spin liquid phase in extended Kitaev model on the Honeycomb lattice

he Kitaev model on the honeycomb lattice offers exactly soluble examples of both gapped and gapless quantum spin liquids (QSLs). In addition to the Kitaev (K) interaction, candidate Kitaev materials also possess Heisenberg (J) and off-diagonal symmetric (Gamma) couplings. We investigate the quantum spin-1/2 K-J-Gamma model on the honeycomb lattice by a variational Monte Carlo (VMC) method. In addition to the Kitaev spin liquid (KSL) phase, we find that there is one proximate Kitaev Spin Liquid (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The KSL is known as a gapless state with 2 Majorana cones and a (almost) gapped spin response. In contrast, the PKSL has 14 Majorana cones and a gapless spin response. In a magnetic field applied normal to the honeycomb plane, it realizes two of Kitaev's gapped chiral spin-liquid states, namely, a non-Abelian phase with Chern number 5 and an Abelian phase with Chern number 2. The experimental characters of the chiral spin liquids are their thermal Hall conductances. Our phase diagram provides a clue in understanding the physics of candidate Kitaev materials.