Takeaway

At low temperatures and high magnetic fields, electrons form correlated states with quantized Hall conductance at fractional values and quasiparticles with anyonic statistics.

The problem (before → after)

  • Before: Integer QHE arises from Landau levels and localization; fractional plateaus require interactions.
  • After: Strong correlations produce incompressible liquids (e.g., Laughlin states); excitations carry fractional charge and braid with anyonic phases.

Mental model first

Imagine a crowded dance floor where dancers move in circular steps set by the music (magnetic field). At special rhythms (fillings), they lock into a coordinated pattern; defects move with fractional steps and unusual exchange rules.

Just-in-time concepts

  • Filling factor ν: Ratio of electrons to flux quanta.
  • Laughlin wavefunction: Ψ ∝ ∏_{i<j} (z_i − z_j)^{m} e^{−∑|z_i|^2/4ℓ^2} for ν = 1/m.
  • Anyons: Exchange multiplies the wavefunction by e^{iθ} with θ ∉ {0, π}.

First-pass solution

Correlations gap the spectrum and pin the Hall conductance to σ_{xy} = ν e^2/h; edge modes carry current; shot-noise experiments detect fractional charge.

Iterative refinement

  1. Hierarchy and composite fermions explain many fractions.
  2. Topological order: Ground-state degeneracy on nontrivial manifolds; robust to local perturbations.
  3. Non-Abelian states (e.g., ν = 5/2) may enable topological quantum computation.

Principles, not prescriptions

  • Topology + interactions yield robust quantization and exotic statistics.
  • Edges encode bulk topology (bulk–edge correspondence).

Common pitfalls

  • Attributing plateaus solely to disorder; interactions are essential for fractions.
  • Ignoring edge reconstruction in interpreting experiments.

Connections and contrasts

  • See also: [/blog/topological-order], [/blog/quantum-error-correction], [/blog/retinex-color-constancy] (perception analogy to edges is only metaphorical).

Quick checks

  1. What sets fractional σ_{xy}? — Filling ν from correlated ground states.
  2. How detect fractional charge? — Shot-noise and interferometry.
  3. Why anyons? — Braiding in 2D changes wavefunction phase continuously.

Further reading

  • Laughlin, 1983; Jain composite fermions
  • PRL discovery paper (source above)