Low-energy properties of two-leg spin-1 antiferromagnetic ladders with   commensurate external fields and their extensions
  This study addresses low-energy properties of 2-leg spin-1 ladders with antiferromagnetic (AF) intrachain coupling under a uniform or staggered external field $H$, and a few of their modifications. The generalization to spin-$S$ ladders is also discussed. In the strong AF rung (interchain)-coupling $J_\perp$ region, degenerate perturbation theory applied to spin-$S$ ladders predicts $2S$ critical curves in the parameter space $(J_\perp, H)$ for the staggered field case, in contrast to $2S$ finite critical regions for the uniform field case. All critical areas belong to a universality with central charge $c=1$. On the other hand, we employ Abelian and non-Abelian bosonization techniques in the weak rung-coupling region. They show that in the spin-1 ladder, a sufficiently strong uniform field engenders a $c=1$ critical state regardless of the sign of $J_\perp$. Whereas, the staggered field is expected not to yield any singular phenomena. From the bosonization techniques, new field-theoretical expressions of string order parameters in the spin-1 systems is also proposed. 
Masahiro Sato