Electron Correlations in Partially Filled Lowest and Excited Landau   Levels
  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 ${\cal R}$ corresponding to positive anharmonicity of the interaction pseudopotential $V({\cal R})$. In the lowest LL, the super-harmonic behavior of $V({\cal R})$ causes Laughlin correlations (avoiding pairs with ${\cal R}=1$) and the Laughlin-Jain series of incompressible ground states. In the first excited LL, $V({\cal R})$ is harmonic at short range and a different series of incompressible states results. Similar correlations occur in the paired Moore-Read $\nu={5\over2}$ state and in the $\nu={7\over3}$ and ${8\over3}$ states, all having small total parentage from ${\cal R}=1$ and 3 and large parentage from ${\cal R}=5$. The $\nu={7\over3}$ and ${8\over3}$ states are different from Laughlin $\nu={1\over3}$ and ${2\over3}$ states and, in finite systems, occur at a different LL degeneracy (flux). The series of Laughlin correlated states of electron pairs at $\nu=2+2/(q_2+2)={8\over3}$, ${5\over2}$, ${12\over5}$, and ${7\over3}$ is proposed, although only in the $\nu={5\over2}$ state pairing has been confirmed numerically. In the second excited LL, $V({\cal R})$ is sub-harmonic at short range and (near the half-filling) the electrons group into spatially separated larger $\nu=1$ droplets to minimize the number of strongly repulsive pair states at ${\cal R}=3$ and 5. 
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