Column 6 — Topology in Crystal Band Structures

Last column introduced the Berry phase — the geometric phase a quantum state picks up when its parameters complete a closed loop. In this column, we will discover how the resulting Berry curvature field produces physical consequences for electrons in crystal. The symbolic quantity within this mechanism is the Chern number — an integer firmly stamped into each energy band by topology, and that cannot be erased without tearing the band structure.

A “Magnetic Field” in Momentum Space

In column 5 we saw that a quantum state undergoing adiabatic evolution accumulates a geometric phase when its parameters trace a closed loop. We introduced the Berry connection A and derived the Berry phase by performing a line integral of A along any closed loop in the parameter space. Converting line integral into area integral gave the Berry curvature.

Relationship between the Berry connection and the Berry curvature is exactly equal to the relationship between the vector potential and the magnetic field. In the classical electrodynamics, the vector potential is gauge-dependent: you can add the gradient of any scalar function to the vector potential without changing the measurable physical quantity. The Berry connection has the same gauge dependence. If you add phase to the eigenstate:

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the Berry connection shifts by:

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Taking the curl of A gives the Berry curvature:

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Since the curl of gradient is always zero, the Berry curvature is evidently a gauge-invariant quantity. Thus, unlike the Berry connection, the Berry curvature is a physical, observable field, playing the role of a “magnetic field” within the parameter space of the Hamiltonian.

The Berry phase around a closed loop is the flux of the Berry curvature through the surface bounded by the loop:

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Hence, the phase is a “Berry flux”, analogous to the magnetic flux Φ.

Also, in three-dimensional parameter space, the Berry curvature can be written as a rank-2 tensor:

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, a direct parallel to the electromagnetic field tensor F_μν.

Into the Brillouin Zone

Recall the k-dependent Schrödinger equation we derived:

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Here the crystal momentum k enters the Hamiltonian as a parameter. In the Berry phase formalism:

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The Berry curvature of the n-th Bloch band is therefore:

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This is a gauge-invariant “magnetic field” in the Brillouin zone.

There is a different way to write the same quantity. The Berry curvature

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Since

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the second term of the Berry curvature formula is zero and it reduces to:

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using the completeness of the basis:

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The n’=n term cancels out in the bracket, so:

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Now let’s use the Schrödinger equation:

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Applying the bra-vector:

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Therefore,

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The denominator of the Berry curvature is the square of the energy gap between the bands. If the two bands are close in energy, the Berry curvature becomes large. Whenever the energy eigenvalues become degenerate, the Berry curvature diverges. Degeneracy points are sources of Berry curvature, where it radiates outward into k-space like magnetic monopoles.

Anomalous Velocity

Using the semiclassical theory of Bloch electron gives us the average velocity:

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As we saw in column 5, when the Berry curvature is non-zero, there is an additional contribution:

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where the anomalous velocity:

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The anomalous velocity is perpendicular to the electric field, thus contributes a current in the transverse direction. This is the mechanism of the anomalous Hall effect — where a Hall voltage is generated without any external magnetic field.

The anomalous velocity is k-dependent, which means electrons at different crystal momenta are deflected by different amounts. For a fully occupied bands, summing the contributions of all electrons gives a Hall conductance:

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where e²/h is the conductance quantum. The Hall conductance of a filled band is proportional to the Berry curvature integrated over the 1st Brillouin zone. At first glance, this may seem like nothing more than a mere intermediate calculation, but it plays a crucial role in our discussion.

The Chern Number

Believe it or not, this is actually a drawing of the World Cup match ball…

First, let’s step back and think about what kind of object we are calculating in the integral. Topology is the study of properties that do not change under continuous deformations. Two objects are topologically equivalent if one can be smoothly deformed into the other one without cutting or gluing. A coffee cup and a donut are equivalent (both have one handle), but neither of them can be continuously deformed into a football. The number of handles is a topological invariant: an integer quantity that cannot change unless you tear the surface and reconnect it. It means that you cannot smoothly deform one integer to another.

If we integrate the Berry curvature over the Brillouin zone — which is a torus — we can get a quantity called the Chern number:

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As long as the n-th band does not touch any other band within the Brillouin zone — as long as it is isolated by an energy gap in the k-space — the Chern number is an integer. Using this definition, we can rewrite the Hall conductance formula:

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This is the TKNN formula, explaining the quantized Hall conductance in units of the conductance quantum.

There is also a geometric picture. As we discussed before, the points that different energy bands become degenerate acts as a magnetic monopoles. The Chern number thus counts the number of monopoles within the Brillouin zone.

Why the Chern Number Must Be an Integer

The claim that the Chern number is always an integer stems from differential geometry. The Gauss-Bonnet theorem states that for any smooth, closed 2D surface M:

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where K is the Gaussian curvature and g_M is the genus — the number of handles. For a sphere:

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we can obtain the equality:

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In the case of donut, the integral equals to 0. The total curvature is a topological invariant: no matter how you stretch, squeeze, or deform it into any shape, the evaluated integral remains unchanged. The genus cannot change under continuous deformation.

The Berry curvature plays the role of K, the Brillouin zone torus plays the role of M, and the Chern number plays the role of 2–2g_M. Smooth changes in the band structure cannot change the total integral over the torus. The Chern number is thus topologically protected, just like a genus of a donut. The only thing that can change the Chern number is a band-closing event. If the energy gap between two separate bands closes, the singularity of the Berry curvature emerges, which leads to a change in the Chern number.

In addition, the Chern number integral can be rewritten as a line integral of the Berry connection around the zone boundary. Since the Brillouin zone in the torus — surface with opposite edges glued together — the boundary is a closed loop, and as k traverses the loop, the Bloch wavefunction returns to itself, picking up the phase γ. For the wavefunction to be single-valued, γ must be an integer multiple of 2π. Therefore, the Chern number is exactly the winding number of phase factor:

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A Glimpse of the Chern Insulator

The Chern number does actually provide a bizarre phenomenon at any boundary of the material. For instance, think of an insulator whose occupied bands carry a total non-zero Chern number C. Threading a magnetic flux quantum through the system adiabatically forces exactly C electrons from one edge of the sample to the opposite side. In order to make the electrons mobile, they should cross the Fermi energy. However the Fermi energy sits in the bulk gap, so no bulk states are available. Hence, the electrons that move are necessarily at the edges. This is the origin of the gapless edge states: conducting channels at the boundary of a topologically non-trivial insulator, which is called a Chern insulator.

The Berry curvature is the source of all of this. It is the “magnetic field” in the Brillouin zone, and the Chern number is the total flux through the closed Brillouin zone torus. So the analogy of a Berry connection as vector potential and Berry curvature as magnetic field, is not just a stretch. It is a core viewpoint of the topology of the band theory, and it will keep reappearing throughout the upcoming columns.

References

• Girvin & Yang, Modern Condensed Matter Physics (2019)
• Sprinkart, Scheer & Di Bernardo, “Tutorial: From Topology to Hall Effects — Implications of Berry Phase Physics” (2024)