Z4 to U(1) crossover of the order-parameter symmetry in a two-dimensional valence-bond solid

We discuss ground-state projector simulations of a modified two-dimensional S=1/2 Heisenberg model in the valence bonds basis. Tuning matrix elements corresponding to the diagonal and off-diagonal terms in the quantum dimer model, we show that there is a quantum phase transition from the antiferromagnet into a columnar valence-bond solid (VBS). There are no signs of discontinuities, suggesting a continuous or very weak first-order transition. The Z(4)-symmetric VBS order parameter exhibits an emergent U(1) symmetry as the phase transition is approached. We extract the associated length-scale governing the U(1)-Z(4) cross-over inside the VBS phase.