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 is 1/n where n is an odd positive integer. It was constructed to explain the observation of one third of an electron charge in the Fractional quantum Hall effect, and Laughlin received one third of the Nobel Prize in Physics in 1998 for his discovery. Being an ansatz, it's not exact, but qualitatively, it reproduces many features of the exact solution and quantitatively, it is pretty good.

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 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 $\psi _{0}$ is the single particle wavefunction of the LLL state with the lowest orbital angular momentum, then the Laughlin ansatz for the multiparticle wavefunction is

$\psi (z_{1},z_{2},z_{3},\ldots ,z_{N})=\left[{1 \over 2\pi \;\left({\sqrt {2}}\right)^{3n}\;{\sqrt {n!}}\;\left({\sqrt {N-1}}\right)^{n+1}}\right]^{N\left(N-1\right) \over 2}\left[\prod _{N\geqslant i>j\geqslant 1}\left(z_{i}-z_{j}\right)^{n}\right]\prod _{k=1}^{N}\psi _{0}(z_{k})$

where

$z={1 \over 2{\mathit {l}}_{B}}\left(x+iy\right)$

and

$\psi _{0}(z_{k})=\exp \left(-\mid z_{k}\mid ^{2}\right)$

and (Gaussian units)

${\mathit {l}}_{B}={\sqrt {\hbar c \over eB}}$

and $x$ and $y$ are coordinates in the xy plane. Here $\hbar$ is Planck's constant, $e$ is the electron charge, $N$ is the total umber of particles, and $B$ 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 $n\hbar$.

See also

• Landau level
• Fractional quantum Hall effect
• Coulomb potential between two current loops embedded in a magnetic field