Quantum kagomé antiferromagnet in a magnetic field: Low-lying nonmagnetic excitations versus valence-bond crystal order

We study the ground state properties of a quantum antiferromagnet on the kagomé lattice in the presence of a magnetic field, paying particular attention to the stability of the plateau at magnetization $1∕3$ of saturation and the nature of its ground state. We discuss fluctuations around classical ground states and argue that quantum and classical calculations at the harmonic level do not lead to the same result in contrast to the zero-field case. For spin $S=1∕2$ we find a magnetic gap below which an exponential number of nonmagnetic excitations are present. Moreover, such non-magnetic excitations also have a (much smaller) gap above the threefold degenerate ground state. We provide evidence that the ground state has long-range order of valence-bond crystal type with nine spins in the unit cell.

Figure 1

Boundaries of the $⟨M⟩=1∕3$ plateau as a function of the anisotropy $\Delta$ for different lattice sizes (see legend) and the transition to saturation $⟨M⟩=1$ for the thermodynamic limit (thin full line). Inset: kagomé lattice with an ordered state of the valence-bond crystal type at $⟨M⟩=1∕3$: circles in certain hexagons indicate local resonances between different Néel configurations on the hexagons, arrows indicate spins which are aligned with the field.

Figure 2

Low-lying excitations above the $⟨M⟩=1∕3$ plateau for the $S=1∕2$ Heisenberg antiferromagnet on the $N=36$ kagomé lattice. (a) Full lines show all excitations with $S^z=6$ in the given energy range, bold dashed lines the lowest excitations with $S^z=5$ and $S^z=7$ as a function of magnetic field $h$. (b) Excitation energy versus number of states with $S^z=6$ below that energy. One observes a total of 100 states below the magnetic gap in the middle of the $⟨M⟩=1∕3$ plateau (corresponding to the largest gap to magnetic excitations).

Figure 3

Main panel: Spectra of the effective Hamiltonian (5) for $\Delta \to \infty$ with $N=36$, 54, 81, 108, and 144. Inset: Scaling of the energy of the $l\text{th}$ excited state with inverse system size $1∕N$ for some selected levels.

Figure 4

Overlap between the ground state of the full $XXZ$ model and the effective Hamiltonian with $N=36$ for different values of the anisotropy parameter $\Delta$ (full line). For comparison, we include the corresponding result for a different model; namely, the $N=36$ triangular lattice (Ref. 17) (dotted line), which exhibits a sharp drop around $\Delta =0.77$.