Magnetic field induced intermediate quantum spin-liquid with a spinon Fermi surface.

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.