In the world of crystals, there is a hidden geometry. When an electron runs through a two-dimensional insulator, its path is twisted by an invisible force — topology. This twist is described by the Chern number, an integer that cannot be changed by smooth deformation. Previously, it had to be calculated like the area under a complex curve: step by step, reconstructing the Berry curvature across the entire Brillouin zone (the space of all possible electron momenta). The process is lengthy and sensitive to noise.

The authors of the study took a different route. They noticed that the Chern number hides not in the area but in points. Where the wave function vanishes, the phase of the wave makes a full turn, like a compass needle at the North Pole. This is a phase vortex. Simply count such vortices in the structure of eigenvectors, and the Chern number pops out by itself — integer and exact. The method works even under strong noise because vortices are rigid objects: they do not disappear from a slight shake.

The second method is even trickier. In classical wave systems (for example, in a lattice of laser beams) modes do not have to be normalized. The amplitude can vary, and this is not a bug but a key. By comparing phases at different edges of the sample, one can extract the Chern number without a full scan of the Brillouin zone. An experiment on a photonic lattice confirmed that both methods give exactly the same integers as the cumbersome old calculations.