  Recent experimental evidence for a field-induced quantum spin liquid (QSL) in
$\alpha$-RuCl$_3$ calls for an understanding for the ground state of honeycomb
Kitaev model under a magnetic field. In this work we address the nature of an
enigmatic gapless paramagnetic phase in the antiferromagnetic Kitave model,
under an intermediate magnetic field perpendicular to the plane. Combining
theoretical and numerical efforts, we identify this gapless phase as a $U(1)$
QSL with spinon Fermi surfaces. We also reveal the nature of continuous quantum
phase transitions involving this $U(1)$ QSL, and obtain a phase diagram of the
Kitaev model as a function of bond anisotropy and perpendicular magnetic field.
