The frustrated XY model on the honeycomb lattice has drawn lots of attention because of the potential emergence of chiral spin liquid (CSL) with the increasing of frustrations or competing interactions. In this paper, we study the extended spin-$\frac{1}{2}$ XY model with nearest-neighbor $\left(J_1\right)$, and next-nearest-neighbor $\left(J_2\right)$ interactions in the presence of a three-spins chiral $\left(J_{\chi }\right)$ term using density matrix renormalization group methods. We obtain a quantum phase diagram with both conventionally ordered and topologically ordered phases. In particular, the long-sought Kalmeyer-Laughlin CSL is shown to emerge under a small $J_{\chi }$ perturbation due to the interplay of the magnetic frustration and chiral interactions. The CSL, which is a nonmagnetic phase, is identified by the scalar chiral order, the finite spin gap on a torus, and the chiral entanglement spectrum described by chiral $SU{\left(2\right)}_1$ conformal field theory.

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• Quantum spin liquid
• Topological phases of matter
• Honeycomb lattice
• Density matrix renormalization group
• Matrix product states
• XY model