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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 $C$.

For filled, separated, noninteracting magnetic Bloch bands, the Berry curvature integral over the magnetic Brillouin zone is an integer Chern number $C$, giving $|{\sigma }_{xy}|=|C|e^2/h$. Patches of Berry connection meet through phases with integer windings; continuous gap-preserving changes cannot change $C$. D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs developed this band argument; it does not establish the interacting fractional case.^[ref]

For $\mathbf{d}\ne 0$, put $\hat{\mathbf{d}}=\mathbf{d}/|\mathbf{d}|$. The lower-band orthogonal projector is $P_{-}=(I-\hat{\mathbf{d}}\cdot \mathbf{\sigma })/2$. Applying the Pauli matrix multiplication law to $i\mathrm{Tr}(P_{-}[{\partial }_{k_x}{P}_{-},{\partial }_{k_y}{P}_{-}])$ givesThus the Chern number is the signed number of times the Brillouin zone covers a two-sphere under $\hat{\mathbf{d}}$. The lower energy eigenstate's actual Bloch vector is $-\hat{\mathbf{d}}$. The Berry curvature has the displayed positive sign for this lower band and the stated Berry connection; switching to the upper band changes its sign.

A useful local calculation exposes what each valley contributes. Write $p=\hslash v|\mathbf{q}|$. The linear Dirac Hamiltonian givesEach unbounded continuum cone gives a half-integer in this calculation. It is not a separately realizable half-integer Chern number of an isolated lattice energy band. The full honeycomb lattice supplies both valleys and the large-momentum completion. Starting with a trivial large-$|M|$ state, each single-valley mass crossing changes the lattice integer by the corresponding unit. There are no other band touchings. This establishes the exact lattice resultfor $t_2>0$, excluding the boundaries. This sign belongs to our explicit bond vectors, orientation and occupied-band convention.

For an original worked example, choose $t_1=1$, $t_2=0.08$, $\varphi =\pi /2$ and $M=0$. Then $\mu \simeq 0.415692$, $(m_{+},m_{-})=(-\mu ,+\mu )$, both valley gaps are $2\mu \simeq 0.831384$, and $C=-1$. Choosing instead $t_2=0$, $M=\mu$ gives exactly the same valley-gap magnitudes but $C=0$. The first state's two valley contributions reinforce; the second state's cancel.

An independent eigenvector-overlap calculation on periodic $31\times 31$, $61\times 61$ and $121\times 121$ momentum grids gives $C=-1$ for the first example and $C=0$ for the second. Reversing $\varphi$ in the first example gives $C=+1$. Direct integration of the analytic lattice Berry curvature agrees. For the overlap calculation one can remove the embedding phase from $f$, replacing it by $1+e^{i\mathbf{k}\cdot {\mathbf{a}}_1}+e^{i\mathbf{k}\cdot {\mathbf{a}}_2}$; this makes the cell-basis Hamiltonian periodic and prevents a false boundary-seam contribution. The local Berry curvature can depend on this basis convention; the integrated Chern number agrees. All mass-zero parameters are excluded before evaluating any integer.

Gregor Jotzu and collaborators realized an artificial Haldane model in 2014. They loaded spin-polarized potassium-40 fermions into a laser-made honeycomb lattice. Piezoelectric mirrors shook the optical lattice along two axes; their relative phase controlled time-reversal symmetry breaking. Optical phase control adjusted the sublattice offset.^[ref]

A magnetic field gradient drove Bloch oscillations. Band mapping measured interband transfer to locate valley-gap closures. Opposite-force drift measurements distinguished cancelling from reinforcing Berry curvature contributions; reversing shaking handedness reversed the drift. Interband transfer and harmonic confinement affected this readout, including drift beyond ideal transition lines. This neutral-atom Floquet Hamiltonian realization did not measure electrical sheet conductivity. Magnetic fields were used for preparation and forcing.^[ref]