Dimerizations in spin-$S$ antiferromagnetic chains with three-spin interaction

We discuss spin-$S$ antiferromagnetic Heisenberg chains with three-spin interactions, next-nearest-neighbor interactions, and bond alternation. First, we prove rigorously that there exist parameter regions of the exact dimerized ground state in this system. This is a generalization of the Majumdar-Ghosh model to arbitrary $S$. Next, we discuss the ground-state phase diagram of the models by introducing several effective field theories and the universality classes of the transitions are described by the level-$2S$ $\mathrm{SU}\left(2\right)$ Wess-Zumino-Witten model and the Gaussian model. Finally, we determine the phase diagrams of $S=1$ and $S=3/2$ systems by using exact diagonalization and level spectroscopy.

Figure 1

A schematic figure of the $J_1$-$J_2$-$J_3$ model with bond alternation $\delta$ (1). Alternating nearest-neighbor interactions $J_1(1\pm \delta )$ are depicted by the red solid and dashed lines. The next-nearest-neighbor interaction $J_2$ is represented by the blue line. The three-spin interaction $J_3$ works on a triangle (green shaded area).

Figure 2

Schematic ground-state phase diagram for the model (1) with $J_2=0$ for $S=1$ and $3/2$. The solid lines correspond to the exact dimer ground-state regions. The dashed lines depict the critical lines between several dimerized phases including the Haldane phase. These critical lines are expected to be of the $c=1$ Gaussian type. The stars denote multicritical points where the critical lines merge as described by $\mathrm{SU}{\left(2\right)}_{2S}$ WZW theory with $c=3S/(S+1)$.

Figure 3

The parameter space of the model (1) for several $S\ge 1$ with (a) $\delta =0$ and (b) $\left|\delta \right|=0.3$. The solid (dotted) lines denote the regions where the dimerized state is the exact ground state (eigenstate).

Figure 4

Ground-state phase diagram of the bond-alternating Heisenberg model with three-spin interactions [Eq. (1) with $J_2=0$] for (a) $S=1$ and (b) $S=3/2$. The multicritical points are located at ${(J_3/J_1)}_c=0.11$ for $S=1$ and at ${(J_3/J_1)}_c=0.062$ for $S=3/2$.

Figure 5

Conformal charge on the Haldane-dimer transition line in the $S=1$ chain obtained from the size scaling of the ground state for $N=8$–18. The value approaches $c=1$ ($c=3/2$) for $\delta \ne 0$ ($\delta =0$).