N-leg integer-spin ladders and tubes in commensurate external fields -Nonlinear sigma model approach-

We investigate the low-energy properties of N-leg integer-spin ladders and tubes with an antiferromagnetic (AF) intrachain coupling. In the odd-leg tubes, the AF rung coupling causes the frustration. To treat all ladders and tubes systematically, we apply S\'en\'echal's method [Phys. Rev. B 52, 15319 (1995)], based on the nonlinear sigma model. This strategy is valid in the weak interchain (rung) coupling regime. We show that all frustrated tubes possess six-fold degenerate spin-1 magnon bands, as the lowest excitations, while other ladders and tubes have a standard triply degenerate bands. We also consider effects of four kinds of Zeeman terms: uniform, staggered only along the rung, only along the chain, or both directions. The above prediction of the no-field case implies that a sufficiently strong uniform field yields a two-component Tomonaga-Luttinger liquid (TLL) due to the condensation of doubly degenerate lowest magnons in frustrated tubes. In contrast, the field induces a standard one-component TLL in all other systems. This is supported by symmetry and bosonization arguments based on the Ginzburg-Landau theory. The bosonization also suggests that the two-component TLL vanishes and a one-component TLL appears, when the uniform field becomes larger for the second lowest magnon bands to touch the zero-energy line. This transition could be observed as a cusp singularity in the magnetization process. All the analyses for the systems with a staggered Zeeman term suggest that the emergence of the doubly degenerate transverse magnons and the single longitudinal one is universal for the one-dimensional AF spin systems with weak staggered field.