  Motivated by the recent experiment on a rare-earth material YbMgGaO$_4$ [Y.
Li \textit{et al.}, Phys. Rev. Lett. \textbf{115}, 167203 (2015)], which found
that the ground state of YbMgGaO$_4$ is a quantum spin liquid, we study the
ground-state phase diagram of an anisotropic spin-$1/2$ model that was proposed
to describe YbMgGaO$_4$. Using the density-matrix renormalization group method
in combination with the exact diagonalization, we calculate a variety of
physical quantities, including the ground-state energy, the fidelity, the
entanglement entropy and spin-spin correlation functions. Our studies show that
in the quantum phase diagram there is a $120^{\circ}$ phase and two distinct
stripe phases. The transitions from the two stripe phases to the $120^{\circ}$
phase are of the first order. However, the transition between the two stripe
phases is not the first order, which is different from its classical
counterpart. Additionally, we find no evidence for a quantum spin liquid in
this model. Our results suggest that additional terms may be also important to
model the material YbMgGaO$_4$. These findings will stimulate further
experimental and theoretical works in understanding the quantum spin liquid
ground state in YbMgGaO$_4$.
