Quantum phase diagram and chiral spin liquid in the extended spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:math> honeycomb XY model

The frustrated XY model on the honeycomb lattice has drawn lots of attentions because potential emergence chiral spin liquid (CSL) with increasing frustrations or competing interactions. In this work, we study extended spin-$\frac{1}{2}$ nearest-neighbor ($J_1$), and next-nearest-neighbor ($J_2$) interactions in presence a three-spins ($J_{\chi}$) term using density matrix renormalization group methods. We obtain quantum phase diagram both conventionally ordered topologically phases. particular, long-sought Kalmeyer-Laughlin CSL is shown to emerge under small $J_{\chi}$ perturbation due interplay magnetic frustration CSL, which non-magnetic phase, identified by scalar order, finite gap torus, entanglement spectrum described $SU(2)_{1}$ conformal field theory.