Numerical study of magnetization plateaus in the spin-$\frac{1}{2}$ kagome Heisenberg antiferromagnet

We clarify the existence of several magnetization plateaus for the kagome $S=\frac{1}{2}$ antiferromagnetic Heisenberg model in a magnetic field. Using approximate or exact localized magnon eigenstates, we are able to describe in a similar manner the plateau states that occur for magnetization per site $m=\frac{1}{3}$, $\frac{5}{9}$, and $\frac{7}{9}$ of the saturation value. These results are confirmed using large-scale exact diagonalization on lattices up to 63 sites.

Figure 1

Visualization of the valence bond crystal type states for the $m=\frac{3}{9}$, $\frac{5}{9}$, and $\frac{7}{9}$ plateaus in the $S=\frac{1}{2}$ Heisenberg antiferromagnet on the kagome lattice.

Figure 2

Magnetization curve of the $S=\frac{1}{2}$ kagome Heisenberg model on various lattices that can accommodate the VBC shown in Fig. 1 (see Appendix for details).

Figure 3

Widths of the $m=\frac{1}{3}$, $\frac{5}{9}$, and $\frac{7}{9}$ plateaus obtained from exact diagonalization on various lattices (see Appendix for a list), plotted as a function of the inverse diameter. Filled symbols correspond to lattices that can accommodate the VBC states, i.e., possess the $K$ point in their Brillouin zone: 27, $36c$, $36d$, $45b$, 54, and $63a$. Open symbols are for the others. Data are consistent with finite values for all three plateaus in the thermodynamic limit.

Figure 4

Energy gaps vs $S^z$ for $N=36d$ lattice labeled with their quantum numbers. For $S^z=6$, 10, and 14 corresponding, respectively, to $m=\frac{1}{3}$, $\frac{5}{9}$, and $\frac{7}{9}$, the lowest excitation corresponds to a twofold-degenerate state at the $K$ point in the Brillouin zone, and then a sizable gap above it (indicated by the encircled symbols and the arrows). These spectra are compatible with the VBC states for such magnetizations. The dashed encircled symbols denote levels which might indicate persistence of symmetry breaking also away from the nominal plateaus. The magnetization sectors with a gray background are not visited in the magnetization curve, i.e., they are obscured by a magnetization jump.

Figure 5

Dimer and spin correlations (see text for definitions) computed either by ED or on the magnon VBC state for various magnetizations: on $N=36d$ for $m=\frac{1}{3}$ and $\frac{5}{9}$, on $N=63a$ for $m=\frac{7}{9}$. (a) Dimer-dimer correlations [cf. Eq. (10)]: positive and negative values are shown, respectively, with filled blue lines (respectively dashed red lines) and width is proportional to the data (see Table ); reference bond is shown in black. (b) Spin correlations [cf. Eq. (9)]: positive and negative values are shown, respectively, with filled blue (resp. red) disks and diameter is proportional to the data (see Table ); the reference site is shown as an empty black circle. In order to have similar amplitudes, scale is multiplied by 2 and 5 for $m=\frac{5}{9}$ and $\frac{7}{9}$ with respect to $m=\frac{1}{3}$ data.