Variational Monte Carlo study of chiral spin liquid in quantum antiferromagnet on the triangular lattice

By using Gutzwiller projected fermionic wave functions and variational Monte Carlo technique, we study the spin-$1/2$ Heisenberg model with the first-neighbor ($J_1$), second-neighbor ($J_2$), and additional scalar chiral interaction $J_{\chi}{\bf S}_i \cdot ({\bf S}_j \times {\bf S}_k)$ on the triangular lattice. In the non-magnetic phase of the $J_1-J_2$ triangular model with $0.08 \lesssim J_2/J_1 \lesssim 0.16$, recent density-matrix renormalization group (DMRG) studies [Zhu and White, Phys. Rev. B {\bf 92}, 041105 (2015); Hu, Gong, Zhu, and Sheng, Phys. Rev. B {\bf 92}, 140403 (2015)] find a possible gapped spin liquid with the signal of a competition between a chiral and a $Z_2$ spin liquid. Motivated by the DMRG results, we consider the chiral interaction $J_{\chi}{\bf S}_i \cdot ({\bf S}_j \times {\bf S}_k)$ as a pertubation for this non-magnetic phase. We find that with growing $J_{\chi}$, the gapless U(1) Dirac spin liquid, which has the best variational energy for $J_{\chi}=0$, exhibits the energy instability towards a gapped spin liquid with non-trivial magnetic fluxes and nonzero chiral order. We calculate topological Chern number and ground-state degeneracy, both of which identify this flux state as the chiral spin liquid with fractionalized Chern number $C=1/2$ and two-fold topological degeneracy. Our results indicate a positive direction to stabilize a chiral spin liquid near the non-magnetic phase of the $J_1-J_2$ triangular model.