| Laughlin wavefunction | |
|---|---|
| Name | Laughlin wavefunction |
| Field | Condensed matter physics |
| Introduced | 1983 |
| Author | Robert B. Laughlin |
| Associated with | Fractional quantum Hall effect |
Laughlin wavefunction
The Laughlin wavefunction is a variational many-body wavefunction proposed to describe the ground state of two-dimensional electrons in a strong perpendicular magnetic field at certain fractional Landau level fillings. It provided the first quantitative explanation of the Fractional quantum Hall effect and introduced conceptual tools for understanding topological order, fractionalization, and emergent quasiparticles in strongly correlated systems.
The Laughlin proposal arose to explain the plateaus observed in the Fractional quantum Hall effect (FQHE) experiments of the early 1980s at filling fractions ν = 1/m (with odd integer m). It concerns electrons confined to two dimensions, subject to a large uniform magnetic field so that single-particle states are organized into highly degenerate Landau levels. The wavefunction emphasizes the role of interparticle correlations and the combination of quantum statistics, magnetic flux quantization, and strong Coulomb interactions in producing incompressible quantum fluids. The Laughlin state contrasts with the Integer quantum Hall effect described by noninteracting electrons and highlights the centrality of electron–electron interactions and collective phenomena in modern condensed matter physics.
The Laughlin wavefunction for N electrons in the lowest Landau level at filling fraction ν = 1/m is Ψ_m(z_1,...,z_N) = ∏_{i
Laughlin's state exemplifies an incompressible quantum fluid with a finite energy gap to excitations, accounting for robust quantization of the Hall conductivity σ_xy = ν e^2/h. It introduced the notion of a quantum phase not characterized by local symmetry breaking but by global, nonlocal properties now termed topological order. Topological degeneracy on manifolds with nontrivial genus and the robustness of edge modes predicted by the bulk–edge correspondence connect Laughlin states to concepts developed in topological phases of matter research. Theoretical frameworks relating Laughlin states to Chern–Simons theory and effective field theories further clarified universal transport coefficients and the role of gauge invariance in FQHE physics.
Localized excitations above the Laughlin ground state carry fractional electric charge e* = e/m and exhibit fractional statistics (anyonic braiding). These quasiparticles were first derived by Laughlin via adiabatic insertion of magnetic flux quanta, a thought experiment building on ideas from Kenneth G. Wilson's renormalization perspective and flux-threading arguments. The braiding phase between quasiparticles is neither bosonic nor fermionic but anyonic, a property with profound implications for statistics in two dimensions and for proposals of fault-tolerant topological quantum computation using nonabelian anyons in related states. While Laughlin quasiparticles are abelian anyons, their fractionalization demonstrates charge conservation in a many-body quantum context and motivates searches for nonabelian generalizations.
Experimental confirmation of Laughlin physics includes precise measurements of plateau values in the Hall conductivity at fractions such as 1/3 and shot-noise experiments that infer fractional charge. Key experimental platforms and groups at institutions like Bell Labs, Columbia University, Princeton University, and IBM helped establish the FQHE. Techniques include high-mobility GaAs/AlGaAs heterostructures, two-dimensional electron gases, and more recently graphene and Moore–Read–related systems enabling exploration of correlated states. Interferometry experiments, tunneling spectroscopy, and single-electron transistor charge sensors have provided signatures consistent with fractional charge and anyonic statistics, though direct unambiguous braiding measurements remain experimentally challenging.
The Laughlin state is a paradigmatic example linking condensed matter and quantum information: it possesses long-range entanglement characteristic of topological order and supports protected edge modes described by conformal field theory (CFT). Quantities such as topological entanglement entropy distinguish Laughlin phases from trivial insulators and are computable in numerical studies using density matrix renormalization group (DMRG) and exact diagonalization. The robustness of information encoded nonlocally in topologically ordered states motivates research into quantum memory and fault-tolerant quantum computation, although Laughlin anyons are abelian and not universal for braiding-based computing; nonetheless, they remain important testbeds for error-resilient encodings.
Generalizations include composite-particle constructions like the Composite fermion theory by Jainendra K. Jain, hierarchical constructions by Benjamin I. Halperin and Horst L. Störmer, and wavefunctions for other filling fractions (e.g., Moore–Read Pfaffian for ν = 5/2). Open problems concern sharper microscopic derivations from realistic Hamiltonians, controlled understanding of disorder and finite-temperature effects, and experimental realization of braiding statistics with unambiguous interferometric protocols. Societal and equity considerations arise in ensuring diverse access to the experimental infrastructure and computational resources needed to advance this field, and in directing funding and collaboration to broaden participation in condensed matter research. Future directions include engineered platforms in cold atom systems, twisted bilayer graphene, and hybrid nanostructures seeking to realize and manipulate fractionalized excitations for both foundational science and technological applications.
Category:Quantum Hall effect Category:Many-body wavefunctions