Semiclassical approach to competing orders in a two-leg spin ladder with ring exchange

We investigate the competition between different orders in the two-leg spin ladder with a ring-exchange interaction by means of a bosonic approach. The latter is defined in terms of spin-1 hard-core bosons which treat the Néel and vector-chirality order parameters on an equal footing. A semiclassical approach of the resulting model describes the phases of the two-leg spin ladder with a ring exchange. In particular, we derive the low-energy effective actions which govern the physical properties of the rung-singlet and dominant vector-chirality phases. As a by-product of our approach, we reveal the mutual induction phenomenon between spin and chirality with, for instance, the emergence of a vector-chirality phase from the application of a magnetic field in bilayer systems coupled by four-spin exchange interactions.

Figure 1

Two-leg ladder (a) and four-spin plaquette (b) on which ring exchange $P_4$ is defined.

Figure 2

Phase diagram of the model (1) ($J=\cos \theta$, $K_4=\sin \theta$) obtained by applying a mean-field approximation to ${\mathcal{H}}_{\text{cl}}$ [Eq. (28a)]. The self-dual point ($K_4=J/2$) separates the NAF-dominant (i.e., $\mathbf{A}$-ordered) phase and the chirality-dominant ($\mathbf{B}$-ordered) one. In $(1+1)$-D, strong quantum fluctuations turn these ordered phases into short-ranged ones.

Figure 3

(Left) Square-root ${\rho }_0$ [Eq. (40)] of the boson density $n^{\text{B}}$ as a function of $K_4/J$. Note that the boson density ${\rho }_0^2$ is always finite. (Right) The low-energy dispersion obtained by a semiclassical (spin-wave-like) calculation ($K_4/J=0.45$). Inset shows the variation of the $z$-mode gap (at $k=\pi$) as a function of $K_4/J$. The staggered ordering of $\mathbf{A}$ in the $z$ direction is assumed. Among the three modes ($x$, $y$, and $z$), the longitudinal one ($z$ direction) may be regarded as the density fluctuation and is massive, while the remaining two are gapless (Goldstone modes) corresponding to the symmetry breaking: O(3)$\to$O(2) (in 1D, these Goldstone modes eventually get gapped by strong quantum fluctuations). Only at the self-dual point $K_4/J=1/2$ does the longitudinal branch ($z$ mode) show a gapless $k$-linear behavior.

Figure 4

A naively expected high-field configuration of $\mathbf{A}\propto {\mathbf{\text{S}}}_1-{\mathbf{\text{S}}}_2$, $\mathbf{B}\propto {\mathbf{\text{S}}}_1\times {\mathbf{\text{S}}}_2$, and $\langle \mathbf{T}\rangle \propto {\mathbf{\text{S}}}_1+{\mathbf{\text{S}}}_2$. Spin ($\mathbf{T}$) canting induces a uniform $z$ component ${({\mathbf{\text{S}}}_1\times {\mathbf{\text{S}}}_2)}^z$. If a similar spin-canted state is realized for large enough $g_4$, we may expect a state where local magnetization is induced asymmetrically in the two chains $\langle S_1^z\rangle \ne \langle S_2^z\rangle$.

Figure 5

Mean-field phase diagram of the ring-exchange ladder in magnetic field ($J=\cos \theta$, $K_4=\sin \theta$). The large portion of the parameter space is occupied by the 1/2-plateau phase. All the phases except the singlet-product phase are collinear and are characterized by parallel or antiparallel alignment of $\mathbf{A}$, $\mathbf{B}$, and $\langle \widehat{\mathbf{T}}\rangle$. The canted phase anticipated in the text does not appear in this parameter space. Inset: magnetization curves for $\theta =0$ and $0.07\pi$ (marked by an arrow head in the main panel).

Figure 6

Three collinear configurations exhibiting smooth magnetization process: (a) collinear I, (b) collinear II, and (c) SDW. The configuration (a) generically appears in the high-field region, while (b) realizes when the field is applied in the F-nematic phase (see Fig. 5). In the configurations (a) and (b), the $z$ component of local magnetization $T^z$ and local boson density $n_{\text{B}}$ are uniform. In the (c) configuration, on the other hand, these quantities alternate along the ladder to form a density-wave state.

Figure 7

Magnetization process exhibiting the anticipated canted configuration: $J_{\text{bl}}=2.0$, $J_{\text{bq}}=0.0$, $(t+u)/2=1.2$, $(t-u)/2=0.01$, $V_{\text{c}}=0.6$, $\mu =-0.5$. At around $H=6.2$, the spin state changes from the canted state to the collinear I (with a kink in $n_{\text{B}}$). The vector chirality ${({\mathbf{\text{S}}}_1\times {\mathbf{\text{S}}}_2)}^z$ is finite in the small-field region where the local magnetization assumes canted configuration (“moment canted”).

Figure 8

Vector chirality correlations obtained on $2\times 32$ ladder with parameters given in the text. Magnetization was fixed at 1/4 of its saturation value.