Condensed matter physics

The quantum Hall effect

A two-dimensional electron gas in a strong magnetic field develops quantized Hall plateaus; edge states help protect transport.

A two-dimensional electron gas in a strong magnetic field develops quantized Hall plateaus; edge states help protect transport.

Physical model

A perpendicular magnetic field quantizes orbital motion into Landau levels. When the Fermi level lies in a gap, the Hall response is stable over a parameter range.

An ideal integer plateau has σ_xy=νe²/h and R_xy=h/(νe²). Chiral edge states carry current along boundaries; bulk-boundary correspondence links a topological index to net channels.

Quick check

Each Landau level has degeneracy eBA/h over area A, so the filling factor ν counts occupied levels. When the Fermi energy lies in a gap, the Chern number of filled bands makes Hall conductance invariant under weak disorder that does not close the gap. Current flows through chiral edge channels, and the quantized Hall resistance provides a precise resistance standard. In the fractional effect, many-body correlations create phases with fractionally charged excitations.

A geometric viewpoint explains the precision: integrating Berry curvature over the Brillouin zone gives an integer Chern number. Because it cannot change under continuous deformations while the gap remains open, the macroscopic response is insensitive to many sample details. Experimental plateaus arise when the Fermi level lies among localized states between Landau levels; their width depends on disorder and temperature. The constant h/e² underlies the quantum resistance standard.

On an integer plateau, the net number of edge channels equals the integer filling factor when degeneracy is counted consistently; local perturbations cannot change this integer while the bulk gap stays open. Fractional phases can host quasiparticles with fractional charge and unusual statistics. Both regimes quantize Hall response but require different microscopic descriptions.