  The weak Mott insulating regime of the triangular lattice Hubbard model
exhibits a rich magnetic phase diagram as a result of the ring exchange
interaction in the spin Hamiltonian. These phases include the Kalmeyer-Laughlin
type chiral spin liquid (CSL) and a valence bond solid (VBS). A natural
question arises regarding the robustness of these phases in the presence of a
weak spin-orbit coupling (SOC). In this study, we derive the effective spin
model for the spin-orbit coupled triangular lattice Hubbard model in the weak
Mott insulting regime, including all SOC-mediated spin-bilinears and
ring-exchange interactions. We then construct a simplified spin model keeping
only the most relevant SOC-mediated spin interactions. Using infinite density
matrix renormalization group (iDMRG) we show that the CSL and VBS phases of the
triangular lattice Hubbard model can be stabilized in the presence of a weak
SOC. The stabilization results from a compensation between the
Dzyaloshinskii-Moriya interaction and a SOC-mediated ring exchange interaction.
We also provide additional qualitative arguments to intuitively understand the
compensation mechanism in the iDMRG quantum phase diagrams. This mechanism for
stabilization can potentially be useful for the experimental realization of
quantum spin liquids.
