Laughlin wavefunction

Laughlin wavefunction. A groundbreaking wavefunction proposed by Robert B. Laughlin in 1983 to explain the fractional quantum Hall effect. It describes the correlated ground state of a two-dimensional electron gas in a strong perpendicular magnetic field at specific fractional Landau level fillings. The wavefunction's success provided the first microscopic theory for fractionally charged quasiparticles and opened the field of topological phases of matter.

Introduction and historical context

The discovery of the fractional quantum Hall effect by Daniel C. Tsui and Horst Störmer under the direction of Arthur Gossard at Bell Laboratories presented a major theoretical challenge. While the integer quantum Hall effect was explained by Klaus von Klitzing using Landau level quantization and localization, the fractional version implied new, strongly correlated electron states. In 1983, Robert B. Laughlin, then at Lawrence Livermore National Laboratory, proposed his celebrated wavefunction as a variational ansatz for the ground state at filling factor one-third. This work, for which Laughlin shared the Nobel Prize in Physics with Tsui and Störmer in 1998, fundamentally altered the understanding of two-dimensional electron systems.

Mathematical form and properties

For a system of N electrons in the lowest Landau level, the Laughlin wavefunction in the symmetric gauge is given by \(\Psi_m = \prod_{jLandau level. The filling factor of the state is \(\nu = 1/m\), corresponding to observed fractions like \(1/3\) and \(1/5\).

Physical interpretation and fractional quantum Hall effect

The wavefunction captures the essence of the fractional quantum Hall effect by modeling a highly correlated quantum fluid. The odd exponent \(m\) ensures the wavefunction is antisymmetric under particle exchange, as required for fermions. The correlation hole minimizes the repulsive Coulomb interaction in the presence of the strong magnetic field, leading to an incompressible liquid state with a gap to excitations. Most profoundly, Laughlin's theory predicted that the fundamental excitations are vortices carrying fractional charge \(e/m\) and obeying fractional statistics, later termed anyons. This explained the observed quantization of the Hall conductance in units of \((e^2/h)/m\).

Generalizations and related wavefunctions

The success of the Laughlin wavefunction inspired numerous extensions to describe other observed fractional states. The hierarchical theory developed by Duncan Haldane and Bertrand Halperin constructed states at other fractions like \(2/5\) by condensing fractionally charged quasiparticles. The composite fermion theory, pioneered by Jainendra K. Jain, maps the problem to one of non-interacting composite particles in an effective field, generating a vast "Jain series" of states. Other related constructs include the Moore-Read Pfaffian state, a candidate for the even-denominator \(5/2\) state, and wavefunctions for Wigner crystal phases at very low filling.

Experimental evidence and significance

Compelling experimental verification of Laughlin's predictions came from measurements of fractionally charged quasiparticles. Pioneering experiments at the Weizmann Institute of Science and later at Princeton University using shot noise and single-electron transistor techniques confirmed charge quantization in units of \(e/3\). Observations of the predicted energy gap and its magnetic field dependence further supported the theory. The Laughlin wavefunction's legacy is immense, establishing the conceptual pillars of topological order, non-abelian statistics, and the possibility of topological quantum computation. It remains a cornerstone in the study of strongly correlated electron systems and quantum many-body physics.

Category:Condensed matter physics Category:Quantum mechanics Category:Wavefunctions