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Example of a first-order Néel to Valence-Bond-Solid transition in two-dimensions

We consider the $S=1/2$ Heisenberg model with nearest-neighbor interaction $J$ and an additional multi-spin interaction $Q_3$ on the square lattice. The $Q_3$ term consists of three bond-singlet projectors and is chosen to favor the formation of a valence-bond solid (VBS) where the valence bonds (singlet pairs) form a staggered pattern. The model exhibits a quantum phase transition from the Néel state to the VBS as a function of $Q_3/J$. We study the model using quantum Monte Carlo (stochastic series expansion) simulations. The Néel-VBS transition in this case is strongly first-order in nature, in contrast to similar previously studied models with continuous transitions into columnar VBS states. The qualitatively different transitions illustrate the important role of an emerging U(1) symmetry in the latter case, which is not possible in the present model due to the staggered VBS pattern (which does not allow local fluctuations necessary to rotate the local VBS order parameter).