Laughlin wavefunction

True ground state in FQHE at ν = 1/3

Consider ${\displaystyle n=3}$ above: resultant ${\displaystyle {\mathrm{\Psi }}_L(z_1,z_2,z_3,\dots ,z_N)\propto {\mathrm{\Pi }}_{i<j}(z_i-z_j{)}^3}$ 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 between any two electrons, that ground state ${\displaystyle {\mathrm{\Psi }}_{ED}}$ can be determined using exact diagonalisation and the overlaps have been calculated to be close to one. Moreover, with short-range interaction (Haldane pseudopotentials for ${\displaystyle m>3}$ set to zero), Laughlin wavefunction becomes exact, i.e. ${\displaystyle ⟨{\mathrm{\Psi }}_{ED}|{\mathrm{\Psi }}_L⟩=1}$.

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

${\displaystyle ⟨V⟩=⟨n,N\mid V\mid n,N⟩,N=2}$

where the screened potential is (see Static forces and virtual-particle exchange § Coulomb potential between two current loops embedded in a magnetic field)

${\displaystyle V\left(r_{12}\right)=\left(\frac{2e^2}{L_B}\right){\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_B^2{r}_B^2}M\left(\mathit{l}+1,1,-\frac{k^2}{4}\right)M\left({\mathit{l}}^{\mathrm{\prime }}+1,1,-\frac{k^2}{4}\right){\mathcal{J}}_0\left(k\frac{r_{12}}{r_B}\right)}$

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 ${\displaystyle \mathit{l}\hslash }$ and ${\displaystyle {\mathit{l}}^{\mathrm{\prime }}\hslash }$ where ${\displaystyle \mathit{l}+{\mathit{l}}^{\mathrm{\prime }}=n}$. The inverse screening length is given by (Gaussian units)

${\displaystyle k_B^2=\frac{4\pi e^2}{\hslash {\omega }_{c}A{L}_B}}$

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:

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

${\displaystyle u_{12}=\frac{z_1-z_2}{\sqrt{2}}}$

and

${\displaystyle v_{12}=\frac{z_1+z_2}{\sqrt{2}}}$

and noted (see Common integrals in quantum field theory)

${\displaystyle \frac{1}{{\left(2\pi \right)}^2{2}^{2n}n!}\int d^2{z}_1{d}^2{z}_2\mid z_1-z_2{\mid }^{2n}\exp \left[-2\left(\mid z_1{\mid }^2+\mid z_2{\mid }^2\right)\right]{\mathcal{J}}_0\left(\sqrt{2}k\mid z_1-z_2\mid \right)=}$

${\displaystyle \frac{1}{{\left(2\pi \right)}^2{2}^{n}n!}\int d^2{u}_{12}{d}^2{v}_{12}\mid u_{12}{\mid }^{2n}\exp \left[-2\left(\mid u_{12}{\mid }^2+\mid v_{12}{\mid }^2\right)\right]{\mathcal{J}}_0\left(2k\mid u_{12}\mid \right)=}$

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

The interaction energy has minima for (Figure 1)

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

and

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

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