Low-energy properties of two-leg spin-1 antiferromagnetic ladders with commensurate external fields and their extensions

This study addresses low-energy properties of two-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 are also proposed.

Figure 1

A schematic gap profile in the spin-1 AF ladder. The symbol ${\Delta }_{\mathrm{Hal}}$ denotes the Haldane gap $(\simeq 0.41J)$. For details, see Ref. 18.

Figure 2

RVB pictures in the GS of the spin-1 AF ladder (1) without external fields. The black point denotes a spin-$\frac{1}{2}$ state. The ellipse including two black points indicates the symmetrization of two spin-$\frac{1}{2}$ states and recovers an original spin-1 site. The black line represents for the singlet bond, which consists of a spin-$\frac{1}{2}$ pair. For $0<J_{\perp }<\infty$, the GS is well described by the PSS state. With increasing the rung-coupling $J_{\perp }$, the weight $a$ $\left(b\right)$ increases (decreases) in the GS wave function, and it approaches the tensor product of singlet dimers [singlet of the spin-1 pair]. For details, see Ref. 18.

Figure 3

RVB pictures of the GS of the spin-$\frac{1}{2}$ ladder. Panels (a) and (b) are typical spin configurations in the AF-rung and FM-rung spin-liquid phases, respectively. The gray line denotes a singlet bond. Up and down arrows represent $S_{l,j}^z=+\frac{1}{2}$ and $-\frac{1}{2}$, respectively. Each number under the dashed loop encircling two sites shows the value of ${\tilde{S}}_j^z=S_{1,j}^z+S_{2,j+1}^z$ in (a), and one of $S_j^z=S_{1,j}^z+S_{2,j}^z$ in (b). Removing all sites of ${\tilde{S}}_j^z=0$ or $S_j^z=0$, one can see a “hidden” Néel order $(+1,-1,+1,-1,\dots )$. For more details, see Refs. 30, 32.

Figure 4

Schematic GS phase diagram of the spin-$\frac{1}{2}$ ladder with the uniform Zeeman term (2a).

Figure 5

Schematic GS phase diagram of the spin-$\frac{1}{2}$ ladder with the staggered Zeeman term (2b). In weak and strong rung-coupling regions, the transition curve follows $h\simeq 0.60\times J_{\perp }^{3∕2}∕J^{1∕2}$ and $h\simeq J_{\perp }-0.50\times J$, respectively, where the factors (0.60 and 0.50) are determined by our numerical calculations (Ref. 51). These results are consistent with analytical ones in Ref. 14.

Figure 6

Eigenstates and eigenenergies of ${\widehat{\mathcal{H}}}_j$ with the strength of the uniform field $H$ varying. The thick dotted line corresponds to the GS. Each vector ∣⋯,⋯) represents a state ${\mid \mathbb{S},{\mathbb{S}}^z⟩}_j$.

Figure 7

Intermediate plateau state in the spin-1 ladder (1) with the uniform field (2a): the case with $J\to 0$ $\left(\alpha \right)$ and the case where $J$ is finite $\left(\beta \right)$. Thin black bonds mean singlet bonds of spin-$\frac{1}{2}$ pair. The dotted bonds mean triplet bonds where both $z$ components of two spins are $+\frac{1}{2}$. In the panel $\left(\beta \right)$, gray plaquettes represent a super-position state drawn in the lowest part of the figure.

Figure 8

Panels $\left(\alpha 1\right)$ and $\left(\alpha 2\right)$ are, respectively, the GS phase diagrams in the uniform-field case (2a) and the staggered-field one (2b). Panels $\left(\beta 1\right)$ and $\left(\beta 2\right)$ are, respectively, the uniform magnetization curve in the case (2a) and the staggered one in the case (2b). In the panel $\left(\beta 2\right)$, we denote the magnetization of the even-$j$ rung.

Figure 9

Expected “RVB” pictures for the plateaux with $⟨S_j^z⟩=1$ (A) and $⟨S_j^z⟩=2$ (B) in the spin-$\frac{3}{2}$ case.

Figure 10

GS phase diagram and the schematic gap behavior in the model (26).

Figure 11

Band structures of the spin-1 AF chain with a uniform field $H$.

Figure 12

Schematic GS phase diagram of the spin-1 AF ladder (1) with the uniform Zeeman term (2a).

Figure 13

Expected GS phase diagram around two decoupled TB chains $(c=3)$.

Figure 14

Expected GS phase diagrams of two-leg spin-$S$ AF ladder with the uniform Zeeman term (2a).