Topological materials have been receiving a lot of attention in recent years because of their potential applications in future electronic and quantum technologies.

These are a type of material whose electrons have properties that stay stable even when the material has small defects, impurities, or imperfections. They often have surface or edge behaviour that is very different from the interior, because of the structure of the material’s quantum states.

Researchers use quantities such as Berry curvature and Chern numbers to describe topological materials. Berry curvature measures how an electron’s quantum state changes as its momentum changes. Adding this curvature across the whole band gives a Chern number, a whole-number label that can show whether the material is topological.

A key question in this area is: when a material has topological properties, how much of that behaviour is actually tied to quantum entanglement?

In a new paper published recently, Kazuki Ikeda and Steven Rayan tackled this question using the spinless Haldane model. This is a widely used theoretical model for understanding topological materials. This model describes electrons moving on a honeycomb lattice, which has two different sublattices, called A and B.

Here, the researchers added an extra layer of analysis. Instead of only calculating whether the model is topological, they were able to determine where the topological signal comes from.

By introducing a filter based on quantum entanglement, they tested whether the Berry curvature is associated with electron states that are spread across both A and B sublattices, or with states that mostly sit on one sublattice.

They also focused on what happens when the system moves between different topological phases, giving a more organised way to describe what changes when the energy gap closes and the material switches phase.

These results will provide invaluable insights into understanding and designing topological materials going forward.

What’s also interesting here is that, in order to obtain these results, the authors employed Langlands-inspired mathematics to keep track of what changes when the material switches between topological phases.

The Langlands programme is an ambitious mathematical dictionary that tries to translate problems about numbers into problems about symmetry, geometry and analysis.

Although the link in this work is purely conceptual, it does show that Langlands-style structures can organise real physics problems in topological matter.