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

Motivated by the possible triple-$\bf Q$ classical order in the Kitaev candidate material Na$_2$Co$_2$TeO$_6$, we investigate microscopic models that may stabilize the triple-$\bf Q$ order by studying an extended Kitaev honeycomb model with ring-exchange interactions (namely, the $K$-$\Gamma$-$\Gamma'$-$J_{\rm R}$ model) using the variational Monte Carlo method. It turns out that with positive ring-exchange interaction ($J_{\rm R}>0$) there indeed appears an exotic non-coplanar triple-$\bf Q$ 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-$\bf Q$ order and induce a gapless quantum spin liquid (QSL) with 8 cones. Furthermore, with the increase of $J_{\rm R}$ 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 $J_{\rm R}$. 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-$\bf Q$ 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-$\bf Q$ order.