Getting to grips with fragile topology

A class of materials known as topological insulators has excited those searching for the materials of the future for more than a decade. 

The reason? They have an uncommon property: their interiors are insulators, where electrons cannot flow, but their surfaces are perfect conductors, where they flow without resistance.

Well, not quite all. Two years ago it was discovered that some topological materials actually can’t conduct current on their surface – a phenomenon now known as fragile topology.

It is “a strange beast”, says Andrei Bernevig, a professor of physics at Princeton University in the US, and “predicted to exist in hundreds of materials”. 

Now he and researchers from Switzerland, China, Israel, Germany and Spain say they have a better understanding of the how and why.

First, they constructed a mathematical theory to explain what is happening inside the materials. Then they tested the theory by building a life-sized topological material from 3D-printed plastics.

The explanation, they found, lies in the connection between the electrons on the surface and those in the interior. 

Electrons can be thought of not as individual particles but as waves that spread out like ripples of water from a pebble tossed in a pond. In this quantum mechanical view, each electron’s location is described by a spreading wave that is called a quantum wavefunction. 

In a topological material, the quantum wavefunction of an electron in the bulk spreads to the edge of the crystal, or surface boundary. This correspondence between the bulk and the boundary leads to a perfectly conducting surface state – except where fragile topology comes into play.

Writing in the journal Science, Bernevig and colleagues provide a theoretical explanation for a new bulk-boundary correspondence to explain fragile topology. 

They show that the electron wavefunction of fragile topology only extends to the surface under specific conditions, which they call a twisted bulk-boundary-correspondence.

They further found that this twisted correspondence can be tuned so that the conducting surface states reappear. 

“Based on the wavefunction shapes, we designed a set of mechanisms to introduce interference on the boundary in such a way that the boundary state necessarily becomes perfectly conducting,” says Luis Elcoro, from the University of the Basque Country.

And that’s not just satisfying to know from a technical perspective. It might have some practical value

“The twisted bulk-boundary-correspondence of fragile topology provides a potential procedure to control the surface state, which might be useful in mechanical, electronic and optical applications,” says Princeton’s Zhi-Da Song.

Central to success was building a large-scale mock topological crystal out of plastic using 3D printed parts – work led by Sebastian Huber from ETH Zurich

They were able to mimic the twisted boundary condition, and by manipulating it could demonstrate that a freely conducting sound wave travels across the surface.

“This was a very left-field idea and realisation,” Huber says. “We can now show that virtually all topological states that have been realised in our artificial systems are fragile, and not stable as was thought in the past. 

“This work provides that confirmation, but much more it introduces a new overarching principle.”