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Reverted edit by 91.90.66.253 (talk) to last version by Citation bot

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In [[condensed matter physics]], the '''Laughlin wavefunction'''<ref name="Laughlin pp. 1395–1398">{{cite journal | last=Laughlin | first=R. B. | title=Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=50 | issue=18 | date=2 May 1983 | issn=0031-9007 | doi=10.1103/physrevlett.50.1395 | pages=1395–1398| bibcode=1983PhRvL..50.1395L }}</ref><ref>{{cite book | author=Z. F. Ezewa | title=Quantum Hall Effects, Second Edition| publisher= World Scientific| year=2008 | isbn=978-981-270-032-2}} pp. 210-213</ref> is an [[ansatz]], proposed by [[Robert B. Laughlin|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 [[Quantum Hall effect|filling factor]] of the [[lowest Landau level]] is <math>\nu=1/n</math> where <math>n</math> is an odd positive integer. It was constructed to explain the observation of the <math>\nu=1/3</math> [[fractional quantum Hall effect]] (FQHE), and predicted the existence of additional <math>\nu = 1/n</math> states as well as quasiparticle excitations with fractional electric charge <math>e/n</math>, 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 <math>\psi_0</math> is the single particle wavefunction of the LLL state with the lowest [[angular momentum operator|orbital angular momenta]], then the Laughlin ansatz for the multiparticle wavefunction is
: <math>
\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)
</math>
where position is denoted by
: <math>z={1 \over 2 \mathit l_B} \left( x + iy\right) </math>
in ([[Gaussian units]])
: <math> \mathit l_B = \sqrt{\hbar c\over e B} </math>
and <math> x </math> and <math> y </math> are coordinates in the x–y plane. Here <math> \hbar </math> is the [[reduced Planck constant]], <math> e </math> is the [[electron charge]], <math> N </math> is the total number of particles, and <math> B </math> 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 [[fermion]]s, n must be an odd integer. This forces the wavefunction to be antisymmetric under particle interchange. The angular momentum for this state is <math> n\hbar </math>.

== True ground state in FQHE at ''ν'' = 1/3 ==

Consider <math>n=3</math> above: resultant <math>\Psi_L(z_1,z_2, z_3, \ldots, z_N)\propto \Pi_{i<j} (z_i-z_j)^3</math> 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 <math>\Psi_{ED}</math> can be determined using exact diagonalisation<ref name="YHL">{{cite journal | last=Yoshioka | first=D. | title=Ground State of Two-Dimensional Electrons in Strong Magnetic Fields | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=50 | issue=18 | date=2 May 1983 | issn=0031-9007 | doi=10.1103/physrevlett.50.1219 | pages=1219 }}</ref> and the
overlaps have been calculated to be close to one. Moreover, with short-range interaction (Haldane pseudopotentials for <math>m>3</math> set to zero),
Laughlin wavefunction becomes exact,<ref name=HaldaneRezayi>{{Cite journal
| doi = 10.1103/PhysRevLett.54.237
| volume = 54
| pages = 237–240
| last = Haldane
| first = F.D.M.
|author2= E.H. Rezayi
| title = Finite-Size Studies of the Incompressible State of the Fractionally Quantized Hall Effect and its Excitations
| journal = Physical Review Letters
| date = 1985
| issue = 3
| pmid = 10031449
| bibcode = 1985PhRvL..54..237H
}}</ref>
i.e. <math>\langle \Psi_{ED}|\Psi_L\rangle=1</math>.

== Energy of interaction for two particles ==
[[Image:101017 Expectation value vs l.jpg|thumb|250px|right|Figure 1. Interaction energy vs. <math>{\mathit l}</math> for <math>n=7</math> and <math>k_Br_B=20</math>. The energy is in units of <math>{e^2 \over L_B}</math>. Note that the minima occur for <math>{\mathit l} =3</math> and <math>{\mathit l} =4</math>. In general the minima occur at <math>{\mathit l \over n} = {1\over 2} \pm {1\over 2n}</math>.]]
The Laughlin wavefunction is the multiparticle wavefunction for [[quasiparticle]]s. The [[expectation value]] of the interaction energy for a pair of quasiparticles is
: <math>
\langle V \rangle
=
\langle n, N \mid V \mid n, N\rangle, \; \; \; N=2
</math>
where the screened potential is (see ''{{slink|Static forces and virtual-particle exchange#Coulomb potential between two current loops embedded in a magnetic field}}'')
: <math>
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)
</math>
where <math>M</math> is a [[confluent hypergeometric function]] and <math>\mathcal J_0</math> is a [[Bessel function]] of the first kind. Here, <math>r_{12}</math> is the distance between the centers of two current loops, <math>e</math> is the magnitude of the [[electron charge]], <math>r_{B}= \sqrt{2} \mathit l_B</math> is the quantum version of the [[Larmor radius]], and <math>L_B</math> is the thickness of the electron gas in the direction of the magnetic field. The [[angular momentum|angular momenta]] of the two individual current loops are <math>\mathit l \hbar</math> and <math>\mathit l^{\prime} \hbar</math> where <math>\mathit l + \mathit l^{\prime} = n</math>. The inverse screening length is given by ([[Gaussian units]])
: <math>
k_B^2 = {4 \pi e^2 \over \hbar \omega_c A L_B}
</math>
where <math>\omega_c </math> is the [[cyclotron frequency]], and <math>A </math> is the area of the electron gas in the xy plane.

The interaction energy evaluates to:
: {|cellpadding="2" style="border:2px solid #ccccff"
|
: <math>
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)
</math>
|}
[[Image:101021 energy vs n.jpg|thumb|250px|right|Figure 2. Interaction energy vs. <math>{n}</math> for <math> {\mathit l\over n}={1\over 2} \pm {1\over 2n}</math> and <math>k_Br_B=0.1,1.0,10</math>. The energy is in units of <math>{e^2 \over L_B}</math>.]]

To obtain this result we have made the change of integration variables
: <math>
u_{12} = {z_1 - z_2 \over \sqrt{2} }
</math>
and
: <math>
v_{12} = {z_1 + z_2 \over \sqrt{2} }
</math>
and noted (see [[Common integrals in quantum field theory#Integration over a magnetic wave function|Common integrals in quantum field theory]])
: <math>
{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)
=
</math>
: <math>
{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)
=
</math>
: <math>
M \left ( n + 1, 1, -{k^2 \over 2 } \right)
.</math>

The interaction energy has minima for (Figure 1)
: <math>{\mathit l \over n} ={1\over 3}, {2\over 5}, {3\over 7}, \mbox{etc.,} </math>
and
: <math>{\mathit l \over n} ={2\over3}, {3\over 5}, {4\over 7}, \mbox{etc.} </math>

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

== References ==
{{reflist}}

== See also ==
* [[Landau level]]
* [[Fractional quantum Hall effect]]
* [[Static forces and virtual-particle exchange#Coulomb potential between two current loops embedded in a magnetic field|Coulomb potential between two current loops embedded in a magnetic field]]

[[Category:Hall effect]]
[[Category:Condensed matter physics]]
[[Category:Quantum phases]]

Laughlin wavefunction - Revision history

imported>Favonian: Reverted edit by 91.90.66.253 (talk) to last version by Citation bot