  We investigate the spin-$\frac{1}{2}$ Heisenberg model on the triangular
lattice in the presence of nearest-neighbor $J_1$ and next-nearest-neighbor
$J_2$ antiferromagnetic couplings. Motivated by recent findings from
density-matrix renormalization group (DMRG) claiming the existence of a gapped
spin liquid with signatures of spontaneously broken lattice point group
symmetry [Zhu and White, Phys. Rev. B 92, 041105 (2015); Hu, Gong, Zhu, and
Sheng, Phys. Rev. B 92, 140403 (2015)], we employ the variational Monte Carlo
(VMC) approach to analyze the model from an alternative perspective that
considers both magnetically ordered and paramagnetic trial states. We find a
quantum paramagnet in the regime $0.08\lesssim J_2/J_1\lesssim 0.16$, framed by
$120^{\circ}$ coplanar (stripe collinear) antiferromagnetic order for smaller
(larger) $J_2/J_1$. By considering the optimization of spin-liquid wave
functions of a different gauge group and lattice point group content as derived
from Abrikosov mean-field theory, we obtain the gapless $U(1)$ Dirac spin
liquid as the energetically most preferable state in comparison to all
symmetric or nematic gapped $\mathbb{Z}_{2}$ spin liquids so far advocated by
DMRG. Moreover, by the application of few Lanczos iterations, we find the
energy to be the same as the DMRG result within error-bars. To further resolve
the intriguing disagreement between VMC and DMRG, we complement our
methodological approach by the pseudofermion functional renormalization group
(PFFRG) to compare the spin structure factors for the paramagnetic regime
calculated by VMC, DMRG, and PFFRG. This model promises to be an ideal test-bed
for future numerical refinements in tracking the long-range correlations in
frustrated magnets.
