Trimerized ground state of the spin-1 Heisenberg antiferromagnet on the kagome lattice

We study the phase diagram of the spin-1 quantum bilinear-biquadratic antiferromagnet on the kagome lattice using exact diagonalization and the density matrix renormalization group algorithm. The $\mathrm{SU}\left(3\right)$-symmetric point of this model Hamiltonian is a spontaneously trimerized state whose qualitative nature persists even at the Heisenberg point, a finding that contrasts previous proposals. We report the ground state energy per site of the Heisenberg model to be $-1.410\left(2\right)$ and establish the presence of a spin gap.

Figure 1

(a) shows a schematic of the simplex solid on the kagome lattice. The bond thicknesses represent the relative magnitude of the bond energy. (b) shows a schematic of the hexagon singlet solid (HSS). Each spin-1 (depicted in blue) fractionalizes into two spin-1/2 (shown by red circles). The spin-1/2's on the hexagons form a singlet, shown by the black lines connecting them. (c) shows the cylindrical geometry used in the DMRG calculation. Periodic boundary conditions in the width direction have not been shown.

Figure 2

(a) The total ground state energy per bond for cylinders of odd lengths and different widths is extrapolated to infinite length by fitting to the functional form, Eq. (4). The bulk energy (see text) is also shown. (b) shows the spin gap for various cylinder widths and lengths. The estimated gap in the infinite length limit is finite.

Figure 3

The low-energy spectrum of the BLBQ model on the 21-site kagome lattice, resolved by lattice momenta, as a function of $J_{bq}$, is shown. On tuning $J_{bq}$ from 1 towards 0, the low-energy features appear adiabatically connected, suggesting the persistence of the trimerized phase to the Heisenberg point. Qualitative changes in the energy spectrum seen at a negative value of $J_{bq}$ indicate a quantum phase transition to a ferroquadrupolar phase.

Figure 4

The fidelity of ground state wave functions from ED of 12- and 21-site clusters is shown as a function of $J_{bq}$ for two reference wave functions. The reference wave function is chosen to be the ground state of (a) the $\mathrm{SU}\left(3\right)$-symmetric model, known to favor a trimerized (simplex solid) phase, and (b) the Heisenberg model, whose qualitative nature remains to be established and is the subject of this study. An abrupt change in fidelity is found to occur in both cases in the range $-0.2<J_{bq}<-0.13$.

Figure 5

The main panel shows the trimerization order parameter for the $8\times 8$ and $14\times 8$ kagome lattice as a function of $J_{bq}$. The dashed line gives an extrapolated estimate of the $J_{bq}^{*}$ at which the trimerization vanishes. Inset: Derivative of the total energy per bond with respect to $J_{bq}$ shows an abrupt change around the same value of $J_{bq}^{*}\approx -0.16$.