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We study the spin-$1$ Heisenberg model on the square lattice with theantiferromagnetic nearest-neighbor $J_1$ and the next-nearest-neighbor $J_2$couplings by using the infinite projected entangled-pair state (iPEPS) ansatzand density matrix renormalization group (DMRG) calculation. The iPEPSsimulation, which studies the model directly in the thermodynamic limit, findsa crossing of the ground state from the N\'eel magnetic state to the stripemagnetic state at $J_2/J_1 \simeq 0.549$, showing a direct phase transition. Inthe finite-size DMRG calculation on the cylinder geometry up to the cylinderwidth $L_y = 10$, we find a very small intermediate regime $\sim 0.005 J_1$between the two magnetic order phases, which may imply the absent intermediatephase. Both calculations identify that the stripe order comes with afirst-order transition at $J_2/J_1 \simeq 0.549$. Our results indicate thatunlike the spin-$1/2$ $J_1-J_2$ square model, quantum fluctuations in thespin-$1$ model may be not strong enough to stabilize an intermediatenon-magnetic phase.

physical review. b./physical review. b

Quantum phase diagram of spin-$1$ $J_1-J_2$ Heisenberg model on the  square lattice: an infinite projected entangled-pair state and density matrix  renormalization group study