Chern Number

Formula

For an isolated band over a two-dimensional Brillouin zone,

$$C_n = \frac{1}{2\pi} \int_{\mathrm{BZ}} \Omega_n(\mathbf k)\,d^2k,$$

where $\Omega_n$ is the Berry curvature component compatible with the chosen orientation.

Equivalently, using the curvature two-form $F_n$,

$$C_n = \frac{1}{2\pi} \int_{\mathrm{BZ}}F_n.$$

With electron charge $q_{\mathrm e}=-e$, the linked $\mathcal A=i\langle u|\boldsymbol\nabla_{\mathbf k}u\rangle$ convention, an oriented measure $dk_x\wedge dk_y$, and $\sigma_{xy}=J_x/E_y$, filled noninteracting bands obey

$$\sigma_{xy} = -\frac{e^2}{h} \sum_{\text{occupied }n}C_n,$$

where $e>0$. A source using the opposite Berry-connection sign reverses both the reported Chern sign and the written Chern-to-Hall bridge. Charge, orientation, Berry-connection, and tensor-index conventions must therefore be translated as one block.

Assumptions

• The Brillouin zone is treated as a closed oriented two-dimensional torus.
• The band, or occupied band subspace, is isolated over the whole Brillouin zone.
• The Berry-curvature normalization follows the $1/(2\pi)$ convention used here.
• The Hall-conductance formula assumes a noninteracting filled-band setting and fixed sign conventions.

Validity

The Chern number is integer-valued and stable under smooth deformations that keep the relevant gap open. It can change only when the assumptions fail, most commonly when bands touch and the isolated-band bundle is no longer defined.

For multiple occupied bands, the gauge-invariant object is the total Chern number of the occupied subspace, not necessarily a unique assignment to individual bands when they are internally degenerate.

Common Mistakes

• Integrating Berry curvature over an open patch and calling the result a Chern number.
• Forgetting the factor $1/(2\pi)$.
• Applying a band Chern number when the band is not isolated over the whole Brillouin zone.
• Ignoring orientation and electron-charge sign conventions in Hall-conductance formulas.
• Treating a nonzero Chern number as something a single smooth global eigenvector gauge can represent.

Quick Check

If an isolated two-dimensional band has curvature $\Omega_n=N/(2\pi)$ on a Brillouin-zone torus with coordinates $0\le k_x,k_y<2\pi$, what is $C_n$?

Solution

The integral is

$$\int_{\mathrm{BZ}}\Omega_n\,d^2k = \frac{N}{2\pi}(2\pi)(2\pi) = 2\pi N.$$

Therefore

$$C_n = \frac{1}{2\pi} (2\pi N) = N.$$

Canonical Links

• Chern Numbers
• Chern Numbers in Quantum Mechanics
• Chern Numbers in Band Theory
• Berry Curvature
• Topological Invariants
• Quantum Hall Effect
• Integer Quantum Hall Effect

References

• S. S. Chern, “Characteristic classes of Hermitian manifolds,” Annals of Mathematics 47, 85-121, 1946.
• D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Physical Review Letters 49, 405-408, 1982.
• B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors, Princeton University Press, 2013.