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

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

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 $n\hbar$.

Energy of interaction for two particles

Figure 1. Interaction energy vs. ${\mathit {l}}$ for $n=7$ and $k_{B}r_{B}=20$. Note that the minima occurs for ${\mathit {l}}=3$ or ${\mathit {l}}=3$. In general the minima occur at ${{\mathit {l}} \over n}={n \over 2}\pm {1 \over 2}$.

The Laughlin wavefunction is the multiparticle wavefunction for quasiparticles. The expectation value of the interaction energy for a pair of quasiparticles is

$\langle V\rangle =\langle n,N\mid V\mid n,N\rangle ,\;\;\;N=2$

where the screened potential is (see Coulomb potential between two current loops embedded in a magnetic field)

$V\left(r_{12}\right)=\left({2e^{2} \over L_{B}}\right)\int _{0}^{\infty }{{k\;dk\;} \over k^{2}+k_{B}^{2}r_{B}^{2}}\;M\left({\mathit {l}}+1,1,-{k^{2} \over 4}\right)\;M\left({\mathit {l}}^{\prime }+1,1,-{k^{2} \over 4}\right)\;{\mathcal {J}}_{0}\left(k{r_{12} \over r_{B}}\right)$

where $M$ is a confluent hypergeometric function. Here, $r_{12}$ is the distance between the centers of two current loops, $e$ is the magnitude of the electron charge, $r_{B}={\sqrt {2}}{\mathit {l}}_{B}$ is the quantum version of the Larmor radius, and $L_{B}$ is the thickness of the electron gas in the direction of the magnetic field. The angular momenta of the two individual current loops are ${\mathit {l}}\hbar$ and ${\mathit {l}}^{\prime }\hbar$ where ${\mathit {l}}+{\mathit {l}}^{\prime }=n$. The inverse screening length is given by (Gaussian units)

$k_{B}^{2}={4\pi e^{2} \over \hbar \omega _{c}AL_{B}}$

where $\omega _{c}$ is the cyclotron frequency, and $A$ is the area of the electron gas in the xy plane.

The interaction energy evaluates to

$E=\left({2e^{2} \over L_{B}}\right)\int _{0}^{\infty }{{k\;dk\;} \over k^{2}+k_{B}^{2}r_{B}^{2}}\;M\left({\mathit {l}}+1,1,-{k^{2} \over 4}\right)\;M\left({\mathit {l}}^{\prime }+1,1,-{k^{2} \over 4}\right)\;M\left(n+1,1,-{k^{2} \over 2}\right)$

To obtain this result we have made the change of integration variables

$u_{12}={z_{1}-z_{2} \over {\sqrt {2}}}$

and

$v_{12}={z_{1}+z_{2} \over {\sqrt {2}}}$

and noted (see Common integrals in quantum field theory)

${1 \over \left(2\pi \right)^{2}\;2^{3n}\;n!}\int d^{2}z_{1}\;d^{2}z_{2}\;\left(z_{1}-z_{2}\right)^{2n}\;\exp \left[-2\left(z_{1}^{2}+z_{2}^{2}\right)\right]\;{\mathcal {J}}_{0}\left({\sqrt {2}}\;{k\mid z_{1}-z_{2}\mid }\right)=$

${1 \over \left(2\pi \right)^{2}\;2^{n}\;n!}\int d^{2}u_{12}\;d^{2}v_{12}\;u_{12}^{2n}\;\exp \left[-2\left(u_{12}^{2}+v_{12}^{2}\right)\right]\;{\mathcal {J}}_{0}\left({2}k\mid u_{12}\mid \right)=$

$M\left(n+1,1,-{k^{2} \over 2}\right).$

The interaction energy has minima at (Figure 1)

${{\mathit {l}} \over n}={1 \over 3},{2 \over 5},{3 \over 7},{\mbox{etc.,}}$

and

${{\mathit {l}} \over n}={2 \over 3},{3 \over 5},{4 \over 7},{\mbox{etc.}}$

See also

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