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 \( u = 1/m\), the ground state is an incompressible quantum liquid whose elementary excitations carry charge \(e/m\) and anyonic exchange statistics \( heta = \pi/m\). 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:
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 (55):
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.
Summary#
Fractionalization is real: at filling \( u = 1/3\), the elementary excitations carry charge \(e/3\), confirmed by experiment.
Statistics is emergent: the exchange phase \( heta = \pi/3\) 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 \( u = 1/m\) anyon carries charge \(e/m\) and one flux quantum, giving \( heta_{\mathrm{br}} = 2\pi/m\) and \( heta_{\mathrm{ex}} = \pi/m\).
Discussion: Is \(e/3\) a Real Charge?
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. State the filling fraction, anyon excitation charge, and statistical angle for the \(\nu = 1/3\) fractional quantum Hall state.
2. Using the charge-flux composite picture: if an anyon excitation of charge \(q\) carries a flux \(\Phi\), and another identical composite orbits around it in a full loop, what is the Aharonov-Bohm (braiding) phase? What is the exchange phase (half-braid)?
3. At filling \(\nu = 1/5\), the anyon excitation has charge \(e/5\) and carries one flux quantum \(\Phi_0 = h/e\). Using the charge-flux picture, compute the braiding phase and the exchange angle \(\theta_{\mathrm{ex}}\).
4. The Laughlin wavefunction at \(\nu = 1/m\) contains the factor \(\prod_{i<j}(z_i - z_j)^m\). Explain in words why this factor vanishes when two electrons approach each other, and what this has to do with the Pauli principle (for odd \(m\)).
5. True or false (explain briefly): (a) Anyons with \(\theta = \pi/3\) are bosons. (b) The fractional charge \(e/3\) has been measured experimentally. (c) The charge-flux picture works because of the Aharonov-Bohm effect. (d) Fractional statistics can occur in 3D bulk materials.
6. If you braid one \(\nu = 1/3\) anyon excitation around another three times (three full loops), what is the total accumulated phase? Express your answer as a multiple of \(\pi\).