf{s}_i(t_1)$ at time $t_1$ dotted with itself at a later time $t_2$. The intuition is that the spins in spin glasses change slowly with time. The authors say that the probability of finding spin $i$ in the state $\mathbf{s}_i^{(1)}$ is given by \[ p(\mathbf{s}_i^{(1)}) = \frac{1}{Z}e^{-\beta E(\mathbf{s}_i^{(1)} )}. \] They follow this up by saying that the probability of finding $\mathbf{s}_i^{(1)}$ at one time and $\mathbf{s}_i^{(2)}$ at some later time is $$P(\mathbf{s}_i^{(1)},\mathbf{s}_i^{(2)}) = p(\mathbf{s}_i^{(1)})p(\mathbf{s}_i^{(2)}),$$ (see equation (2.2)) which suggests the probabilities are independent. But if $\mathbf{s}_i^{(1)}$ and $\mathbf{s}_i^{(2)}$ are correlated, the probability of observing $\mathbf{s}_i^{(2)}$ depends on the value of $\mathbf{s}_i^{(1)}$.

Please let me know of any resources which may explain the shortcomings in this paper. It seems that there may be several mistakes in the manuscript, but for some reason or another I cannot find any mention of this. Thanks for any help.

asked Jun 4 in General Physics by oscarafone (10 points) [ no revision ]

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