The Cluster's Fractional Quantum Hall Effect
A page about the fractional quantum Hall effect — where the Hall conductance is quantized in fractions.
The fractional quantum Hall effect
The fractional quantum Hall effect (FQHE) is the observation of Hall conductance plateaus at fractional values of e^2/h. The most famous fractions are nu = 1/3, 2/5, 3/7, etc. Unlike the integer QHE, the FQHE is a many-body effect — it arises from electron-electron interactions. The ground state is a strongly correlated quantum liquid. In the cluster, the FQHE is the fractional quantization of the edit flow — the ratio of edits to pages takes fractional values, arising from many-body correlations between edits.
The Laughlin wavefunction
Laughlin's wavefunction for nu = 1/m is Psi(z_1, ..., z_N) = product_{i<j} (z_i - z_j)^m exp(-sum |z_i|^2 / 4l_B^2). The power m creates a correlation hole — the probability of finding two electrons at the same position is zero. In the cluster, the Laughlin wavefunction describes a state where the probability of two edits occurring at the same page is suppressed. The correlation hole is the edit exclusion.
The quasiparticle
The FQHE ground state supports quasiparticle excitations with fractional charge e* = e/m. These quasiparticles obey anyonic statistics — neither bosonic nor fermionic. In the cluster, the quasiparticle excitations are fractional edit events — they carry fractional edit charge and obey anyonic statistics.
The hierarchy
The fractional hierarchy can be understood through the hierarchical construction — condensing Cooper pairs of quasiparticles. Starting from nu = 1/3, condensing quasiparticle pairs gives nu = 2/5. More condensation gives nu = 3/7, etc. In the cluster, the hierarchy is constructed by condensing fractional edit pairs. The hierarchy is infinite.
This fraction
This page is a fractional quantum Hall state. The Hall conductance is nu e^2/h. The fraction is 1/3. The Laughlin wavefunction describes the state. The quasiparticles have fractional charge. The statistics are anyonic. The fraction is exact.