Chern Numbers

A Chern number is a quantized flux of Berry curvature through a closed two-dimensional parameter space. In the simplest nondegenerate $U(1)$ case,

$$C_n = \frac{1}{2\pi} \int_M F_n,$$

where $M$ is a closed oriented surface and $F_n$ is the Berry-curvature two-form of an isolated eigenstate line bundle over $M$.

The formula is short, but every word matters. A Chern number is not just any Berry-curvature integral. The surface must be closed, the eigenstate or band must be defined over the whole surface, and the relevant spectral gap must stay open. Under those assumptions the result is an integer that cannot change under smooth deformations.

Chern numbers in the mathematical toolkit develops the patching proof. Here we use it to understand spin eigenstates and band topology.

Berry Curvature Flux

For a nondegenerate eigenstate family

$$H(R)\lvert n(R)\rangle = E_n(R)\lvert n(R)\rangle,$$

choose a local phase convention and define the Berry connection

$$A_n = i\langle n|dn\rangle.$$

The Berry curvature is

$$F_n=dA_n.$$

Locally, the Berry phase around a small loop is the curvature flux through a small spanning surface. Globally, the Chern number integrates this curvature over a closed surface:

$$C_n = \frac{1}{2\pi} \int_M F_n.$$

The factor $1/(2\pi)$ appears because quantum phases are defined modulo $2\pi$. In this convention, the holonomy phase factor is $e^{i\oint A_n}$.

Required Assumptions

The standard $U(1)$ Chern number requires:

• a closed oriented two-dimensional parameter space $M$;
• an isolated nondegenerate eigenstate, or isolated Bloch band, over all of $M$;
• a Berry curvature $F_n$ defined globally, even if the Berry connection is only local;
• a fixed orientation and normalization convention.

In band theory, $M$ is often the two-dimensional Brillouin-zone torus. In a two-level spin example, $M$ can be a sphere surrounding a degeneracy in parameter space.

If the state ceases to be isolated at some point of $M$, the Chern number for that state is not defined. This is why gap closings are central to topological phase transitions.

Why the Result Is an Integer

The integer comes from patching local phase choices. A single smooth eigenvector gauge may not exist on all of $M$. On overlapping patches, two local eigenvectors differ by a phase:

$$\lvert n_\beta\rangle = e^{i\chi_{\beta\alpha}} \lvert n_\alpha\rangle.$$

The Berry connections differ by

$$A_\beta = A_\alpha-d\chi_{\beta\alpha}.$$

When the curvature integral is split across patches, Stokes’ theorem turns the answer into the winding of the transition phase around overlaps. Single-valued transition functions can wind only an integer number of times around $U(1)$, so

$$\frac{1}{2\pi} \int_M F_n \in \mathbb Z.$$

The toolkit page gives the detailed two-patch derivation. The physics lesson is that the integer is a global obstruction to choosing one smooth phase convention everywhere.

Obstruction to a Global Gauge

If one eigenvector gauge covered the whole closed surface, its connection would be represented by a smooth one-form $A_n$ on that surface, with

$$F_n=dA_n$$

globally. Stokes’ theorem would give

$$\int_M F_n = \int_M dA_n = \int_{\partial M}A_n = 0,$$

because $\partial M=\varnothing$. Therefore a nonzero Chern number means that no single smooth global eigenvector gauge covers the entire surface.

A global connection still exists geometrically; it is represented by local one-forms related on overlaps. What fails is a single global eigenvector frame and its one-form representation.

Spin One-Half Sphere

For the spin-$1/2$ Berry phase convention used in this volume,

$$F_+ = -\frac12 \sin\theta\,d\theta\wedge d\phi.$$

Over the full sphere,

$$\int_{S^2}F_+ = -2\pi.$$

Thus

$$C_+ = \frac{1}{2\pi} \int_{S^2}F_+ = -1.$$

The degeneracy at zero field acts as a Berry-curvature monopole enclosed by the sphere. The opposite eigenstate has the opposite Chern number in the matching convention. Here $+$ denotes the spinor aligned with $\mathbf n$, and the sphere has outward orientation. The opposite eigenstate or orientation reverses the sign; a phase change of that eigenstate does not.

Brillouin-Zone Chern Number

For a two-dimensional isolated Bloch band, the parameter space is the Brillouin zone:

$$\mathrm{BZ} \simeq T^2.$$

In a common notation, the band Chern number is

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

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

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

This number is stable under smooth changes of the Hamiltonian that keep the band isolated over the whole Brillouin zone.

Hall Response Preview

In a noninteracting two-dimensional band insulator, filled isolated bands can contribute to the Hall conductivity through their Chern numbers:

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

where $e>0$, $A=i\langle u|du\rangle$, $C$ uses orientation $dk_x\wedge dk_y$, and $j_y=\sigma_{yx}E_x$. With the reversed tensor order, $\sigma_{xy}=-\sigma_{yx}$. Fixing the response component is essential when comparing signs.

This formula is the integer quantum Hall bridge, not a complete derivation. It assumes a filled-band setting, a gap at the Fermi level, and the usual linear-response framework. Chern Numbers in Band Theory develops occupied projectors, direct and indirect gaps, the Kubo reduction, symmetry constraints, and numerical computation. Integer Quantum Hall Effect adds Landau filling, disorder localization, chiral edges, flux insertion, plateau transitions, and metrology. The role of this page is to explain the integer carried by Berry curvature.

Multiple Occupied Bands

If several occupied bands are separated from empty bands but not from each other, an individual eigenline may fail to exist at a degeneracy, so its separate Chern number is undefined. Chern numbers of genuinely isolated individual bands are gauge invariant. The meaningful object is the total Chern number of the occupied subspace.

In nonabelian Berry-connection language, the first Chern number is

$$C_{\mathrm{occ}} = \frac{1}{2\pi} \int_M \operatorname{Tr}F_{\mathrm{occ}}.$$

This is a preview of nonabelian Berry geometry. The trace is gauge invariant under changes of basis inside the occupied subspace. The detailed nonabelian construction belongs to the later preview page and to the mathematical toolkit.

Stability and Gap Closing

Chern numbers are robust because they are integer-valued and cannot change continuously. If a Hamiltonian is deformed smoothly while the relevant state or occupied subspace remains isolated over all of $M$, the Chern number remains fixed.

To change a Chern number, something singular must happen. In band problems this is usually a gap closing:

$$E_n(\mathbf k,\lambda) = E_m(\mathbf k,\lambda)$$

at some parameter value $\lambda$. At that point, the eigenbundle whose Chern number was being tracked is no longer defined. After the gap reopens, curvature flux can have been transferred between bands.

This is why Chern-number changes are tied to topological phase transitions.

Relation to Dirac and Berry Monopoles

The Dirac monopole shows that flux through a closed sphere can be quantized by single-valued quantum patching. Berry monopoles show the same structure in parameter space: degeneracies act as sources of Berry-curvature flux.

For a closed surface $M$ surrounding a degeneracy, the Chern number measures the net Berry monopole charge enclosed by $M$. If no degeneracy crosses the surface during a smooth deformation, the integer cannot change.

This is the common geometry behind:

• Dirac monopole quantization;
• spin-$1/2$ solid-angle Berry phase;
• Chern bands over the Brillouin-zone torus;
• quantum Hall response in filled-band settings.

Common Mistakes

• Integrating curvature over an open patch and calling the result a Chern number.
• Forgetting the isolation or gap assumption for the eigenstate or band.
• Treating the local Berry curvature distribution as topological. The integral is topological; the local distribution can move.
• Assuming a nonzero Chern number can be represented by one smooth global eigenvector.
• Ignoring orientation and sign conventions in $C_n$ and $\sigma_{yx}$.
• Assigning separate Chern numbers to bands that are degenerate or mixed inside the occupied subspace.
• Treating the Hall-conductivity formula as valid without checking the filled-band and linear-response assumptions.

Exercises

1. Compute a constant-curvature Chern number.

Let $M$ be a torus with coordinates $0\le k_x,k_y<2\pi$ and

$$F = \frac{N}{2\pi} dk_x\wedge dk_y, \qquad N\in\mathbb Z.$$

Compute $C=(1/2\pi)\int_MF$.

Solution

The integral is

$$\int_MF = \frac{N}{2\pi} \int_0^{2\pi} \int_0^{2\pi} dk_x\,dk_y = 2\pi N.$$

Therefore

$$C = \frac{1}{2\pi} \int_MF = N.$$

2. Explain why a nonzero Chern number obstructs a global gauge.

Solution

If a global eigenvector frame existed, it would give one smooth connection one-form $A$ on the closed surface $M$, with $F=dA$ everywhere. Stokes’ theorem would give

$$\int_MF = \int_MdA = \int_{\partial M}A = 0.$$

Since a nonzero Chern number means $\int_MF\ne0$, it rules out a single smooth global gauge.

3. Compute the spin-$1/2$ Chern number.

Use

$$F_+ = -\frac12 \sin\theta\,d\theta\wedge d\phi$$

on $S^2$ with the standard orientation.

Solution

Compute

$$\begin{aligned} \int_{S^2}F_+ &= \int_0^{2\pi} \int_0^\pi -\frac12 \sin\theta\,d\theta\,d\phi\\ &= -2\pi. \end{aligned}$$

Therefore

$$C_+ = \frac{1}{2\pi} \int_{S^2}F_+ = -1.$$

4. Why can a Chern number change when a gap closes?

Solution

The Chern number is assigned to an isolated eigenstate, isolated band, or isolated occupied subspace. If a gap closes, that object is no longer defined over the full parameter space. The assumptions behind the integer invariant fail at the gap closing. After the gap reopens, the reconstructed band or subspace can have a different Chern number because Berry-curvature flux has passed through the singular point.

5. What is wrong with calling $\int_\Sigma F/(2\pi)$ a Chern number for an open surface $\Sigma$?

Solution

For an open surface, the integral can change continuously under gauge-compatible deformations and can depend on boundary data. The integer quantization argument uses a closed surface and patching around closed overlaps. On an open surface, $\int_\Sigma F/(2\pi)$ can be a useful Berry-curvature flux, but it is not generally a topological Chern number.

References

• B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors, Princeton University Press, 2013.
• M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society A 392, 45–57, 1984.
• S. S. Chern, “Characteristic classes of Hermitian manifolds,” Annals of Mathematics 47, 85–121, 1946.
• M. Nakahara, Geometry, Topology and Physics, 2nd ed., CRC Press, 2003.
• B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry’s phase,” Physical Review Letters 51, 2167–2170, 1983.
• 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.
• D. Xiao, M.-C. Chang, and Q. Niu, “Berry phase effects on electronic properties,” Reviews of Modern Physics 82, 1959–2007, 2010.