  We study the spin-$1/2$ Heisenberg model on the square lattice with the first
and second nearest-neighbor antiferromagnetic couplings $J_1$, $J_2$, 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 $0 \leq J_2/J_1 \leq 1.0$ and $0 \leq J_{\chi}/J_1 \leq 1.5$. We
identify the N\'eel 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 spin system. In the vicinity of the N\'eel and CSL phase
boundary, our numerical results do not find evidence to support the phase
coexistence of N\'eel 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 magnetic order parameter, we find the vanished magnetic
order suggesting a magnetic disorder phase, whose nature needs further studies.
Different from the spin-$1$ $J_1$-$J_2$-$J_\chi$ 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.
