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arXiv:cond-mat/0009046 · Physical review. B, Condensed matter

Electron correlations in partially filled lowest and excited Landau levels

Arkadiusz Wójs

Published Mar 7, 2001• 40 citations
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Abstract

The electron correlations near the half-filling of the lowest and excited Landau levels (LL's) are studied using numerical diagonalization. It is shown that in the low-lying states electrons avoid pair states with relative angular momenta $\mathcal{R}$ corresponding to positive anharmonicity of the interaction pseudopotential $V(\mathcal{R}).$ In the lowest LL, the superharmonic behavior of $V(\mathcal{R})$ causes Laughlin correlations (avoiding pairs with $\mathcal{R}=1)$ and the Laughlin-Jain series of incompressible ground states. In the first excited LL, $V(\mathcal{R})$ is harmonic at short range and a different series of incompressible states results. Similar correlations occur in the paired Moore-Read $\ensuremath{\nu}=\frac{5}{2}$ state and in the $\ensuremath{\nu}=\frac{7}{3}$ and $\frac{8}{3}$ states, all having small total parentage from $\mathcal{R}=1$ and 3 and large parentage from $\mathcal{R}=5.$ The $\ensuremath{\nu}=\frac{7}{3}$ and $\frac{8}{3}$ states are different from Laughlin $\ensuremath{\nu}=\frac{1}{3}$ and $\frac{2}{3}$ states and, in finite systems, occur at a different LL degeneracy (flux). The series of Laughlin-correlated states of electron pairs at $\ensuremath{\nu}{=2+2/(q}_{2}+2)=\frac{8}{3},$ $\frac{5}{2},$ $\frac{12}{5},$ and $\frac{7}{3}$ is proposed, although only in the $\ensuremath{\nu}=\frac{5}{2}$ state pairing has been confirmed numerically. In the second excited LL, $V(\mathcal{R})$ is subharmonic at short range and (near the half-filling) the electrons group into spatially separated larger $\ensuremath{\nu}=1$ droplets to minimize the number of strongly repulsive pair states at $\mathcal{R}=3$ and 5.

DOI
10.1103/physrevb.63.125312

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The electron correlations near the half-filling of the lowest and excited Landau levels (LL's) are studied using numerical diagonalization. It is shown that in the low-lying states electrons avoid pair states with relative angular momenta R\mathcal{R}R corresponding to positive anharmonicity of the interaction pseudopotential V(R).V(\mathcal{R}).V(R). In the lowest LL, the superharmonic behavior of V(R)V(\mathcal{R})V(R) causes Laughlin correlations (avoiding pairs with R=1)\mathcal{R}=1)R=1) and the Laughlin-Jain series of incompressible ground states. In the first excited LL, V(R)V(\mathcal{R})V(R) is harmonic at short range and a different series of incompressible states results. Similar correlations occur in the paired Moore-Read \ensuremathν=52\ensuremath{\nu}=\frac{5}{2}\ensuremathν=25​ state and in the \ensuremathν=73\ensuremath{\nu}=\frac{7}{3}\ensuremathν=37​ and 83\frac{8}{3}38​ states, all having small total parentage from R=1\mathcal{R}=1R=1 and 3 and large parentage from R=5.\mathcal{R}=5.R=5. The \ensuremathν=73\ensuremath{\nu}=\frac{7}{3}\ensuremathν=37​ and 83\frac{8}{3}38​ states are different from Laughlin \ensuremathν=13\ensuremath{\nu}=\frac{1}{3}\ensuremathν=31​ and 23\frac{2}{3}32​ states and, in finite systems, occur at a different LL degeneracy (flux). The series of Laughlin-correlated states of electron pairs at \ensuremathν=2+2/(q2+2)=83,\ensuremath{\nu}{=2+2/(q}_{2}+2)=\frac{8}{3},\ensuremathν=2+2/(q2​+2)=38​, 52,\frac{5}{2},25​, 125,\frac{12}{5},512​, and 73\frac{7}{3}37​ is proposed, although only in the \ensuremathν=52\ensuremath{\nu}=\frac{5}{2}\ensuremathν=25​ state pairing has been confirmed numerically. In the second excited LL, V(R)V(\mathcal{R})V(R) is subharmonic at short range and (near the half-filling) the electrons group into spatially separated larger \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1 droplets to minimize the number of strongly repulsive pair states at R=3\mathcal{R}=3R=3 and 5.
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).
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.
We report on results of finite-size numerical studies of partially filled lowest Landau level, at low electron filling factors. We find convincing evidence suggesting that electrons form Wigner crystals at sufficiently low filling factors, and the critical filling factor is ${\ensuremath{\nu}}_{c}\ensuremath{\approx}1/7.$ At $\ensuremath{\nu}=1/7$ we find the system undergoes a phase transition from the Wigner crystal to the incompressible Laughlin state when the short-range part of the Coulomb interaction is modified slightly. This transition is either continuous or very weakly first order.
New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry ±1/4\pm1/4±1/4 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta NϕN_ϕNϕ​ is shown to be related to the number of electrons NNN by Nϕ=2N−(5−q)N_ϕ = 2N-(5-q)Nϕ​=2N−(5−q) with qqq being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at Nϕ=2N−3N_ϕ = 2N-3Nϕ​=2N−3. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.
We use exact numerical diagonalization in the search of fractional quantum Hall states with non-Abelian quasiparticle statistics. For the (most promising) states in a partially filled second Landau level, the search is narrowed to the range of filling factors 7/3 < v e < 8/3. In this range, the analysis of energy spectra and correlation functions, calculated including finite width and Landau level mixing, supports the prominent non-Abelian candidates at v e = 5/2 (paired Moore–Read "pfafian" state) and 12/5 (clustered Read–Rezayi "parafermion" state). Outside of this range, the noninteracting composite fermion model with four attached flux quanta is validated, yielding the family of quantum liquids with fractional, but Abelian statistics. The borderline v e = 7/3 state is shown to be adiabatically connected to the Laughlin liquid, but its short-range correlations are significantly different.
A system of electrons in a half-filled Landau level is investigated in spherical geometry. For systems of size from N=1 to 14 electrons with flux ${\mathit{N}}_{\mathrm{\ensuremath{\varphi}}}$=2(N-1) the angular momentum of the ground state is as predicted by Hund's second rule for composite fermions of one electron and two vortices at zero magnetic field. Low lying excitations also fit this interpretation and trial wave functions give excellent overlaps. The two-particle correlation function shows a significant correlation hole at short distances and suggests an asymptotically oscillating form at long distances, as in a Fermi liquid.