2.3.2 Fractional Quantum Hall Anyons#

Prompts

What is the fractional quantum Hall effect, and why can it not be explained by single-particle physics?
At filling fraction \(\nu = 1/3\), what are the charge and exchange phase of the elementary excitation?
Walk through the charge-flux calculation: given charge \(q^* = e/3\) and one flux quantum \(\Phi_0\), derive the braiding and exchange phases.
In what sense are the anyonic statistics of FQHE quasiparticles ‘emergent’ rather than fundamental?

Lecture Notes#

Overview#

The fractional quantum Hall effect (FQHE) is the premier experimental realization of anyons: at filling fraction \(\nu = 1/m\), the ground state is an incompressible quantum liquid whose elementary excitations carry charge \(e/m\) and anyonic exchange statistics \(\theta = \pi/m\). This lesson applies the charge-flux composite picture to explain both the fractional charge and the fractional statistics from a single Aharonov-Bohm phase calculation.

The Fractional Quantum Hall Effect#

When a 2D electron gas is placed in a strong perpendicular magnetic field at low temperature, the Hall conductance is quantized. At certain fractional filling factors, the system forms an incompressible quantum liquid:

Filling Fractions and the Hierarchy

The fractional quantum Hall effect (FQHE) occurs at filling fractions \(\nu = 1/3, 2/5, 1/5, 2/3, \ldots\) The most prominent is \(\nu = 1/3\), discovered by Tsui, Störmer, and Gossard in 1982 (Nobel Prize 1998).

The FQHE cannot be explained by single-particle physics — it arises from strong electron-electron correlations. The ground state is described by the Laughlin wavefunction:

\[\psi(z_1, \ldots, z_N) = \prod_{i < j} (z_i - z_j)^m \, \mathrm{e}^{-\sum_k |z_k|^2 / 4\ell_B^2}\]

where \(z_k = x_k + \mathrm{i} y_k\) are complex coordinates, \(m\) is an odd integer, \(\ell_B = \sqrt{\hbar/eB}\) is the magnetic length, and \(\nu = 1/m\).

Anyon Excitation Properties at \(\nu = 1/3\)

The elementary excitations of the \(\nu = 1/3\) state are anyons with:

Fractional charge: \(q = e/3\) (one-third of the electron charge)
Fractional statistics: statistical angle \(\theta = \pi/3\) (exchange phase \(\mathrm{e}^{\mathrm{i}\pi/3}\))

These properties have been confirmed experimentally through shot noise measurements (charge) and interferometry (statistics).

Applying the Charge-Flux Picture#

The charge-flux composite picture developed in §2.3.1 explains why FQHE anyon excitations have fractional statistics. At \(\nu = 1/m\), each electron binds \(m\) flux quanta, and the anyon excitation carries charge \(q^* = e/m\) and one flux quantum \(\Phi^* = \Phi_0 = h/e\).

Example: \(\nu = 1/3\) anyons

An anyon excitation at \(\nu = 1/3\) has charge \(q^* = e/3\) and carries one flux quantum \(\Phi^* = \Phi_0 = h/e\). Applying the braiding phase (53):

\[ \theta_{\mathrm{br}} = \frac{q^*\,\Phi^*}{\hbar} = \frac{(e/3)(h/e)}{\hbar} = \frac{2\pi}{3} \]

The exchange phase is half the braiding phase: \(\theta_{\mathrm{ex}} = \theta_{\mathrm{br}}/2 = \pi/3\). The spin quantum number is \(s = \theta_{\mathrm{ex}}/(2\pi) = 1/6\).

The charge-flux picture reduces the exotic phenomenon of fractional statistics to the Aharonov-Bohm effect — a charge moving around a flux picks up a phase proportional to the enclosed flux.

Emergent Statistics#

Statistics Is Emergent

The anyonic statistics of FQHE quasiparticles is not a fundamental property of elementary particles — it emerges from the collective behavior of many strongly correlated electrons in a magnetic field. A single electron in vacuum is always a fermion. Only when \(\sim 10^{10}\) electrons form an incompressible quantum liquid does the effective low-energy description produce quasiparticles with fractional charge and fractional statistics. This is a hallmark of topological order: the statistics is encoded in the global entanglement pattern of the ground state, not in any local property.

Poll: Anyonic exchange phase

Anyons in 2D satisfy fractional exchange statistics. When you exchange two anyons at filling factor \(\nu = 1/3\), the state is multiplied by phase \(\mathrm{e}^{\mathrm{i}\pi\nu}\). What phase do you get?

(A) \(\mathrm{e}^{\mathrm{i}\pi} = -1\) (like fermions).

(B) \(\mathrm{e}^{\mathrm{i}\pi/3}\) (a fraction between 1 and -1).

(D) An undefined phase (fractionalization is not well-defined).

Summary#

Fractionalization is real: at filling \(\nu = 1/3\), the elementary excitations carry charge \(e/3\), confirmed by experiment.
Statistics is emergent: the exchange phase \(\theta = \pi/3\) is not fundamental — it arises from many-body correlations in the Laughlin ground state.
Topology protects: fractional charge and statistics are topological invariants, robust against local perturbations.
The charge-flux composite picture (§2.3.1) reduces fractional statistics to the Aharonov-Bohm effect: each \(\nu = 1/m\) anyon carries charge \(e/m\) and one flux quantum, giving \(\theta_{\mathrm{br}} = 2\pi/m\) and \(\theta_{\mathrm{ex}} = \pi/m\).
An anyon excitation in the \(\nu = 1/3\) state carries charge \(e/3\), but it is not literally one-third of an electron — it is a collective excitation of many electrons. In what sense is the fractional charge “real”? Can you measure \(e/3\) in a single experiment, or only statistically? What would Millikan’s oil drop experiment show if you could trap a single anyon excitation?

Homework#

1. Anyon properties. State the filling fraction, anyon excitation charge, and statistical angle for the \(\nu = 1/3\) fractional quantum Hall state. Verify that the exchange phase \(\mathrm{e}^{\mathrm{i}\theta}\) with \(\theta = \pi/3\) is neither bosonic nor fermionic.

2. Braiding phase. An anyon excitation at \(\nu = 1/m\) carries charge \(q^* = e/m\) and one flux quantum \(\Phi_0 = h/e\).

(a) Using the Aharonov-Bohm formula \(\theta_\text{br} = q^*\Phi_0/\hbar\), compute the braiding phase for \(\nu = 1/3\), \(\nu = 1/5\), and \(\nu = 1/7\).

(b) For each case, find the exchange phase \(\theta_\text{ex} = \theta_\text{br}/2\) and the spin quantum number \(s = \theta_\text{ex}/(2\pi)\).

3. Laughlin wavefunction. The Laughlin wavefunction at \(\nu = 1/m\) contains the Jastrow factor \(\prod_{i<j}(z_i - z_j)^m\) where \(z_k = x_k + \mathrm{i}y_k\) are complex coordinates.

(a) What happens to this factor when two electrons approach each other (\(z_i \to z_j\))? How does this enforce the Pauli principle for odd \(m\)?

(b) The exponent \(m\) controls the vanishing order. Explain why larger \(m\) (lower filling \(\nu = 1/m\)) means electrons avoid each other more strongly.

4. Multiple braiding. One \(\nu = 1/3\) anyon is braided around another (full loop) \(n\) times.

(a) Show that the accumulated phase after \(n\) full braidings is \(2\pi n/3\).

(b) After how many full braidings does the system first return to its original state?

(c) Compare with \(\nu = 1/5\) anyons. For general \(\nu = 1/m\), what is the minimum number of full braidings needed to return to the original state?

5. Conceptual questions. Answer true or false with a brief explanation:

(a) Anyons with \(\theta_\text{ex} = \pi/3\) are bosons.

(b) The fractional charge \(e/3\) of \(\nu = 1/3\) anyon excitations has been measured experimentally.

(d) Fractional statistics can occur in 3D bulk materials without special boundary conditions.

6. Emergent vs fundamental. The anyonic statistics of FQHE quasiparticles is emergent, not fundamental.

(a) Explain what this means: a single electron in vacuum is always a fermion, but when many electrons form a quantum Hall liquid, the collective excitations behave as anyons.

(b) The fractional charge and statistics are said to be “topologically protected.” What does this mean in terms of robustness to local perturbations?

(c) Compare with phonons in a crystal: phonons are bosonic quasiparticles emerging from atomic vibrations. In what sense are FQHE anyons a more exotic form of emergence than phonons?

7. Charge-flux at general filling. At filling \(\nu = p/q\) (with \(q\) odd), the elementary anyon excitation carries charge \(e\nu = pe/q\) and the exchange statistics depends on the specific hierarchy state.

(a) For \(\nu = 2/5\), the theory predicts anyon charge \(q^* = e/5\). If each anyon still binds one flux quantum \(\Phi_0\), compute the braiding and exchange phases using the charge-flux formula.

(b) Explain why the \(\nu = 2/5\) state can be understood as a “daughter” of the \(\nu = 1/3\) state in the Haldane-Halperin hierarchy: the anyonic excitations of the \(\nu = 1/3\) state themselves form a quantum Hall liquid at an effective filling factor.