An electron inside a crystal is like a traveler walking through a city with winding streets. Its direction of motion constantly changes depending on where it is. Physicists call this feature the geometric Berry phase, and it determines how a material conducts electricity or responds to a magnetic field.

To describe such curved paths, scientists use a mathematical object — the Berry connection. It's like a map showing how sharply the road turns at each point. Previously, the map was drawn approximately: the city was divided into small squares, and turns were measured only at their boundaries. This introduced inaccuracies, especially in areas with sharp turns — in materials with a small energy gap.

The authors of the new work derived an exact rule for constructing the map. It relies not on the boundaries of squares but on the properties of the road itself at each point. The formula was tested on one-dimensional and three-dimensional models: calculations of the Berry phase and the Chern-Simons angle matched reference results.

Now theorists have a reliable tool for calculating the geometric properties of materials. This will accelerate the search for new compounds for electronics, spintronics, and quantum computers.