Bond operators and triplon analysis for spin-$S$ dimer antiferromagnets

The mean-field triplon analysis is developed for spin-$S$ quantum antiferromagnets with dimerized ground states. For the spin-1/2 case, it reduces to the well-known bond-operator mean-field theory. It is applied to a columnar dimer model on square lattice and to a model on honeycomb lattice with spontaneous dimerization in the ground state. Different phases in the ground state are investigated as a function of spin. It is found that under suitable conditions (such as strong frustration), a quantum ground state (dimerized singlet phase in the present study) can survive even in the limit $S\to \infty$. Two quick extensions of this representation are also presented. In one case, it is extended to include the quintet states. In another, a similar representation is worked out on a square plaquette. A convenient procedure for evaluating the total-spin eigenstates for a pair of quantum spins is presented in the appendix.

Figure 1

Quantum spin-$S$ coupled dimer model. As in Eq. (8), the exchange interactions are: thick $\text{bonds}=J$, thin horizontal $\text{lines}={\eta }_{x}J$, not-so-thin vertical $\text{lines}={\eta }_{y}J$, and thin dashed $\text{lines}={\eta }^{\prime }J$, where $J$, ${\eta }_x$, ${\eta }_y$, and ${\eta }^{\prime }>0$. Also shown are the primitive translations and the spin labeling on a dimer.

Figure 2

(Color online) The mean-field quantum phase diagram of $H_{\text{I}}$ for case 1 $\left({\eta }^{\prime }=0\right)$. The left hand side panel shows the phase boundaries between the columnar-dimer and Néel-ordered phases for different spins. For a given $S$, the region between the axes and the phase boundary is the spin-gapped dimer phase, and on the other side of the boundary is the Néel-ordered phase. Note the quantum phase boundary in the classical limit $\left(S=\infty \right)$. Surprisingly, the gapped dimer phase survives even in the classical limit for sufficiently weak ${\eta }_x$ (coupled two-leg ladders) or ${\eta }_y$ (coupled chains). In the plane of rescaled couplings (right hand panel), the quantum phase boundaries for different spins collapse onto a single line.

Figure 3

(Color online) The mean-field quantum phase diagram of $H_{\text{I}}$ for case 2 (${\eta }^{\prime }\ne 0$ and ${\eta }_x={\eta }_y=\eta$). Here, $\eta =1$ corresponds to the $J_1\text{−}{J}_2$ model. Left hand side panel. For a given $S$, the region above the corresponding upper phase boundary, and bounded by the axes, is the collinear phase while that below the lower phase boundary is the Néel phase. In between the two transition lines lies the columnar-dimer phase. For $S=\infty$, the phase boundaries are given by the equations, ${\eta }^{\prime }=0.408\eta$ and ${\eta }^{\prime }=0.525\eta$. Right hand side panel. The phase boundaries of for different spins collapse onto two lines for two different transitions.

Figure 4

The model of Eq. (14). The lines indicate the nearest-neighbor Heisenberg exchange, $J$. The multiple spin exchange proportional to $K$ is represented by a hexagon itself, with six spins labeled as 1 to 6. This multiple spin-exchange interaction is present on every hexagonal plaquette of the honeycomb. Moreover, $J$, $K>0$.

Figure 5

(Top) The singlet state for a pair of spin $S$. It can be viewed as a “sea” of $2S$ valence bonds. (Bottom) The three triplet states are created by replacing one valence bond by three different symmetric bonds. See the text for details.

Figure 6

The quintet states. See Eqs. (A10a, A10b, A10c).