4.3.3 Quantum Hall Effect#

Prompts

What is the filling factor \(\nu\)? How does it relate the number of electrons to flux quanta?
Explain Laughlin’s charge pumping argument: how does threading a flux quantum through a Corbino disk quantize the Hall conductance?
At fractional filling, why does the Hall conductance become \(\sigma_{xy} = (p/q) e^2/h\), and how does this connect to anyonic quasiparticles from Chapter 2?
Why is the quantum Hall conductance robust to disorder and sample impurities, despite the apparent delicacy of quantum effects?

Lecture Notes#

Overview#

The quantum Hall effect reveals one of the most robust and universal phenomena in condensed matter physics. When a 2D electron gas is placed in a strong perpendicular magnetic field, the Hall conductance becomes quantized in units of \(e^2/h\). Integer quantum Hall effect (IQHE) occurs when Landau levels are completely filled; fractional quantum Hall effect (FQHE) occurs at rational filling factors, where interactions dominate. The charge pumping argument elegantly shows why quantization emerges and connects gauge physics to topological properties. At fractional fillings, quasiparticles carry fractional charge and obey fractional statistics—the anyons we encountered in Chapter 2.

Filling Landau Levels#

In a perpendicular magnetic field \(B\), electron energy levels organize into Landau levels, each with degeneracy \(N_\phi = BA\Phi_0^{-1} = BAe/h\), where \(A\) is the sample area. The key parameter is the filling factor.

Filling Factor

The filling factor is

\[ \nu = \frac{N_e}{N_\phi} \]

where \(N_e\) is the number of electrons and \(N_\phi\) is the number of flux quanta piercing the sample. When \(\nu\) is an integer, exactly \(\nu\) Landau levels are completely filled, and the Fermi level lies in a gap.

Integer filling (\(\nu = 1, 2, 3, \ldots\)) means all electrons fit neatly into \(\nu\) filled Landau levels with empty levels above. The energy cost to excite an electron across the gap is \(\hbar\omega_c\) (the cyclotron energy). This gap is crucial: it prevents disorder from mixing states between filled and empty levels, protecting the quantization.

Charge Pumping Argument#

The most elegant derivation of quantized Hall conductance uses Laughlin’s charge pumping picture. Consider a Corbino disk geometry: a 2D sample shaped like an annulus (ring), with inner and outer edges.

Setup: Imagine slowly threading a single flux quantum \(\Phi_0 = h/e\) through the hole of the Corbino disk, adiabatically, while maintaining filled Landau levels. As the flux increases, the number of flux quanta \(N_\phi\) in the sample increases by exactly one.

Key insight: The total charge in the system is conserved. If \(N_e\) is fixed and \(N_\phi\) increases by 1, the system must adjust. By the Pauli exclusion principle, if one Landau level is completely filled and a new flux quantum appears, one electron must be “pumped” out to make room. This pumping occurs adiabatically at the edge.

Charge Pumping and Quantized Conductance

When one flux quantum \(\Phi_0\) is threaded through the sample at filling \(\nu\), exactly \(\nu\) electrons are pumped out of the sample. This pumping is adiabatic: the Landau level gap remains open, preventing level mixing.

For each pumped charge \(\nu e\), the Lorentz force on the edge drives a current \(I = (d Q)/(d t) = \nu e \cdot (d\Phi)/(dt \Phi_0)\). The Hall voltage is set by the force balance and the edge geometry. Integrating over the full pumping cycle (\(\Delta\Phi = \Phi_0\)), the Hall conductance is:

\[ \sigma_{xy} = \frac{I}{E_y} = \nu \frac{e^2}{h} \]

This is exact and depends only on \(\nu\), \(e\), \(h\)—not on disorder, sample geometry, or microscopic details.

The charge pumping picture is the key to understanding topological protection: the quantization emerges from a global constraint (flux conservation and filling factor) combined with a gap that prevents local scattering from destroying the state.

Discussion

The charge pumping argument assumes the Landau level gap remains open throughout the adiabatic threading of the flux quantum.

(a) Why is adiabaticity essential? What would happen if we threaded the flux quickly instead of slowly?

(b) Suppose the sample has strong disorder that reduces the gap \(\hbar\omega_c\) in some regions. Can disorder nucleate a scattering channel that would violate charge quantization?

(c) In real devices, the sample has boundaries (edges). Explain how the charge being pumped actually leaves: does it escape through the bulk, or along the edge?

(d) This argument applies equally well to fractional fillings (e.g., \(\nu = 1/3\)). What is the physical meaning of pumping \(\nu = 1/3\) of an electron in one adiabatic cycle?

Linear Response and Hall Conductance#

The charge pumping argument is elegant but topological — it does not invoke the microscopic dynamics. A complementary approach uses linear response theory: apply a weak electric field and compute the induced current to first order in perturbation theory. The result is the Kubo formula for the Hall conductivity, which gives the same quantization \(\sigma_{xy} = \nu\, e^2/h\) from a completely different starting point.

Setup. Apply an oscillatory electric field \(\boldsymbol{E}(t) = \boldsymbol{E}\,\mathrm{e}^{-\mathrm{i}(\omega + \mathrm{i}0^+)t}\) that is adiabatically switched on from the infinite past. This perturbs the vector potential by \(\delta\boldsymbol{A}(t) = -\boldsymbol{E}/(\mathrm{i}\omega)\,\mathrm{e}^{-\mathrm{i}(\omega + \mathrm{i}0^+)t}\), which in turn perturbs the Hamiltonian: \(\delta\hat{H}(t) = -\hat{\boldsymbol{\pi}} \cdot \delta\boldsymbol{A}(t)\), where \(\hat{\boldsymbol{\pi}} = \hat{\boldsymbol{p}} - e\boldsymbol{A}/c\) is the kinetic momentum.

First-order time-dependent perturbation theory (Dyson series) gives the induced current. Taking the DC limit \(\omega \to 0\) yields the Kubo formula for the Hall conductivity:

Kubo Formula for Hall Conductivity

(71)#\[ \sigma_H = \frac{1}{\mathrm{i} N_\phi} \sum_{\substack{n,m \in \text{occ}}} \sum_{n' \neq n} \frac{\langle n,m \vert \hat{\pi}_x \vert n',m \rangle \langle n',m \vert \hat{\pi}_y \vert n,m \rangle - \text{h.c.}}{(E_{n'} - E_n)^2} \]

where \(\vert n,m\rangle\) are Landau level states (with Landau level index \(n\) and guiding center label \(m\)), \(E_n = \hbar\omega_c(n + \tfrac{1}{2})\) are the Landau level energies, \(\hat{\boldsymbol{\pi}}\) is the kinetic momentum operator, and the sum runs over all occupied states.

Evaluating this using the ladder-operator representation \(\hat{\pi}_x = \sqrt{\hbar\omega_c m/2}\,(\hat{a}^\dagger + \hat{a})\), \(\hat{\pi}_y = \mathrm{i}\sqrt{\hbar\omega_c m/2}\,(\hat{a}^\dagger - \hat{a})\) gives

(72)#\[ \sigma_H = \frac{1}{N_\phi} \sum_{\substack{n,m \in \text{occ}}} 1 = \nu \frac{e^2}{h} \]

Each occupied state contributes exactly \(1/N_\phi\) (in units of \(e^2/h\)) to the Hall conductivity. When \(\nu\) Landau levels are completely filled, the sum yields \(\sigma_H = \nu\, e^2/h\) — the same quantization obtained from the charge pumping argument.

Derivation: Kubo Formula for Hall Conductivity

We work in units \(m = e = \hbar = 1\) throughout and restore physical units at the end. In these units, the current density operator per electron is \(\hat{\boldsymbol{j}} = (B/N_\phi)\hat{\boldsymbol{\pi}}\), and the Landau level Hamiltonian is \(\hat{H} = \tfrac{1}{2}\hat{\boldsymbol{\pi}}^2\) with energies \(E_n = B(n + \tfrac{1}{2})\).

Step 1: Perturbation from the electric field.

An oscillatory electric field \(\boldsymbol{E}(t) = \boldsymbol{E}\,\mathrm{e}^{-\mathrm{i}(\omega + \mathrm{i}0^+)t}\) (adiabatically turned on) corresponds to a vector potential perturbation

\[ \delta\boldsymbol{A}(t) = \int_{-\infty}^t \boldsymbol{E}(t')\,\mathrm{d}t' = -\frac{\boldsymbol{E}}{\mathrm{i}\omega}\,\mathrm{e}^{-\mathrm{i}(\omega + \mathrm{i}0^+)t}. \]

Since \(\hat{H} = \tfrac{1}{2}(\hat{\boldsymbol{p}} - \boldsymbol{A})^2\), the perturbation to \(\hat{H}\) from \(\boldsymbol{A} \to \boldsymbol{A} + \delta\boldsymbol{A}\) is

\[ \delta\hat{H}(t) = \frac{\partial \hat{H}}{\partial \boldsymbol{A}} \cdot \delta\boldsymbol{A}(t) = -\hat{\boldsymbol{\pi}} \cdot \delta\boldsymbol{A}(t) = \frac{1}{\mathrm{i}\omega}\,\boldsymbol{E} \cdot \hat{\boldsymbol{\pi}}\,\mathrm{e}^{-\mathrm{i}(\omega + \mathrm{i}0^+)t}. \]

Step 2: First-order perturbation theory (interaction picture).

Switch to the interaction picture. Define \(\hat{\pi}_i(t) := \hat{G}_0(-\infty, t)\,\hat{\pi}_i\,\hat{G}_0(t, -\infty)\) where \(\hat{G}_0(t, t') = \sum_{n,m} \vert n,m\rangle\,\mathrm{e}^{-\mathrm{i}E_n(t-t')}\langle n,m\vert\) is the unperturbed propagator.

The Dyson series to first order gives the current density

\[ \langle \boldsymbol{j}(t)\rangle = \frac{B}{N_\phi} \sum_{n,m \in \text{occ}} \langle n,m\vert\left(\hat{\boldsymbol{\pi}}(t) + \mathrm{i}\int_{-\infty}^t \mathrm{d}t'\,[\delta\hat{H}_{\mathcal{I}}(t'), \hat{\boldsymbol{\pi}}(t)] + \cdots\right)\vert n,m\rangle. \]

The zeroth-order term vanishes: \(\langle n,m\vert\hat{\boldsymbol{\pi}}\vert n,m\rangle = 0\) because \(\hat{\pi}_x \propto (\hat{a}^\dagger + \hat{a})\) and \(\hat{\pi}_y \propto \mathrm{i}(\hat{a}^\dagger - \hat{a})\) are off-diagonal in the Landau level index.

The first-order term yields

\[ \langle j_i(t)\rangle = E_j\,\sigma_{ji}(\omega)\,\mathrm{e}^{-\mathrm{i}(\omega + \mathrm{i}0^+)t}, \]

with the conductivity tensor

\[ \sigma_{ji}(\omega) = \frac{B}{\omega N_\phi} \int_{-\infty}^0 \mathrm{d}t\,\mathrm{e}^{-\mathrm{i}(\omega + \mathrm{i}0^+)t} \sum_{n,m \in \text{occ}} \langle n,m\vert [\hat{\pi}_j(t), \hat{\pi}_i(0)]\vert n,m\rangle, \]

where we used time-translation invariance to shift the integration variable.

Step 3: Insert a complete set and do the time integral.

Expanding the commutator by inserting a complete set \(\sum_{n',m'} \vert n',m'\rangle\langle n',m'\vert\):

\[ \langle n,m\vert [\hat{\pi}_j(t), \hat{\pi}_i(0)]\vert n,m\rangle = \sum_{n'} \left(\langle n,m\vert \hat{\pi}_j\vert n',m\rangle\langle n',m\vert \hat{\pi}_i\vert n,m\rangle\,\mathrm{e}^{\mathrm{i}(E_n - E_{n'})t} - \text{h.c.}\right), \]

where \(m' = m\) because \(\hat{\pi}_i\) only changes the Landau level index \(n\), not the guiding center label \(m\).

Performing the time integral and combining denominators:

\[ \sigma_{ji}(\omega) = \frac{\mathrm{i}B}{\omega N_\phi} \sum_{n,m \in \text{occ}} \sum_{n' \neq n} \left(\frac{\langle n,m\vert \hat{\pi}_j\vert n',m\rangle\langle n',m\vert \hat{\pi}_i\vert n,m\rangle}{\omega - E_n + E_{n'}} - \frac{\langle n,m\vert \hat{\pi}_i\vert n',m\rangle\langle n',m\vert \hat{\pi}_j\vert n,m\rangle}{\omega + E_n - E_{n'}}\right). \]

Step 4: DC limit \(\omega \to 0\).

Expand each denominator to first order in \(\omega\):

\[ \frac{1}{\omega + E_{n'} - E_n} = \frac{1}{E_{n'} - E_n} - \frac{\omega}{(E_{n'} - E_n)^2} + \cdots \]

The \(O(1/\omega)\) piece (from the leading terms) is symmetric in \(i \leftrightarrow j\) and contributes only to the longitudinal (dissipative) conductivity. The Hall conductivity \(\sigma_{xy}\) comes from the antisymmetric \(O(1)\) piece:

\[ \sigma_H = \lim_{\omega \to 0} \sigma_{xy}(\omega) = \frac{B}{\mathrm{i}N_\phi} \sum_{n,m \in \text{occ}} \sum_{n' \neq n} \frac{\langle n,m\vert \hat{\pi}_x\vert n',m\rangle\langle n',m\vert \hat{\pi}_y\vert n,m\rangle - \text{h.c.}}{(E_{n'} - E_n)^2}. \]

This is the Kubo formula for the Hall conductivity.

Step 5: Evaluate using ladder operators.

Using \(\hat{\pi}_x = \sqrt{B/2}\,(\hat{a}^\dagger + \hat{a})\) and \(\hat{\pi}_y = \mathrm{i}\sqrt{B/2}\,(\hat{a}^\dagger - \hat{a})\):

\[ \langle n,m\vert \hat{\pi}_x\vert n',m\rangle\langle n',m\vert \hat{\pi}_y\vert n,m\rangle = \frac{\mathrm{i}B}{2}\Big((n+1)\,\delta_{n', n+1} - n\,\delta_{n', n-1}\Big). \]

This expression is purely imaginary, so it equals its own “h.c.” with a sign flip. Substituting into the Kubo formula:

\[ \sigma_H = \frac{B^2}{N_\phi} \sum_{n,m \in \text{occ}} \left(\frac{n+1}{(E_{n+1} - E_n)^2} - \frac{n}{(E_{n-1} - E_n)^2}\right). \]

Since \(E_{n'} - E_n = B(n' - n)\) (recall \(E_n = B(n + \tfrac{1}{2})\)), both denominators equal \(B^2\):

\[ \sigma_H = \frac{1}{N_\phi} \sum_{n,m \in \text{occ}} \Big((n+1) - n\Big) = \frac{1}{N_\phi} \sum_{n,m \in \text{occ}} 1. \]

Each occupied state contributes \(1/N_\phi\). For \(\nu\) completely filled Landau levels (\(N_\phi\) states each):

\[ \sigma_H = \frac{\nu N_\phi}{N_\phi} = \nu. \]

Restoring units: \(\sigma_H = \nu\, e^2/h\). \(\checkmark\)

Discussion

The Kubo formula and the charge pumping argument give the same answer \(\sigma_H = \nu\, e^2/h\), but from completely different reasoning.

The charge pumping argument is topological: it counts how many electrons are transferred when a flux quantum is threaded, and never refers to the microscopic Hamiltonian. The Kubo formula is dynamical: it sums matrix elements of the current operator between all pairs of occupied and unoccupied states. Why do these two very different calculations agree?
In the Kubo formula, each Landau level contributes exactly \(e^2/h\) to \(\sigma_H\) regardless of the magnetic field strength \(B\). What property of the Landau level spectrum guarantees this cancellation?
The Kubo formula involves a sum over all states \(n' \neq n\), yet only adjacent levels (\(n' = n \pm 1\)) contribute. What enforces this selection rule?

Fractional Quantum Hall Effect and Anyons#

At fractional fillings \(\nu = p/q\) (where \(p, q\) are coprime integers with \(q\) odd), the Landau level picture breaks down: many-body electron-electron interactions become essential. The ground state is a highly entangled quantum fluid with no simple mean-field description.

Laughlin Wavefunction

Laughlin proposed an ansatz for the ground state at \(\nu = 1/m\) (odd \(m\)):

\[ \Psi_{\text{Laughlin}} = \prod_{i<j} (z_i - z_j)^m \exp\left(-\frac{1}{4\ell_B^2}\sum_i |z_i|^2\right) \]

The factor \((z_i - z_j)^m\) enforces strong repulsion between electrons—the closer two electrons, the more suppressed the amplitude. This creates an incompressible quantum fluid with a gap to excitations.

The charge pumping argument extends to FQHE: when one flux quantum is added at filling \(\nu = p/q\), exactly \(p/q\) of an electron is pumped out. This fractional pumping is possible because the ground state is a quantum fluid that can redistribute charge non-locally.

Most remarkably, the quasiparticles (hole excitations) in the FQHE carry fractional charge \(e/q\) and obey fractional statistics—they are the anyons introduced in 2.3.2 Fractional Quantum Hall Anyons. The FQHE thus sits at the intersection of gauge physics, topology, and many-body quantum mechanics, unifying ideas from Chapters 2, 3, and 4.

Hall Conductance at Fractional Filling

At fractional filling \(\nu = p/q\), the Hall conductance is quantized as:

\[ \sigma_{xy} = \nu \frac{e^2}{h} = \frac{p}{q} \frac{e^2}{h} \]

Quasiparticles carry charge \(\pm e/q\) and obey fractional Bragg statistics: exchanging two quasiparticles introduces a phase \(\exp(2\pi \mathrm{i} p/q)\).

Summary#

Filling factor \(\nu = N_e/N_\phi\) counts filled Landau levels; integer \(\nu\) means a gap protects the state.
Charge pumping argument: threading a flux quantum through an adiabatic cycle pumps out \(\nu\) electrons, giving exact quantization \(\sigma_{xy} = \nu e^2/h\).
Robustness comes from the Landau level gap and topological protection: disorder cannot mix states within a filled level, and quantization depends only on global filling, not local physics.
Fractional QHE at \(\nu = p/q\) produces incompressible quantum fluids with fractional charge and fractional statistics.
The QHE unifies gauge physics (Landau levels and flux quantization), topology (quantized invariants), and many-body physics (Laughlin wavefunction and anyons).

Homework#

1. A 2D electron gas contains \(N_e = 2.4 \times 10^{11}\) electrons in a square sample of side \(L = 1\,\text{mm}\). A perpendicular magnetic field \(B = 5\,\text{T}\) is applied.

(a) Calculate the number of flux quanta \(N_\phi = \Phi_{\text{total}}/\Phi_0\) through the sample, where \(\Phi_0 = h/e\) is the flux quantum.

(b) Find the filling factor \(\nu = N_e/N_\phi\).

2. At filling factor \(\nu = 3\) (three completely filled Landau levels), the sample exhibits the quantum Hall effect.

(a) Calculate the Hall conductance \(\sigma_H\) in SI units (S = siemens). Use \(e = 1.602 \times 10^{-19}\,\text{C}\) and \(h = 6.626 \times 10^{-34}\,\text{J s}\).

(b) Express your answer as a multiple of the conductance quantum \(e^2/h \approx 77.5\,\mu\text{S}\).

3. Consider two samples at \(\nu = 1\): one ultra-pure (mean free path \(\ell \gg \ell_B\), where \(\ell_B = \sqrt{\hbar/(eB)}\) is the magnetic length) and one with strong disorder (\(\ell \sim \ell_B\)).

(a) Explain qualitatively why both samples exhibit the same quantized Hall conductance \(\sigma_H = e^2/h\), despite vastly different disorder levels.

(b) In the pure sample, do bulk electrons contribute to Hall transport? Why or why not?

(c) In the disordered sample, explain how edge states remain delocalized even though bulk Landau level states become localized.

4. When \(\nu = 2\) Landau levels are completely filled, the sample has two chiral edge state branches propagating along its boundary.

(a) Each edge branch acts as a 1D quantum wire. What is the conductance of a single edge branch in units of \(e^2/h\)?

(b) If there are two independent edge branches, what is the total Hall conductance? Write it as \(\sigma_H = \nu \times (e^2/h)\).

5. Charge Pumping and Flux Insertion. A sample at filling \(\nu = 1\) has total area \(A = 10^{-4}\,\text{m}^2\) and is placed in a magnetic field \(B = 1\,\text{T}\).

(a) Calculate the initial number of flux quanta \(N_\phi\) through the sample.

(b) If the magnetic field is increased to \(B' = 1.01\,\text{T}\) (adiabatically), by how many flux quanta does \(N_\phi\) increase?

(c) According to the charge pumping argument, how many electrons are pumped out of the sample during this adiabatic change?

(d) If this happens over a time interval \(\Delta t = 10\,\text{s}\), what average current is driven by the charge pumping?

6. At fractional filling \(\nu = 1/3\), the ground state is described by Laughlin’s wavefunction. The quasiparticles (holes) carry charge \(e/3\).

(a) Use the charge pumping argument to explain: when one flux quantum is threaded through the sample at \(\nu = 1/3\), what fractional charge is pumped out?

(b) For \(\nu = 2/5\) filling, what would be the charge of a quasiparticle excitation?

7. Landau Level Gap and Robustness. The cyclotron frequency is \(\omega_c = eB/m\), where \(m\) is the electron effective mass.

(a) For \(B = 10\,\text{T}\) and \(m = 0.067\,m_e\) (GaAs), calculate \(\hbar\omega_c\) in meV.

(b) If thermal energy \(k_B T \approx 1\,\text{meV}\) at a certain temperature, can thermal fluctuations excite electrons across the Landau level gap? What does this imply for observing the IQHE?

(c) If disorder introduces potential fluctuations of magnitude \(U_0 \sim 0.1 \times \hbar\omega_c\), can such disorder scatter electrons across the gap at filling \(\nu = 1\)?

8. Fractional Statistics and FQHE. At filling \(\nu = 1/3\), each quasihole carries charge \(e/3\) and obeys fractional Bragg statistics: exchanging two quasiholes introduces a phase \(\exp(2\pi \mathrm{i}/3)\).

(a) Compare this to the spin-statistics theorem for fermions (exchange phase = \(\pi\)) and bosons (exchange phase = 0). Is the quasihole a fermion, boson, or something else?

(b) The quasihole is called an anyon. Briefly explain how fractional statistics arises in a 2D system with magnetic fields and strong interactions. (See 2.3.2 Fractional Quantum Hall Anyons.)

(c) At \(\nu = 1/2\) (special case: no quasihole gap), what interesting phenomenon occurs at the edge? Does the bulk stay incompressible? (This leads to the composite fermion picture.)