Cut a spin chain, and particles without an energy gap appear at the edges. They are called zero modes, and they live only at the boundary. Previously, they were counted using complex topological invariants, but now there is a simpler way.
The authors proposed using higher Berry curvature — a mathematical object that senses the geometry of the quantum state. They constructed a smooth interpolation from the real chain to a simple product state — the so-called Néel state (where spins alternate strictly up-down). The integral of HBC along this interpolation turned out to equal the ordinary Berry phase of half the chain. This phase is acquired by the half in response to a tiny magnetic field. And the Berry phase directly reveals the spin of the zero modes at the cut edge.
The method was tested on several examples. In spin-Peierls chains with spin 1/2, 1, and 3/2, a transition occurs between different dimerization patterns — the so-called "singlet flop." In the bilinear-biquadratic chain, the approach clarified the connection with the rigorous classification of symmetry-protected phases. And in the staggered chain J1–J2, the pattern of singlets depends on the sign of the interaction: for some signs, nearest neighbors pair up; for others, third neighbors do.
Thus, cutting the chain turns it into a measuring device: edge modes tell us what is happening inside the material.
The Berry phase is a quantum system's memory of the path it has taken in parameter space. If you slowly change the magnetic field and return to the starting point, the wave function can "remember" the route and acquire an additional phase shift, even if the field itself has not changed.