The TKNN Formula and Quantized Hall Conductivity

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to the next lesson in our exploration of topological matter.

In the previous lesson, we investigated the Integer Quantum Hall Effect (IQHE). We saw that the quantization of Hall conductivity is incredibly robust, a fact explained by two complementary pictures: the presence of dissipationless chiral edge states, and a profound property of the insulating bulk captured by Laughlin's gauge invariance argument. Laughlin's argument proved that the Hall conductivity must be quantized in units of $e^{2} / h$ , but it didn't provide a microscopic formula based on the material's electronic structure.

Today, we will bridge that gap. Our goal is to derive the celebrated TKNN formula, named after Thouless, Kohmoto, Nightingale, and den Nijs. This formula provides the missing link, explicitly relating the macroscopic Hall conductivity $σ_{x y}$ to a microscopic topological property of the bulk electronic bands: the Chern number. This will solidify our understanding of the IQHE as a topological phenomenon rooted in the geometry of quantum states in the Brillouin zone.

1. From Geometry to Transport: A Refresher on Berry Curvature

Before diving into the derivation, let's refresh our memory on the key geometric concept: the Berry curvature. In previous lessons, we defined the Berry connection $A_{n} (k)$ and the Berry curvature $Ω_{n} (k)$ for the Bloch states of a given band $n$ . We established that the Berry curvature is a gauge-invariant quantity that acts like an effective magnetic field in momentum space.

This "momentum-space magnetic field" has real physical consequences for electron transport. It introduces an anomalous velocity component transverse to the applied electric field, which is the microscopic origin of the Hall effect in topological systems.

The following video section provides an excellent conceptual overview of this connection.

Deriving Berry phase, Berry connection & Berry curvature

This clip from the 'Professor Nano' channel provides a superb analogy between Berry curvature and the magnetic field, and explicitly states the connection to the Hall effect and the Chern number.

Please watch from 24:39 to 29:32. Focus on the analogy between the magnetic vector potential/magnetic field and the Berry connection/Berry curvature. Pay close attention to how the integral of the Berry curvature over the Brillouin zone is defined as the Chern number and linked to quantized Hall conductivity.

This video sets the stage perfectly. Our task now is to make this connection quantitative by deriving the precise relationship between $σ_{x y}$ and the Chern number.

2. The Kubo Formula: A Starting Point for Conductivity

To calculate a transport coefficient like conductivity from a microscopic quantum model, we need a general framework. This is provided by linear response theory, and the result is the Kubo formula.

The physical idea is straightforward:

Start with a system in thermal equilibrium.
Apply a small, time-dependent external perturbation. In our case, this will be an electric field $E$ .
Calculate the system's response to this perturbation to first order. The response is the induced electric current $j$ .
The conductivity tensor $σ_{ij}$ is the constant of proportionality relating the current to the field: $j_{i} = \sum_{j} σ_{ij} E_{j}$ .

The full derivation of the Kubo formula involves time-dependent perturbation theory and some statistical mechanics. For our purposes, we will take the final result as our starting point. For a system of non-interacting electrons at zero temperature, the DC Hall conductivity $σ_{x y}$ is given by:

σ_{x y} = i ℏ n \in occ \sum m \in unocc \sum \int_{B Z} \frac{d ^{2} k}{( 2 π ) ^{2}} \frac{⟨ u _{n k} ∣ v _{x} ∣ u _{m k} ⟩ ⟨ u _{m k} ∣ v _{y} ∣ u _{n k} ⟩ - ( x \leftrightarrow y )}{( E _{n k} - E _{m k} ) ^{2}}

Here, the sum is over all occupied initial states $∣ u_{n k} ⟩$ and unoccupied final states $∣ u_{m k} ⟩$ , $v_{x}, v_{y}$ are the velocity operators, and $E_{n k}$ are the band energies.

This expression, while correct, looks quite formidable. Our goal is to show how this simplifies dramatically into an integral of the Berry curvature.

3. The Derivation of the TKNN Formula

The key to simplifying the Kubo formula lies in expressing the velocity matrix elements in terms of the Bloch Hamiltonian, $H (k)$ . Recall that the Bloch states $∣ u_{n k} ⟩$ are the eigenstates of $H (k)$ , which is obtained from the full Hamiltonian $H$ via $H (k) = e^{- i k \cdot r} H e^{i k \cdot r}$ .

The velocity operator can be shown to be related to the gradient of this Bloch Hamiltonian: $v = \frac{1}{ℏ} \nabla_{k} H (k)$ . Using this, and a result from perturbation theory known as the "off-diagonal Hellmann-Feynman theorem", the velocity matrix elements can be written as:

⟨ u_{n k} ∣ v_{i} ∣ u_{m k} ⟩ = \frac{1}{ℏ} ⟨ u_{n k} ∣ \partial_{k_{i}} H (k) ∣ u_{m k} ⟩ = \frac{E _{m k} - E _{n k}}{ℏ} ⟨ u_{n k} ∣ \partial_{k_{i}} u_{m k} ⟩ (for n \neq = m)

When we substitute this back into the Kubo formula, the energy denominators $(E_{n k} - E_{m k})$ cancel out, leaving a much cleaner expression. After some algebra and using the completeness relation $\sum_{m} ∣ u_{m k} ⟩ ⟨ u_{m k} ∣ = 1$ , the expression for $σ_{x y}$ transforms into a sum over only the occupied bands.

The derivation is a beautiful piece of theoretical physics. I would like you to study it carefully in the following resource.

2. The Integer Quantum Hall Effect

The following text from David Tong's lecture notes on the Quantum Hall Effect provides a concise and elegant derivation of the TKNN formula, starting from the Kubo formula.

Please read section 2.3.1, 'TKNN Invariants' (pages 61-65). Start from the list of assumptions and follow the derivation down to the final TKNN formula. Focus on how the velocity matrix elements are rewritten and how the sum over unoccupied states is eliminated, leading to an expression involving only derivatives of the Bloch wavefunctions of the occupied bands.

As you read through the derivation, notice the crucial step where the Hall conductivity is expressed as:

σ_{x y} = \frac{i e ^{2}}{ℏ} n \in filled \sum \int_{B Z} \frac{d ^{2} k}{( 2 π ) ^{2}} (⟨ \partial_{k_{x}} u_{n k} ∣ \partial_{k_{y}} u_{n k} ⟩ - ⟨ \partial_{k_{y}} u_{n k} ∣ \partial_{k_{x}} u_{n k} ⟩)

The term in the parenthesis is exactly the definition of the Berry curvature for band $n$ , $Ω_{n, z} (k)$ ! (Note that the lecture notes use $F_{x y}$ for the curvature).

This confirms our intuition: the Hall conductivity is the integral of the Berry curvature over the Brillouin zone, summed over all filled bands.

4. The Final Result: Topology Quantizes Conductance

We are now at the final step. By identifying the integrand as the Berry curvature, we can write:

σ_{x y} = \frac{e ^{2}}{ℏ} n \in filled \sum \int_{B Z} \frac{d ^{2} k}{( 2 π ) ^{2}} Ω_{n, z} (k)

From our earlier lesson on the Chern number, we know that it is defined as the integral of the Berry curvature over the entire Brillouin zone, normalized by $2 π$ :

C_{n} = \frac{1}{2 π} \int_{B Z} d^{2} k Ω_{n, z} (k)

Substituting this definition into our expression for $σ_{x y}$ and using $ℏ = h /2 π$ , we arrive at the celebrated TKNN formula:

Topology of IQHE: TKNN Formula and Chern Number — This slide summarizes the final result. The Hall conductivity $\sigma_{xy}$ is proportional to the sum of integer topological invariants, the Chern numbers $C_n$, for all filled bands.

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

This is a remarkable result. It states that the Hall conductivity, a quantity measured in a transport experiment, is determined by a topological invariant of the ground state wavefunction. Since the Chern number $C_{n}$ for any gapped band must be an integer, the Hall conductivity is perfectly quantized in units of the conductance quantum, $e^{2} / h$ . This provides the deep, microscopic explanation for the robustness of the IQHE plateaus.

To connect this back to the physical system of the IQHE, one can show that each filled Landau level, when viewed as a band in the magnetic Brillouin zone, has a Chern number $C_{n} = 1$ . Therefore, if $ν$ Landau levels are filled, the total Hall conductivity is $σ_{x y} = ν \frac{e ^{2}}{h}$ , exactly as observed.

Test your understanding!

Consider a 2D insulator with a band structure consisting of two filled bands and several empty bands, all separated by energy gaps. Through a band structure calculation, you find that the lower filled band has a Chern number of $C_{1} = + 2$ , and the upper filled band has a Chern number of $C_{2} = - 1$ . What value would you predict for the Hall conductivity $σ_{x y}$ ? What does this imply about the edge states of this material?

Show answer

According to the TKNN formula, the Hall conductivity is the sum of the Chern numbers of all filled bands, multiplied by the conductance quantum.

σ_{x y} = \frac{e ^{2}}{h} n \in filled \sum C_{n} = \frac{e ^{2}}{h} (C_{1} + C_{2}) = \frac{e ^{2}}{h} (+ 2 - 1) = \frac{e ^{2}}{h}

The predicted Hall conductivity is $σ_{x y} = + e^{2} / h$ .

The bulk-boundary correspondence tells us that the total Chern number of the bulk determines the net number of chiral edge states. A total Chern number of +1 implies that there is one more edge state moving in the "positive" direction (e.g., clockwise) than in the "negative" direction (counter-clockwise). So, despite the complex bulk topology with multiple bands and Chern numbers, the net effect at the boundary is a single, chiral, conducting edge mode, leading to the quantized Hall conductance of $e^{2} / h$ .

Conclusion

In this lesson, we have forged the final link in our understanding of the Integer Quantum Hall Effect's bulk topology. By following a formal derivation starting from the Kubo formula, we have seen how the Hall conductivity is beautifully expressed in terms of the Berry curvature.

Key Takeaways:

The Kubo formula from linear response theory provides a general starting point for calculating conductivity from a quantum mechanical model.
For a band insulator, the Kubo formula for Hall conductivity can be transformed into an integral of the Berry curvature over the Brillouin zone for all occupied bands.
This leads directly to the TKNN formula: $σ_{x y} = (e^{2} / h) \sum C_{n}$ , which relates the Hall conductivity to the sum of the integer-valued Chern numbers of the filled bands.
This formula provides the microscopic, bulk topological explanation for the precise quantization and robustness of the Integer Quantum Hall Effect.

Preview of the Next Lesson

So far, our primary example of a topological phase, the IQHE, has required a large external magnetic field. A natural question arises: is it possible to have a non-zero Chern number, and thus a quantized Hall effect, without any magnetic field? The answer is yes. In our next lesson, we will analyze the Haldane model, a pioneering tight-binding model on a honeycomb lattice. It serves as the archetypal example of a Chern insulator (or Quantum Anomalous Hall insulator), demonstrating that intricate hopping patterns can generate the necessary complex phases in the Bloch wavefunctions to produce a topological phase at zero net magnetic field.