  We use the finite projected entangled pair state (PEPS) method to investigate
the global phase diagram of the spin-1/2 square-lattice $J_1$-$J_2$-$J_3$
antiferromagnetic (AFM) Heisenberg model. The ground state phase diagram is
established with a rich variety of phases: AFM, gapless quantum spin liquid,
valence-bond solid (VBS), stripe, and incommensurate spiral phases. The nature
of the VBS region is revealed, containing a plaquette VBS and a mixed
columnar-plaquette VBS, with the emergence of short-range incommensurate spin
correlations in some region. The long-range incommensurate magnetic phase is
also explicitly characterized as a planar spiral with incommensurate spatial
periodicities. Most interestingly, there exists several multicritical points
connecting different phases. These findings elucidate the true nature of the
long-standing square-lattice $J_1$-$J_2$-$J_3$ antiferromagnet at
zero-temperature. Our results also pave the way to accurately simulate complex
two-dimensional quantum systems that may host nonuniform features by means of
finite PEPS.
