The World in a Spin
What a Difference a D Makes
Once Onsager had shown the way, the two-dimensional Ising model was solved again by several more methods. Solutions were also discovered for certain other two-dimensional models, which share a basic conceptual framework with the Ising model but differ in details of the lattice or the spin-spin interaction. But, significantly, all of these solutions are confined to flatland. In the three-dimensional world, not one Ising-like model has been solved exactly. This is a bit disappointing for three-dimensional physicists who would like to understand three-dimensional matter.
What is the essential difference between a 2D and a 3D model? Obviously, one is flat and the other isn't, but there's more to it than that. Geometry per se never enters the Ising model; distances and angles are simply not part of the problem description. All that matters is the topology—the pattern of connections between nearest-neighbor sites. If you were to build a three-dimensional cubic framework out of infinitely stretchy rubber, you could flatten this lattice by sliding layers of sites apart and then smashing them down into a plane. Now the 3D lattice would be geometrically two-dimensional, and yet it would still differ topologically from a true 2D lattice. The difference is that many of the bonds would be nonplanar—they would cross over one another—whereas a 2D lattice can always be drawn without crossings. This topological distinction seems to be at the root of the difficulty of 3D Ising models.
Tullio Regge of the Politecnico di Torino and Riccardo Zecchina of the Abdus Salam International Centre for Theoretical Physics in Trieste have recently looked at what happens to the partition function when you start with a two-dimensional lattice and add nonplanar bonds one by one. In this way they explore the territory between 2D and 3D. Each added nonplanar bond doubles the labor needed to solve the model—an exponential increase. Whether or not this approach leads to a practical method for calculating the 3D partition function, it shows more clearly why that task is so hard.
Curiously, adding still more dimensions beyond three makes the Ising model simpler again rather than harder. A four-dimensional lattice is so densely connected that each spin responds to the averaged magnetic field of all the other spins, and analysis is easy. Thus it seems there's something special about three dimensions; perhaps the world we live in was created explicitly as a vexation to Ising modelers.
I do not want to leave the impression that nothing at all is known about the 3D Ising model. An exact mathematical solution is still lacking, but the region near the critical point has been explored by various methods of approximation. One approach, called series expansion, would have been familiar to Laplace. The idea is to start with the known solutions at very high and very low temperature and extrapolate into the more problematic region between. Another approximation method has a name that makes it sound like an organization of diplomats or economists: the renormalization group. The simplest version of this algorithm gathers sets of spins into blocks, replaces each block with a single new spin, and finally adjusts the couplings between spins to compensate for the coarsening of the lattice.
Still another important technique for Ising studies is the Monte Carlo method, which relies on a random process to approximate the probability distribution of the spin states. I shall say a little more about Monte Carlo methods below.