two electron spins line up in opposite direction and the Ising model

Question

What cofuses me with the Ising Model is the claim that these nearby little magnets want to face the same direction. facing the same direction, they have lower energy. This seems to conflict with the fact that north poles of magnets repel each other.

So little magnets act different from big everyday magnets?

But then, it's said that if you bring two electrons near enough that they can feel each other's magnetic field, then if you wait enough, a photon will be emitted and they will get into an entangled state such that their spins are facing opposite directions. This seems to conflict with the claim that little magnets facing the same direction have lower energy.

I am confused. It just looks like there are two kinds of little magnets and one kind's north poles repel each other, and another kind's north poles attract each other. What is really going on?

Answer 1

The Ising model is a simple model not meant to describe the behavior of electrons in a real material. In the Ising model adjacent spins affect each other only. Thus if $s_i$ is one spin and $s_j$ is a nearest neighbor spin, the interaction energy for that pair is given by $-Js_i s_j$. From this you can see that if the spins are the same, the product of the $s$'s is positive (plus one in the usual notation), and if they are different the product is negative (usually taken to be minus 1). $J$ is basically the size of the interaction energy and is usually taken to be positive.

Noting the minus sign above, you can see that aligned spins have lower energies in this model than antialigned spins.

The spins involved in this simple, but interesting model, do not have to be electrons. Thus the second part of your question is both correct and has little or nothing to do with the Ising model.