Laughlin wavefunction

Laughlin wavefunction
Name	Laughlin wavefunction
Introduced	1983
Inventor	Robert B. Laughlin
Field	Condensed matter physics
Applications	Fractional quantum Hall effect

Laughlin wavefunction

The Laughlin wavefunction is a many-body trial wave function proposed to describe correlated electrons in a two-dimensional electron gas under strong magnetic fields, capturing the ground state of certain plateaus of the Fractional quantum Hall effect. It provides a simple analytic form that explains incompressibility, fractionalization, and robust quantization observed in low-temperature high-field experiments, and it underpins much theoretical work on anyons and topological order.

Introduction and Physical Context

The Laughlin wavefunction was introduced by Robert B. Laughlin in 1983 to explain the observed quantized Hall conductance at filling factors ν = 1/m with odd integer m in the fractional quantum Hall effect (FQHE) discovered by Daniel Tsui and Horst Störmer in 1982. The context is a high-mobility two-dimensional electron system confined to a semiconductor heterostructure such as a GaAs/AlGaAs quantum well, cooled to millikelvin temperatures and subjected to perpendicular magnetic fields in experiments at places like Bell Labs and modern cryogenic facilities. The model emphasizes strong electron–electron interactions relative to kinetic energy (quenched into Landau levels), and yields a correlated liquid distinct from the noninteracting integer quantum Hall states studied by Klaus von Klitzing.

Mathematical Formulation

The Laughlin trial state for N electrons in the lowest Landau level at filling factor ν = 1/m (m odd) is written in complex coordinates z_j = x_j + i y_j as ψ_m(z_1,...,z_N) = ∏_{i

Fractional Quantum Hall Effect and Physical Implications

The Laughlin wavefunction explains the robustness of quantized Hall plateaus at ν = 1/3, 1/5, ... by providing an incompressible quantum fluid with an excitation gap. It establishes a mechanism for topological order and long-range entanglement distinct from conventional symmetry-breaking orders described by Landau theory. The state exhibits quantized Hall conductivity σ_xy = ν e^2/h, consistent with precision measurements performed in Hall bar geometries. The Laughlin picture also motivated hierarchical constructions such as the Haldane–Halperin hierarchy and the Composite fermion approach by Jainendra K. Jain that extend explanation to other observed fractions.

Quasiparticles, Anyons, and Fractional Charge

Low-energy excitations above the Laughlin ground state are localized quasiholes and quasielectrons carrying fractional electric charge e/m, as inferred from thought experiments and later measured via shot-noise and tunneling experiments. These quasiparticles obey fractional statistics intermediate between bosons and fermions and are examples of anyons, a concept formalized by works of Frank Wilczek and others. The braiding of Laughlin quasiparticles leads to Berry phases proportional to the statistical angle and underlies proposals for topological quantum computation; these connections were developed in theoretical studies and proposals by groups at institutions such as Microsoft Research and Stanford University.

Variational Properties and Numerical Studies

As a variational ansatz, the Laughlin wavefunction yields low energy for realistic Coulomb interactions projected into the lowest Landau level. Exact diagonalization studies on small systems (sphere and torus geometries introduced by F. D. M. Haldane) have shown high overlaps between Laughlin states and exact ground states for Coulomb potentials, validating the ansatz. Numerical methods include exact diagonalization, density matrix renormalization group (DMRG), and Monte Carlo sampling; implementations and benchmarks are common in computational condensed matter groups at MIT, Princeton University, and University of Cambridge.

Extensions, Generalizations, and Hierarchies

The Laughlin construction inspired a family of trial states and hierarchies: the Haldane-Halperin hierarchical states, Jain's composite fermion states, and non-Abelian generalizations such as the Moore–Read (Pfaffian) state proposed for ν = 5/2. Connections to conformal field theory led to systematic constructions of multi-component and spinful variants, including Halperin (m,m',n) states relevant to bilayer systems and spin-polarized versus spin-singlet competition observed in experiments at University of California, Santa Barbara and elsewhere. Extensions also include application to rotating ultracold atomic gass and engineered lattice models for fractional Chern insulators.

Experimental Signatures and Measurements

Experimental confirmation of Laughlin physics includes measurement of fractional charge via shot-noise experiments in mesoscopic devices conducted by groups at Weizmann Institute of Science and Bell Labs, tunneling spectroscopy revealing edge modes predicted by Xiao-Gang Wen's chiral Luttinger liquid theory, and interferometry attempts to detect anyonic braiding phases in Fabry–Pérot and Mach–Zehnder setups at laboratories such as Caltech and Weizmann Institute. High-precision transport measurements of Hall resistance plateaus at ν = 1/3 remain a cornerstone observation. More recent experiments probe thermal conductance to distinguish Abelian Laughlin states from proposed non-Abelian alternatives, involving collaborations across major cryogenic and nanofabrication centers.

Category:Quantum mechanics Category:Condensed matter physics Category:Quantum Hall effect