| fractional quantum Hall effect | |
|---|---|
| Name | Fractional quantum Hall effect |
| Discovered | 1982 |
| Discoverer | Daniel Tsui and Horst L. Störmer (experimental), Robert B. Laughlin (theoretical) |
| Field | Condensed matter physics |
| Related | Quantum Hall effect, Topological order |
fractional quantum Hall effect
The fractional quantum Hall effect is a quantum phenomenon in two-dimensional electron systems subjected to low temperatures and strong perpendicular magnetic fields, in which the Hall conductance becomes quantized at fractional values of e^2/h. It revealed new types of correlated many-body states with emergent quasiparticles carrying fractional electric charge and unusual exchange statistics, reshaping concepts in Condensed matter physics and Quantum Physics.
The effect is a variant of the Quantum Hall effect first observed in high-mobility two-dimensional electron gases realized in GaAs/AlGaAs heterostructures. The integer quantum Hall effect was discovered by Klaus von Klitzing in 1980; soon after, in 1982, experiments by Daniel Tsui and Horst L. Störmer at Bell Labs revealed plateaus at fractional filling factors, most prominently at filling factor 1/3. The theoretical explanation was supplied by Robert B. Laughlin in 1983 via a many-body trial wavefunction; the combined experimental and theoretical advances earned Nobel Prize in Physics recognition for Tsui, Störmer, and Laughlin in 1998. The discovery catalyzed research into topological order, correlated electrons, and low-dimensional systems.
Tsui and Störmer reported precise Hall resistivity plateaus and concomitant minima in longitudinal resistivity in modulation-doped GaAs/AlGaAs semiconductor heterostructures cooled to millikelvin temperatures under magnetic fields of several tesla. Key observables include quantized Hall conductivity σ_xy = ν e^2/h at rational filling factor ν (e.g., 1/3, 2/5), vanishing longitudinal conductivity σ_xx at plateaus, and activation gaps measured via temperature dependence. Subsequent experiments employed high-mobility two-dimensional electron gases, graphene, and oxide interfaces to probe robustness. Techniques such as Shubnikov–de Haas effect measurements, tunneling spectroscopy, shot-noise detection for fractional charge, and interferometry in mesoscopic devices provided detailed data. Experimental groups at institutions including Bell Labs, Columbia University, Princeton University, and MIT have been central to progress.
Laughlin proposed a variational wavefunction for filling ν = 1/m (m odd integer) capturing strong correlations and an energy gap to excitations. The Laughlin wavefunction explained fractional quantization, incompressibility, and the appearance of quasiparticles. Building on this, the composite fermion theory of Jainendra K. Jain maps interacting electrons at fractional fillings to weakly interacting composite fermions—electrons bound to an even number of flux quanta—moving in an effective magnetic field, producing a unified sequence of observed fractions (Jain sequences). Field-theoretic descriptions employ flux attachment transformations and mean-field approximations. Numerical methods—exact diagonalization, density-matrix renormalization group (DMRG), and Monte Carlo—have validated candidate ground states and excitation spectra. Concepts from topological quantum field theory and Chern–Simons theory underpin low-energy effective descriptions.
A hallmark is the existence of quasiparticles with fractional electric charge (e.g., e/3) predicted by theory and observed via shot-noise and tunneling experiments. These quasiparticles are examples of anyons: excitations in two dimensions with exchange statistics interpolating between bosons and fermions. Some fractional states, notably at ν = 5/2 and proposed non-Abelian candidates, are predicted to host non-Abelian anyons described by conformal field theories such as the Moore–Read Pfaffian state. Experimental probes of statistics include Fabry–Pérot and Mach–Zehnder interferometers, quasiparticle braiding proposals, and noise cross-correlation measurements. Demonstrating non-Abelian braiding remains a central experimental challenge.
Beyond primary Laughlin fractions, hierarchical constructions (Haldane–Halperin hierarchy) and composite fermion filling sequences generate the observed zoo of rational ν. Low-energy effective theories capture topological properties via abelian and non-abelian Chern–Simons theory, encoded in K-matrix formulations that predict ground-state degeneracy on closed manifolds, edge theories, and quasiparticle braiding statistics. These effective descriptions connect to broader notions of topological order and ground-state entanglement, and are used to compute electromagnetic responses, Hall viscosity, and thermal Hall conductance linked to chiral central charge in associated edge conformal field theories.
Bulk-edge correspondence implies gapless one-dimensional edge modes at sample boundaries, described by chiral Luttinger liquid theory for Laughlin states and by multi-channel edge theories for hierarchical states. Edge transport measurements—two-terminal and four-terminal conductance, tunneling exponents, and shot noise—probe Luttinger parameters and quasiparticle tunneling. Mesoscopic interferometers (Fabry–Pérot, Mach–Zehnder) fabricated in high-mobility heterostructures aim to detect quasiparticle phase accumulation and exchange statistics; results are influenced by edge reconstruction, Coulomb interactions, and dephasing. Experimental control over edge structure remains critical for unambiguous tests of anyonic behavior.
Fractional quantum Hall systems, especially proposals with non-Abelian anyons (e.g., Ising anyons, Fibonacci anyons), are candidates for fault-tolerant topological quantum computation via braiding operations; prominent platforms include ν = 5/2 and engineered heterostructures coupling superconductors to quantum Hall edges. Practical realization faces challenges: isolating non-Abelian states, coherent quasiparticle control, and scalable architectures. Open theoretical problems include the microscopic origin of certain observed fractions, role of disorder and Landau level mixing, nature of edge reconstruction, and thermal Hall measurements matching predicted chiral central charge. Continued advances in materials (e.g., moiré heterostructures, high-mobility two-dimensional systems), nanofabrication, and numerical methods promise to illuminate unresolved aspects and potential technological applications.
Category:Quantum Hall effect Category:Condensed matter physics