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 $N$ sitees $i=1,2,\dots ,N$. The enregy is writteen 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 indicate the avergage over the couplings called disorder average, with an overline:

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

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

• Gaussian couplings: $\pi (J)=\exp (-J^2/2)/\sqrt{2\pi }$
• 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:

$E=-\sum _{i,j}{J}_{ij}2N{\sigma }_i{\sigma }_j$

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

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

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

Bibliography

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