Two-and three-body correlations of incompressible quantum liquids are studied numerically. Pairing of composite fermions (CFs) in the 1/3-filled second CF Landau level is found at νe = 4/11. It is explained by reduced short-range repulsion due to ring-like single-particle charge distribution. Although Moore-Read state of CFs is unstable in the 1/2-filled second CF level, condensation of its quasiholes is a possible origin of incompressibility at νe = 4/11. Electron pairing occurs at νe = 7/3 and 13/3, but with different pair-pair correlations. Signatures of triplets are found at higher fillings.

Pair-distribution functions g(r) of Laughlin quasielectrons (composite fermions in their second Landau level) are calculated in the fractional quantum Hall states at electron filling factors νe = 4/11 and 3/8. A shoulder in g(r) is found, supporting the idea of cluster formation. The intra-and intercluster contributions to g(r) are identified, largely independent of νe. The average cluster sizes are estimated; pairs and triplets of quasielectrons are suggested at νe = 4/11 and 3/8, respectively.

Interaction between spin waves (or excitons) moving in the lowest Landau level is studied using numerical diagonalization. Becuse of complicated statistics obeyed by these composite particles, their effective interaction is completely different from the dipole-dipole interaction predicted in the model of independent (bosonic) waves. In particular, spin waves moving in the same direction attract one another which leads to their dynamical binding. The interaction pseudopotentials V ↑↑ (k) and V ↑↓ (k) for two spin waves with equal wavevectors k and moving in the same or opposite directions have been calculated and shown to obey power laws V (k) ∝ k α at small k. A high value of α ↑↑ ≈ 4 explains the occurrence of linear bands in the spin excitation spectra of quantum Hall droplets.

We compare quantum Hall systems at filling factors ν = 2 to ν = 2 3 and 2 5 , corresponding to the exact filling of two lowest electron or composite fermion (CF) Landau levels. The two fractional states are examples of CF liquids with spin dynamics. There is a close analogy between the ferromagnetic (spin polarization P = 1) and paramagnetic (P = 0) incompressible ground states that occur in all three systems in the limits of large and small Zeeman spin splitting. However, the excitation spectra are different. At ν = 2, we find spin domains at half-polarization (P = 1 2 ), while antiferromagnetic order seems most favorable in the CF systems. The transition between P = 0 and 1, as seen when e.g. the magnetic field is tilted, is also studied by exact diagonalization in toroidal and spherical geometries. The essential role of an effective CF-CF interaction is discussed, and the experimentally observed incompresible half-polarized state is found in some models.