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We study the spin-$1/2$ Heisenberg model on the triangular lattice with theantiferromagnetic first ($J_1$) and second ($J_2$) nearest-neighborinteractions using density matrix renormalization group. By studying the spincorrelation function, we find a $120^{\circ}$ magnetic order phase for $J_2\lesssim 0.07 J_1$ and a stripe antiferromagnetic phase for $J_2 \gtrsim 0.15J_1$. Between these two phases, we identify a spin liquid region characterizedby the exponential decaying spin and dimer correlations, as well as the largespin singlet and triplet excitation gaps on finite-size systems. We find twonear degenerating ground states with distinct properties in two sectors, whichindicates more than one spin liquid candidates in this region. While the sectorwith spinon is found to respect the time reversal symmetry, the even sectorwithout a spinon breaks such a symmetry for finite-size systems. Furthermore,we detect the signature of the fractionalization by following the evolution ofdifferent ground states with inserting spin flux into the cylinder system.Moreover, by tuning the anisotropic bond coupling, we explore the nature of thespin liquid phase and find the optimal parameter region for the gapped $Z_2$spin liquid.

Competing Spin Liquid States in the Spin-$1/2$ Heisenberg Model On Triangular Lattice