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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 ${\mathcal{A}}_j=i⟨u_{-}|{\partial }_{k_j}{u}_{-}⟩$, and orient the momentum plane by $(k_x,k_y)$. 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 $-e$ and conventional electric current $j_i={\sigma }_{ij}{E}_j$, these conventions give ${\sigma }_{yx}=C{e}^2/h$ and ${\sigma }_{xy}=-C{e}^2/h$. 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 $C$.

At large inverse field, $B$ and the cyclotron frequency are small, so the orbital spacing $\hslash {\omega }_c$ becomes comparable with both disorder broadening and the Fermi occupation's thermal width. Scattering of quantum lifetime ${\tau }_q$ gives a typical half-width $\Gamma =\hslash /(2{\tau }_q)$; overlapping levels lose their density-of-states contrast. The fundamental Dingle factor and thermal damping of quantum oscillations areBoth shrink rapidly as $B$ falls. Well-resolved oscillations require a sufficiently large ${\omega }_c{\tau }_q$ and a temperature small compared with the orbital spacing. The scattering time in this broadening argument is not automatically the transport time inferred from mobility; small-angle events affect the two differently.

The transverse Hall resistance is approximately $|R_{xy}|=B/(n_{s}e)$ in the classical weak-field regime. At stronger fields the integer quantum Hall effect produces plateausaccompanied by nearly vanishing longitudinal resistance. Localized bulk states between extended Landau states allow plateau intervals, while current can be carried by quantum Hall edge states. At still larger fields and sufficiently low temperature, the cleanest samples show the fractional quantum Hall effect, with interaction-generated incompressible states at fractional fillings such as $\nu =1/3$. The sign of $R_{xy}$ depends on the contact and magnetic-field orientation; the sketch shows its magnitude.

A quantum Hall edge state is confined across a boundary but can extend along it and carry electric current. B. I. Halperin related these current-carrying boundary states to quantized Hall transport in a disordered two-dimensional system.^[ref]