# 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.