Revision as of 11:22, 22 January 2024

Goal: Spin glass trasnsition. From the expeirments with the anomaly on the magnetic susceptibility to order parameter of the transition. We will discuss the arguments linked to extreme value statistics

Spin glass Transition

Experiments

Spin glass behviour was first reported in non-magnetic metals (Cu, Fe, Au,...) doped with a few percent of a magnetic impurities, typically Mn. At low doping, Mn magnetic moments feel the Ruderman–Kittel–Kasuya–Yosida (RKKY) inetraction which has a random sign because of the random location of Mn atoms in the non-magnetic metal. A freezing temparature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_f} seprates the high-temperature paramagnetic phase from the low temeprature spin glass phase:

Above Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_f} the susceptibility obeys to the standard Curie law Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(T) \sim 1/T} .
Below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_f} , a strong metastability is observed: at the origin of the difference between the field cooled (FC) and the zero field cooled (ZFC) protocols. In zero field cooled ZFC, the susceptibility decays with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} . In FC, the susceptibility freezes at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_f} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_{FC}(T<T_f)=\chi(T_f)}

Edwards Anderson model

The first important theoretical attempt for spin glasses in the Edwards Anderson model. We consider for simplicity the Ising version of this model.

Ising spins takes two values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma=\pm 1} and live on a lattice of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } sites $i=1,2,\ldots ,N$ . The enregy is written as a sum between the nearest neighbours <i,j>:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E= - \sum_{ <i, j> } J_{ij} \sigma_i \sigma_j }

Edwards and Anderson proposed to study this model for couplings Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J } that are i.i.d. random variables with zero mean. We set $\pi (J)$ the coupling distribution and we indicate the average over the couplings, called disorder average, with an overline:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{J} \equiv \int d J \, J \, \pi(J)=0 }

We will discuss two distributions:

Gaussian couplings: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi(J) =\exp\left(-J^2/2\right)/\sqrt{2 \pi}}
Coin toss couplings, $J=\pm 1$ , selected with probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2 } .

Edwards Anderson order parameter

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{ J}=0 } , the model does not display spatial magnetic order, such as ferro/antiferro order. The idea is to distinguish:

a paramagnet that explores configurations with all possible orientations
a glass where the orientation are random, but frozen (i.e.immobile).

The glass phase is then characterized by long range correlation in time without any long range correlation in space. The order parameters is

q_{EA}=\lim _{t\to \infty }\lim _{N\to \infty }{\frac {1}{N}}\sum _{i}\sigma _{i}(0)\sigma _{i}(t)

In the paramagnetic phase Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{EA}=0 } , in the glass phase Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{EA}>0 } . One can wonder is this transition is thermodynamic. For example, the magnetic susceptibility does not diverge at the freezing temperature, but the magnetization is not the order parameter $M=\sum _{i}S_{i}=0$ . Here the order parameter is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{EA} } and it can be proved that its susceptibility is the non-linear susceptibility.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{M}{H}= \chi + a_3 H^2 +a_5 H^4+ \ldots }

Experiments showed that $a_{3},a_{5}$ are indeed singular. This means that agreement the existence of a thermodynamic transition at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_f} is an experimental fact.

The SK model

Sherrington and Kirkpatrik considered the fully connected version of the model with Gaussian couplings:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E= - \sum_{i,j} \frac{J_{ij}}{2 \sqrt{N}} \sigma_i \sigma_j }

At the inverse temperature $\beta$ , the partion function of the model is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z= \sum_{\alpha=1}^{2^N} z_{\alpha}, \quad \text{with}\; z_{\alpha}= e^{-\beta E_\alpha} }

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_\alpha } is the energy associated to the configuration $\alpha$ . This model presents a thermodynamic transition.

Random energy model

The solution of the SK is difficult. To make progress we first study the radnom energy model (REM) introduced by B. Derrida. This model neglects the correlations between the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=2^N } configurations and assumes the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\alpha} } as iid variables.

Show that the energy distribution is

p(E_{\alpha })={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {E_{\alpha }^{2}}{2\sigma _{M}^{2}}}}

and determine Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_M^2= N= \log M/\log 2}

We provide different solutions of the Random Energy Model (REM). The first one focus on the statistics of the smallest energies among the ones associated to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=2^N} configurations. For this, we need to become familiar with the main results of extreme value statistic of iid variables.

Extreme value statistics

Consider the $M=2^{N}$ energies: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_1,...,E_M} as iid variables, drawn from the distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E)} (Gaussian for the REM). It is useful to introduce the probability to find an energy smaller than E:

P^{<}(E)=\int _{-\infty }^{E}dxp(x)

.

Hence the probability to find an energy larger than E is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^>(E)=\int_E^{+\infty} dx p(x) = 1- P^<(E) } . We denote

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\min}=\min(E_1,...,E_M)}

Our goal is to compute the cumulative distribution $Q_{M}(\epsilon )\equiv {\text{Prob}}(E_{\min }>\epsilon )$ for large M. To achieve this we need three key relations:

first relation (exact):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_M(\epsilon) = \left(P^>(\epsilon)\right)^M }

Second relation (estimation): typical value of the minimum, namely Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\min}^{\text{typ}} } :

P^{<}(E_{\min }^{\text{typ}})={\frac {1}{M}}

.

Third relation (approximation) valid for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\to \infty}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_M(\epsilon) = e^{M\log\left(1-P^<(\epsilon)\right)} \sim \exp\left(-M P^<(\epsilon)\right) }

Gaussian case and beyond

The asymptotic tail of $P^{<}(E)$ is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^<(E)=\int_{-\infty}^E dx p(x) \sim \frac{\sigma}{\sqrt{2 \pi}|E|}e^{-\frac{E^2}{2 \sigma_M^2}} \; } for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \to -\infty}

Hence, the typical value of the minimum is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\min}^{\text{typ}} = -\sigma \sqrt{2 \log M}+\frac{1}{2}\sqrt{\log(\log M)} +O(1) }

Let us to be more general and consider tails

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^<(E) \sim e^{-\frac{|E|^\gamma}{2 \sigma_M^2}} \; . }

In the spirit of the central limit theorem we are looking for a scaling form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\min}=a_M + b_M z } . The constants Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_M, b_M} are M-dependent while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} is a random variable of order one drawa from the M-independent distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(z)} . Shows that

at the leading order $a_{M}=\left[2\sigma _{M}^{2}\log M\right]^{1/\gamma }$
$b_{M}={\frac {2\sigma _{M}^{2}}{\gamma a_{M}^{\gamma -1}}}$
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(z) = e^{z} e^{-e^{z}}} which is the Gumbel distribution

Ground state fluctuations

Depending on the distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E)} we have a different dependence of M of both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_M, b_M } . It is convenient to emphasize the N dependence we define

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_M \equiv 1/y_N \propto N^{\omega} }

Note that the typical fluctuations of the minimum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim 1/y_N} . In particular we can write:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{ \left(E_{\min} - \overline {E_{\min}}\right)^2 }\propto N^{2\omega}}

We will see that three different scenarios occur depending on the sign of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega } .

Density of states above the minimum

For a given disorder realization, we compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n(x) } , the number of configurations above the minimum with an energy smaller than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\min}+x} . The key relation for this quantity is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Prob}[d(x) = k] = M \binom{M-1}{k}\int_{-\infty}^\infty dE \; p(E) [P^>(E) - P^>(E+x) ]^{k} P^>(E+x)^{M - k - 1} }

Taking the average, we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{n(x)} = \sum_k k \, \text{Prob}[d(x) = k] } . We use the following identity

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=0}^{M-1} k \binom{M-1}{k} (A-B)^k B^{M-1-k} = (A-B)\frac{d}{d A} \sum_{k=0}^{M-1} \binom{M-1}{k} (A-B)^k B^{M-1-k}= (M-1)(A-B)A^{M-2} }

we arrive to the final form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{n(x)} = M (M-1) \int dE \; p(E) \left[P^>(E) - P^>(E+x) \right] P^>(E)^{M-2} }

Replace Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=a_M + b_M z } and obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{n(x)} = \left(e^{y_N x}-1\right) \int_{-\infty}^{\infty} dz e^{2 z -e^{z} } = \left(e^{y_N x}-1\right)\quad \text{with} \; y_N \sim N^{-\omega} }

The Glass phase

In the Glass phase the measure is concentrated in few configurations which has a finite occupation probability while in the paramagnetic phase the occupation probability is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim 1/M } . As a consequence the entropy is extensive in the paramagnetic phase and sub-extensive in the glass phase. It is useful to evaluate the occupation probability of the ground state in the infinite system:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{z_{\alpha_{\min}}}{\sum_{\alpha=1}^M z_\alpha}= \frac{1}{1+\sum_{\alpha\ne \alpha_{\min}} z_\alpha}\sim \frac{1}{1+\int_0^\infty dx\, e^{-\beta x} \left(e^{y_N x}-1\right) } }

In the high temperature phase, for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta<y_N } , the occupation probability is close to zero, meaning that the ground state is not deep enough to make the system glassy
In the low temperature phase, for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta>y_N } , the above integral is finite. Hence, setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta=1/T, T_f=1/y_N} you can find

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{z_{\alpha_{\min}}}{\sum_{\alpha=1}^M z_\alpha}= \frac{1}{1+ \frac{T^2}{T_f-T} } }

This means that below the freezing temperature, the ground state is occupied with a finite probability as in Bose-Einstein Condensation.

Take home message

Let us recall Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_N \sim N^{-\omega}} , so that three situations can occur

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega<0} , there is no freezing transition as there are too many states just above the minimum. This is the situation of many low-dimensional systems such as the Edwards Anderson model is two dimensions.
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega>0} there are two important features: (i) there is only the glass phase, (ii) the system condensate only in the ground state because the excited states have very high energy. We will see that in real systems (i) is not always the case and that the exponent Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} can change with temperature. This situation can be realistic (there is a very deep groud sate), but it is not revolutionary as the following one.
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega=0} there is for sure a freezing transition. One important feature of this transition that we will see in the next tutorial is that the condensation does not occur only in the ground state but in a large (but not extensive) number of low energy exctitations.

Exercise L1-A: More on extreme values

For a large set of iid random variables there are only three scaling form for the distribution of the minimum. In this lecture we studied the Gumbel case for fast decaying functions. The other two possibilities are

Frechet case: for a power law tail Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E) \sim c/|E|^{\gamma+1} } with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma>0 }
Weidbul case: for a bounded distribution with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E) \sim c(E-E_0)^\gamma \; \text{when } E\to E_0^+ }

Compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_M, b_M} in both cases as well as the limiting distribution.

References

"Spin glasses: Experimental signatures and salient outcomes", E. Vincent and V. Dupuis, Frustrated Materials and Ferroic Glasses 31, 2018
Theory of spin glasses, S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5 965, 1975
"Spin glass i-vii" P.W. Anderson, Physics Today, 1988

@@ Line 36: / Line 36: @@
 The glass phase is then characterized by long range correlation in time without any long range correlation in space.  The order parameters is
 <center> <math>
-  q_{EA}= \lim_{t\to \infty} \lim_{N\to \infty} \frac{1}{N}\sum_{i} S_i(0) S_i(t)
+  q_{EA}= \lim_{t\to \infty} \lim_{N\to \infty} \frac{1}{N}\sum_{i} \sigma_i(0) \sigma_i(t)
 </math></center>
 In the paramagnetic phase <math>

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