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We use the finite projected entangled pair state (PEPS) method to investigatethe 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 isestablished with a rich variety of phases: AFM, gapless quantum spin liquid,valence-bond solid (VBS), stripe, and incommensurate spiral phases. The natureof the VBS region is revealed, containing a plaquette VBS and a mixedcolumnar-plaquette VBS, with the emergence of short-range incommensurate spincorrelations in some region. The long-range incommensurate magnetic phase isalso explicitly characterized as a planar spiral with incommensurate spatialperiodicities. Most interestingly, there exists several multicritical pointsconnecting different phases. These findings elucidate the true nature of thelong-standing square-lattice $J_1$-$J_2$-$J_3$ antiferromagnet atzero-temperature. Our results also pave the way to accurately simulate complextwo-dimensional quantum systems that may host nonuniform features by means offinite PEPS.

Tensor network study of the spin-1/2 square-lattice $J_1$-$J_2$-$J_3$  model: incommensurate spiral order, mixed valence-bond solids, and  multicritical points