= Chern insulator
{c}
{title2=$C\ne0$}
{wiki}
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 $\mathcal A_j=i\langle u_-|\partial_{k_j}u_-\rangle$, and orient the momentum plane by $(k_x,k_y)$. Then
$$
\Omega_-=\partial_{k_x}\mathcal A_y-\partial_{k_y}\mathcal A_x,
\qquad C=\frac{1}{2\pi}\int_{\mathrm{BZ}}\Omega_-\,d^2k.
$$
This 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 $\sigma_{yx}=Ce^2/h$ and $\sigma_{xy}=-Ce^2/h$. 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 $C$.
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