Finite-size spectrum, magnon interactions, and magnetization ofS=1Heisenberg spin chains

We report on density-matrix renormalization-group and analytical work on $S=1\mathrm{}$ antiferromagnetic Heisenberg spin chains. We study the finite-size behavior within the framework of the nonlinear $\sigma$ model. We study the effect of magnon-magnon interactions on the finite-size spectrum and on the magnetization curve close to the critical magnetic field, determine the magnon scattering length, and compare it to the prediction from the nonlinear $\sigma$ model.