Chern number
/ CHURN NUM-ber /
Imagine wrapping a sheet of stretchy fabric over a ball. No matter how you smooth it, the fabric must wrap the ball a whole number of times — once, twice — never one-and-a-half times. You can't have a fractional wrap without a tear. The Chern number is the physicist's measure of exactly this kind of whole-number wrapping, applied to the quantum states of electrons.
More precisely, the Chern number counts how many times the Berry phase — the geometric twist electrons accumulate — winds around as you sweep through all the possible momentum states of a filled energy band. It is always a whole number: positive, negative, or zero. A band with a nonzero Chern number is genuinely twisted in a way that cannot be undone smoothly, marking the material as topologically nontrivial. In the quantum Hall effect, the integer labeling each conductance plateau is precisely the total Chern number of the filled bands.
This matters because the Chern number is the single number behind the breathtaking precision of the quantum Hall effect — its integer nature is why the plateaus are exact to a billionth. A subtle caveat: a nonzero Chern number requires breaking time-reversal symmetry, which a magnetic field provides; the time-reversal-symmetric topological insulators are classified by a different, two-valued invariant instead. So the Chern number is central, but it is not the only flavor of topological label.
In 1982, the theorists Thouless, Kohmoto, Nightingale and den Nijs showed that the integer in each quantum Hall plateau equals a Chern number of the filled electronic states. This identification — that a measured conductance equals an abstract topological count — was a turning point that later shared in a Nobel Prize.
The 1982 link between a measured Hall plateau and an abstract Chern number launched topological band theory.
The Chern number is named after the mathematician Shiing-Shen Chern, whose geometry the physics borrows; in materials it is computed from the Berry phase summed over the Brillouin zone, the full space of an electron's possible momenta.