  We study the ground-state properties of the quantum spin liquid (QSL) phases
of the spin-$1/2$ antiferromagnetic Heisenberg model on the triangular lattice
with nearest- ($J_1$), next-nearest- ($J_2$), and third-neighbor ($J_3$)
interactions by using density-matrix renormalization group (DMRG) method. By
combining parallel DMRG with $SU(2)$ spin rotational symmetry, we are able to
obtain accurate results on large cylinders with length up to $L_x=48$ and
circumference $L_y=6 - 12$. Our results suggest that the QSL phase of the
$J_1$-$J_2$ Heisenberg model is gapped which is characterized by the absence of
gapless mode, short-range spin-spin and dimer-dimer correlations. In the
presence of $J_3$ interaction, we find that a new critical QSL with a single
gapless mode emerges. While both spin-spin and scalar chiral-chiral
correlations are short-ranged, dimer-dimer correlations are quasi-long-ranged
which decays as a power-law at long distances.
