Chiral spin liquid and quantum phase diagram of spin-$\frac{1}{2} J_1\text{−}{J}_2\text{−}{J}_{\chi }$ model on the square lattice

We study the spin-$1/2$ Heisenberg model on the square lattice with the first- and second-nearest-neighbor antiferromagnetic couplings $J_1$ and $J_2$, as well as the three-spin scalar chiral coupling $J_{\chi }$. Using density matrix renormalization group calculations, we obtain a quantum phase diagram of this system for $0\le J_2/J_1\le 1.0$ and $0\le J_{\chi }/J_1\le 1.5$. We identify the Néel and stripe magnetic order phase at small-$J_{\chi }$ coupling. With growing $J_{\chi }$, we identify the emergent chiral spin liquid (CSL) phase characterized by the quantized spin Chern number $C=1/2$ and entanglement spectrum with the quasidegenerate group of levels agreeing with chiral SU(2)${}_1$ conformal field theory, which is an analog of the $\nu =1/2$ Laughlin state in the spin system. In the vicinity of the Néel and CSL phase boundary, our numerical results do not find evidence to support the coexistence of Néel order and topological order that was conjectured by mean-field calculations. In the larger-$J_2$ and -$J_{\chi }$ coupling regime, the entanglement spectrum of the ground state also exhibits the chiral quasidegeneracy consistent with a CSL, but the adiabatic flux insertion simulations fail to obtain the quantized Chern number. By analyzing the finite-size scaling of the magnetic order parameter, we find the vanished magnetic order suggesting a magnetic disorder phase, whose nature needs further studies. Different from the spin-$1J_1\text{−}{J}_2\text{−}{J}_{\chi }$ model, we do not find the coexistent stripe magnetic order and topological order. We also investigate the $J_{\chi }$ dominant regime and find a strong tendency of the system to develop a dimer order rather than the chiral spin magnetic order observed in the spin-1 model. Our results unveil interesting quantum phases in this spin-$1/2$ model and also demonstrate the drastic differences between the spin-$1/2$ and spin-1 cases.