# Origin of magnetic frustration in Bi$_3$Mn$_4$O$_{12}$(NO$_3$)

## TL;DR

This study uses ab initio calculations and Monte Carlo simulations to understand the magnetic frustration in Bi$_3$Mn$_4$O$_{12}$(NO$_3$), revealing competing spin configurations and a phase boundary preventing magnetic order.

## Contribution

It provides a detailed effective spin Hamiltonian for BMNO and identifies the phase boundary between two collinear spin states caused by interlayer exchange couplings.

## Key findings

- Identified competing collinear spin configurations in BMNO.
- Discovered the phase boundary preventing magnetic order.
- Matched theoretical results with experimental Curie-Weiss temperature.

## Abstract

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.