Haldane, Kosterlitz and Thouless did work at the "tip of a branch that joins the core of physics very close to the trunk"by Philip Ball / October 5, 2016 / Leave a comment
If it weren’t for the fact that my fellow science communicators relish a challenge, I would say that the physics Nobel committee has done them no favours. The work of this year’s laureates—Duncan Haldane, John Michael Kosterlitz and David Thouless, all of them British but all now working in the USA—concerns processes called topological phase transitions, and you could spend precious column inches explaining both the adjective and compound noun here without inching much closer at all to an explanation of what they are doing in conjunction or what the subject is all about.
I am duty bound to have a crack at it. But there is, frankly, no point in doing so without first digging down to a deeper stratum. I could end up saying vague things about weird electrical behaviour and offering a probably false promise that some fancy new electronic devices might emerge from it all, and I would only thereby be colluding with the pernicious myth that the value of science lies with its applications. (I call it pernicious not so much because it is what we are often told but because funding criteria increasingly force scientists to maintain the pretense too, which is far worse.)
There’s no point in denying that the field that Haldane, Kosterlitz and Thouless have helped to excavate lies far from everyday experience, and could thereby easily seem as esoteric as the frozen-nitrogen plains of Pluto. But this is the tip of a branch that joins the core of physics very close to the trunk. This trunk is rarely glimpsed, because there is a tendency to portray physics as a process of dissection and reduction whose ultimate quest is the fundamental particles of reality. It is not.
Undoubtedly, the search for new fundamental particles by CERN-scale proton-smashing is vital to the whole enterprise, and discovering fragments such as the celebrated Higgs boson lays bare some of nature’s innermost workings. But the reason it does so is not so much because we have found a new building block, but because these building blocks can embody and illuminate new principles. That is even more the case for the current hope (right now looking a little desperate) that CERN’s Large Hadron Collider will give us a glimpse of so-called supersymmetric particles.
But here’s the point. Regardless of what these building blocks are, there is one thing you can say about them for sure even now: there is an awful lot of them. And the reason why the centre of gravity of physics is widely misperceived is that we tend to cleave to the idea that collections of these fundamental particles are just the same thing over and over again.
They are not. If you want to catch sight of the trunk of this sprouting tree of knowledge called physics, you must embrace the slogan coined by the 1977 Nobel laureate Philip Anderson: “More is Different.” What Anderson meant was that you can’t deduce how lots of particles will behave simply from understanding how one of them behaves. Never mind recondite bosons—just think of a swallow. Biologists knew a vast amount about the anatomy, physiology and phylogeny of a swallow several decades ago, and they could for good measure have known also its genome down to the last piece of DNA, and still they would have been as perplexed as they were about how on earth flocks of swallows at dusk execute those extraordinary synchronized ballets called murmurations. They didn’t know this not because they were poor scientists, but because murmurations are a property of many swallows—and because they are a question of physics (though not necessarily of the kind that must be conducted within a building tendentiously labeled “Department of Physics.”)
A murmuration is what physicists call a many-body effect, which means it arises from the mutual interactions of many bodies—and doesn’t depend on the fine details of what those bodies are (starlings make murmurations too) but is much more a question of the mere fact of their interaction. The properties we observe in matter are almost entirely and inevitably many-body effects of their atoms and other constituent particles. What makes it tractable to study them mathematically is that the numbers of particles are typically so vast that one can approximate the collective behaviour from statistical averages of the behaviour of the components. That is why the field is known as statistical physics. (The intermediate stage between one and many—that is, few-body systems, where averages are less reliable—is arguably the hardest to handle, and is still something of an uncharted frontier.)
Most of what happens in the universe comes from many-body effects. They are what make the sky blue, and steam condense into water, and water swirl into eddies, and our heart cells beat in synchrony, and ecosystems maintain a stable flux of life and death, extinction and speciation. And among their constructions are topological phase transitions.
The beautiful discovery by Kosterlitz and Thouless in the 1970s was part of the lore of my research discipline long ago, and so I’m rather delighted to see it honoured this way. It concerns a specific kind of collective, many-body effect among interacting particles, exemplified by a simple model of a magnetic material. Think of many magnetic atoms stacked together on a regular grid, like tiny compasses arranged on the squares of a chessboard. The magnetic poles of each rotating needle feel the magnetic fields created by the poles of its near neighbours, and if the poles aren’t too shaken about by heat, we might expect them all to line up in the same direction, all the atomic norths and souths pointing the same way as the others. That’s what happens in a piece of magnetised iron, where all the atoms’ magnetic fields line up and produce a collective north and south pole in the material as a whole.
For the chessboard of compasses, however, this perfect alignment over the entire array is never possible. The system acquires flaws where neighbouring needles become aligned not in a single direction but around a loop, rather like the aligned grooves of your fingerprint twisting into a whorl on the pad of the finger. (Actually the formation of fingerprints, as growing skin buckles and creases, is an analogous problem to the magnet.) These flaws, called vortex defects, mar the ordering.
Nevertheless, Kosterlitz and Thouless (and independently the deceased Russian physicist Vadim Berezinskii) showed that at low temperatures the vortices can become stuck together in oppositely circulating pairs, keeping them tethered and limiting their disruptiveness. But warm up the system and the pairs can get unstuck, and you’re left with a disorderly mess of wandering vortices. This switch from rather orderly vortex pairs to disorderly lone vortices is called the Kosterlitz-Thouless transition, and it happens sharply at a specific temperature. The unpairing of vortex defects turned out to be an important feature of materials called superconductors, which, thanks to quantum-mechanical effects, can conduct electricity without any electrical resistance if they are made very cold.
The reason a block of iron can become magnetically ordered but the chessboard of compasses can’t is that the first is three-dimensional—the atoms are stacked top to bottom as well as side to side—whereas the second is two-dimensional: a flat sheet of atoms. This difference in dimensionality is crucial, and it is what makes edges different. One might even invoke an adjunct principle to Anderson’s: “Edges are Different.” They are places where you lose a dimension: the edge (surface) of a three-dimensional sphere is a two-dimensional shell, while the edge of a two-dimensional circle is a one-dimensional (bent) line. The loss of perfect 2D ordering that Kosterlitz and Thouless studied underlines the importance of dimensionality in many-body physics: it matters for the global behaviour of a system like this how many dimensions it has to play in.
Haldane’s work in the 1980s reiterated that fact. It was concerned with one-dimensional collections of particles, called spin chains, which you might loosely think of as a single row of magnetic atoms. I say loosely, because Haldane’s analysis of these systems brought in detailed quantum-mechanical considerations that take us well beyond what it might be wise to pursue here. The crucial point is this: these chains can maintain a certain kind of orientational ordering purely because of the geometric constraints arising from their symmetry, or one might say, their topology. You can no more break up the ordering than you can smooth a hemispherical shell into a perfectly flat sheet. The topology won’t allow it. And it’s a topological property of the entire array: a many-body effect.
This so-called “topological protection” has now become a lively field of study in the form of so-called topological insulators: materials that are insulating (not electrically conducting) as a whole, but can carry a current at their edges, where the dimensionality is reduced. In some topological insulators, the edge states can display the exotic property not just of conductivity but of superconductivity. The same principle can apply in other kinds of systems involving circulating, vortex-like motions. It’s possible to make shallow baths of liquid that are stirred into many vortices, and the half-vortices at the edges can carry sound in one direction but not the other.
Uses might be found for such structures in electronics, acoustics and beyond. But don’t hold your breath for that. They are mostly studied because they are interesting examples of many-body physics: a prime example of the unexpected things that the laws of nature and symmetry make possible. Topological phase transitions reveal a deep connection between collective many-body effects, shape, defects and dimensionality that also turns up in fields as diverse as liquid crystals and cosmology. It’s an example of how ideas in one area of physics can help explain effects in an entirely different one, and thereby illuminate laws that are based in abstract concepts such as symmetry, beyond the distractions of minute particulars—or particles, for that matter. I mentioned bird flocking not because it offers a nice visual metaphor for many-body physics, but because it demonstrably embodies a version of precisely the same physics as that of the 2D magnetic model studied by Kosterlitz and Thouless. And the extraordinariness of that, and not the prospect of Nobel prizes, is why most physicists do what they do.