Chern Number Calculation: Berry Curvature Integration

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Hello! Welcome to the third lesson in our module on topological matter.

In our last lesson, we built a crucial bridge from the abstract idea of a geometric phase to the concrete reality of electronic band structures. We saw how any two-band model can be mapped to a spin-1/2 Hamiltonian $H (k) = d (k) \cdot σ$ , and we derived the Berry curvature, $Ω_{n} (k)$ , a gauge-invariant quantity that acts as a kind of "magnetic field" in momentum space. This local property measures the geometric "twist" of the Bloch wavefunctions at each point $k$ in the Brillouin zone.

Today, we will take the final step to define a true topological invariant. Our learning outcome is to calculate the Chern number as the integral of the Berry curvature over the Brillouin zone. We will see that this integral yields a quantized integer that is insensitive to small changes in the material, classifying the entire band structure into distinct topological classes. This integer has a profound physical meaning, directly determining the quantized Hall conductivity.

1. From Local Curvature to a Global Invariant: An Analogy

Before diving into the quantum mechanics, let's build our intuition with a powerful analogy from differential geometry, which your background in mathematics should make quite accessible.

Imagine a closed, two-dimensional surface, like a sphere or a torus (a donut shape). At every point on the surface, we can define a local property called the Gaussian curvature, which tells us how the surface is bent at that point (e.g., positive for a sphere, zero for a cylinder, negative for a saddle).

A remarkable result, the Gauss-Bonnet theorem, states that if you integrate this local Gaussian curvature over the entire closed surface, the result is always an integer multiple of $2 π$ . This integer is directly related to a global topological property of the surface: its number of "holes" or genus, $g$ .

\int_{Surface} K d A = 2 π (2 - 2 g)

The key idea is that a global, quantized, topological property (the genus $g$ ) is determined by the integral of a local, continuous, geometric property (the curvature $K$ ).

The following video provides an excellent visual introduction to this concept, setting the stage for our main topic.

Berry phases and topological properties - QuanSheng Wu

The video 'Berry phases and topological properties' by QuanSheng Wu introduces the idea of topological invariants using the Gauss-Bonnet theorem as a starting point. This analogy is extremely helpful for understanding the Chern number.

Please watch the segment from 04:29 to 08:44. Focus on the connection between the local Gaussian curvature and the global, quantized genus of a surface.

We are going to perform the exact same procedure, but in momentum space. The Brillouin zone of a 2D crystal is topologically a torus. The role of the Gaussian curvature will be played by the Berry curvature $Ω (k)$ . The resulting integer will be the Chern number.

2. Defining the Chern Number

For a given energy band $n$ , the (first) Chern number, $C_{n}$ , is defined as the integral of its Berry curvature over the entire 2D Brillouin zone (BZ), normalized by a factor of $2 π$ .

C_{n} = \frac{1}{2 π} \int_{BZ} Ω_{x y}^{(n)} (k) d^{2} k

Here, $Ω_{x y}^{(n)} (k) = \partial_{k_{x}} A_{y}^{(n)} - \partial_{k_{y}} A_{x}^{(n)}$ is the $x y$ -component of the Berry curvature for the $n$ -th band.

Just like the integral in the Gauss-Bonnet theorem, this integral is guaranteed to be an integer, provided the band $n$ is separated from all other bands by a finite energy gap everywhere in the BZ.

The proof of this quantization is elegant and relies on Stokes' theorem and the gauge structure of the Berry connection. For a deeper, more formal look at this derivation, the following lecture notes are an excellent resource.

Topological Phases of Matter

These lecture notes, 'Topological Phases of Matter' by Nicolai Lang, provide a concise and rigorous derivation of the Chern number and the reason for its quantization.

Please read pages 26-27, starting from 'Berry curvature' and ending at the definition of the Chern number, Eq. (1.56). Pay close attention to the argument on page 27 that uses Stokes' theorem and the gauge transformation properties of the Berry connection to show that the integral must be an integer multiple of 2π (which leads to an integer Chern number after normalization).

The argument in the notes shows that you cannot define a single, continuous gauge for the Bloch states over the entire toroidal BZ. By dividing the BZ into two patches and requiring consistency on their boundary, the integer quantization of the integral emerges naturally. This is deeply analogous to Dirac's argument for the quantization of magnetic charge, where the Berry curvature plays the role of the magnetic field of a monopole.

3. The Chern Number of a Two-Band Model

Let's connect this definition back to the two-band model from our last lesson, $H (k) = d (k) \cdot σ$ . We found the Berry curvature for the two bands ( $\pm$ ) to be:

Ω_{x y}^{(\pm)} (k) = \pm \frac{1}{2} d^\cdot (\frac{\partial d ^}{\partial k _{x}} \times \frac{\partial d ^}{\partial k _{y}})

where $d^= d /∣ d ∣$ . Plugging this into the definition of the Chern number gives:

C_{\pm} = \pm \frac{1}{4 π} \int_{BZ} d^\cdot (\frac{\partial d ^}{\partial k _{x}} \times \frac{\partial d ^}{\partial k _{y}}) d k_{x} d k_{y}

This expression has a beautiful geometric interpretation. The term inside the integral is precisely the element of solid angle that the unit vector $d^(k)$ sweeps out on the surface of a unit sphere as the momentum $k$ traces an infinitesimal area $d k_{x} d k_{y}$ in the BZ.

Therefore, the Chern number $C_{\pm}$ is the total number of times the vector $d^(k)$ wraps around the unit sphere as $k$ scans the entire Brillouin zone. It is the winding number of the map from the BZ (a torus, $T^{2}$ ) to the target space of the $d^$ vector (a sphere, $S^{2}$ ).

Chern Number as Winding Number

As shown in this diagram, the map from the Brillouin zone torus to the unit sphere of \hat{d}_k vectors has a winding number, which is the Chern number. It counts how many times the surface of the sphere is covered by the mapping.

Test your understanding!

Let's revisit the 2D massive Dirac model from the last lesson, with $d (k) = (v_{F} k_{x}, v_{F} k_{y}, m)$ . Calculate the Chern number for the lower band, $C_{-}$ . For this calculation, assume the model is valid for all $k$ and integrate over the entire $R^{2}$ plane instead of a finite BZ. What do you notice about the result?

Hint: In the previous lesson, we found the curvature for this model to be $Ω_{x y}^{(-)} (k) = - \frac{m v _{F}^{2}}{2 ( v _{F}^{2} ( k _{x}^{2} + k _{y}^{2} ) + m ^{2} ) ^{3/2}}$ . You will need to integrate this function.

Show answer

We need to compute $C_{-} = \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} Ω_{x y}^{(-)} (k) d k_{x} d k_{y}$ .
The integral is best done in polar coordinates, where $k^{2} = k_{x}^{2} + k_{y}^{2}$ and $d^{2} k = k d k d θ$ .

C_{-} = \frac{1}{2 π} \int_{0}^{2 π} d θ \int_{0}^{\infty} k d k [- \frac{m v _{F}^{2}}{2 ( v _{F}^{2} k ^{2} + m ^{2} ) ^{3/2}}]

The integral over $θ$ gives $2 π$ .

C_{-} = - \frac{m v _{F}^{2}}{2} \int_{0}^{\infty} \frac{k}{( v _{F}^{2} k ^{2} + m ^{2} ) ^{3/2}} d k

Let $u = v_{F}^{2} k^{2} + m^{2}$ , so $d u = 2 v_{F}^{2} k d k$ . The integral becomes:

C_{-} = - \frac{m v _{F}^{2}}{2} \int_{m^{2}}^{\infty} \frac{1}{2 v _{F}^{2}} u^{- 3/2} d u = - \frac{m}{4} [- 2 u^{- 1/2}]_{m^{2}}^{\infty}

C_{-} = \frac{m}{2} [\frac{1}{\infty} - \frac{1}{m ^{2}}] = \frac{m}{2} (0 - \frac{1}{∣ m ∣}) = - \frac{m}{2∣ m ∣} = - \frac{1}{2} sgn (m)

The result is $C_{-} = - 1/2$ if $m > 0$ and $+ 1/2$ if $m < 0$ . This is a half-integer! This famous result demonstrates that for a continuum model not defined on a compact Brillouin zone, the topological invariant is not guaranteed to be an integer. The quantization to an integer is a specific feature of lattice models, like the Haldane model we will see later, where the BZ is a compact torus. This half-quantization is a signature of the "parity anomaly" in 2+1 dimensions.

4. The Physical Manifestation: The TKNN Formula

So far, the Chern number is a mathematical abstraction. Its profound physical importance was revealed in 1982 by Thouless, Kohmoto, Nightingale, and den Nijs (TKNN). They showed that the Chern number directly determines the quantized Hall conductivity.

For an insulator at zero temperature (where some bands are completely full and others are completely empty), the Hall conductivity $σ_{x y}$ is given by:

σ_{x y} = \frac{e ^{2}}{h} n \in occupied \sum C_{n}

where the sum is over all filled bands. Since the Chern numbers are integers, the Hall conductivity is perfectly quantized in integer multiples of the fundamental conductance quantum, $e^{2} / h$ . This explained the extraordinary precision of the Integer Quantum Hall Effect.

The formal derivation of this result, starting from the Kubo formula for linear response, is a cornerstone of modern condensed matter theory. Given your background, following the main steps of this derivation will be very insightful.

Topological Phases of Matter

We return to Lang's 'Topological Phases of Matter' for the derivation of the TKNN formula.

Please read pages 32-36, titled 'The TKNN invariant'. You don't need to reproduce every line, but focus on the key steps: How the Kubo formula for \sigma_{xy} (Eq. 1.80) is transformed by using properties of Bloch states and the momentum-space Hamiltonian ilde{H}_0(k) (Steps 4-7). How the final expression for conductivity (Eq. 1.98) is shown to be an integral involving terms that look exactly like the Berry curvature. The final connection made in Step 10, where the expression for \sigma_{xy} is identified as the sum of Chern numbers of the filled bands (Eq. 1.106). Note that the text uses a convention with a minus sign and \hbar; the result is equivalent to the standard \sigma_{xy} = (e^2/h) \sum C_n.

While the Kubo formula derivation is rigorous, a more intuitive semi-classical picture also exists. It relates the Chern number to a quantized pumping of charge. As an electric field drives an electron across the BZ in one direction, the non-zero Berry curvature induces an "anomalous velocity" in the perpendicular direction. For a filled band, this motion corresponds to a net transport of one unit of charge across the sample for each unit of Chern number, leading to the quantized Hall current. This picture is beautifully explained in Sec. V.B of the notes by Grushin (LINK) if you are interested.

Conclusion

Today we have defined a robust integer invariant that classifies 2D gapped band structures. This is a pivotal moment in our course, transitioning from geometric concepts to topological classification.

Key Takeaways:

The Chern number, $C_{n} = \frac{1}{2 π} \int_{BZ} Ω_{n} (k) d^{2} k$ , is the integral of the Berry curvature of a band over the Brillouin zone.
For a gapped band on a lattice, $C_{n}$ is a quantized integer.
Geometrically, for a two-band model, $C_{n}$ counts the number of times the characteristic vector $d (k)$ wraps around the unit sphere as $k$ covers the BZ.
Physically, the sum of Chern numbers of the occupied bands determines the quantized Hall conductivity via the TKNN formula: $σ_{x y} = (e^{2} / h) \sum C_{n}$ .

Preview of the Next Lesson

The TKNN formula was a theoretical tour de force that explained the existing experimental data from the Integer Quantum Hall Effect (IQHE). This effect is observed in a two-dimensional electron gas (2DEG) subjected to a strong perpendicular magnetic field. To fully understand the IQHE, we must first understand the electronic structure of a 2DEG in a magnetic field. In the next lesson, we will step back from topology momentarily to solve for the Landau level spectrum of a 2D electron gas in a perpendicular magnetic field. Later, we will bring these two threads together and explicitly calculate the Chern numbers for the Landau levels.