A Chern insulator is a two-dimensional band insulator whose occupied energy bands have a nonzero total Chern number. It has a bulk band gap, yet an interface with a topologically trivial band insulator carries a net excess of quantum Hall edge states propagating in one direction. Time-reversal symmetry must be broken. The Haldane model shows that a nonzero average magnetic flux is unnecessary.
Why distinguish two solids that both have a band gap? The energy eigenvalues tell us whether an excitation costs energy; the energy eigenstates also describe how the state changes around the Brillouin zone. Their Berry curvature can integrate to a nonzero integer. A small deformation changes the Berry curvature continuously, but cannot continuously change an integer. A Chern insulator and a trivial band insulator therefore cannot be connected through a path of isolated occupied energy bands with a common bulk band gap. The distinction survives sufficiently weak local perturbations. It does not make every microscopic quantity or every boundary state immune to every disturbance.
For a single filled energy band, set the Berry connection to , and orient the momentum plane by . ThenThis is the First Chern class evaluated on the Brillouin zone, a torus. Smooth eigenvectors may require multiple patches; their transition phases have integer winding numbers. For charge and conventional electric current , these conventions give and . The measurable magnitude agrees with Chern number and integer Hall conductivity. Reversing the momentum orientation or the sign of the Berry connection reverses the displayed .
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