The simple 2D Ising model, shown schematically on the left, can be made equivalent to any other more complicated spin model, such as those on the right.

The simple 2D Ising model, shown schematically on the left, can be made equivalent to any other more complicated spin model, such as those on the right.

Christian Hackenberg

‘Shocking’ unification reduces a lot of tough physics problems to just one

It's the sort of physics advance that Sauron might appreciate. The villain in J. R. R. Tolkien's fantasy trilogy, The Lord of the Rings, gives the kings of men, elves, and dwarves magic rings, but then forges a single ring that controls all the others. In a similar way, a duo of theoretical physicists has come up with a way to transform all the disparate members of a vast family of complex systems known as spin models into different shades of a single simple model, which now serves as the one to rule them all.

That "Ising model" is the simplest spin model and already has a legendary history. The advance could have implications well beyond physics, as spin models have been used to simulate everything from stock markets to protein folding. "I find it pretty shocking," says David Perez, a mathematician at the Complutense University of Madrid (UCM), who was not involved with the work. "What is surprising is not that there is a universal model, but that it is so simple."

Spin models were invented to explain magnetic materials, such as iron and nickel. Those metals can be magnetized because each of their atoms acts like a tiny bar magnet. At high temperatures, the jiggling atoms point in random directions and their magnetic fields cancel one another. However, below the so-called Curie temperature, the material undergoes a "phase transition" much like water freezing into ice, and all the atoms suddenly point in the same direction. That alignment reduces the atoms' total energy and makes their magnetic fields add together. Because each atom's magnetism originates from the spin of an unpaired electron within it, models of how magnetism arises are known as spin models.

The Ising model was the first spin model, invented in 1920 by German physicist Wilhelm Lenz, who gave it to his student Ernst Ising to analyze. In it, each atom is a simple object that can point either up or down. Each spin flips randomly with thermal energy, but it interacts with its neighbors so that each pair of spins can lower its energy by pointing in the same direction. Each spin can also lower its energy by aligning with an externally applied magnetic field. The coupling between each pair of spins can be different, as can be the external field applied to each spin.

Ising hoped to show that below a certain temperature the spins would undergo a magnetic phase transition. However, he could "solve" only the 1D Ising model—a single string of spins—and found it had no phase transition. Ising speculated that the 2- and 3D cases wouldn't, either. Then in 1944 the enigmatic Norwegian-American chemist Lars Onsager solved the Ising model with uniform couplings and no external fields on a 2D square pattern of spin. The famously incomprehensible Onsager, who won the 1968 Nobel Prize in Chemistry for earlier work but also lost two faculty jobs, showed that the 2D Ising model does have a phase transition—the first seen in a theoretical model. Onsager's tour de force calculation is now legendary, although he published it only 2 years after the fact. The 3D Ising model is still unsolved.

Meanwhile, spurred in part by Ising's difficulties, physicists invented plenty of other spin models. Instead of up and down, the spins can have, say, five possible settings, or like compass needles can point in any direction. The spins might also interact in groups larger than pairs and with spins far beyond their neighbors. Spin models have found use outside physics. For example, the spread of an epidemic might be simulated on a spin model with spins having three states corresponding to well, sick, and recovered. "Spin model is a really bad name for something that's a lot more general," says Gemma De las Cuevas, a theoretical physicist at the Max Planck Institute of Quantum Optics in Garching, Germany.

But all those disparate spin models can be transformed into the good old 2D Ising model, De las Cuevas and Toby Cubitt, a theorist of University College London, report online today in Science. Crudely their proof works as follows. First, the two scientists note that the up-or-down Ising spin resembles the true-or-false character of a logical statement such as "the car is white." They then prove that any particular 2D Ising model—i.e., with a particular set of coupling and external fields -- is equivalent to an instance of a logical problem called the satisfiability, or SAT, problem, in which the goal is to come up with a set of logical statements, A,B,C, … that satisfy a long logical formula such as "A and not (B or C) …" The theorists present a way to map the SAT problem onto the 2D Ising model.

Next, they show how any other spin model can also be translated into a SAT problem. That SAT problem can then be translated onto the 2D Ising model, thus making the two spin models equivalent. There is a price to pay, however. The 2D Ising model must have more spins than the original spin model. But De las Cuevas says that the computational demands of the Ising model are only modestly bigger than those of the original model. "If you could explain all of the parameter regions of the 2D Ising model with fields, that would be equivalent to probing all possible spin models," she says.

That's a big “if.” Although Onsager solved the 2D Ising model with uniform couplings and no external fields, the general problem with nonuniform couplings and external fields remains unsolved and is among the most computationally demanding problems there is—with the number of computational steps exploding exponentially with the number of spins. "It's surprising that you can map any model onto this simple model," says Miguel Angel Martin-Delgado, a theoretical physicist at UCM. "But you can take the other side, which is that this simple-seeming model is as complex as any other."

De las Cuevas agrees and says that the value of the advance may come in practical computations. It provides a recipe for translating any spin model, no matter how baroque, into a 2D Ising model, with the complexity of the original model encoded in the couplings between the Ising spins and the magnetic fields. If that recipe can be optimized, then it may be easier to simulate on a computer the Ising model instead of the original model, De las Cuevas says. "I think there's a lot of room for thinking, 'Hey, now I can study this model that I couldn’t before, using this universal model.'"

The advance might also make into the textbooks. The Ising model is introduced in statistical mechanics courses as the simplest spin model. Future texts might also note that in spite of its simplicity, it can reproduce all other spin models. In a sense, it's all you need to know.