Effects of randomness on the antiferromagnetic spin-1 chain

We investigate the effect of weak randomness on the antiferromagnetic anisotropic spin-1 chain. We use Abelian bosonization to construct the low-energy effective theory. A renormalization group calculation up to second order in the strength of the disorder is performed on this effective theory. We observe in this framework the destruction of the antiferromagnetic ordered phase in the manner of Imry and Ma. We predict the effects of a random magnetic field along the $z$ axis and a random field in the $\mathrm{XY}$ plane as well as random exchange with and without $\mathrm{XY}$ symmetry. Instabilities of massless phases appear in general by mechanisms different from the case of the two-leg spin ladder.