  Searching for spin liquid states has long been attracting both
experimentalists and theorists. In this article, we review recent density
matrix renormalization group studies of the spin-1/2 XY model on the honeycomb
lattice, with first-neighbor ($J_1 = 1$) and frustrating second-neighbor
($J_2>0$) interactions. For the intermediate frustration regime $0.22\lesssim
J_2\lesssim0.36$, there exists a surprising antiferromagnetic Ising phase, with
ordered moments pointing along the $z$ axis, despite the absence of any $S_z
S_z$ interactions in the Hamiltonian. Surrounding this phase as a function of
$J_2$ are antiferromagnetic phases with the moments pointing in the
$\textit{x-y}$ plane for small $J_2$ and a close competition between an
$\textit{x-y}$ plane magnetic collinear phase and a dimer phase for large
values of $J_2$. No spin-liquid phases was found in the XY model even with the
third neighbor ($J_3>0$) interactions.
