The energy spectra and wave functions of up to 14 interacting quasielectrons (QE's) in the Laughlin $\ensuremath{\nu}=\frac{1}{3}$ fractional quantum Hall (FQH) state are investigated using exact numerical diagonalization. It is shown that at sufficiently high density the QE's form pairs or larger clusters. This behavior, opposite to Laughlin correlations, invalidates the (sometimes invoked) reapplication of the composite fermion picture to the individual QE's. The series of finite-size incompressible ground states are identified at the QE filling factors ${\ensuremath{\nu}}_{\mathrm{QE}}=\frac{1}{2},$
31, 32, corresponding to the electron fillings $\ensuremath{\nu}=\frac{3}{8},$
114, 135. The equivalent quasihole (QH) states occur at ${\ensuremath{\nu}}_{\mathrm{QH}}=\frac{1}{4},$
51, 72, corresponding to $\ensuremath{\nu}=\frac{3}{10},$
134, 175. All these six novel FQH states were recently discovered experimentally. Detailed analysis indicates that QE or QH correlations in these states are different from those of well-known FQH electron states (e.g., Laughlin or Moore--Read states), leaving the origin of their incompressibility uncertain. Halperin's idea of Laughlin states of QP pairs is also explored, but is does not seem adequate.
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 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta
Nϕ is shown to be related to the number of electrons
N by
Nϕ=2N−(5−q) with
q 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−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.
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 corresponding to positive anharmonicity of the interaction pseudopotential
V(R). In the lowest LL, the superharmonic behavior of
V(R) causes Laughlin correlations (avoiding pairs with
R=1) and the Laughlin-Jain series of incompressible ground states. In the first excited LL,
V(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
38 states, all having small total parentage from
R=1 and 3 and large parentage from
R=5. The $\ensuremath{\nu}=\frac{7}{3}$ and
38 states are different from Laughlin $\ensuremath{\nu}=\frac{1}{3}$ and
32 states and, in finite systems, occur at a different LL degeneracy (flux). The series of Laughlin-correlated states of electron pairs at $\ensuremath{\nu}{=2+2/(q}_{2}+2)=\frac{8}{3},$
25, 512, and
37 is proposed, although only in the $\ensuremath{\nu}=\frac{5}{2}$ state pairing has been confirmed numerically. In the second excited LL,
V(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
R=3 and 5.
We study 'second generation' of fractional quantum Hall states which contain a partially filled composite fermion (CF) Landau level (LL). We explain the role of CF-CF interaction in the incompressibility of the underlying quantum electron liquid. In particular, using exact diagonalization on a Haldane sphere, we determine two- and three-body CF correlation functions for these liquids and show that the CFs form a paired state (instead of a Laughlin liquid) when filling v = 1/3 of their second LL. At v = 1/2 we show that the CFs group into even larger clusters rather than form a Moore—Read paired state. We also address the problem of spin polarization of interacting CFs and predict transition to the partially unpolarized ground state in sufficiently narrow quantum wells and/or weak magnetic fields.
Two-and three-body correlation functions (number of pairs or triplets vs. relative angular momentum) of electrons or Laughlin quasielectrons (i.e., composite fermions in their first excited Landau level) are studied numerically in several fractional quantum Hall liquids.It is shown directly that the νe = 4/11 liquid (corresponding to a ν = 1/3 filling of composite fermions in their first excited Landau level) is a paired state of quasielectrons, hence interpreted as a condensate of "second-generation" quasiholes of Moore-Read ν = 1/2 state of composite fermions.
The authors discuss the existence of fractional quantised Hall states in other than the lowest (n=0) Landau level. They present the results of numerical studies of systems with up to nine particles in the n=1 Landau level. They find that a fractional quantised Hall state is likely for filling fractions of the n=1 Landau level, nu 1 , with nu 1 -1 =5. At nu 1 -1 =3 they find that the Laughlin trial wavefunctions is not a good candidate ground-state wavefunction. If there is a gap at nu 1 -1 it is predicted to be small. At nu 1 -1 = 7 / 2 they predict the existence of a fractional quantised Hall state despite the small or non-existent gap at nu 1 -1 =3. They also show that introducing truncated pseudopotential interactions is not always a physical procedure outside of the lowest Landau level.