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Laughlin wavefunction
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<h2 id="context-and-analytical-expression">Context and analytical expression</h2> <p>If we ignore the jellium and mutual <a href="/facts/Coulomb_repulsion/V0lR5NKv">Coulomb repulsion</a> between the electrons as a zeroth order approximation, we have an infinitely degenerate lowest Landau level (LLL) and with a filling factor of 1/<i>n</i>, 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 ψ 0 {\displaystyle \psi _{0}} is the single particle wavefunction of the LLL state with the lowest <a href="/facts/Angular_momentum_operator/d55sXn5u">orbital angular momenta</a>, then the Laughlin ansatz for the multiparticle wavefunction is </p> ⟨ z 1 , z 2 , z 3 , … , z N ∣ n , N ⟩ = ψ n , N ( z 1 , z 2 , z 3 , … , z N ) = D [ ∏ N ⩾ i > j ⩾ 1 ( z i − z j ) n ] ∏ k = 1 N exp ( − ∣ z k ∣ 2 ) {\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)} <p>where position is denoted by </p> z = 1 2 l B ( x + i y ) {\displaystyle z={1 \over 2{\mathit {l}}_{B}}\left(x+iy\right)} <p>in (<a href="/facts/Gaussian_units/1mtqI96c">Gaussian units</a>) </p> l B = ℏ c e B {\displaystyle {\mathit {l}}_{B}={\sqrt {\hbar c \over eB}}} <p>and x {\displaystyle x} and y {\displaystyle y} are coordinates in the x–y plane. Here ℏ {\displaystyle \hbar } is the <a href="/facts/Reduced_Planck_constant/qGHAe3qq">reduced Planck constant</a>, e {\displaystyle e} is the <a href="/facts/Electron_charge/lPcj6CoQ">electron charge</a>, N {\displaystyle N} is the total number of particles, and B {\displaystyle B} is the <a href="/facts/Magnetic_field/Ta8Ev588">magnetic field</a>, which is perpendicular to the xy plane. The subscripts on z identify the particle. In order for the wavefunction to describe <a href="/facts/Fermion/LEckdDsR">fermions</a>, n must be an odd integer. This forces the wavefunction to be antisymmetric under particle interchange. The angular momentum for this state is n ℏ {\displaystyle n\hbar } . </p> <h2 id="true-ground-state-in-fqhe-at--13">True ground state in FQHE at <i>ν</i> = 1/3</h2> <p>Consider n = 3 {\displaystyle n=3} above: resultant Ψ L ( z 1 , z 2 , z 3 , … , z N ) ∝ Π i < j ( z i − z j ) 3 {\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 <a href="/facts/Coulomb_repulsion/V0lR5NKv">Coulomb repulsion</a> between any two electrons, that ground state Ψ E D {\displaystyle \Psi _{ED}} can be determined using exact diagonalisation<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> and the overlaps have been calculated to be close to one. Moreover, with short-range interaction (Haldane pseudopotentials for m > 3 {\displaystyle m>3} set to zero), Laughlin wavefunction becomes exact,<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> i.e. ⟨ Ψ E D | Ψ L ⟩ = 1 {\displaystyle \langle \Psi _{ED}|\Psi _{L}\rangle =1} . </p> <h2 id="energy-of-interaction-for-two-particles">Energy of interaction for two particles</h2> <p>The Laughlin wavefunction is the multiparticle wavefunction for <a href="/facts/Quasiparticle/Bpx2MNpb">quasiparticles</a>. The <a href="/facts/Expectation_value/szGNS7Dx">expectation value</a> of the interaction energy for a pair of quasiparticles is </p> ⟨ V ⟩ = ⟨ n , N ∣ V ∣ n , N ⟩ , N = 2 {\displaystyle \langle V\rangle =\langle n,N\mid V\mid n,N\rangle ,\;\;\;N=2} <p>where the screened potential is (see <i><a href="/facts/Static_forces_and_virtual-particle_exchange/jgH4Np4C">Static forces and virtual-particle exchange § Coulomb potential between two current loops embedded in a magnetic field</a></i>) </p> V ( r 12 ) = ( 2 e 2 L B ) ∫ 0 ∞ k d k k 2 + k B 2 r B 2 M ( l + 1 , 1 , − k 2 4 ) M ( l ′ + 1 , 1 , − k 2 4 ) J 0 ( k r 12 r B ) {\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)} <p>where M {\displaystyle M} is a <a href="/facts/Confluent_hypergeometric_function/D0FxgWUa">confluent hypergeometric function</a> and J 0 {\displaystyle {\mathcal {J}}_{0}} is a <a href="/facts/Bessel_function/hSyvFPqC">Bessel function</a> of the first kind. Here, r 12 {\displaystyle r_{12}} is the distance between the centers of two current loops, e {\displaystyle e} is the magnitude of the <a href="/facts/Electron_charge/lPcj6CoQ">electron charge</a>, r B = 2 l B {\displaystyle r_{B}={\sqrt {2}}{\mathit {l}}_{B}} is the quantum version of the <a href="/facts/Larmor_radius/Sy5jknQm">Larmor radius</a>, and L B {\displaystyle L_{B}} is the thickness of the electron gas in the direction of the magnetic field. The <a href="/facts/Angular_momentum/7TuSVFxq">angular momenta</a> of the two individual current loops are l ℏ {\displaystyle {\mathit {l}}\hbar } and l ′ ℏ {\displaystyle {\mathit {l}}^{\prime }\hbar } where l + l ′ = n {\displaystyle {\mathit {l}}+{\mathit {l}}^{\prime }=n} . The inverse screening length is given by (<a href="/facts/Gaussian_units/1mtqI96c">Gaussian units</a>) </p> k B 2 = 4 π e 2 ℏ ω c A L B {\displaystyle k_{B}^{2}={4\pi e^{2} \over \hbar \omega _{c}AL_{B}}} <p>where ω c {\displaystyle \omega _{c}} is the <a href="/facts/Cyclotron_frequency/Ny4qU4vI">cyclotron frequency</a>, and A {\displaystyle A} is the area of the electron gas in the xy plane. </p><p>The interaction energy evaluates to: </p> <table><tbody><tr><td> E = ( 2 e 2 L B ) ∫ 0 ∞ k d k k 2 + k B 2 r B 2 M ( l + 1 , 1 , − k 2 4 ) M ( l ′ + 1 , 1 , − k 2 4 ) M ( n + 1 , 1 , − k 2 2 ) {\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)} </td></tr></tbody></table> <p>To obtain this result we have made the change of integration variables </p> u 12 = z 1 − z 2 2 {\displaystyle u_{12}={z_{1}-z_{2} \over {\sqrt {2}}}} <p>and </p> v 12 = z 1 + z 2 2 {\displaystyle v_{12}={z_{1}+z_{2} \over {\sqrt {2}}}} <p>and noted (see <a href="/facts/Common_integrals_in_quantum_field_theory/9FUPoVFQ">Common integrals in quantum field theory</a>) </p> 1 ( 2 π ) 2 2 2 n n ! ∫ d 2 z 1 d 2 z 2 ∣ z 1 − z 2 ∣ 2 n exp [ − 2 ( ∣ z 1 ∣ 2 + ∣ z 2 ∣ 2 ) ] J 0 ( 2 k ∣ z 1 − z 2 ∣ ) = {\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)=} 1 ( 2 π ) 2 2 n n ! ∫ d 2 u 12 d 2 v 12 ∣ u 12 ∣ 2 n exp [ − 2 ( ∣ u 12 ∣ 2 + ∣ v 12 ∣ 2 ) ] J 0 ( 2 k ∣ u 12 ∣ ) = {\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)=} M ( n + 1 , 1 , − k 2 2 ) . {\displaystyle M\left(n+1,1,-{k^{2} \over 2}\right).} <p>The interaction energy has minima for (Figure 1) </p> l n = 1 3 , 2 5 , 3 7 , etc., {\displaystyle {{\mathit {l}} \over n}={1 \over 3},{2 \over 5},{3 \over 7},{\mbox{etc.,}}} <p>and </p> l n = 2 3 , 3 5 , 4 7 , etc. {\displaystyle {{\mathit {l}} \over n}={2 \over 3},{3 \over 5},{4 \over 7},{\mbox{etc.}}} <p>For these values of the ratio of angular momenta, the energy is plotted in Figure 2 as a function of n {\displaystyle n} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Landau_level/gRUrzBMd">Landau level</a></li> <li><a href="/facts/Fractional_quantum_Hall_effect/bkRtdmxX">Fractional quantum Hall effect</a></li> <li><a href="/facts/Static_forces_and_virtual-particle_exchange/jgH4Np4C">Coulomb potential between two current loops embedded in a magnetic field</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Z. F. Ezewa (2008). Quantum Hall Effects, Second Edition. World Scientific. ISBN 978-981-270-032-2. pp. 210-213 <a href="978-981-270-032-2" target="_blank">978-981-270-032-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Haldane, F.D.M.; E.H. Rezayi (1985). "Finite-Size Studies of the Incompressible State of the Fractionally Quantized Hall Effect and its Excitations". Physical Review Letters. 54 (3): 237–240. Bibcode:1985PhRvL..54..237H. doi:10.1103/PhysRevLett.54.237. PMID 10031449. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>