Quantum phases of the frustrated XY models on the honeycomb lattice

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.