Another exact ground state of a two-dimensional quantum antiferromagnet

We present the exact dimer ground state of a quantum antiferromagnet on the maple-leaf lattice. A coupling anisotropy for one of the three inequivalent nearest-neighbor bonds is sufficient to stabilize the dimer state. Together with the Shastry-Sutherland Hamiltonian, we show that this is the only other model with an exact dimer ground state for all two-dimensional lattices with uniform tilings.

Figure 1

Maple-leaf lattice (left) and the dimer eigenstate of the Hamiltonian in (1) (right). According to (1) there are three inequivalent nearest-neighbor couplings denoted red (dotted), blue (dashed), and green (double line). The singlets (yellow ellipses) reside on the green bonds. As the red and blue couplings do not contribute to the energy, it is an eigenstate of (1).

Figure 2

Canted ${120}^{\circ }$ order: Spins on red triangles (brown) show ${120}^{\circ }$ order, while those connected by blue bonds are canted by an angle $\mathrm{\Phi }\left(\alpha \right)$. As a function of $\alpha$, spins across blue bonds are antiparallel for $\alpha =0$ (orange), uniform ${120}^{\circ }$ order appears for $\alpha =1$ (light green), and decoupled green dimer bonds are formed for $\alpha \to \infty$ (light blue).

Figure 3

Dimer phase diagram of MLM as a function of bond coupling $\alpha$ and spin $S$. ${\alpha }_{b1}$ denotes a lower bound above which the MLM is exactly solvable. ${\alpha }_{b2}$ is an upper bound of $\alpha$ below which $\mathrm{c}{120}^{\circ }$ serves as a better lower bound of energy than (6). ${\alpha }_c$ is the numerically obtained critical value of $\alpha$ above which (5) stabilizes.