  It is widely believed that the $U(1)$ spin liquid with a large spinon Fermi
surface(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 $J_{4}$ is
sufficiently strong to suppress the 120 degree magnetic ordered state. This
belief is supported by many variational studies on this model and seems to be
consistent with the observations on the organic spin liquid materials such as
$\kappa$-(BEDT-TTF)$_{2}$Cu$_{2}$(CN)$_{3}$ and
EtMe$_{3}$Sb[Pd(dmit)$_{2}$]$_{2}$, which are systems close to their Mott
transition and thus have large $J_{4}$. Here we show through systematic
variational 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 symmetry
is favored in the most probable parameter regime for the SFS state and has an
energy much lower than that of the SFS state and other proposed variational
states. More specifically, we find that for $J_{4}\ge 0.09J_{1}$, the model
favors a valence bond solid state with a $4\times6$ period in its local spin
correlation pattern and has a variational energy that is about $5\%$ lower than
that 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 zigzag
pattern in its local spin correlation. We find that the variational phase
diagram we got is in qualitative agreement with that obtained from exact
diagonalization on a $6\times6$ cluster.
