Spin triplet excitations for a valence bond solid on the kagome lattice

One of the most promising candidate ground states for the quantum antiferromagnetic Heisenberg model on the kagome lattice is the valence bond solid (VBS) with a 36-site unit cell. We present a theory of triplet excitation spectra about this ground state using bond operator formalism. In particular we obtain dispersions of all 18 triplet modes in the reduced Brillouin zone. In the bond operator mean-field theory, it is found that a large number of triplet modes are nondispersive. In particular, the lowest triplet excitation is nondispersive and degenerate with a dispersive mode at the zone center. Away from the zone center, the lowest triplet is separated from two other flat modes by a small energy gap. Quantum fluctuations are considered by taking into account scattering processes of two triplets and their bound-state formation, which leads to a downward renormalization of the lowest spin triplet gap. The dispersion of the lowest triplet excitation in the VBS state is compared with the dispersive lower bound of the triplet continuum expected in competing spin liquid phases. Implications to future neutron-scattering experiments are discussed.

Figure 1

(Color online) Dimer covering pattern for the valence bond solid phase with a 36-site unit cell. The unit cell with 36 sites (or 18 dimers) is indicated by the enclosing dotted line. Thick solid bars denote singlet dimers. We use numbers between one and nine to represent surrounding dimers (light-blue bars) while core dimers (dark-blue bars) are denoted by numbers between 10 and 18. Red circles indicate the locations of perfect hexagons. Note that perfect hexagons are made of three neighboring dimers, (1,4,7) or (3,6,9). The dimers 2, 5, and 8 become bridges connecting perfect hexagons.

Figure 2

(Color online) Mean-field order parameters, $P$ and $Q$, as a function of $\lambda$. The top figure depicts dimer covering of the valence bond solid state in the vicinity of a perfect hexagon, which also shows how various $Q$s are defined. Note that each $Q$ describes an off-diagonal correlation between two dimers connected by corresponding arrows. Diagonal correlations, $P_H$, $P_{Br1}$, and $P_{Br2}$, are defined in the same way. It is interesting to note that $Q_{Br1}$ $\left(P_{Br1}\right)$ and $Q_{Br2}$ $\left(P_{Br2}\right)$ have opposite signs.

Figure 3

(Color online) Energy dispersion of the triplet excitations when quartic interaction effects are ignored. The dotted line describes the ninefold degenerate triplet states from core dimers. The solid lines denote the nine triplet states from surrounding dimers.

Figure 4

(Color online) Real-space representation of the eigenvectors of the flat $A$ mode at the $\Gamma$ point. The size of ovals and the sign inside indicate relative weights and phases of the amplitude at each dimer, respectively. (a) One of the flat $A$ modes with the energy eigenvalue ${\Omega }_A$. Triplets around each perfect hexagon have the same phases and weights. Two different flat $A$ modes are distinguished by relative phases between neighboring perfect hexagons. (b) Local geometry around a perfect hexagon. Here red dots indicate the position of $R$ spins within a dimer. See Eq. (3) for the definition of the $R$ spin.

Figure 5

(Color online) Real-space representation of the eigenvectors of the flat $E$ modes. To avoid degeneracy, eigenvectors are computed slightly away from the $\Gamma$ point to the direction of the $K$ point, for (a) $E_1$, (b) $E_2$, and (c) $E_3$ modes. Note that the size of ovals and the sign indicate relative weights and phases of the amplitude at each dimer, respectively.

Figure 6

(Color online) Energy dispersion of the triplet excitations when quartic interaction effects are included in the self-consistent mean-field theory. Away from the zone center, the lowest flat mode is separated from doubly degenerate flat $A$ modes by a small gap, ${\Delta }_0\cong 0.069\text{J}$. Eigenenergies of the flat modes are shown on the right-hand side of the plot, where subscripts indicate the irreducible representation of each state at the $\Gamma$ point.

Figure 7

Renormalization of the triplet pair emission (or absorption) amplitude by quantum fluctuations. Vertices come from the singlet $S=0$ scattering channels of two triplets.

Figure 8

(Color online) Spin triplet excitation spectra for three candidate ground states of the kagome-lattice Heisenberg antiferromagnet: (i) the valence bond solid (VBS) with a 36-site unit cell, (ii) the U(1) Dirac spin liquid, and (iii) the $Z_2$ spin liquid. In the case of the VBS state, we plot the renormalized spectrum of the lowest triplet mode including quantum fluctuation effects as described in Sec. 4D. For these two spin liquid states, we plot the lower bound of the two-spinon continuum. The first Brillouin zone for the spin liquid states that has a three-site triangular unit cell is depicted on the right-hand side. The small dotted hexagon shows the first Brillouin zone for the VBS state with a 36-site unit cell. Two red dots lying in the $y$ axis indicate the positions (gauge dependent) of the Dirac nodes for the U(1) Dirac spin liquid state.

Figure 9

(Color online) Real-space representation of localized eigenstates (enclosed by dotted lines) and noncontractible loop states (located on the left side of each panel) constructed in the flat $E_1$, $E_2$, and $E_3$ bands.