Laughlin wavefunction

Figure 1. Interaction energy vs. ${\displaystyle {\mathit {l}}}$ for ${\displaystyle n=7}$ and ${\displaystyle k_{B}r_{B}=20}$. The energy is in units of ${\displaystyle {e^{2} \over L_{B}}}$. Note that the minima occur for ${\displaystyle {\mathit {l}}=3}$ and ${\displaystyle {\mathit {l}}=4}$. In general the minima occur at ${\displaystyle {{\mathit {l}} \over n}={1 \over 2}\pm {1 \over 2n}}$.

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

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

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

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

where ${\displaystyle M}$ is a confluent hypergeometric function and ${\displaystyle {\mathcal {J}}_{0}}$ is a Bessel function of the first kind. Here, ${\displaystyle r_{12}}$ is the distance between the centers of two current loops, ${\displaystyle e}$ is the magnitude of the electron charge, ${\displaystyle r_{B}={\sqrt {2}}{\mathit {l}}_{B}}$ is the quantum version of the Larmor radius, and ${\displaystyle L_{B}}$ is the thickness of the electron gas in the direction of the magnetic field. The angular momenta of the two individual current loops are ${\displaystyle {\mathit {l}}\hbar }$ and ${\displaystyle {\mathit {l}}^{\prime }\hbar }$ where ${\displaystyle {\mathit {l}}+{\mathit {l}}^{\prime }=n}$. The inverse screening length is given by (Gaussian units)

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

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

The interaction energy evaluates to:

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

Figure 2. Interaction energy vs. ${\displaystyle {n}}$ for ${\displaystyle {{\mathit {l}} \over n}={1 \over 2}\pm {1 \over 2n}}$ and ${\displaystyle k_{B}r_{B}=0.1,1.0,10}$. The energy is in units of ${\displaystyle {e^{2} \over L_{B}}}$.

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

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

and

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

and noted (see Common integrals in quantum field theory)

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

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

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

The interaction energy has minima for (Figure 1)

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

and

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

For these values of the ratio of angular momenta, the energy is plotted in Figure 2 as a function of ${\displaystyle n}$.

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