Instabilities Toward Charge Density Wave and Paired Quantum Hall State of Half-Filled Landau Levels

We study the stability of spin-resolved Landau levels at electron filling factor $\nu=n+1/2$, where n is a positive integer. Representing the half-filled topmost Landau level by fermions and the n filled inner Landau levels by n bosons, coupled to the Chern-Simons gauge fields, we show that the ground states exhibit charge density wave order for $n(n+1/2)>15.54 \pi(a_B^*)^2 n_e$, where $n_e$ is the 2D carrier density and $a_B^*$ is the effective Bohr radius. We find that the pairing interaction mediated by the fluctuating gauge field is enhanced near the charge density wave instability such that p-wave pairing of the Chern-Simons fermions prevails for $3.49(\pi{\ab}^2)n_e<n^2(n+1/2)<20.22\pi(a_B^*)^2 n_e$. The competition between charge density wave order and paired quantum Hall state is discussed in connection to recent experiments.