Laughlin state

Laughlin state
Name	Laughlin state
Creator	Robert Laughlin
Introduced	1983
Field	Condensed matter physics
Phenomenon	Fractional quantum Hall effect
Notable for	Trial wavefunction for correlated electrons

Laughlin state The Laughlin state is a trial many-body wavefunction proposed to describe correlated electrons in the Fractional quantum Hall effect regime. It provides an explanation for the emergence of incompressible quantum fluids at certain rational filling factors and predicts fractionally charged quasiparticles and anyonic statistics, linking experiments in GaAs/AlGaAs heterostructures to theoretical constructs from Paul Dirac's and Lev Landau's quantum frameworks. The construction by Robert Laughlin catalyzed research across Princeton University, Bell Labs, IBM, and inspired work by Daniel Tsui, Horst Störmer, and later developments by Xiao-Gang Wen and Frank Wilczek.

Introduction

The Laughlin state addresses the observation of plateaus in the Hall effect at fractional fillings, explaining the 2D electron gas behavior in high magnetic fields found at facilities such as Bell Laboratories and Columbia University. Its compact analytic form captures correlations central to phenomena observed in GaAs heterostructures, graphene, and engineered platforms at Harvard University and MIT. The proposal earned recognition including the Nobel Prize in Physics awarded to experimentalists whose findings motivated it and stimulated theoretical advances by groups at Princeton and Stanford University.

Historical background and motivation

The 1982 experiments by Daniel Tsui and Horst Störmer at Bell Labs showed a Hall conductance plateau that could not be reconciled with existing Landau level theory, motivating Laughlin's 1983 ansatz at Princeton University. The context included earlier work on correlated electrons by John Bardeen, Lev Landau, and field-theory approaches by Kenneth Wilson and Philip Anderson. Laughlin's solution connected to concepts from Yang–Mills theory, Chern–Simons theory, and ideas explored in conferences at Les Houches and institutes such as the Institute for Advanced Study. Subsequent theoretical responses involved contributors like S. Das Sarma, Ady Stern, and Nobel laureate Frank Wilczek on anyons.

Wavefunction and mathematical formulation

The Laughlin wavefunction for N particles at filling fraction 1/m (m odd integer) on the plane is given in first-quantized form and constructed using complex coordinates in the lowest Landau level basis developed from Paul Dirac's magnetic monopole formalism and Haldane's sphere geometry. Its Jastrow factor structure echoes correlations studied by Robert Jastrow and links to conformal blocks in conformal field theory explored by Belavin, Polyakov, and Zamolodchikov. Mathematical analysis connects the state to modular invariants considered by Andrew Wiles in arithmetic contexts and to matrix model techniques used by Migdal and Brezin. Edge descriptions use chiral bosonization methods from Tomonaga and Luttinger models, while quasiparticle operators relate to vertex operators in the work of Victor Kac and Igor Frenkel.

Physical properties and excitations

The Laughlin state is an incompressible quantum fluid with a gap to bulk excitations, exhibiting fractionally charged quasiholes and quasielectrons whose statistics interpolate between bosons and fermions, a concept formalized by Frank Wilczek as anyons. Edge modes predicted by the state are chiral and map to theories by Xiao-Gang Wen and experiments guided by Clifford Hicks-style transport setups at National High Magnetic Field Laboratory. Theoretical descriptions use Chern–Simons theory, topological quantum field theory frameworks developed by Edward Witten, and modular tensor category language refined by Michael Freedman and Alexei Kitaev. Numerical studies by teams at Caltech, University of Tokyo, and ETH Zurich validated energetics and correlations via exact diagonalization and density-matrix-renormalization-group methods pioneered by Steven White.

Experimental signatures and realizations

Signatures include fractional plateaus in Hall conductance measured in GaAs/AlGaAs heterostructures by groups at Bell Labs and Princeton, shot-noise detection of fractional charge in experiments by Marcelo Reznikov and Leo Glazman, and interferometry searches for anyonic braiding by collaborations at Weizmann Institute and Microsoft Station Q. Realizations extend to monolayer and bilayer graphene devices studied at Columbia University and University of Manchester, and to cold-atom analogs in optical lattices at Harvard and Max Planck Institute for Quantum Optics employing synthetic gauge fields developed by Jakub Dalibard. Precise energy gaps and activation behavior were measured in high-mobility samples grown by molecular beam epitaxy groups at Sandia National Laboratories and characterized with techniques from National Institute of Standards and Technology.

Generalizations and related states

Generalizations include hierarchical constructions by Horst Störmer-inspired models and the composite fermion approach formulated by Jainendra Jain, leading to Jain states and series that explain many observed fractions. Non-Abelian extensions such as the Moore–Read (Pfaffian) state proposed by Gregory Moore and Nicholas Read describe paired states relevant to the 5/2 plateau and connect to Ising conformal field theory studied by Alexander Belavin. Multicomponent and spinful generalizations involve Halperin states from Bertrand Halperin and bilayer constructions examined by Eugene Demler and Ludwig Balents. Further links exist to lattice fractional Chern insulators researched by Shou-Cheng Zhang's school and to topological order classifications by Xiao-Liang Qi and Shou-Cheng Zhang.

Applications and theoretical significance

The Laughlin state established paradigms for topological order, fractionalization, and fault-tolerant ideas influencing proposals for topological quantum computation by Alexei Kitaev and Michael Freedman. It informs quantum information theory developments at Caltech and Microsoft Research and shapes approaches to entanglement spectrum analysis by Haldane and F. D. M. Haldane. The state’s impact spans condensed matter, quantum field theory, and mathematical physics, inspiring cross-disciplinary efforts at institutes like Perimeter Institute, CERN, and Kavli Institute.

Category:Quantum Hall effect