  We study the spin-$1/2$ Heisenberg model on the triangular lattice with the
antiferromagnetic first ($J_1$) and second ($J_2$) nearest-neighbor
interactions using density matrix renormalization group. By studying the spin
correlation function, we find a $120^{\circ}$ magnetic order phase for $J_2
\lesssim 0.07 J_1$ and a stripe antiferromagnetic phase for $J_2 \gtrsim 0.15
J_1$. Between these two phases, we identify a spin liquid region characterized
by the exponential decaying spin and dimer correlations, as well as the large
spin singlet and triplet excitation gaps on finite-size systems. We find two
near degenerating ground states with distinct properties in two sectors, which
indicates more than one spin liquid candidates in this region. While the sector
with spinon is found to respect the time reversal symmetry, the even sector
without a spinon breaks such a symmetry for finite-size systems. Furthermore,
we detect the signature of the fractionalization by following the evolution of
different ground states with inserting spin flux into the cylinder system.
Moreover, by tuning the anisotropic bond coupling, we explore the nature of the
spin liquid phase and find the optimal parameter region for the gapped $Z_2$
spin liquid.
