The mathematics hidden in materials keeps getting more exotic. Topological states of matter — which derive exotic properties from their electrons’ ‘knotty’ quantum states — have shot from rare curiosity to one of the hottest fields in physics. Now, theorists are finding that topology is ubiquitous — and recognizing it as one of the most significant ways in which solid matter can behave.

In the past few years, physicists have identified a ‘fragile’ version of topology that might occur in almost all crystalline solids, according to a preprint posted in May1. Another study, published2 last month in Nature, describes hints of a fragile state in the electrons of a carbon-based device — which, if confirmed, would be the first experimental evidence for fragile topology.

The strange topology that is reshaping physics

It is too early to say whether these discoveries will have a major impact on practical materials. But researchers have found that they might help to explain certain kinds of superconductivity, and say that the phenomenon is also likely to be important in photonics, technologies that carry information in light pulses rather than electrons. Fragile topology could also have implications for researchers who use supercomputers to simulate the behaviour of materials.

The latest studies show that fragile topology “is not just a radical, academic rabbit hole that people are going down”, says Ashvin Vishwanath, a theoretical physicist specializing in condensed matter at Harvard University in Cambridge, Massachusetts. “I am having a hard time keeping up with the field, even though it’s just a year old.”

Loop the loop

Topology is the branch of mathematics that deals with deformations that reshape objects continuously — as opposed to those that cut or break objects, in the way that cutting two linked loops unlinks them. In some materials, electrons can exist in ‘knotty’ quantum states, and these can, for example, keep an electron moving in a particular direction, because altering course would require an abrupt change of its state, akin to cutting a knot.

As a result, the physical qualities are ‘topologically protected’. The most celebrated example of this is the quantum Hall effect discovered in 1980 in certain 2D materials that conduct electricity, in which resistance is unaffected by small changes in variables such as temperature. The effect is so robust that it was taken as the basis for defining the ohm — the unit of measurement of resistance — in the reformed SI that went into effect in May. A similar effect in 3D systems allows some materials called topological insulators to be — despite their names — perfect conductors on their outer edges while the bulk of the material is insulating.

‘Strongly topological’ materials that harbour these robust effects are seen to be promising as thermoelectric materials, which convert heat into electricity. And some physicists hope that the materials will provide the basis of future topological quantum computers, which could solve certain problems exponentially faster than classical computers.

Strong topological properties come from quirks in the quantum states of electrons: rather than crowding around individual atoms as they do in a typical insulator such as solid rock salt, some electrons in a topological material are ‘delocalized’ and share collective quantum states that stretch over the bulk of a material.

How ‘magic angle’ graphene is stirring up physics

But theorists have calculated that there are some materials that have delocalized electrons, yet don’t have strongly topological properties. In other words, strongly topological materials make up only one category in a vast taxonomy of delocalized states. Among them are electron states that are protected from small perturbations, but aren’t quite as robust as strongly topological states. They can be made normal, for example, by slightly changing the impurities mixed into the crystal. In a 2018 study3, Vishwanath’s team dubbed the phenomenon fragile topology.

Twisted discovery

At first, physicists weren’t sure whether fragile topology was important, but that changed after a surprising discovery in March 2018. Physicists revealed5,6 that two stacked layers of graphene, the single-atom-thick form of carbon, become superconducting when they are misaligned at particular ‘magic’ angles, carrying electricity with zero resistance. Vishwanath and others soon calculated that certain electron states in this twisted graphene display fragile topology. That was “amazing”, Vishwanath says. “We had thought this had no application. Then there was this huge deal.”

It is unclear so far whether fragile states actually play a part in making twisted graphene superconductive. Whereas strong topology manifests in known, measurable ways, fragile topology might be subtler.

Still, fragile topology is bound to affect materials’ behaviour, some physicists say, because it is even more ubiquitous than strong topology. Studies have suggested that about one-quarter of materials are strongly topological. But in a preprint posted in May in the arXiv repository1, physicists found that almost all materials have some electrons in fragile topological states. They systematically searched for fragile topology across a database of known crystals, and found hundreds of thousands of examples of the phenomenon. “It seems that basically every material has something topological in it” when fragile topology is taken into account, says Andrei Bernevig, a theoretical physicist at Princeton University in New Jersey and lead author of that paper.

Now, the first experimental hints of fragile topology are emerging. A June study in Nature2 found evidence of fragile topology in a non-twisted double layer of graphene. The researchers, led by Joshua Island at the University of California, Santa Barbara, were trying to concoct a graphene-based strong topological insulator, which could be used to store information in future topological quantum computers. They sandwiched the graphene between layers of another 2D material, tungsten diselenide, and applied an electric field. As the field varied, they recorded electrons moving at the edge of the device, as expected in topological insulators. “Once we saw that new phase, then we were racing to figure out what it was,” says Island.

Surprise graphene discovery could unlock secrets of superconductivity

But other measurements showed that it could not be a conventional topological insulator. So Island turned to a theorist colleague who realized that this was a first experimental hint of a fragile state7.

Spanner in simulations

Fragile topology could affect numerical simulations of the physics of materials. To make supercomputer calculations about materials more manageable, researchers often adopt simplifying assumptions, but those might not be valid in the presence of fragile states, says Jennifer Cano, a theoretical condensed-matter physicist at Stony Brook University in New York State, who has worked4 on fragile topology.

Fragile topology might be easier to observe experimentally in devices that carry light than in solid materials. Its effects might be more consequential in these, too. Thomas Christensen, a physicist at the Massachusetts Institute of Technology in Cambridge, says that according to his preliminary calculations, many ‘topological’ devices that have been proposed in photonics might in fact display examples of fragile topology.

Although it remains to be seen whether fragile topology will have a lot of applications, to a theorist it is interesting, says Barry Bradlyn, a theoretical physicist at the University of Illinois at Urbana–Champaign who co-authored an early paper on the subject4. It “defies the conventional lore” of how electron states in materials are supposed to work, he says.