Spin-1 antiferromagnetic Heisenberg chains in an external staggered field

We present in this paper a nonlinear σ-model analysis of a spin-1 antiferromagnetic Heisenberg chain in an external commensurate staggered magnetic field. After rediscussing briefly and extending previous results for the staggered magnetization curve, the core of the paper is a calculation at the tree level, of the Green functions of the model. We obtain precise results for the elementary excitation spectrum and in particular for the spin gaps in the transverse and longitudinal channels. It is shown that, while the spectral weight in the transverse channel is exhausted by a single magnon pole, in the longitudinal one, besides a magnon pole a two-magnon continuum appears as well whose weight is a steadily increasing function of the applied field, while the weight of the magnon decreases correspondingly. The balance between the two is governed by a sum rule that is derived and discussed. A detailed comparison with the present experimental and numerical state of the art as well as with previous analytical approaches is also made.