Theory of spin glasses – Theory of spin glasses. II asserted that the essential mechanism underlying spin glass behavior was the presence of both ferromagnetic and antiferromagnetic interactions that were quenched randomly in place. video

place all the spins on a three-dimensional cubic lattice. Then the Edwards-Anderson (hereafter EA) Hamiltonian is

$\mathcal{H}_\mathcal{J} = -\sum\limits_{\langle x, y \rangle} J_{xy} \mathbf{m}_x \cdot \mathbf{m}_y - \mathbf{h} \cdot \sum\limits_x \mathbf{m}_x$

_{where $x,y$ denote sites in the cubic lattice and the notation
$\langle x,y \rangle$ means that the sum is over nearest-neighbor (i.e., adjacent) sites only; $\mathbf{m}_x$ denotes the localized magnetic moment (or, more simply, spin), which for now we treat as a classical vector, at site $x$; $J_{xy}$ is the magnetic coupling between nearest-neighbor sites $x$ and $y$;and $\mathbf{h}$ is a uniform external field (which could be zero) acting on the system.}

An even simpler formulation is to treat the spins not as vectors, as in (4.2), but as simple binary variables taking on the values $\pm 1$ only. This is known as the EA Ising model:

$\mathcal{H}_\mathcal{J} = -\sum\limits_{\langle x, y \rangle} J_{xy} \sigma_x \cdot \sigma_y - \mathbf{h} \cdot \sum\limits_x \sigma_x$

The difference with the Heisenberg Hamiltonian is that the couplings $J_{xy}$ depend on the particular edge $\langle x, y \rangle$ and are chosen randomly and independently from a probability distribution, at the initial time, and stay fixed thereafter (Quenched disorder). On the $\pm J$ model, there are only two possible values of $J_{xy}$. An important question is which properties depend on the specific realization and which don’t?

_{The EA Hamiltonian is a very simplified model. If you wanted to faithfully model a dilute magnetic alloy, for example, you would use the RKKY coupling between all pairs of spins with the spins situated at random locations (which would then be quenched).}

Properties of the model

Frustration

Ground States of the 2D Edwards-Anderson Spin Glass - Michael Damron

Spin glass phase transition

EA order parameter

Actually, it turned out there is an infinity of order parameters, see Replica symmetry breaking

(silent video)

Some properties