Partial filled Landau Level at even denominator, a vortex metal with Berry phase

We develop a vortex metal theory for partial filled Landau Level at $\nu=\frac{1}{2n}$, whose ground state contains a composite Fermi surface(FS) formed by the vortex of electrons. In the projected Landau Level limit, the composite Fermi surface contains $\frac{-\pi}{n}$ Berry phase. Such fractional Berry phase is a consequence of LL projection which produces the GMP guiding center algebra and embellishes an anomalous velocity to the equation of motion for the vortex metal. Further, we investigate a particle-hole symmetric bilayer system with $\nu_1=\frac{1}{2n}$ and $\nu_2=1-\frac{1}{2n}$ at each layer, and demonstrate that the $\frac{-\pi}{n}$ Berry phase on the composite Fermi surface leads to the suppression of $2k_f$ back-scattering between the PH partner bilayer, which could be a smoking gun to detect the fractional Berry phase. We also mention various instabilities and competing orders in such bilayer system including a $Z_{4n}$ topological order phase driven by quantum criticality.