Unveiling magnetic interactions of ruthenium trichloride via constraining direction of orbital moments: Potential routes to realize a quantum spin liquid

Recent experiments reveal that the honeycomb ruthenium trichloride $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$ is a prime candidate of the Kitaev quantum spin liquid (QSL). However, there is no theoretical model which can properly describe its experimental dynamical response due to the lack of a full understanding of its magnetic interactions. Here, we propose a general scheme to calculate the magnetic interactions in systems (e.g., $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$) with nonnegligible orbital moments by constraining the directions of orbital moments. With this scheme, we put forward a minimal ${J}_{1}\ensuremath{-}{K}_{1}\ensuremath{-}{\mathrm{\ensuremath{\Gamma}}}_{1}\ensuremath{-}{J}_{3}\ensuremath{-}{K}_{3}$ model for $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$ and find that: (I) The third nearest neighbor (NN) antiferromagnetic Heisenberg interaction ${J}_{3}$ stabilizes the zigzag antiferromagnetic order; (II) The NN symmetric off-diagonal exchange ${\mathrm{\ensuremath{\Gamma}}}_{1}$ plays a pivotal role in determining the preferred direction of magnetic moments and generating the spin wave gap. An exact diagonalization study on this model shows that the Kitaev QSL can be realized by suppressing the NN symmetric off-diagonal exchange ${\mathrm{\ensuremath{\Gamma}}}_{1}$ and the third NN Heisenberg interaction ${J}_{3}$. Thus, we not only propose a powerful general scheme for investigating the intriguing magnetism of ${J}_{\mathrm{eff}}=1/2$ magnets, but also point out future directions for realizing the Kitaev QSL in the honeycomb ruthenium trichloride $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$.