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

Context and analytical expression

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

${\displaystyle \langle z_{1},z_{2},z_{3},\ldots ,z_{N}\mid n,N\rangle =\psi _{n,N}(z_{1},z_{2},z_{3},\ldots ,z_{N})=D\left[\prod _{N\geqslant i>j\geqslant 1}\left(z_{i}-z_{j}\right)^{n}\right]\prod _{k=1}^{N}\exp \left(-\mid z_{k}\mid ^{2}\right)}$

where position is denoted by

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

in (Gaussian units)

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

and ${\displaystyle x}$ and ${\displaystyle y}$ are coordinates in the x–y plane. Here ${\displaystyle \hbar }$ is the reduced Planck constant, ${\displaystyle e}$ is the electron charge, ${\displaystyle N}$ is the total number of particles, and ${\displaystyle B}$ 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 ${\displaystyle n\hbar }$.

True ground state in FQHE at ν = 1/3

Consider ${\displaystyle n=3}$ above: resultant ${\displaystyle \Psi _{L}(z_{1},z_{2},z_{3},\ldots ,z_{N})\propto \Pi _{i<j}(z_{i}-z_{j})^{3}}$ 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 ${\displaystyle \Psi _{ED}}$ can be determined using exact diagonalisation^[3] 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,^[4] i.e. ${\displaystyle \langle \Psi _{ED}|\Psi _{L}\rangle =1}$.

Energy of interaction for two particles

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 Static forces and virtual-particle exchange § 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 ${\displaystyle n}$.

References

1. Laughlin, R. B. (2 May 1983). "Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations". Physical Review Letters. 50 (18). American Physical Society (APS): 1395–1398. Bibcode:1983PhRvL..50.1395L. doi:10.1103/physrevlett.50.1395. ISSN 0031-9007.
2. Z. F. Ezewa (2008). Quantum Hall Effects, Second Edition. World Scientific. ISBN 978-981-270-032-2. pp. 210-213
3. Yoshioka, D. (2 May 1983). "Ground State of Two-Dimensional Electrons in Strong Magnetic Fields". Physical Review Letters. 50 (18). American Physical Society (APS): 1219. doi:10.1103/physrevlett.50.1219. ISSN 0031-9007.
4. Haldane, F.D.M.; E.H. Rezayi. "Finite-Size Studies of the Incompressible State of the Fractionally Quantized Hall Effect and its Excitations". Physical Review Letters. 54: 237. doi:10.1103/PhysRevLett.54.237.

See also

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