The spin-$\frac{1}{2}$ Heisenberg model on the kagome lattice, which is closely realized in layered Mott insulators such as ${\mathrm{ZnCu}}_3{\left(\mathrm{OH}\right)}_6{\mathrm{Cl}}_2$, is one of the oldest and most enigmatic spin-$\frac{1}{2}$ lattice models. While the numerical evidence has accumulated in favor of a quantum spin liquid, the debate is still open as to whether it is a $Z_2$ spin liquid with very short-range correlations (some kind of resonating valence bond spin liquid), or an algebraic spin liquid with power-law correlations. To address this issue, we have pushed the program started by Rokhsar and Kivelson in their derivation of the effective quantum dimer model description of Heisenberg models to unprecedented accuracy for the spin-$\frac{1}{2}$ kagome, by including all the most important virtual singlet contributions on top of the orthogonalization of the nearest-neighbor valence bond singlet basis. Quite remarkably, the resulting picture is a competition between a $Z_2$ spin liquid and a diamond valence bond crystal with a 12-site unit cell, as in the density-matrix renormalization group simulations of Yan et al. Furthermore, we found that, on cylinders of finite diameter $d$, there is a transition between the $Z_2$ spin liquid at small $d$ and the diamond valence bond crystal at large $d$, the prediction of the present microscopic description for the two-dimensional lattice. These results show that, if the ground state of the spin-$\frac{1}{2}$ kagome antiferromagnet can be described by nearest-neighbor singlet dimers, it is a diamond valence bond crystal, and, a contrario, that, if the system is a quantum spin liquid, it has to involve long-range singlets, consistent with the algebraic spin liquid scenario.

• Received 29 October 2017
• Revised 20 February 2018

DOI:https://doi.org/10.1103/PhysRevB.97.104401