L-1

Spin glass Transition

Experiments

Parlare dei campioni di rame dopati con il magnesio, marino o no: trovare due figure una di suscettivita e una di calore specifico, prova della transizione termodinamica.

Edwards Anderson model

We consider for simplicity the Ising version of this model.

Ising spins takes two values $\sigma =\pm 1$ and live on a lattice of $N$ sites $i=1,2,\ldots ,N$. The enregy is written as a sum between the nearest neighbours <i,j>:

$E=-\sum _{<i,j>}J_{ij}\sigma _{i}\sigma _{j}$

Edwards and Anderson proposed to study this model for couplings $J$ 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:

${\overline {J}}\equiv \int dJ\,J\,\pi (J)=0$

It is crucial to assume ${\bar {J}}=0$, otherwise the model displays ferro/antiferro order. We will discuss two distributions:

• Gaussian couplings: $\pi (J)=\exp \left(-J^{2}/2\right)/{\sqrt {2\pi }}$
• Coin toss couplings, $J=\pm 1$, selected with probability $1/2$.

The SK model

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

$E=-\sum _{i,j}{\frac {J_{ij}}{2{\sqrt {N}}}}\sigma _{i}\sigma _{j}$

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

$Z=\sum _{\alpha =1}^{2^{N}}z_{\alpha },\quad {\text{with}}\;z_{\alpha }=e^{-\beta E_{\alpha }}$

Here $E_{\alpha }$ is the energy associated to the configuration $\alpha$. This model presents a thermodynamic transition at $\beta _{c}=??$.

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 $M=2^{N}$ configurations and assumes the $E_{\alpha }$ as iid variables.

• Show that the energy distribution is

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

and determine $\sigma ^{2}$

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 $M=2^{N}$ 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: $E_{1},...,E_{M}$ as iid variables, drawn from the distribution $p(E)$ (Gaussian, but we remain general in this section). It is useful to use the following notations:

• $P^{<}(E)=\int _{-\infty }^{E}dxp(x)$, it is the probability to find an energy smaller than E.
• $P^{>}(E)=\int _{E}^{+\infty }dxp(x)=1-P^{<}(E)$, it is the probability to find an energy larger than E.

We denote

$E_{\min }=\min(E_{1},...,E_{M})$

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 two key relations:

• The first relation is exact:

$Q_{M}(\epsilon )=\left(P^{>}(\epsilon )\right)^{M}$

• The second relation identifies the typical value of the minimum, namely $a_{M}$:

$P^{<}(a_{M})={\frac {1}{M}}$

.

Let us consider the limit, $\lim _{M\to \infty }(1-{\frac {k}{M}})^{M}=\exp(-k)$, which allow to re-write the first relation:

$Q_{M}(\epsilon )\sim \exp \left(-MP^{<}(\epsilon )\right)$

This relation holds only when $\epsilon \approx a_{M}$ and one hase to expand around this value. However, a direct Taylor expansion does not ensures that probabilities remain positive. Hence, we define $P^{<}(\epsilon )=\exp(-A(\epsilon ))$ and remark that $A(\epsilon )$ is a decreasing function. We propose the following Taylor expansion

$A(\epsilon )=a_{M}+A'(a_{M})(\epsilon -a_{M})=a_{M}-y_{N}(\epsilon -a_{M})$

Depending on the distribution $p(E)$ we have a different dependence on N or M of both $a_{M},y_{N}$. It is convenient to define

$y_{N}\propto N^{-\omega }$

We will see that three different scenarios occur depending on the sign of $\omega$. Using this expansion we derive:

• The famous Gumbel distribution:

$Q_{M}(\epsilon )\sim \exp \left(-e^{y_{N}(\epsilon -a_{M})}\right)$

• the typical fluctuations of the minimum $\sim 1/y_{N}$. In particular we can write:

${\overline {\left(E_{\min }-{\overline {E_{\min }}}\right)^{2}}}\propto N^{2\omega }$

Density of states above the minimum

For a given disorder realization, we compute $d(x)$, the number of configurations above the minimum with an energy smaller than $E_{\min }+x$. The key relation for this quantity is:

${\text{Prob}}(d(x)=k)=M{\binom {M-1}{k}}\int dE\;p(E)[P^{>}(E)-P^{>}(E+x)]^{k}P^{>}(E+x)^{M-k-1}$

Taking the average ${\overline {d(x)}}=\sum _{k}k{\text{Prob}}(d(x)=k)$, we derive

${\overline {d(x)}}=M(M-1)\int dE\;p(E)\left[P^{>}(E)-P^{>}(E+x)\right]P^{>}(E)^{M-2}$

In the above integral, $E$ is the energy of the minimum. Hence, we can use the Taylor expansion $A(E)=a_{M}-y_{N}(E-a_{M})$. In particular we can write

• $p(E)={\frac {d}{dE}}P^{<}(E)=-A'(E)e^{-A(E)}\sim y_{N}e^{y_{N}(E-a_{M})}/M$
• $P^{>}(E)-P^{>}(E+x)=e^{-A(E+x)}-e^{-A(E)}\sim e^{y_{N}(E-a_{M})}\left(e^{y_{N}x}-1\right)/M$
• $P^{>}(E)^{M-2}=Q_{M-2}(E)\sim \exp \left(-e^{y_{N}(E-a_{M})}\right)$

Calling $u=y_{N}(E-a_{M})$ we obtain

${\overline {d(x)}}=\left(e^{y_{N}x}-1\right)\int _{-\infty }^{\infty }due^{2u-e^{u}}=\left(e^{y_{N}x}-1\right)\quad {\text{with}}\;y_{N}\sim N^{-\omega }$

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 $\sim 1/M$. 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:

${\frac {z_{\alpha _{\min }}}{\sum _{\alpha =1}^{M}z_{\alpha }}}={\frac {1}{1+\sum _{\alpha \neq \alpha _{\min }}z_{\alpha }}}\sim {\frac {1}{1+\int _{0}^{\infty }dx\,e^{-\beta x}\left(e^{y_{N}x}-1\right)}}$

• In the high temperature phase, for $\beta <y_{N}$, 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 $\beta >y_{N}$, the above integral is finite. Hence, setting $\beta =1/T,T_{f}=1/y_{N}$ you can find

${\frac {z_{\alpha _{\min }}}{\sum _{\alpha =1}^{M}z_{\alpha }}}={\frac {1}{1+{\frac {T^{2}}{T_{f}-T}}}}$

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

Let us recall $y_{N}\sim N^{-\omega }$, so that three situations can occur

• For $\omega <0$, 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 $\omega >0$ 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 $\omega$ 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 $\omega =0$ 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: the Gaussian case

Specify these results to the Guassian case and find $P^{<}(E)=\int _{-\infty }^{E}dxp(x)\sim {\frac {\sigma }{{\sqrt {2\pi }}|E|}}e^{-{\frac {E^{2}}{2\sigma ^{2}}}}\;$ for $x\to -\infty$

• the typical value of the minimum

%

$a_{M}=\sigma {\sqrt {2\log M}}-{\frac {1}{2}}{\sqrt {\log(\log M)}}+O(1)$

• The expression $A(\epsilon )={\frac {\epsilon ^{2}}{2\sigma ^{2}}}-{\frac {\sqrt {2\pi }}{\sigma }}\log |\epsilon |+\ldots$
• The expression of the Gumbel distribution for the Gaussian case

$Q_{M}(\epsilon )\sim \exp \left(-e^{-{\frac {\sqrt {2\log M}}{\sigma }}(\epsilon -a_{M})}\right)$

Bibliography

• Theory of spin glasses, S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5 965, 1975