The strange topology that is reshaping physics

Topological effects might be hiding inside perfectly ordinary materials, waiting to reveal bizarre new particles or bolster quantum computing.

Charles Kane never thought he would be cavorting with topologists. “I don't think like a mathematician,” admits Kane, a theoretical physicist who has tended to focus on tangible problems about solid materials. He is not alone. Physicists have typically paid little attention to topology — the mathematical study of shapes and their arrangement in space. But now Kane and other physicists are flocking to the field.


In the past decade, they have found that topology provides unique insight into the physics of materials, such as how some insulators can sneakily conduct electricity along a single-atom layer on their surfaces.

Some of these topological effects were uncovered in the 1980s, but only in the past few years have researchers begun to realize that they could be much more prevalent and bizarre than anyone expected. Topological materials have been “sitting in plain sight, and people didn't think to look for them”, says Kane, who is at the University of Pennsylvania in Philadelphia.


Now, topological physics is truly exploding: it seems increasingly rare to see a paper on solid-state physics that doesn’t have the word topology in the title. And experimentalists are about to get even busier. A study on page 298 of this week’s Nature unveils an atlas of materials that might host topological effects1, giving physicists many more places to go looking for bizarre states of matter such as Weyl fermions or quantum-spin liquids.

Scientists hope that topological materials could eventually find applications in faster, more efficient computer chips, or even in fanciful quantum computers. And the materials are already being used as virtual laboratories to test predictions about exotic and undiscovered elementary particles and the laws of physics. Many researchers say that the real reward of topological physics will be a deeper understanding of the nature of matter itself. “Emergent phenomena in topological physics are probably all around us — even in a piece of rock,” says Zahid Hasan, a physicist at Princeton University in New Jersey.

Some of the most fundamental properties of subatomic particles are, at their heart, topological. Take the spin of the electron, for example, which can point up or down. Flip an electron from up to down, and then up again, and you might think that this 360° rotation would return the particle to its original state. But that’s not the case.


In the strange world of quantum physics, an electron can also be represented as a wavefunction that encodes information about the particle, such as the probability of finding it in a particular spin state. Counterintuitively, a 360° rotation actually shifts the phase of the wavefunction, so that the wave’s crests become troughs and vice versa. It takes another full 360° turn to finally bring the electron and its wavefunction back to their starting states.


This is exactly what happens in one of mathematicians’ favourite topological oddities: the Möbius strip, formed by giving a ribbon a single twist and then gluing its ends together. If an ant crawled one full loop of the ribbon, it would find itself on the opposite side from where it started. It must make another full circuit before it can return to its initial position.

The ant’s situation is not just an analogy for what happens to the electron’s wavefunction — it actually occurs within an abstract geometric space made of quantum waves. It’s as if each electron contains a tiny Möbius strip that carries a little bit of interesting topology. All kinds of particles that share this property, including quarks and neutrinos, are known as fermions; those that do not, such as photons, are bosons.

Most physicists study quantum concepts such as spin without worrying about their topological meaning. But in the 1980s, theorists such as David Thouless of the University of Washington in Seattle began to suspect that topology might be responsible for a surprising phenomenon called the quantum Hall effect, which had just been discovered. This effect sees the electrical resistance in a single-atom-thick layer of a crystal jump in discrete steps when the material is placed in magnetic fields of different intensities. Crucially, the resistance remains unchanged by fluctuations in temperature, or by impurities in the crystal. Such robustness was unheard of, says Hasan, and it is one of the key attributes of topological states that physicists are now eager to exploit.

Physics with a twist
In 1982, Thouless and his colleagues2 unravelled the topology behind the quantum Hall effect, which ultimately helped to win Thouless a share of last year’s Nobel Prize in Physics. Like the electron’s spin, this topology occurs in an abstract space. But in this case, the underlying shape is not a Möbius strip, but the surface of a doughnut. As the magnetic field ramps up and down, vortices can form and disappear on the surface, like the wind pattern around the eye of a hurricane (see ‘All wound up’).




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