Paramagnetic ground states and field-driven N\'eel order in S=3/2 Heisenberg antiferromagnets on a honeycomb lattice

We study the spin-3/2 Heisenberg antiferromagnet on a honeycomb lattice with exchange interactions which frustrate N\'eel order. Our motivation stems from the recent synthesis of $Bi_3 Mn_4 O_{12} (NO_3 )$, a spin-3/2 bilayer honeycomb lattice antiferromagnet which remains paramagnetic to the lowest temperature, but shows a field-induced N\'eel transition. We use a combination of spin wave theory, exact diagonalization, and bond operator theory to study the effects of quantum and thermal fluctuations, second-neighbor exchange, biquadratic exchange and bilayer coupling. Biquadratic terms give rise an AKLT valence bond solid ground state, and bilayer coupling leads to an interlayer dimer solid. Upon applying a magnetic field, both these states undergo a phase transition into a N\'eel long range ordered state. We comment on experimental consequences and disorder effects.