  The Kitaev model with an applied magnetic field in the $H||[111]$ direction
shows two transitions: from a non-abelian gapped quantum spin liquid (QSL) to a
gapless QSL at $H_{c1} \simeq 0.2K$ and a second transition at a higher field
$H_{c2} \simeq 0.35K$ to a gapped partially polarized phase, where $K$ is the
strength of the Kitaev exchange interaction. We identify the intermediate phase
to be a gapless U(1) QSL and determine the spin structure function $S({\bf k})
$ and the Fermi surface $\epsilon_F^S({\bf k})$ of the gapless spinons using
the density matrix renormalization group (DMRG) method for large honeycomb
clusters. Further calculations of static spin-spin correlations, magnetization,
spin susceptibility, and finite temperature specific heat and entropy,
corroborate the gapped and gapless nature of the different field-dependent
phases. In the intermediate phase, the spin-spin correlations decay as a power
law with distance, indicative of a gapless phase.
