  Bi$_3$Mn$_4$O$_{12}$(NO$_3$) (BMNO) is a honeycomb bilayers anti-ferromagnet,
not showing any ordering down to very low temperatures despite having a
relatively large Curie-Weiss temperature. Using ab initio density functional
theory, we extract an effective spin Hamiltonian for this compound. The
proposed spin Hamiltonian consists of anti-ferrimagnetic Heisenberg terms with
coupling constants ranging up to third intra-layer and fourth inter-layer
neighbors. Performing Monte Carlo simulation, we obtain the temperature
dependence of magnetic susceptibility and so the Curie-Weiss temperature and
find the coupling constants which best matches with the experimental value. We
discover that depending on the strength of the interlayer exchange couplings,
two collinear spin configurations compete with each other in this system. Both
states have in plane N{\'e}el character, however, at small interlayer coupling
spin directions in the two layers are antiparallel (N$_1$ state) and
discontinuously transform to parallel (N$_2$ state) by enlarging the interlayer
couplings at a first order transition point. Classical Monte Carlo simulation
and density matrix renormalization group calculations confirm that exchange
couplings obtained for BMNO are in such a way that put this material {at the
phase boundary of a first order phase transition}, where the trading between
these two collinear spin states prevents it from setting in a magnetically
ordered state.
