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The frustrated XY model on the honeycomb lattice has drawn lots of attentionsbecause of the potential emergence of chiral spin liquid (CSL) with theincreasing of frustrations or competing interactions. In this work, we studythe extended spin-$\frac{1}{2}$ XY model with nearest-neighbor ($J_1$), andnext-nearest-neighbor ($J_2$) interactions in the presence of a three-spinschiral ($J_{\chi}$) term using density matrix renormalization group methods. Weobtain a quantum phase diagram with both conventionally ordered andtopologically ordered phases. In particular, the long-sought Kalmeyer-LaughlinCSL is shown to emerge under a small $J_{\chi}$ perturbation due to theinterplay of the magnetic frustration and chiral interactions. The CSL, whichis a non-magnetic phase, is identified by the scalar chiral order, the finitespin gap on a torus, and the chiral entanglement spectrum described by chiral$SU(2)_{1}$ conformal field theory.

Quantum phase diagram and chiral spin liquid in the extended spin-$\frac{1}{2}$ honeycomb XY model