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

Abstract

Motivated by the possible triple-$\mathbf{Q}$ classical order in the Kitaev candidate material ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$, we investigate microscopic models that may stabilize the triple-$\mathbf{Q}$ order by studying an extended Kitaev honeycomb model with ring-exchange interactions (namely, the $K\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}\text{\ensuremath{-}}{J}_{\mathrm{R}}$ model) using the variational Monte Carlo method. It turns out that with a positive ring-exchange interaction (${J}_{\mathrm{R}}>0$) there indeed appears an exotic noncoplanar triple-$\mathbf{Q}$ 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-$\mathbf{Q}$ order and induce a gapless quantum spin liquid (QSL) with eight cones. Furthermore, with the increase of ${J}_{\mathrm{R}}$ a proximate Kitaev spin liquid with eight Majorana cones labeled ``PKSL8'' is found, which is very stable over a large range of $\mathrm{\ensuremath{\Gamma}}$ interactions. The PKSL8 state shares the same projective symmetry group with the Kitaev spin liquid (KSL) which is located at small $\mathrm{\ensuremath{\Gamma}}$ and ${J}_{\mathrm{R}}$. In a weak magnetic field applied normal to the honeycomb plane, the PKSL8 turns into an Abelian chiral spin liquid with Chern number $\ensuremath{\nu}=\ensuremath{-}4$, unlike the KSL, which yields a chiral spin liquid with $\ensuremath{\nu}=1$. Since the triple-$\mathbf{Q}$ 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-$\mathbf{Q}$ order.

Topics & Concepts