Quantum dimer model for the spin-$\frac{1}{2}$ kagome $Z_2$ spin liquid

We revisit the description of the low-energy singlet sector of the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on kagome in terms of an effective quantum dimer model. With the help of exact diagonalizations of appropriate finite-size clusters, we show that the embedding of a given process in its kagome environment leads to dramatic modifications of the amplitudes of the elementary loop processes, an effect not accessible to the standard approach based on the truncation of the Hamiltonian to the nearest-neighbor valence-bond basis. The resulting parameters are consistent with a $Z_2$ spin liquid rather than with a valence-bond crystal, in agreement with the last density matrix renormalization group results.

Figure 1

(a) Two NNVB states on the kagome, $|{\varphi }_b\rangle$ and $|{\varphi }_r\rangle$, that are everywhere the same except around a hexagon (ovals denote spin singlets). (b)–(c) The VB coverings $|{\chi }_{\mathrm{in}}\rangle$ and $|{\chi }_{\mathrm{out}}\rangle$ (dashed green ovals) contain long-range singlets that can be virtually excited around the defect triangle “D” of $|{\varphi }_b\rangle$.

Figure 2

Finite clusters used to extract the dominant tunneling parameters $t_{\mathrm{ED}}$ on the kagome. Ovals denote singlets and arrows indicate our singlet orientation [37] convention (clockwise in all hexagons [7]); dashed (blue) lines are reflection symmetries. $L$ denotes the length of the loop formed by the transition graph of the two NNVB states involved in the tunneling process.

Figure 3

Low-energy spectra of the “$R=1$” clusters ${\mathrm{B}}_1$–D of Fig. 2 in the magnetization sectors $S_z=0$ and 1. Legends: parity under reflection (for ${\mathrm{B}}_1$–${\mathrm{C}}_3$), where “e” and “o” denote even and odd parity, respectively, or angular momenta under sixfold rotations (for D). Spin-inversion parity (for $S_z=0$) is labeled by “Sze” and “Szo”. The two NNVB low-lying singlets are designated by double vertical arrows and the third singlet by horizontal arrows.

Figure 4

(a) Evolution of $\left|t\right|$ as we include further-range virtual singlet fluctuations, quantified by the length $R$ of the sawtooth chains attached to the minimal clusters of the first row of Fig. 2. The NNVB parameters are placed at the origin for comparison. Inset: The ratio of the two strongest processes. The horizontal dashed (gray) line separates the 36-site VBC from the $Z_2$ spin liquid phase [33].