Spontaneous dimerization and moment formation in the Hida model of the spin-1 kagome antiferromagnet

The Hida model, defined on a honeycomb lattice, is a spin-1/2 Heisenberg model of aniferromagnetic hexagons (with nearest-neighbor interaction, $J_A>0)$ coupled via ferromagnetic bonds (with exchange interaction, $J_F<0)$. It applies to the spin-gapped organic materials, $\mathrm{m}\text{−}\mathrm{MPYNN}·X$ (for $X=\mathrm{I},{\mathrm{BF}}_4,{\mathrm{ClO}}_4)$, and for $|J_F|\gg J_A$, it reduces to the spin-1 kagome Heisenberg antiferromagnet (KHA). Motivated by the recent finding of the trimerized singlet (TS) ground state for spin-1 KHA, we investigate the evolution of the ground state of the Hida model from weak to strong $J_F/J_A$ using mean-field triplon analysis and Schwinger boson mean-field theory. Our triplon analysis of the Hida model shows that its uniform hexagonal singlet (HS) ground state (for weak $J_F/J_A)$ gives way to the dimerized hexagonal singlet (D-HS) ground state for $|J_F|/J_A\gtrsim 1.26$ (which for strong $J_F/J_A$ approaches the TS state). From the Schwinger boson calculations, we find that the evolution from the uniform HS phase for spin-1/2 Hida model to the TS phase for spin-1 KHA happens through two quantum phase transitions: (1) the spontaneous dimerization transition at $J_F/J_A\sim -0.28$ from the uniform HS to D-HS phase and (2) the moment formation transition at $J_F/J_A\sim -1.46$, across which the pair of spin-1/2's on every FM bond begins to appear as a bound moment that tends to spin-1 for large negative $J_F$'s. The TS ground state of spin-1 KHA is thus adiabatically connected to the D-HS ground state of the Hida model. Our calculations imply that the $\mathrm{m}\text{−}\mathrm{MPYNN}·X$ salts realize the D-HS phase at low temperatures, which can be ascertained through neutron diffraction.

Figure 1

The Hida model with ferromagnetic Heisenberg exchange, $J_F$, shown as blue links, and the antiferromagnetic exchange interaction, $J_A$, shown in red color. The ${\mathbf{a}}_1=2\widehat{x}$ and ${\mathbf{a}}_2=-\widehat{x}+\sqrt{3}\widehat{y}$ are two primitive vectors of the lattice. Moreover, we define ${\mathbf{a}}_3={\mathbf{a}}_1+{\mathbf{a}}_2$.

Figure 2

Three possible ground states of the Hida model. (a) The hexagonal singlet (HS) state with a uniform singlet amplitude per bond on every antiferromagnetic (red) hexagon. (b) The D-HS state with dimerized AFM bonds (shown as thick and thin red bonds). (c) Trimerized singlet (TS) state.

Figure 3

The triplon mean-field energies of the HS (hexagonal singlet), D-HS (dimerized HS), and TS (trimerized singlet) states of the Hida model.

Figure 4

The static structure factor $S\left(\mathbf{q}\right)$ in the HS and D-HS ground states of the Hida model. The color-code bar on the bottom-left applies to the plots (a) and (b), and that on the bottom-right applies to (c) and (d). The extended Brillouin zone (up to fourth Brillouin zone) is shown in solid-black lines.

Figure 5

The static structure factor, $S\left(\mathbf{q}\right)$, plotted along the high-symmetry lines of the first and the extended Brillouin zone in the HS and D-HS ground states of the Hida model. The high-symmetry lines are shown in blue over the Brillouin zones above the actual plots.

Figure 6

Comparing the $S\left(\mathbf{q}\right)$ in the D-HS and TS states of the Hida model (at $J_F/J_A=-10)$ with that in the TS ground state of the spin-1 kagome Heisenberg antiferromagnet.

Figure 7

Variation of the mean-field parameters on the AFM bonds with $J_F$. The ${\alpha }_A$ and ${\varphi }_A$ stay constant at $1/\sqrt{2}$ and 0, respectively, for all $J_F$. The ${\alpha }_A^{\prime }$ and ${\varphi }_A^{\prime }$ compete to dominate each other with a marked change in their relative strengths across $J_F/J_A\sim -2.33$. Notably, ${\alpha }_A^{\prime }\ne {\alpha }_A$ (except when $J_F/J_A$ is closer to zero), which implies spontaneous dimerization of the AFM bonds in the Hida model.

Figure 8

The dimerization order parameter $O_D$ vs $J_F/J_A$. The horizontal (dot-dashed) line at 0.05 corresponds to the value of the trimerization order parameter for spin-1 kagome antiferromagnet (see Appendix). The inset zooms in to show the dimerization transition at $J_F/J_A=-0.28$.

Figure 9

The average total spin per FM bond vs $J_F/J_A$. For large negative $J_F$, it approaches the value of a spin-1 moment. However, for $-J_F/J_A<1.46$, it is (nearly) a constant at 1.5, which corresponds to having two uncorrelated spin-1/2's. It implies that only when $J_F/J_A$ is stronger than a critical value $(-1.46)$ that the pair of spin-1/2's on a FM bond start to behave as a bound moment.

Figure 10

The kagome lattice with primitive vectors ${\mathbf{a}}_1=2\widehat{x}$ and ${\mathbf{a}}_2=-\widehat{x}+\sqrt{3}\widehat{y}$.

Figure 11

The trimerization order parameter $O_T$ vs $S$ from a Schwinger boson mean-field calculation of the spin-S KHA.