Fractional quantum Hall effect

Fractional quantum Hall effect. The fractional quantum Hall effect is a remarkable physical phenomenon observed in two-dimensional electron systems subjected to low temperatures and strong perpendicular magnetic fields. It is characterized by the quantization of the Hall conductance into precise fractions of the fundamental constant \(e^2/h\), where \(e\) is the electron charge and \(h\) is Planck's constant. This effect, discovered in 1982, revealed new states of matter governed by strong electron correlations and topological order, fundamentally distinct from the integer quantum Hall effect. Its discovery and explanation led to the awarding of the Nobel Prize in Physics in 1998 to Robert Laughlin, Horst Störmer, and Daniel Tsui.

Overview

The phenomenon occurs in high-mobility two-dimensional electron gases, typically fabricated in semiconductor heterostructures like those made from gallium arsenide and aluminum gallium arsenide. Under extreme conditions of low temperature, often below 1 kelvin, and high magnetic fields, the Hall resistance exhibits plateaus at values given by \(R_{xy} = h/(\nu e^2)\), where the filling factor \(\nu\) is a simple rational fraction such as 1/3 or 2/5. Concurrently, the longitudinal resistance drops nearly to zero. This behavior signifies the emergence of a new quantum fluid state where electrons lose their individual identity. The discovery was made at Bell Labs by a team including Horst Störmer and Daniel Tsui, with theoretical interpretation provided by Robert Laughlin of Stanford University.

Theoretical explanation

The theoretical foundation was established by Robert Laughlin, who proposed a many-body wavefunction to describe the incompressible quantum fluid state at filling factor \(\nu = 1/m\), where \(m\) is an odd integer. The Laughlin wavefunction successfully accounts for the fractional charge and statistics of the excitations. This theory introduced the concept of quasiparticles and quasiholes with charges that are fractions of the electron charge, such as \(e/3\). Further theoretical advances involved the development of Chern-Simons theory and the recognition of topological order, a concept later expanded by Xiao-Gang Wen. The Jainendra Jain composite fermion theory provided a complementary and highly successful framework for explaining a wide series of observed fractions.

Experimental observations

Initial experiments on high-mobility GaAs/AlGaAs heterostructures at Bell Labs revealed the first fractional state at \(\nu = 1/3\). Subsequent work, often using dilution refrigerators and high-field magnets at facilities like the National High Magnetic Field Laboratory, observed a vast hierarchy of fractions described by series such as the Jain sequence. Key experimental signatures include the precise quantization of the Hall plateau and the vanishing of longitudinal resistance, indicating dissipationless flow. The direct measurement of fractional charge, for instance through shot noise experiments, provided crucial confirmation of Laughlin's predictions. Observations of the \(5/2\) state sparked significant interest due to its potential for hosting non-Abelian anyons.

Composite fermions

The composite fermion model, pioneered by Jainendra Jain, offers a powerful conceptual picture where electrons bind an even number of quantum vortices of the many-body wavefunction to form composite particles. These composite fermions experience a reduced effective magnetic field. At certain filling factors, they form their own effectively non-interacting Landau levels, leading to integer quantum Hall effects for composite fermions, which manifest as fractional quantum Hall effects for the original electrons. This theory elegantly explains the observed sequences of fractions and has been supported by numerous experiments, including geometric resonance measurements.

Applications and related phenomena

While direct technological applications are still emerging, the fractional quantum Hall effect has profoundly influenced modern physics. It is a primary platform for studying topological quantum computation, particularly through the proposed use of non-Abelian anyons like those in the Moore-Read state at \(\nu = 5/2\). Related phenomena include the study of Wigner crystal phases at very low filling factors and connections to other topological states like topological insulators. Research in this area continues at major institutions worldwide, including the Weizmann Institute of Science, Princeton University, and the Massachusetts Institute of Technology, driving advances in quantum matter. Category:Condensed matter physics Category:Quantum mechanics Category:Physical phenomena