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We investigate thermal phase transitions to a valence-bond solid phase inSU(N) Heisenberg models with four- or six-body interactions on a square orhoneycomb lattice, respectively. In both cases, a thermal phase transitionoccurs that is accompanied by rotational symmetry breaking of the lattice. Weperform quantum Monte Carlo calculations in order to clarify the criticalproperties of the models. The estimated critical exponents indicate that theuniversality classes of the square- and honeycomb-lattice cases are identicalto those of the classical $XY$ model with a $Z_4$ symmetry-breaking field andthe 3-state Potts model, respectively. In the square-lattice case, the thermalexponent, $\nu$, monotonically increases as the system approaches the quantumcritical point, while the values of the critical exponents, $\eta$ and$\gamma/\nu$, remain constant. From a finite-size scaling analysis, we findthat the system exhibits weak universality, because the $Z_4$ symmetry-breakingfield is always marginal. In contrast, $\nu$ in the honeycomb-lattice caseexhibits a constant value, even in the vicinity of the quantum critical point,because the $Z_3$ field remains relevant in the SU(3) and SU(4) cases.

Thermal Phase Transition of Generalized Heisenberg Models for SU(N) Spins on Square and Honeycomb Lattices