Integer-spin Heisenberg Chains in a Staggered Magnetic Field. A   Nonlinear $\sigma$-Model Approach
  We present here a nonlinear sigma-model (NL$\sigma$M) study of a spin-1 antiferromagnetic Heisenberg chain in an external commensurate staggered magnetic field. We find, already at the mean-field level, excellent agreement with recent and very accurate Density Matrix Renormalization Group (DMRG) studies, and that up to the highest values of the field for which a comparison is possible, for the staggered magnetization and the transverse spin gap. Qualitative but not quantitative agreement is found between the NL$\sigma$M predictions for the longitudinal spin gap and the DMRG results. The origin of the discrepancies is traced and discussed. Our results allow for extensions to higher-spin chains that have not yet been studied numerically, and the predictions for a spin-2 chain are presented and discussed. Comparison is also made with previous theoretical approaches that led instead to predictions in disagreement with the DMRG results. 
Elisa Ercolessi, Giuseppe Morandi, Pierbiagio Pieri, Marco Roncaglia