# The 2016 Nobel Prize in Physics goes to mathematicians working in Flatland

The 2016 Nobel Prize in Physics has been awarded to David J. Thouless, Duncan Haldane and J. Michael Kosterlitz for “theoretical discoveries of topological phase transitions and topological phases of matter.” Their work represents a set of mathematical insights about exotic phases of matter like superfluids and superconductors, but it also applies to regular old semiconductors too. In short, these guys wrote the mathematical statements that line up our theories with what we see.

To explain these discoveries, a member of the Nobel committee seriously held up baked goods to the audience.

It all stems from the properties observed in sheets of a single-atom-thick, electrically conductive material, sandwiched between layers of semiconducting material and held at cryogenic temperatures. While it’s an infuriating misuse of language, for purposes of this discussion, let’s go along with the conceit of calling such a thin sheet of material two-dimensional.

When cooled to just a few degrees above absolute zero, and subjected to a perpendicular magnetic field of 15 Tesla (just barely weaker than the magnetic field required to levitate a frog), that conductive 2D layer started acting very differently. The electrons permeating it started acting like a superfluid, with zero viscosity. Within the sheet, there developed pairs of linked, stable electron vortices that hovered politely in place (PDF) when the conducting material was kept cold — but suddenly decoupled and zoomed away from one another when the temperature rose.

The disorder left in the wake of these vortices as they zoom apart is the physical avatar of matter undergoing a topological phase change: It’s what phase change looks like in the flatlands. That phase transition is now called the KT (Kosterlitz-Thouless) or BKT transition, where the B stands for Berezinskii, now deceased, who had presented very similar ideas, a little like Newton and Leibniz, or Rosalind, Watson and Crick.

Zooming out a little gives more detail on other quantum metamaterial wackiness. Electrical conductance describes the collective motion of electrons, and in this 2D quantum electron fluid, electrical conductance behaves topologically. Heating the system induced a wave of these vortices that propagated outward across the plane, inducing a phase change and a quantized leap in electrical conductance as the electrons were freed from their relatively ordered state. Cooling the system back through the phase transition resulted in a symmetrical, quantized leap in electrical impedance as order was imposed on the system.

Most things in nature lie along a normal distribution; like mathematics in general, the bell curve is everywhere and you just can’t get away from it. That’s what makes the behavior of these metamaterials so fundamentally different. They don’t always exhibit their behavior along a bell curve, where there’s a smooth statistical slide along a continuum.

Reduce the strength of the magnetic field applied to the chilled semiconductor sandwich, and the 2D sheet’s electrical conductance snaps to precise integer multiples of itself: It doubles, triples, quadruples, but it’s never 2.2x itself. It acts stepwise, in a way that’s better described by drawing a comparison with the set of real numbers versus integers. Or maybe pastries.

Topological behavior like what these researchers describe isn’t limited to superconductors and superfluids. Haldane observed this property in thin slices of semiconductors, even when there’s no magnetic field being applied.

Beyond integer-like behaviors, topology is also useful for considering properties that remain constant when a thing is deformed, like when it’s stretched or bent, but change when the thing is torn or glued. That’s the real reason why the Nobel committee explained the concepts behind topology using baked goods as an analogy. It wasn’t just because scientists are donut-lovin’ dorks. In terms of their topological properties, a bowl and a cinnamon bun belong in the same class, because they’re both one contiguous solid with zero holes. A coffee mug and a donut are also in the same topological class, because they have one hole. A pair of glasses has two, and a pretzel has three. It can be any number of holes, but the number has to be an integer; you can have part of a whole, but you can’t have part of a hole.

Math based on this logical framework (here’s the Nobel Committee’s equation-dense PDF, not for the faint of heart) turned out to be really well suited to dealing with the materials the trio was studying, because they behave in this strange, integer-like fashion. Topology is that math: It’s a branch of mathematics concerned with properties that change stepwise. Where you can have one and a half pies, you can’t have one and a half dimensions, say, or 1.29 topological properties.

How does this all relate to PCs?

With all this talk about semiconductors, you may be wondering when we’ll see these developments writ into real-world PCs and other consumer hardware. The answer is that it’s pretty unlikely, at least for now. In 1985 the Nobel in physics went to Klaus von Klitzing for observing this phenomenon in the flatlands in the first place. These three researchers will split their prize for bringing theory into agreement with what von Klitzing and others had observed *in situ*.

However, and this is a big however, the prize-winning research made it easier to handle and model some novel topological materials: things with quantum-informed properties, like conductance, that we can see and interact with on the macroscale.

So-called “odd” and “even” magnets are one such metamaterial. Separate entirely from the poles of a magnet, the oddness or evenness of a magnet has to do with its spin parity; “even” magnets behave topologically. “Odd” magnets don’t. The researchers found that, rather than being properties of the material as a whole, these properties reveal themselves at the “edges” — the ends of a “one-dimensional” chain of singly entrained magnetic atoms, for example, or at the places where a magnetic field stops permeating a slice of superconductor.

All this has implications for both semiconductors and superconductors. If we can externally manipulate the quantum properties of a system, harnessing that gives us an inroad to quantum computing. But this is definitely an example of a development that mostly benefits researchers in the field. It’s not yet ready for commercial deployment.

*Title image is of F. Duncan Haldane of Princeton. Photo: Princeton University, Office of Communications, Denise Applewhite*