Effect of ring-exchange interactions in the extended Kitaev honeycomb model

Motivated by the possible triple-Q classical order in the Kitaev candidate material Na2Co2TeO6, we investigate microscopic models that may stabilize the triple-Q order by studying an extended Kitaev honeycomb model with ring-exchange interactions (namely, the K-Gamma-Gamma'-J(R) model) using the variational Monte Carlo method. It turns out that with a positive ring-exchange interaction (J(R) > 0) there indeed appears an exotic noncoplanar triple-Q ordered state featured by three Bragg peaks at symmetry-related M points in the crystallographic Brillouin zone. A magnetic field in the honeycomb plane can suppress the triple-Q order and induce a gapless quantum spin liquid (QSL) with eight cones. Furthermore, with the increase of J(R) a proximate Kitaev spin liquid with eight Majorana cones labeled "PKSL8" is found, which is very stable over a large range of Gamma interactions. The PKSL8 state shares the same projective symmetry group with the Kitaev spin liquid (KSL) which is located at small Gamma and J(R). In a weak magnetic field applied normal to the honeycomb plane, the PKSL8 turns into an Abelian chiral spin liquid with Chern number nu = -4, unlike the KSL, which yields a chiral spin liquid with nu = 1. Since the triple-Q phase is adjacent to two QSLs in the phase diagram, our work suggests that it is more hopeful to experimentally realize the exotic QSL phases starting from the triple-Q order.