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 $i=1,2,\ldots ,N$. The enregy is writteen as a sum between the nearest neighbours <i,j>:

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

The key point of the model proposed by Edwards and Anderson is that the couplings $J_{ij}$ are i.i.d. random variables with zero mean. We set $\pi (J_{ij})$ the coupling distribution indicate the avergage over the couplings called disorder average, with an overline:

${\bar {J_{ij}}}\equiv \int dJ_{ij}\,J_{ij}\pi (J_{ij})=0$

It is crucial to assume ${\bar {J_{ij}}}=0$, otherwise the model displays ferro/antiferro order. We sill 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) = \frac{\exp\left({-J_{ij}^2/2}\right)/\sqrt{2 \pi}}
• Coin toss couplings $J_{ij}=\pm 1$, selected with probability $1/2$.

Bibliography

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