Edwards Anderson model

The first significant theoretical attempt to describe spin glasses is the Edwards-Anderson model. For simplicity, we will consider the Ising version of this model. Ising spins take two values, $\sigma _{i}=\pm 1$, and are located on a lattice with $N$ sites, indexed by $i=1,2,\ldots ,N$. The energy of the system is expressed as a sum over nearest neighbors $\langle i,j\rangle$:

$E=-\sum _{\langle i,j\rangle }J_{ij}\sigma _{i}\sigma _{j}.$

Edwards and Anderson proposed studying this model with couplings $J_{ij}$ that are independent and identically distributed (i.i.d.) random variables with a zero mean. The coupling distribution is denoted by $\pi (J)$, and the average over the couplings, referred to as the disorder average, is indicated by an overline:

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

We will consider two specific coupling distributions:

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