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We perform a systematic DMRG investigation of the two natural spin-1generalizations of the spin-1/2 Majumdar-Ghosh chain, the spin-1 $J_1-J_2$Heisenberg chain, where $J_2$ is a next-nearest neighbor Heisenberg coupling,and the spin-1 $J_1-J_3$ model, where $J_3$ refers to a three-site interactiondefined by $J_3\left[({\bf S}_{i-1}\cdot {\bf S}_i)({\bf S}_i\cdot {\bfS}_{i+1})+\mathrm {H.c.}\right]$. Although both models are rigorouslyequivalent to the Majumdar-Ghosh chain for spin-1/2, their physics appears tobe quite different for spin 1. Indeed, when all couplings areantiferromagnetic, the spin-1 $J_1-J_2$ model undergoes an effective decouplinginto two next-nearest neighbour (NNN) Haldane chains upon increasing $J_2$,while the $J_1-J_3$ chain undergoes a spontaneous dimerization similar to thespin-1/2 Majumdar-Ghosh chain upon increasing $J_3$. By extending the phasediagram to all signs of the couplings, we show that both the dimerized and theNNN-Haldane phase are actually present in the $J_1-J_3$ model, the former oneadjacent to the Haldane one, the latter one to the ferromagneric one, with anIsing transition between them. By contrast, the $J_1-J_2$ chain only has aNNN-Haldane phase between the Haldane phase and the ferromagnetic phase forpositive $J_2$. In both cases, our DMRG data are consistent with a continuousKosterlitz-Thouless transition between the NNN-Haldane and the ferromagneticphases.

Dimerization and effective decoupling in two spin-1 generalizations of the spin-1/2 Majumdar-Ghosh chain