  We investigate the $2k_F$ density-wave instability of non-Fermi liquid states
by combining exact diagonalization with renormalization group analysis. At the
half-filled zeroth Landau level, we study the fate of the composite Fermi
liquid in the presence of the mass anisotropy and mixed Landau level form
factors. These two experimentally accessible knobs trigger a phase transition
towards a unidirectional charge-density-wave state with a wavevector equal to
$2k_F$ of the composite Fermi liquid. Based on exact diagonalization, we
identify such a transition by examining both the energy spectra and the static
structure factor of charge density-density correlations. Moreover, the
renormalization group analysis reveals that gauge fluctuations render the
non-Fermi liquid state unstable against density-wave orders, consistent with
numerical observations. Possible experimental probes of the density-wave
instability are also discussed.
