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Motivated by recent experiments on the spin-3/2 frustrated bilayer honeycombantiferromagnet Bi$_3$Mn$_4$O$_{12}$(NO$_3$), we study the spin-S Heisenbergmodel on the honeycomb lattice with various additional exchange interactionswhich frustrate N\'eel order. Using spin wave theory, exact diagonalization,and bond operator theory, we consider the effects of (i) second-neighborexchange, (ii) biquadratic exchange for $S = 3/2$ which leads to an AKLTvalence bond solid, and (iii) bilayer coupling which leads to an interlayerdimer solid. We show that the resulting paramagnetic states undergo atransition to N\'eel order beyond a critical magnetic field. We discussexperimental implications for Bi$_3$Mn$_4$O$_{12}$(NO$_3$).

Quantum paramagnets on the honeycomb lattice and field-induced N\'eel order: Possible application to Bi$_3$Mn$_4$O$_{12}$(NO$_3$)