Valence bond solid order and phase transitions of honeycomb lattice models

We investigate the ground states of generalized SU($N$) Heisenberg models on honeycomb lattices. From large-scale quantum Monte Carlo simulations, we confirm the columnar valence bond solid (c-VBS) orders for $N \ge 5$ at low temperatures, which corresponds to Kekul\'e distortion. It is consistent with Read and Sachdev's prediction[N.~Read and S.~Sachdev, Phys Rev B {\bf 42}, 4568 (1990)]. If we introduce the designed six-body interactions on hexagonal plaquettes, the c-VBS order occurs even in the SU(2) case. While the c-VBS state on a square lattice breaks $Z_4$ rotational symmetry, $Z_3$ rotational symmetry breaks on a honeycomb lattice. The difference may changes the nature of c-VBS phase. In particular, we will report phase transitions from a c-VBS phase to a paramagnetic or N\'eel phase in details. These results give us insight for deconfinement critical phenomena.