Optimizing Density Functional Theory for Strain-Dependent Magnetic Properties of MnBi$_2$Te$_4$ with Diffusion Monte Carlo

In this study, we evaluate the predictive power of density functional theory (DFT) for the magnetic properties of MnBi$_2$Te$_4$ (MBT), an intrinsically magnetic topological insulator with potential applications in spintronics and quantum computing. Our theoretical understanding of MBT has been challenged by discrepancies between experimental results and \textit{ab initio} calculations, particularly with respect to its electronic and magnetic properties. Our results show that the magnetic phase diagram of MBT varies significantly depending on the Hubbard $U$ parameter in the DFT framework, highlighting the importance of benchmark calculations. To address these challenges, we establish an optimized Hubbard $U$ approach derived from Diffusion Monte Carlo (DMC) calculations, which directly solves the many-body Schr\"{o}dinger equation based on the stochastic process, and implement it in the DFT framework. Once the optimized $U$ value is determined as a function of strain, we apply it to achieve DMC-level accuracy within our DFT framework. This approach is instrumental in accurately describing the magnetic states of MBT and understanding the underlying mechanisms governing its magnetic properties and their dependence on external factors.