Competing valence bond crystals in the kagome quantum dimer model

The singlet dynamics, which plays a major role in the physics of the spin-1/2 quantum Heisenberg antiferromagnet (QHAF) on the kagome lattice, can be approximately described by projecting onto the nearest-neighbor valence bond (NNVB) singlet subspace. We revisit here the effective quantum dimer model, which originates from the latter NNVB-projected Heisenberg model via a nonperturbative Rokhsar-Kivelson-like scheme. By using Lanczos exact diagonalization on a 108-site cluster supplemented by a careful symmetry analysis, it is shown that a previously found 36-site valence bond crystal (VBC) in fact competes with a new type of 12-site “resonating-columnar” VBC. Interestingly, these two VBCs “emerge” in different topological sectors. The exceptionally large degeneracy of the ground-state multiplets (144 on our 108-site cluster) and the proximity of a ${\mathbb{Z}}_2$ dimer liquid have implications for the interpretation of numerical results on the QHAF, which are outlined. The possibility of a chiral VBC (i.e., spontaneously breaking time-reversal symmetry) is also discussed.

Figure 1

Kagome lattices. The 108-site clusters delimited by the (red) dashed lines are made of $6\times 6$ unit cells of three-site up triangles. Periodic boundaries are used. The four topological sectors are determined by the parities of the number of dimers (black thick bonds) cut by the three $a-a$, $b-b$, and $c-c$ closed loops along the three lattice directions shown (see text). For clarity, the hard-core dimers are only shown along the three cuts. “Defect” triangles with no dimers are left empty. (a) $2\sqrt{3}\times 2\sqrt{3}$ (36-site) HVBC showing resonating hexagons (shaded in green) and stars (shaded in blue) in the [1,1,1] topological sector. (b) $2\times 1$ (six-site) CVBC with two types of up and two types of down triangles (all colored differently) in the [0, 0,1] topological sector. The red stars indicate the inversion ($\pi$-rotation) centers.

Figure 2

VBC with $2\times 2$ supercells in the [1,1,1] topological sector of the periodic $N=108$-site cluster. (a) VBC${}_0$ with four types of up (and down) triangles displayed with different colors except the defect triangles left empty (degeneracy $g=8$). (b) The resonances of DVBC (colored diamonds) restore reflection symmetry with respect to the vertical and horizontal directions but suppress the $\pi /3$-rotation symmetry ($g=12$). (c) SVBC has higher symmetry than VBC${}_0$, resonating stars restoring reflection symmetry with respect to the crystallographic axes ($g=4$).

Figure 3

36 available momenta of the 108-site cluster ($6\times 6$ up triangles) in the first Brillouin zone. Topological sectors have to be distinguished (as done on graph) because of their different point groups (see text). Equivalent momenta are represented by the same (filled) symbols. Translation by a reciprocal-lattice vector (thin lines) of the momenta at the border of the first BZ are shown by open symbols (correspond to the same momenta). The dashed lines in (b) represent the two (orthogonal) reflection symmetry axes of the $C_{2v}$ group.

Figure 4

Resonant VBC with $2\times 2$ supercells in the [0, 0,1] topological sector of the periodic $N=108$-site cluster. Resonating ten-dimer (a),(c) and eight-dimer (b) loops (see text) are represented by shaded areas. (a) When $\alpha =1$ (see text) a symmetry axis perpendicular to the $c$ direction is restored (light and dark green triangles become equivalent). (b) The pattern exhibits a reflection symmetry with respect to the $c$ direction. (c) Inversion symmetry with respect to the centers of all the hexagons is present. Dimers beyond NNs have been used.

Figure 5

(a) Low-energy spectrum of the 108-site cluster in the $T_0=[1,1,1]$ and $T_3=[0,0,1]$ (degenerate with $[0,1,0]$ and $[1,0,0]$) topological sectors. Different symbols distinguish different momenta and topological sectors [see Figs. 3a and 3(b)] and IRREPs labeled as “$(r_3,r_2,\sigma )$” (see text). Only a few levels corresponding to the onsets of the two continua (shaded areas) are shown. (b) Zoom in of the (quasi)ground states. An Ising degeneracy (see text) is reflected by the (almost) perfect equal spacings between the HVBC ground states.