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Concept
Laughlin wavefunction
Summary
In condensed matter physics, the Laughlin wavefunction is an ansatz, proposed by Robert Laughlin for the ground state of a two-dimensional electron gas placed in a uniform background magnetic field in the presence of a uniform jellium background when the filling factor of the lowest Landau level is where is an odd positive integer. It was constructed to explain the observation of the fractional quantum Hall effect, and predicted the existence of additional states as well as quasiparticle excitations with fractional electric charge , both of which were later experimentally observed. Laughlin received one third of the Nobel Prize in Physics in 1998 for this discovery. Being a trial wavefunction, it is not exact, but qualitatively, it reproduces many features of the exact solution and quantitatively, it has very high overlaps with the exact ground state for small systems.
If we ignore the jellium and mutual Coulomb repulsion between the electrons as a zeroth order approximation, we have an infinitely degenerate lowest Landau level (LLL) and with a filling factor of 1/n, we'd expect that all of the electrons would lie in the LLL. Turning on the interactions, we can make the approximation that all of the electrons lie in the LLL. If is the single particle wavefunction of the LLL state with the lowest orbital angular momenta, then the Laughlin ansatz for the multiparticle wavefunction is
where position is denoted by
in (Gaussian units)
and and are coordinates in the xy plane. Here is the reduced Planck's constant, is the electron charge, is the total number of particles, and is the magnetic field, which is perpendicular to the xy plane. The subscripts on z identify the particle. In order for the wavefunction to describe fermions, n must be an odd integer. This forces the wavefunction to be antisymmetric under particle interchange. The angular momentum for this state is .
The Laughlin wavefunction is the multiparticle wavefunction for quasiparticles.
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Static forces and virtual-particle exchange
Static force fields are fields, such as a simple electric, magnetic or gravitational fields, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces arising from the interactions between two bodies mediated by virtual particles, particles that exist for only a short time determined by the uncertainty principle. The virtual particles, also known as force carriers, are bosons, with different bosons associated with each force.
Quasiparticle
In physics, quasiparticles and collective excitations are closely related phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum. For example, as an electron travels through a semiconductor, its motion is disturbed in a complex way by its interactions with other electrons and with atomic nuclei. The electron behaves as though it has a different effective mass travelling unperturbed in vacuum.