L-1: Difference between revisions

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This model neglects the correlations between the <math> M=2^N </math> configurations and assumes the <math> E_{\alpha} </math> as iid variables.
This model neglects the correlations between the <math> M=2^N </math> configurations and assumes the <math> E_{\alpha} </math> as iid variables.
* Show that the energy distribution is  
* Show that the energy distribution is  
<center><math> p(E_\alpha) =\frac{1}{\sqrt{2 \pi \sigma^2}}e^{-\frac{E_{\alpha}^2}{2 \sigma^2}}</math></center>
<center><math> p(E_\alpha) =\frac{1}{\sqrt{2 \pi \sigma^2}}e^{-\frac{E_{\alpha}^2}{2 \sigma_M^2}}</math></center>
and determine <math>\sigma^2</math>
and determine <math>\sigma_M^2</math>




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===The Gaussian case ===
===The Gaussian case ===
The asymptotic tail of<math>P^<(E)</math> is
The asymptotic tail of<math>P^<(E)</math> is
<center><math>P^<(E)=\int_{-\infty}^E dx p(x)  \sim \frac{\sigma}{\sqrt{2 \pi}|E|}e^{-\frac{E^2}{2 \sigma^2}} \; </math> for  <math>x \to -\infty</math></center>
<center><math>P^<(E)=\int_{-\infty}^E dx p(x)  \sim \frac{\sigma}{\sqrt{2 \pi}|E|}e^{-\frac{E^2}{2 \sigma_M^2}} \; </math> for  <math>x \to -\infty</math></center>
Hence, the typical value of the minimum is
Hence, the typical value of the minimum is
<center><math>E_{\min}^{\text{typ}} = -\sigma \sqrt{2 \log M}+\frac{1}{2}\sqrt{\log(\log M)} +O(1) </math> </center>
<center><math>E_{\min}^{\text{typ}} = -\sigma \sqrt{2 \log M}+\frac{1}{2}\sqrt{\log(\log M)} +O(1) </math> </center>




Let us to be more general and consider tails <math>P^<(E) \sim e^{-\frac{E^\alpha}{2 \sigma^2}} \; </math>. In the spirit of the central limit theorem we are looking   
Let us to be more general and consider tails <math>P^<(E) \sim e^{-\frac{|E|^\alpha}{2 \sigma_M^2}} \; </math>. In the spirit of the central limit theorem we are looking for a scaling form  <math>E_{\min}=a_M + b_M z <math>. The constants <math>a_M, b_M</math> are M-dependent while <math>z</math> is a random variable of order one.
Shows that
* at the leading order <math>a_M= \left[2 \sigma^2_M \log M\right]^{1/alpha}</math>
* <math>b_M= \frac{2 \sigma^2_M}{a_M^{\alpha-1}} </math>
* <math>P(z) = e^{-z} e^{-e^{-z}}</math> which is the Gumbel distribution




 
Depending on the distribution <math>p(E)</math> we have a different dependence of ''M'' of both <math>a_M, y_M </math>. It is convenient to emphasize the ''N'' dependence we  define
<math>\lim_{M\to \infty} (1-\frac{k}{M})^M =\exp(-k)</math>,
  <center><math>  b_M\equiv y_N \propto N^{-\omega}  </math></center>
<center><math>Q_M(\epsilon) \sim \exp\left(-M  P^<(\epsilon)\right)</math> </center>
Note that the  typical fluctuations of the minimum <math> \sim 1/y_N</math>. In particular we can write:  
This relation holds only when <math> \epsilon \approx a_M </math> and one hase to expand around this value.
<center><math> \overline{ \left(E_{\min} -  \overline {E_{\min}}\right)^2 }\propto N^{2\omega}</math></center> We will see that three different scenarios occur depending on the sign of  <math>  \omega  </math>. 
However, a direct Taylor  expansion does not ensures that probabilities remain positive. Hence, we define <math>  P^<(\epsilon)=\exp(-A(\epsilon)) </math> and remark that <math>  A(\epsilon) </math>  is a decreasing function. We propose the following Taylor expansion 
<center><math>  A(\epsilon) =a_M + A'(a_M)(\epsilon -a_M) = a_M - y_N(\epsilon -a_M) </math></center>
Depending on the distribution <math>p(E)</math> we have a different dependence on ''N'' or ''M'' of both <math>a_M, y_N </math>. It is convenient to define
  <center><math>  y_N \propto N^{-\omega}  </math></center>
We will see that three different scenarios occur depending on the sign of  <math>  \omega  </math>. Using this expansion we derive:
* The famous Gumbel distribution:
<center><math>Q_M(\epsilon) \sim \exp\left(-e^{ y_N (\epsilon-a_M)}\right)  </math> </center>
* the  typical fluctuations of the minimum <math> \sim 1/y_N</math>. In particular we can write:  
<center><math> \overline{ \left(E_{\min} -  \overline {E_{\min}}\right)^2 }\propto N^{2\omega}</math></center>
 


===Density of states above the minimum===
===Density of states above the minimum===
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     \overline{n(x)} = M (M-1) \int  dE \; p(E) \left[P^>(E) -  P^>(E+x)  \right] P^>(E)^{M-2}   
     \overline{n(x)} = M (M-1) \int  dE \; p(E) \left[P^>(E) -  P^>(E+x)  \right] P^>(E)^{M-2}   
</math></center>
</math></center>
In the above integral, <math> E </math> is the energy of the minimum. Hence, we can use the Taylor expansion <math> A(E) = a_M -y_N (E -a_M)</math>. In particular we can write
In the above integral, <math> E </math> is the energy of the minimum. Hence, we can scaling form  <math> A(E) = a_M -y_N (E -a_M)</math>. In particular we can write
* <math>  p(E) = \frac{d}{d E} P^<(E)= -A'(E) e^{-A(E)} \sim y_N e^{y_N (E -a_M)} /M</math>
* <math>  p(E) = \frac{d}{d E} P^<(E)= -A'(E) e^{-A(E)} \sim y_N e^{y_N (E -a_M)} /M</math>
* <math> P^>(E) -  P^>(E+x)  = e^{-A(E+x)}-e^{-A(E)}\sim  e^{y_N (E -a_M)} \left(e^{y_N x}-1\right)/M </math>
* <math> P^>(E) -  P^>(E+x)  = e^{-A(E+x)}-e^{-A(E)}\sim  e^{y_N (E -a_M)} \left(e^{y_N x}-1\right)/M </math>
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* For  <math> \omega=0</math> 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.
* For  <math> \omega=0</math> 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: the Gaussian case ==
== Exercise L1-A: More on extreme values ==
Specify these results to the Guassian case and find
<math>P^<(E)=\int_{-\infty}^E dx p(x)  \sim \frac{\sigma}{\sqrt{2 \pi}|E|}e^{-\frac{E^2}{2 \sigma^2}} \; </math> for  <math>x \to -\infty</math>
* the typical value of the minimum
%<center><math>a_M = \sigma \sqrt{2 \log M}-\frac{1}{2}\sqrt{\log(\log M)} +O(1) </math> </center>
* The expression <math>  A(\epsilon) =\frac{\epsilon^2}{2\sigma^2} -\frac{\sqrt{2 \pi}}{\sigma} \log|\epsilon|+\ldots </math>
*The expression of the Gumbel distribution for the Gaussian case
<center><math>Q_M(\epsilon) \sim \exp\left(-e^{- \frac{\sqrt{2 \log M}}{\sigma} (\epsilon-a_M)}\right)  </math> </center>


=References=
=References=

Revision as of 18:14, 20 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 seprates the high-temperature paramagnetic phase from the low temeprature spin glass phase:

  • Above the susceptibility obeys to the standard Curie law .
  • 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 \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 sites 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 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 that are i.i.d. random variables with zero mean. We set 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)} 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:
  • Coin toss 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= \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 , 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

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}= \lim_{t\to \infty} \lim_{N\to \infty} \frac{1}{N}\sum_{i} S_i(0) S_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 . 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 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= \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.

Experiments showed that 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_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:

At the inverse temperature 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 } , 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 is the energy associated to the configuration 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 \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 as iid variables.

  • Show that the energy distribution 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_\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}


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 configurations. For this, we need to become familiar with the main results of extreme value statistic of iid variables.

Specify these results to the Guassian case and find

  • The expression 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(\epsilon) =\frac{\epsilon^2}{2\sigma^2} -\frac{\sqrt{2 \pi}}{\sigma} \log|\epsilon|+\ldots }
  • The expression of the Gumbel distribution for the Gaussian case
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) \sim \exp\left(-e^{- \frac{\sqrt{2 \log M}}{\sigma} (\epsilon-a_M)}\right) }

Extreme value statistics

Consider the 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 use the following notations:

  • , it is the probability to find an energy smaller than E.
  • 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) } , it is the probability to find an energy larger than 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 for large M. To achieve this we need three key relations:

  • The first relation is 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 }
  • The second relation identifies the 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}} } :

.

  • The third is an approximation valid only if 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) }

The Gaussian case

The asymptotic tail ofFailed 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)} 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|^\alpha}{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 <math>. The constants <math>a_M, b_M} are M-dependent while is a random variable of order one. Shows that

  • at the leading order
  • 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= \frac{2 \sigma^2_M}{a_M^{\alpha-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


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, y_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 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 dE \; p(E) [P^>(E) - P^>(E+x) ]^{k} P^>(E+x)^{M - k - 1} }

Taking the average 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 derive

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} }

In the above integral, 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 } is the energy of the minimum. Hence, we can 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 A(E) = a_M -y_N (E -a_M)} . 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 p(E) = \frac{d}{d E} P^<(E)= -A'(E) e^{-A(E)} \sim y_N e^{y_N (E -a_M)} /M}
  • 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) - P^>(E+x) = e^{-A(E+x)}-e^{-A(E)}\sim e^{y_N (E -a_M)} \left(e^{y_N x}-1\right)/M }
  • 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)^{M-2}= Q_{M-2} (E) \sim \exp\left(-e^{ y_N (E-a_M)}\right) }

Calling 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 u=y_N (E -a_M) } we 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} du e^{2 u -e^{u} } = \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.

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

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
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