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Waves in Random and Complex Media

Constructing trial wave functions for a many electron system confined to a quantum well in a strong magnetic field

John J. Quinn

Published Apr 24, 2014• 2 citations
Publisher

Abstract

Incompressible quantum liquid (IQL) states occur when the single particle angular momentum l and the number N of electrons in the partially filled Landau level satisfy the equation 2l = ν−1N − Cν. ...
DOI
10.1080/17455030.2014.898867

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A trial wave function Ψ(1, 2,..., N ) of an N electron system can always be written as the product of an antisymmetric Fermion factor F { z ij } = Π i<j z ij , and a symmetric correlation factor G { z ij }. F results from the Pauli principle, and G is caused by Coulomb interactions. One can represent G diagrammatically [1] by distributing N points on the circumference of a circle, and drawing appropriate lines representing correlation factors (cfs) z ij between pairs. Here, of course, z ij = z i – z j , where z i is the complex coordinate of the i th electron. Laughlin correlations for the ν = 1/3 filled incompressible quantum liquid (IQL) state contain two cfs connecting each pair ( i,j ). For the Moore-Read state of the half-filled excited Landau level (LL), with ν = 2 + 1/2, the even value of N for the half-filled LL is partitioned into two subsets A and B , each containing N /2 electrons [2]. For any one partition ( A,B ), the contribution to G is given by G AB = Π i<j∈A z ij 2 Π k<ι∈B z kι 2 . The full G is equal to the symmetric sum of contributions G AB over all possible partitions of N into two subsets of equal size. For Jain states at filling factor ν = p/q < 1/2, the value of the single particle angular momentum ι satisfies the equation 2 ι = ν -1 N - C ν , with C ν = q + 1 - p . The values of (2 ι, N ) define the function space of G { z ij } , which must satisfy a number of conditions. For example, the highest power of any z i cannot exceed 2 ι + 1 - N . In addition, the value of the total angular momentum L of the lowest correlated state must satisfy the equation L = ( N /2)(2 ι +1 - N ) - K G , where K G is the degree of the homogeneous polynomial generated by G . Knowing the values of L for IQL states (and for states containing a few quasielectrons or a few quasiholes) from Jain's mean field CF picture allows one to determine K G . The dependence of the pair pseudopotential V(L 2 ) on pair angular momentum L 2 suggests a small number of correlation diagrams for a given value of the total angular momentum L . Correlation diagrams and correlation functions for the Jain state at ν = 2/5 and for the Moore-Read states will be presented as examples. The generalizations of the method of selecting G from small to larger systems will be discussed.

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A trial wave function Ψ(1, 2,..., N ) of an N electron system can always be written as the product of an antisymmetric Fermion factor F { z ij } = Π i<j z ij , and a symmetric correlation factor G { z ij }. F results from the Pauli principle, and G is caused by Coulomb interactions. One can represent G diagrammatically [1] by distributing N points on the circumference of a circle, and drawing appropriate lines representing correlation factors (cfs) z ij between pairs. Here, of course, z ij = z i – z j , where z i is the complex coordinate of the i th electron. Laughlin correlations for the ν = 1/3 filled incompressible quantum liquid (IQL) state contain two cfs connecting each pair ( i,j ). For the Moore-Read state of the half-filled excited Landau level (LL), with ν = 2 + 1/2, the even value of N for the half-filled LL is partitioned into two subsets A and B , each containing N /2 electrons [2]. For any one partition ( A,B ), the contribution to G is given by G AB = Π i<j∈A z ij 2 Π k<ι∈B z kι 2 . The full G is equal to the symmetric sum of contributions G AB over all possible partitions of N into two subsets of equal size. For Jain states at filling factor ν = p/q < 1/2, the value of the single particle angular momentum ι satisfies the equation 2 ι = ν -1 N - C ν , with C ν = q + 1 - p . The values of (2 ι, N ) define the function space of G { z ij } , which must satisfy a number of conditions. For example, the highest power of any z i cannot exceed 2 ι + 1 - N . In addition, the value of the total angular momentum L of the lowest correlated state must satisfy the equation L = ( N /2)(2 ι +1 - N ) - K G , where K G is the degree of the homogeneous polynomial generated by G . Knowing the values of L for IQL states (and for states containing a few quasielectrons or a few quasiholes) from Jain's mean field CF picture allows one to determine K G . The dependence of the pair pseudopotential V(L 2 ) on pair angular momentum L 2 suggests a small number of correlation diagrams for a given value of the total angular momentum L . Correlation diagrams and correlation functions for the Jain state at ν = 2/5 and for the Moore-Read states will be presented as examples. The generalizations of the method of selecting G from small to larger systems will be discussed.
A partially filled shell (Landau level) of Laughlin quasiparticles (QP's) gives rise to an incompressible daughter state if the QP's themselves are Laughlin correlated. This occurs only if the pseudopotential V QP (L′) describing the interaction energy of a QP pair as a function of the total pair angular momentum L′ satisfies special conditions. V QP (L′) can be obtained quite accurately from numerical studies of small systems. It does not always satisfy these conditions (e.g. for quasielectrons of the Laughlin ν=1/3 state at their ν QE =1/3 filling). In such cases, formation of pairs or larger clusters may explain the recently observed incompressible states (like ν=4/11).
We present numerical evidence in support of the view that the physical origin of the fractional quantum Hall effect at \ensuremath{\nu}=n/(2n+1) is associated with the fact that this is the largest filling factor at which an n-component system of electrons can avoid having any pair of electrons occur in a state of relative angular momentum 1. Our calculations, for n=2, are based on a model in which the Hilbert space is truncated to the two lowest Landau levels and the Landau-level separation, \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, is a parameter of the model. For \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=0 and short-range repulsive interactions, a zero-energy incompressible ground state occurs at \ensuremath{\nu}=2/5. Our numerical results suggest that, in agreement with arguments advanced by J. K. Jain, this incompressibility survives to the strong-field limit \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=\ensuremath{\infty}.