We study the effect of Dzyaloshinskii-Moriya (DM) interaction on the triangular lattice U(1) quantum spin liquid (QSL) which is stabilized by ring-exchange interactions. A weak DM interaction introduces a staggered flux to the U(1) QSL state and changes the density of states at the spinon Fermi surface. If the DM vector contains in-plane components, then the spinons gain nonzero Berry phase. The resultant thermal conductances κ_{xx} and κ_{xy} qualitatively agree with the experimental results on the material EtMe_{3}Sb[Pd(dmit)_{2}]_{2}. Furthermore, owing to perfect nesting of the Fermi surface, a spin density wave state is triggered by larger DM interactions. On the other hand, when the ring-exchange interaction decreases, another antiferromagnetic (AFM) phase with 120° order shows up which is proximate to a U(1) Dirac QSL. We discuss the difference of the two AFM phases from their static structure factors and excitation spectra.
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