The Ising model is an idealized statistical mechanics model of ferromagnetism—the general mechanism that explains how materials can become permanent magnets.
An Ising model represents atoms as variables that can occupy just two spin states, +1 and -1, which are arranged in a graph (usually a simple lattice) where every variable interacts only with its nearest neighbors.
Let’s start by imagining a 1-D Ising model system, so we just have a line of some finite amount of atoms N, labeled start to end as 1 to N. The system is analogous to a mathematical bracelet, where N+1 will “curve” back to the beginning. In other words, even though there are only N atoms, N+1 is actually defined in the system—it’s the same as atom 1. N+2 is then the same as 2, and so on (so generally speaking, kN+x is the same as x, where k is an integer).
The equation for the Hamiltonian H of this system (if you don’t know what that is yet, just think of it as the equation for the total energy) can be represented like this:
Where atoms are represented as S and their subscript denotes the position. Another way of writing the first sum would be:
Notice that every atom is represented in the sum exactly twice (remember, S_N+1 is the same as S_1), so that every relevant interaction is represented–we earlier required that each atom only interacts with its nearest neighbor, and in a 1-D system each atom would have exactly two. So, having two interactions per atom makes sense.
There are some consequences to this type of system, which are neat but not exclusive to the model:
First, that opposite spins cost more energy. If we have a system with every spin aligned (i.e. all spins are either +1 or -1), then switching one atom to -1 will “cost” 2J energy, since it interacts with the two atoms around it.
The effect of this is that the system will tend towards a uniform spin over time, since alignment is always a lower energy state than nonalignment.
The second consequence is that this type of Ising model system will tend to become ferromagnetic, since every spin wants to be the same. However, while it’s easy to see this trend in the theoretical model, in real life, as you might expect, systems tend to be a lot more complicated, with less regular, predictable structures and neighbors, as well as a heaping of outside sources of energy (predominantly heat).
But that’s okay! It’s not what the model is for. It’s useful as an ideal model, and as a demonstration for why systems become (and remain) ferromagnetic if they are arranged in such a way as to make the lowest energy state be where their spins are relatively aligned.
Note: this should also help demonstrate why magnets have Curie temperatures, a temperature point past which they no longer are magnetic; it’s because there is a point where the energy provided by heat exceeds the associated energy cost of nonalignment.
More Ising Math
Let’s calculate the probability of any given state in the 1-D Ising model.
The amount of possible states should be obvious. Since each of the N atoms can be in two possible states, the number of configurations is 2^N. If every state were equally likely, our answer would trivially be 1/2^N for all states.
But the probability of a given state is actually more complicated. Remember that it tends to want to be more aligned, since that is the lowest energy state. So, the actual configuration probability of a state should be some function incorporating the energy—it’s Hamiltonian.
It turns out to be this guy:
Where beta is the reciprocal of the temperature and Z is just a normalization constant (to ensure the probabilities sum to 1) given by:
The solution for the one-dimensional case follows from these calculations and the Hamiltonian. So far, physicists have solved the two-dimensional case, but the three-dimensional Ising model is an unsolved problem! Pretty surprising for an ideal system.