We study two-dimensional Heisenberg antiferromagnets with additional multispin interactions which can drive the system into a valence-bond-solid state. For standard SU(2) spins, we consider both four- and six-spin interactions. We find continuous quantum phase transitions with the same critical exponents. Extending the symmetry to
SU(N), we also find continuous transitions for
N=3 and 4. In addition, we also study quantitatively the crossover of the order-parameter symmetry from
Z4 deep inside the valence-bond-solid phase to
U(1) as the phase transition is approached.
We study thermal transitions of the generalized SU(N) Heisenberg models with four-body interactions on a square lattice and with six-body interactions on a honeycomb lattice. In both models for the N=3 and 4 cases, a singlet-dimer state is stabilized at a very low temperature, where a rotational symmetry of lattice is broken spontaneously. We discuss the universality class of thermal transition to the singlet dimer phases, performing quantum Monte Carlo calculations. From the finite-size scaling analysis, we find that the criticality for the square lattice case is well explained by the 2D weak Ising universality, while the 2D three-state Potts universality is observed in the honeycomb lattice case.
The
SU(N) symmetric antiferromagnetic Heisenberg model with multicolumn representations on the two-dimensional square lattice is investigated by quantum Monte Carlo simulations. For the representation of a Young diagram with two columns, we confirm that a valence-bond solid (VBS) order appears as soon as the N\'eel order disappears at
N=10, indicating no intermediate phase. In the case of the representation with three columns, there is no evidence for either the N\'eel or the VBS ordering for $N\ensuremath{\ge}15$. This is actually consistent with the large-
N theory, which predicts that the VBS state immediately follows the N\'eel state, because the expected spontaneous order is too weak to be detected.