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It is widely believed that the $U(1)$ spin liquid with a large spinon Fermisurface(SFS state) can be realized in the spin-$\frac{1}{2}$ $J_{1}-J_{4}$model on the triangular lattice, when the ring exchange coupling $J_{4}$ issufficiently strong to suppress the 120 degree magnetic ordered state. Thisbelief is supported by many variational studies on this model and seems to beconsistent with the observations on the organic spin liquid materials such as$\kappa$-(BEDT-TTF)$_{2}$Cu$_{2}$(CN)$_{3}$ andEtMe$_{3}$Sb[Pd(dmit)$_{2}$]$_{2}$, which are systems close to their Motttransition and thus have large $J_{4}$. Here we show through systematicvariational search that such a state is never favored in the $J_{1}-J_{4}$model on the triangular lattice. Instead, a state with broken spatial symmetryis favored in the most probable parameter regime for the SFS state and has anenergy much lower than that of the SFS state and other proposed variationalstates. More specifically, we find that for $J_{4}\ge 0.09J_{1}$, the modelfavors a valence bond solid state with a $4\times6$ period in its local spincorrelation pattern and has a variational energy that is about $5\%$ lower thanthat of the SFS state. This state is separated from the $\pi$-flux state for$J_{4}\le 0.045J_{1}$ by an intermediate symmetry breaking phase with a zigzagpattern in its local spin correlation. We find that the variational phasediagram we got is in qualitative agreement with that obtained from exactdiagonalization on a $6\times6$ cluster.

Instability of the $U(1)$ spin liquid with a large spinon Fermi surface in the Heisenberg-ring exchange model on the triangular lattice