Motivated by recent numerical results, we study the quantum phase transitions between $Z_2$ spin-liquid, Néel-ordered, and various valence-bond solid (VBS) states on the honeycomb and square lattices, with emphasis on the staggered VBS. In contrast to the well-understood columnar VBS order, the staggered VBS is not described by an $\mathit{XY}$-order parameter with $Z_N$ anisotropy close to these quantum phase transitions. Instead, we demonstrate that on the honeycomb lattice, the staggered VBS is more appropriately described as an O(3)- or CP(2)-order parameter with cubic anisotropy, while on the square lattice it is described by an O(4)- or CP(3)-order parameter.

• Received 31 March 2011

DOI:https://doi.org/10.1103/PhysRevB.84.014402