Effective quantum dimer model for the kagome Heisenberg antiferromagnet: Nearby quantum critical point and hidden degeneracy

The low-energy singlet dynamics of the quantum Heisenberg antiferromagnet on the kagome lattice is described by a quantitative quantum dimer model. Using advanced numerical tools, the latter is shown to exhibit valence-bond crystal (VBC) order with a large 36-site unit cell and hidden degeneracy between even odd parities. Evidences are given that this ground state lies in the vicinity of a ${\mathbb{Z}}_2$ dimer-liquid region separated by a quantum critical point (QCP). Implications regarding numerical analysis and experiments are discussed.

Figure 1

(Color online) (a) Sketch of the 36-site VBC on the kagome lattice showing “perfect hexagons” and (yellow) star resonances. Its unit cell is delimited by dashed (red) lines. The extension of the 108-site cluster (which fits exactly three 36-site cells) is (lightly) shaded and delimited by long-dashed (blue) lines. Hardcore dimers (formed by two neighboring spin-$\frac{1}{2}$) are located on the bonds (only shown on the shaded hexagons). (b) First Brillouin zone and available momenta (labeling consistent with Ref. 18).

Figure 2

(Color online) (a) Complete low-energy excitation spectrum of the 48-site cluster as a function of the parameter $\lambda$ (see text). The lowest 25 levels are displayed in each TS (see Ref. 27) using different symbols/colors. The 2-vison (blue lines) and 4-vison (dashed black line) gaps vanish around $\lambda =1$. (b) Close up around $\lambda =1$ showing the excitation energy of the lowest level of the most relevant IR (see Table ) of the 108-site cluster: a QCP is identified from (i) a collapse of these excitations (b) and (ii) a very sharp peak in ${\chi }_{\lambda }$ (c) (data for $N=108$ divided by 200 to fit the scale). Note the transition is far more abrupt for $N=108$ than for $N=48$ (c).

Figure 3

(Color online) Complete low-energy excitation spectrum of the 48-site cluster as a function of $J_{12}$. The lowest 25 levels are displayed in each TS [same symbols as Fig. 2a]. The quantum numbers of the four lowest-energy states for $J_{12}<0$ and $J_{12}>0$ are provided using different line types.

Figure 4

(Color online) (a) GS energies per site versus $J_{12}$ for $K=K_{\Gamma }$ and $\sigma =\pm$. The minimum defines the absolute GS energy $E_N^{\text{GS}}/N$. Data for 48 sites (lines) and 108 sites (circles) are shown. An energy shift and scale transformation $E_N\to \frac{3}{4}\left(E_N-N/2\right)$ allow a direct comparison with twice the bond energy of the QHAF. (b) Same as Fig. 2b but as a function of $J_{12}$ and compared to the $N=48$ excitation at the $\Gamma$ point (full lines). Note the splittings between $\Gamma$, A, and B levels of a few ${10}^{-8}\text{J}$ are invisible at this scale.