We study two planar square lattice Heisenberg models with explicit dimerization or quadrumerization of the couplings in the form of ladder and plaquette arrangements. We investigate the quantum critical points of those models by means of (stochastic series expansion) quantum Monte Carlo simulations as a function of the coupling ratio $\alpha =J^{\prime }/J$. The critical point of the order-disorder quantum phase transition in the ladder model is determined as ${\alpha }_c=1.9096\left(2\right)$ improving on previous studies. For the plaquette model, we obtain ${\alpha }_c=1.8230\left(2\right)$ establishing a first benchmark for this model from quantum Monte Carlo simulations. Based on those values, we give further convincing evidence that the models are in the three-dimensional classical Heisenberg universality class. The results of this contribution shall be useful as references for future investigations on planar Heisenberg models such as concerning the influence of nonmagnetic impurities at the quantum critical point.

• Received 11 August 2008

DOI:https://doi.org/10.1103/PhysRevB.79.014410