Monopoly

There's a lot of buzz about two reports in Science (here and here) on monopoles in magnetic solids. (The best story I saw is by Adrian Cho in ScienceNOW, the online news section of Science.) Much of the excitement comes because some candidates for a grand unified theory predict such particles. But although the results are fun, to my mind the enthusiasm is misplaced.

Eugenie Samuel Reich (author of Plastic Fantastic) got a jump on the coverage in her May story in New Scientist, which also illustrates the problem. The magazine cover screams: "THE MYSTERIOUS MONOPOLE: Predicted by theory; Hunted for decades; FOUND AT LAST." Her closing quote is more accurate: "These might not be exactly the monopoles that Dirac dreamed of, but that doesn't mean they're not remarkable." In what sense, then, was the predicted and hunted particle "found"?

I need to be careful here: last October, I wrote in New Scientist about nonabelian anyons, predicted particles for which outwardly identical arrangements can be distinct quantum states. Researchers are hopeful that these states could stably save quantum information for use in quantum computing. An earlier draft of that story, which I rather liked, began: "When it comes to unveiling fundamental particles, the Large Hadron Collider gets all the press. But in a few, much smaller labs around the world, some physicists suspect they've already found particles as exotic as any the new accelerator is likely to create."

So why do I think finding nonabelian anyons in a solid is more profound and fundamental than finding monopoles in a solid? To answer that question, I need to distinguish three levels of description of a complex system: how the particles interact, how they arrange themselves, and how they deviate from that arrangement. The descriptions are interconnected, but sometimes, one level is more interesting than another.

To be concrete, let's think about a regular lattice of atoms on a square lattice, each of which has a magnetic moment, which acts like a tiny bar magnet that can point in different directions. As it happens, this is somewhat similar to the spin system where the monopoles were seen.

To mathematically describe the interactions, physicists use an expression--unhelpfully called the Hamiltonian--for the total energy of any particular arrangement. The spins, for example, might favor arrangements with neighbors pointing in opposite directions. If they point in the same direction, it takes extra energy. The Hamiltonian just adds up the energy of every pair of neighbors. Fundamental theories, like proposed grand unified theories, aim at this basic level of the Hamiltonian, which for the universe as a whole is unknown. Experiments only probe how particles arrange themselves in response.

The arrangement that gives the lowest total energy for a particular Hamiltonian is the ground state. Sometimes it's obvious what this will be. On a square lattice, for example, spins forming a checkerboard pattern will always have neighbors pointing in opposite directions. On a triangular lattice, though, some neighbors must point the same way. In other cases, physicists don't know what the ground state is, even when they know the Hamiltonian. This is the situation for the two-dimensional electron systems, where nonabelian anyons may exist: the possible arrangements are very complicated and physicists aren't sure what the lowest-energy state is. (It might also depend on neglected details in the Hamiltonian.)

At temperatures above absolute zero, a system doesn't have to be in its ground state. Instead, it will include excitations away from the ground state (which necessarily require energy). For our spin system, for example, one of the spins on the checkerboard could be flipped so that it points the same direction as its neighbors. Although some specialized experiments can measure the ground state directly, many observable properties depend on these excitations. The excitations that can occur depend very much on the underlying ground state, although it's not always easy to tell what the most important excitations are going to be.

So where do monopoles fit in? In the recent experiments, spins in a crystal arrange so that their excitations act like these isolated magnetic poles. This is certainly unusual and interesting. But there is no reason to think that the Hamiltonian that describes the spin has any relationship to the one that describes the fundamental particles of the universe, any more than two equations, both solved by x=2, are necessarily related to each other. People have looked for monopoles in exotic places like cosmic rays because seeing them would give a profound clue about the Hamiltonian describing elementary particles. The monopoles seen here give no such clue.

Nonabelian anyons, by contrast, are inherently interesting particles. They may or may not be the excitations for certain two-dimensional electron systems studied in labs. We do know they can only exist in an effectively two-dimensional world, so they're not likely to turn up in the three-dimensional world of the Large Hadron Collider. But their quantum memory, which may even be useful, is a surprising property not shared by any known particle. In this case, the excitations (which reflect an unusual ground state) are the interesting thing, not the Hamiltonian.

To be fair, the new monopoles also have unique properties, not shared by any known particle. At that level, they are quite interesting in their own right. They just don't tell us much about grand unified field theories.