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We investigate the effect of dimensional crossover in the ground state of theantiferromagnetic spin-$1$ Heisenberg model on the anisotropic triangularlattice that interpolates between the regime of weakly coupled Haldane chains($J^{\prime}\! \!\ll\!\! J$) and the isotropic triangular lattice($J^{\prime}\!\!=\!\!J$). We use the density-matrix renormalization group(DMRG) and Schwinger boson theory performed at the Gaussian correction levelabove the saddle-point solution. Our DMRG results show an abrupt transitionbetween decoupled spin chains and the spirally ordered regime at$(J^{\prime}/J)_c\sim 0.42$, signaled by the sudden closing of the spin gap.Coming from the magnetically ordered side, the computation of the spinstiffness within Schwinger boson theory predicts the instability of the spiralmagnetic order toward a magnetically disordered phase with one-dimensionalfeatures at $(J^{\prime}/J)_c \sim 0.43$. The agreement of these complementarymethods, along with the strong difference found between the intra- and theinterchain DMRG short spin-spin correlations; for sufficiently large values ofthe interchain coupling, suggests that the interplay between the quantumfluctuations and the dimensional crossover effects gives rise to theone-dimensionalization phenomenon in this frustrated spin-$1$ Hamiltonian.

One dimensionalization in the spin-1 Heisenberg model on the anisotropic triangular lattice