Condensed-matter physics and theories of everything

One of the most exciting themes running through condensed-matter physics is the close mathematical association between how electrons behave in certain crystals and how particles behave in certain field theories.

I first noticed the association in 2005 when I wrote a news story about topological quantum computing. The basic currency of quantum computation, topological or otherwise, is the qubit. Until it’s read out, a qubit exists in a superposition of states that encompasses not only the classical bit’s 0 and 1 but also nonclassical states in between. If you could operate on k entangled qubits, a potentially vast space of 2^k values becomes available for computation.

To work, a quantum computer must protect the qubits’ all-important superposition and entanglement from heat, vibration, stray electric fields, and other environment intrusions—at least for long enough to perform a calculation step. Topological qubits are more robust against those intrusions than other qubits because their entanglement abides not in their geometric disposition, which can be perturbed by moving just one qubit, but in their topological disposition, which can’t.

Field theory, to my surprise, was involved in both the conception of topological computing and in ideas for its implementation. Solving a certain class of problems, which includes optimizing a traveling salesman’s route , is mathematically equivalent to calculating a topological invariant that characterizes the tangledupness of a three-dimensional knot. The invariant looks like a certain field theory in two spatial dimensions plus time, which, it turns out, might describe a certain state in the fractional quantum Hall effect (FQHE).

The upshot of that chain of mathematical resemblances is that you can, in principle, solve the traveling salesman problem and others by manipulating FQHE quasiparticles in a sliver of gallium arsenide. Researchers in labs in the US and elsewhere are trying to build an FQHE computer.

Skyrmions and Majorana fermions

Since writing about topological quantum computing, I’ve encountered other areas in which mathematical structures from field theory appear in condensed-matter contexts. Ettore Majorana’s 1932 description of the neutrino could turn out to fit the quasiparticles in superconducting strontium ruthenate, even if it proves inapt for neutrinos themselves. Skyrmions, the elementary particles that Tony Skyrme postulated in 1962 to account for the diversity of baryons, resemble vortices observed in manganese silicide.

What do these and other resemblances mean? They could be manifestations of shared symmetries. In two dimensions, it’s mathematically possible for a particle to be neither a fermion nor a boson but something in between—an anyon, as Frank Wilczek called the possibility. If, as some theorists assert, certain FQHE quasiparticles are anyons, shared symmetries would be confirmed, but perhaps nothing else more profound.

On the other hand, the resemblances could, as Xiao-Gang Wen proposed , indicate that electrons, photons, and other particles in the universe emerge from elementary structures whose mathematical origin lies in condensed-matter physics, not string theory.

Whether they connote shared symmetries or something deeper, the mathematical resemblances between condensed matter and field theory have proven fruitful. Yoichiro Nambu won his share of the 2008 Nobel Prize in Physics for discovering spontaneous symmetry breaking in field theory. The path to that discovery lay through previous work by Philip Anderson and others on superconductivity.

Thanks to Ravi Chathuranga and Maire Evans, fans of Physics Today‘s Facebook page , for suggesting I tackle theories of everything.