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

We investigate the low-energy properties, especially the low-energy excitation structures, 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 Sénéchal’s method [Phys. Rev. B 52, 15319 (1995)], based on the nonlinear sigma model, together with a saddle-point approximation. This strategy is valid in the weak interchain (rung) coupling regime. We show that all frustrated tubes possess sixfold 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. When the field is staggered only along the rung direction, it is implied that the lowest doubly-degenerate bands fall down with the field increasing in all systems. For final two cases where the fields are staggered along the chain, it is showed that at least in the weak rung-coupling region, the lowest-excitation gap grows with the field increasing, and no critical phenomena occurs. Furthermore, for the ladders of the final two cases, we predict that the inhomogeneous magnetization along the rung occurs, and the frustration between the field and the rung coupling can induce the magnetization pointing to the opposite direction to the field. All the analyses 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 a weak staggered field.

Figure 1

Normalized staggered magnetizations $m_s^z∕S$ in spin-1, 2, and 3 chains with the staggered field at $T=0$.

Figure 2

Transverse and longitudinal gaps in spin-1, 2, and 3 chains with the staggered field. We set $J=1$.

Figure 3

Cross section of the 5-leg tube and the reflecting operation for the rung direction.

Figure 4

(Color online) Band splitting induced by the rung coupling $J_{\perp }$. “Three- or sixfold” means the degree of the band degeneracy.

Figure 5

Gaps ${\Delta }_1$ and ${\Delta }_2={\Delta }_{\min }$ in 2-leg AF-rung spin-1, -2, or -3 ladders with $J=1$.

Figure 6

All band gaps ${\Delta }_r$ in $N$-leg AF-rung spin-1 ladders with $J=1$.

Figure 7

Lowest magnon gaps ${\Delta }_{\min }$ in $N$-leg AF-rung spin-1 ladders and FM-rung tubes (non-frustrated systems) with $J=1$. The gap-reduction speed of the $N$-leg tube is larger a little than that of the ladder. The 2-leg “tube” means the 2-leg ladder.

Figure 8

Lowest magnon gaps ${\Delta }_{\min }$ and band gaps ${\Delta }_r$ in $N$-leg AF-rung spin-1 tubes with $J=1$.

Figure 9

Gaps ${\Delta }_{\min }$ of spin-1 ladders and tubes with $J=1$. Marks “SPA” and “QMC” mean “derived by the SPA” and “derived by the QMC simulation” respectively. The QMC data, which all achieve their thermodynamic-limit values, are calculated by Munehisa Matsumoto (Ref. 66). The method of determining the gap and the correlation length in the QMC simulation is, for example, explained in Refs. 55, 63.

Figure 10

Spin correlation lengths in spin-1 ladders and tubes determined by the SPA or the QMC simulation. The QMC data is provided by Munehisa Matsumoto (Ref. 66).

Figure 11

(Color online) Expected magnetization curves in frustrated integer-spin tubes.

Figure 12

(Color online) Expected GS phase diagram of odd-leg integer-spin tubes around $(J_{\perp },H)=(0,0)$.

Figure 13

(Color online) Predicted band splitting in the $[0,\pi ]$-field case.

Figure 14

(Color online) Staggered magnetizations of $N$-leg spin-1 FM-or AF-rung tubes with the $[\pi ,0]$ field, $M_s^{N\text{−}\text{leg}}$. The dotted curve $M_{\text{chain}}$ stands for the staggered magnetization $m_s^z$ of the single AF chain with the staggered field (see Fig. 1). The 2-leg tube means the 2-leg ladder. A relation $M_s^{p\text{−}\text{leg}}\left(H\right)<M_s^{(p+1)\text{−}\text{leg}}\left(H\right)[M_s^{p\text{−}\text{leg}}\left(H\right)>M_s^{(p+1)\text{−}\text{leg}}\left(H\right)]$ is realized for the FM [AF]-rung case. Magnetizations $M_s^{4,5,6,7\text{−}\text{leg}}$ almost overlap.

Figure 15

Staggered magnetizations of $N$-leg spin-1 FM-rung ladders with the $[\pi ,0]$ field and $J_{\perp }∕J=-0.05$. For $l<[N∕2]$, $M_{l+1}\left(H\right)>M_l\left(H\right)$ is realized. Similarly to Fig. 14, the symbol $M_{\text{chain}}$ means $m_s^z$.

Figure 16

Staggered magnetizations $M_l$ of $N$-leg spin-1 AF-rung ladders with the $[\pi ,0]$ field.

Figure 17

Transverse and longitudinal gaps of $N$-leg spin-1 tubes $(N⩾3)$ with the $[\pi ,0]$ field and $J=1$. As a comparison, we also draw the gaps of the spin-1 chain with staggered field (see Fig. 2) in the left upper panel.

Figure 18

Transverse and longitudinal gaps of $N$-leg spin-1 ladders with the $[\pi ,0]$ field and $J=1$. The gaps ${\Delta }_3^T$ and ${\Delta }_3^L$ almost overlap.