$\mathbb{Z}_2$ gauge theory for valence bond solids on the kagome lattice

We present an effective $\mathbb{Z}_2$ gauge theory that captures various competing phases in spin-1/2 kagome lattice antiferromagnets: the topological $\mathbb{Z}_2$ spin liquid (SL) phase, and the 12-site and 36-site valence bond solid (VBS) phases. Our effective theory is a generalization of the recent $\mathbb{Z}_2$ gauge theory proposed for SL phases by Wan and Tchernyshyov. In particular, we investigate possible VBS phases that arise from vison condensations in the SL. In addition to the 12-site and 36-site VBS phases, there exists 6-site VBS that is closely related to the symmetry-breaking valence bond modulation patterns observed in the recent density matrix renormalization group simulations. We find that our results have remarkable consistency with a previous study using a differnt $\mathbb{Z}_2$ gauge theory. Motivated by the lattice geometry in the recently reported vanadium oxyfluoride kagome antiferromagnet, our gauge theory is extended to incorporate lowered symmetry by inequivalent up- and down-triangles. We investigate effects of this anisotropy on the 12-site, 36-site, and 6-site VBS phases. The 12-site VBS is stable to anisotropy while the 36-site VBS undergoes severe dimer melting. Interestingly, any analogue of the 6-site VBS is not found in this approach. We discuss the implications of these findings and also compare the results with a different type of $\mathbb{Z}_2$ gauge theory used in previous studies.