Wikipedia / Laughlin wavefunction
Wikipedia / Laughlin wavefunction
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 fractional quantum Hall effect, both of which were later experimentally observed. Laughlin received one third of the Nobel Prize in Physics in 1998 for this discovery.
Context and analytical expression
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 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)
is the reduced Planck constant is the electron charge 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.
True ground state in FQHE at ν = 1/3
is 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. Assuming Coulomb repulsion can be determined using exact diagonalisation set to zero), Laughlin wavefunction becomes exact,.
Parent Hamiltonian and Haldane Pseudopotentials
While the Laughlin wave function was initially proposed as a highly successful ansatz, its central role in the theory of the fractional quantum Hall effect was cemented by F. Duncan Haldane, who demonstrated that it is the unique, exact zero-energy ground state of a specific "parent" Hamiltonian. This approach reverse-engineers the Hamiltonian from the known properties of the wave function, providing a powerful theoretical framework and a benchmark for numerical studies.
The construction is based on the properties of interacting particles in the lowest Landau level. In a strong magnetic field, the kinetic energy is quenched, and the physics is dominated by the interaction potential. The states of two interacting particles can be decomposed into states of definite relative angular momentum, l, the probability of finding two electrons with a relative angular momentum l less than m is exactly zero. The wave function is constructed to keep particles far apart in a very specific way, encoding correlations in the relative angular momentum channels.
Haldane's idea was to build a Hamiltonian that penalizes any pair of particles that has a relative angular momentum less than m. This is achieved using Haldane pseudopotentials, which can be thought of as a projection of the interaction potential onto states of definite relative angular momentum. The parent Hamiltonian is constructed as a sum of projection operators:
where:
The sum is over all pairs of particles (i,j).
is the operator that projects the pair (i,j) onto a state with relative angular momentum l.
) representing the energy cost for a pair to be found in the l need to be non-zero.
This Hamiltonian is a sum of positive semi-definite operators, so its energy eigenvalues are always non-negative. A ground state with zero energy can only exist if it is annihilated by every term in the sum. This means the ground state wavefunction |Ψ0⟩ must satisfy:
The Laughlin ν = 1/m state, by its very construction, perfectly satisfies this condition. It contains no components where any pair of particles has relative angular momentum less than m. Therefore, the Laughlin state is an exact zero-energy eigenstate of this parent Hamiltonian. Furthermore, for a given number of particles, it can be shown that the Laughlin state is the unique, densest state (i.e., the state with the most particles per unit of magnetic flux) that satisfies this set of conditions. This makes it the unique ground state of this idealized Hamiltonian.
This formalism is extremely powerful. It proves that there exists a local Hamiltonian for which the Laughlin state is the exact ground state, solidifying its physical relevance. It also provides a crucial tool for exact diagonalization studies. The ground state of a more realistic interaction, like the Coulomb potential, can be computed numerically and its overlap with the ideal Laughlin state can be calculated. A large overlap indicates that the Laughlin state is an excellent approximation to the true ground state of the system.
Energy of interaction for two particles
The Laughlin wavefunction is the multiparticle wavefunction for quasiparticles. The expectation value of the interaction energy for a pair of quasiparticles is
where the screened potential is (see )
is a confluent hypergeometric function is a Bessel function is the magnitude of the electron charge is the quantum version of the Larmor radius is the thickness of the electron gas in the direction of the magnetic field. The angular momenta. The inverse screening length is given by (Gaussian units)
is the cyclotron frequency is the area of the electron gas in the xy plane.
The interaction energy evaluates to:
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To obtain this result we have made the change of integration variables
and
and noted (see Common integrals in quantum field theory)
The interaction energy has minima for (Figure 1)
and
.
References
See also