Magnon bands of $N$-leg integer-spin antiferromagnetic systems in the weak-interchain-coupling regime

Using the exact results of the $O\left(3\right)$ nonlinear sigma model (NLSM) and a few quantitative numerical data for integer-spin antiferromagnetic (AF) chains, we systematically estimate all magnon excitation energies of $N$-leg integer-spin AF ladders and tubes in the weak-interchain-coupling regime. Our method is based on a first-order perturbation theory for the strength of the interchain coupling. It can deal with any kind of interchain interactions, in principle. We confirm that results of the perturbation theory are in good agreement with those of a quantum Monte Carlo simulation and with our recent study based on a saddle-point approximation of the NLSM [Phys. Rev. B 72, 104438 (2005)]. Our theory further supports the existence of a Haldane (gapped) phase even in a $d$-dimensional $(d⩾2)$ spatially anisotropic integer-spin AF model, if the exchange coupling in one direction is sufficiently strong compared with those in all the other directions. The strategy in this paper is applicable to other $N$-leg systems consisting of gapped chains which low-energy physics is exactly or quantitatively known.

Figure 1

(Color online) Gaps ${\Delta }_r$ versus rung coupling $J_{\perp }$ in the $N$-leg spin-1 AF systems (1) with $J=1$. The quantity $\Delta$ is the Haldane gap of the spin-1 AF chain ${\widehat{\mathcal{H}}}_0$.

Figure 2

(Color online) Expected lowest-gap profile in the $N$-leg integer-spin system (1). The symbol $\Delta$ means the Haldane gap of the spin-$S$ AF chain ${\widehat{\mathcal{H}}}_0$. In the two-leg spin-1 case, the gap profile have already been predicted in Refs. 20, 32, 33.

Figure 3

Lowest excitation gap and critical point $J_{\perp }^c$ for the $N\to \infty$, spin-1 case predicted by perturbation theory. The quantity $\Delta$ is the Haldane gap of the spin-1 AF chain ${\widehat{\mathcal{H}}}_0$.

Figure 4

(Color online) Decorated rung couplings.