Laughlin wavefunction

In condensed matter physics, the Laughlin wavefunction[1][2] 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 (Quantum Hall effect) of the lowest Landau level is $\nu=1/n$ where $n$ is an odd positive integer. It was constructed to explain the observation of the $\nu=1/3$ fractional quantum Hall effect, and predicted the existence of additional $\nu = 1/n$ states as well as quasiparticle excitations with fractional electric charge $e/n$, both of which were later experimentally observed. Laughlin received one third of the Nobel Prize in Physics in 1998 for this discovery. Being 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.

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, 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

$\langle z_1,z_2,z_3,\ldots , z_N \mid n,N\rangle = \psi_{n,N}(z_1,z_2, z_3, \ldots, z_N ) = D \left[ \prod_{N \geqslant i > j \geqslant 1}\left( z_i-z_j \right)^n \right] \prod^N_{k=1}\exp\left( - \mid z_k \mid^2 \right)$

where position is denoted by

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

and normalization[citation needed]

$D=\left[ {1 \over 2\pi \;\left( \sqrt2\right)^{2n} \; \sqrt{n!} \; \left( \sqrt{ N-1 }\right)^{n+1} } \right]^{N\left( N-1\right) \over 2}$

in (Gaussian units)

$\mathit l_B = \sqrt{\hbar c\over e B}$

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 number 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 $k_Br_B=20$. The energy is in units of ${e^2 \over L_B}$. Note that the minima occur for ${\mathit l} =3$ and ${\mathit l} =4$. In general the minima occur at ${\mathit l \over n} = {1\over 2} \pm {1\over 2n}$.

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

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

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

$V\left( r_{12}\right) = \left( { 2 e^2 \over L_B}\right) \int_0^{\infty} {{k\;dk \;} \over k^2 + k_B^2 r_{B}^2 } \; M \left ( \mathit l + 1, 1, -{k^2 \over 4} \right) \;M \left ( \mathit l^{\prime} + 1, 1, -{k^2 \over 4} \right) \;\mathcal J_0 \left ( k{r_{12}\over r_{B}} \right)$

where $M$ is a confluent hypergeometric function and $\mathcal J_0$ is a Bessel function of the first kind. 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)

$k_B^2 = {4 \pi e^2 \over \hbar \omega_c A L_B}$

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:

 $E= \left( { 2 e^2 \over L_B}\right) \int_0^{\infty} {{k\;dk \;} \over k^2 + k_B^2 r_{B}^2 } \; M \left ( \mathit l + 1, 1, -{k^2 \over 4} \right) \;M \left ( \mathit l^{\prime} + 1, 1, -{k^2 \over 4} \right) \;M \left ( n + 1, 1, -{k^2 \over 2} \right)$
Figure 2. Interaction energy vs. ${n}$ for ${\mathit l\over n}={1\over 2} \pm {1\over 2n}$ and $k_Br_B=0.1,1.0,10$. The energy is in units of ${e^2 \over L_B}$.

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)

${1 \over \left( 2 \pi\right)^2\; 2^{2n} \; n! } \int d^2z_1 \; d^2z_2 \; \mid z_1 - z_2 \mid^{2n} \; \exp \left[ - 2 \left( \mid z_1 \mid^2 + \mid z_2\mid^2 \right) \right] \;\mathcal J_0 \left ( \sqrt{2}\; { k\mid z_1 - z_2 \mid } \right) =$
${1 \over \left( 2 \pi\right)^2\; 2^{n} \; n! } \int d^2u_{12} \; d^2v_{12} \; \mid u_{12}\mid^{2n} \; \exp \left[ - 2 \left( \mid u_{12}\mid^2 + \mid v_{12}\mid^2 \right) \right] \;\mathcal J_0 \left ( {2} k\mid u_{12} \mid \right) =$
$M \left ( n + 1, 1, -{k^2 \over 2 } \right) .$

The interaction energy has minima for (Figure 1)

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

and

${\mathit l \over n} ={2\over3}, {3\over 5}, {4\over 7}, \mbox{etc.}$

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

References

1. ^
2. ^ Z. F. Ezewa (2008). Quantum Hall Effects, Second Edition. World Scientific. ISBN 981-270-032-3. pp. 210-213