The S = 1/2 and S = 1 two-dimensional quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice are investigated by the quantum Monte Carlo method. By finite-size-scaling analyses on the correlation lengths, the ground-state phase diagram parametrized by strengths of the dimerization and of the spatial anisotropy is determined much more accurately than the previous works. It is confirmed that the quantum critical phenomena on the phase boundaries belong to the same universality class as that of the classical three-dimensional Heisenberg model. Furthermore, for S = 1, we show that all the spin-gapped phases, such as the Haldane and dimer phases, are adiabatically connected in the extended parameter space, though they are classified into different classes in terms of the string order parameter in the one-dimensional, i.e., the zero-interchaincoupling, case.

Monte Carlo simulations of the SU(2)-symmetric deconfined critical point action reveal strong violations of scale invariance for the deconfinement transition. We find compelling evidence that the generic runaway renormalization flow of the gauge coupling is to a weak first-order transition, similar to the case of U(1) x U(1) symmetry. Our results imply that recent numeric studies of the Nèel antiferromagnet to valence bond solid quantum phase transition in SU(2)-symmetric models were not accurate enough in determining the nature of the transition.

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Ground-state properties of the spin-1 two-leg antiferromagnetic ladder are investigated precisely by means of the quantum Monte Carlo method. It is found that the correlation length along the chains and the spin gap both remain finite regardless of the strength of interchain coupling, i.e., the Haldane state and the spin-1 dimer state are connected smoothly without any quantum phase transitions between them. We propose a plaquette-singlet solid state, which qualitatively describes the ground state of the spin-1 ladder quite well, and also a corresponding topological hidden order parameter. It is shown numerically that the new hidden order parameter remains finite up to the dimer limit, though the conventional string order defined on each chain vanishes immediately when infinitesimal interchain coupling is introduced.