Fractional Quantum Hall Effect

Core Idea

The fractional quantum Hall effect (FQHE) occurs at fractional Landau level fillings nu = p/q (most prominently nu = 1/3, 2/5, 3/7, ...) where electron-electron interactions in a partially filled Landau level create incompressible quantum liquid states. The Laughlin wavefunction Psi = product_{i<j} (z_i - z_j)^m exp(-sum|z_k|^2/4l_B²) for nu = 1/m describes a state with no single-particle analog — it is an intrinsically many-body phenomenon. The quasiparticle excitations carry fractional charge e/m and obey fractional (anyonic) statistics, neither bosonic nor fermionic. The FQHE was the first example of topological order and remains the most dramatic manifestation of strong correlations in condensed matter.

Explainer

The fractional quantum Hall effect, discovered by Tsui, Stormer, and Gossard in 1982 (Nobel Prize 1998 with Laughlin), is one of the most remarkable phenomena in all of physics. At certain fractional Landau level fillings — most notably nu = 1/3, 2/5, 3/7, and their particle-hole conjugates — the Hall conductance is quantized at sigma_{xy} = (p/q)(e²/h) with the same extraordinary precision as the integer effect. But unlike the IQHE, no single-particle picture can explain it: the FQHE is a purely interaction-driven phenomenon.

The physical setup is the same as the IQHE — a 2DEG in a strong magnetic field — but at fractional filling, a Landau level is partially occupied. Since all electrons in a Landau level have the same kinetic energy (the level is massively degenerate), the kinetic energy is "quenched" and the Coulomb interaction alone determines the ground state. Laughlin (1983) proposed a variational wavefunction for nu = 1/m: Psi = product_{i<j} (z_i - z_j)^m exp(-sum|z_k|^2/4l_B²), where z_i = x_i + iy_i are complex coordinates. The (z_i - z_j)^m factor ensures that electrons avoid each other (each electron has an m-th order zero when another approaches), while the exponential confines them to the lowest Landau level. For m = 3 (nu = 1/3), this wavefunction has overlap >0.99 with the exact ground state.

The excitations of the Laughlin state are extraordinary. Creating a quasihole (by inserting a flux quantum) produces an excitation with fractional charge e/m = e/3 at nu = 1/3. This fractional charge has been directly measured through shot noise experiments. Even more remarkably, these quasiparticles obey anyonic statistics: exchanging two quasiholes multiplies the wavefunction by a phase e^{i pi/m}, intermediate between bosons (phase 1) and fermions (phase -1). Anyonic statistics are possible only in two spatial dimensions, where the topology of particle exchanges is richer than in 3D.

The broader significance of the FQHE is that it introduced the concept of topological order — a kind of quantum order that is not described by any local order parameter or symmetry breaking. The Laughlin state has a topological ground state degeneracy (m-fold on a torus), long-range quantum entanglement, and edge excitations described by a chiral Luttinger liquid. Composite fermion theory (Jain, 1989) extended the Laughlin picture to explain the full hierarchy of observed fractions: at nu = p/(2sp+1), electrons bind with 2s flux quanta to form composite fermions that then fill p integer Landau levels. The FQHE remains the most compelling example of emergent phenomena in condensed matter — properties of the collective state (fractional charge, anyonic statistics) that no individual electron possesses.