The role of electron correlations in the electronic structure of putative Chern magnet TbMn$_6$Sn$_6$ using correlated methods

A member of the RMn$_6$Sn$_6$ rare-earth family materials, TbMn$_6$Sn$_6$, recently showed experimental signatures of the realization of a quantum-limit Chern magnet. Despite the promising experimental results, theoretical studies with accurate electron correlations which probe these observations have been lacking. In this work, we use quantum Monte Carlo (QMC) and density functional theory with Hubbard $U$ (DFT$+U$) calculations to examine the electronic structure of TbMn$_6$Sn$_6$. To do so, we optimize accurate, correlation-consistent pseudopotentials for Tb and Sn using coupled-cluster and configuration-interaction (CI) methods. We find that DFT$+U$ and single reference QMC calculations suffer from the same overestimation of the magnetic moments as meta-GGA and hybrid density functional approximations. Our findings point to the need for improved orbitals/wavefunctions for this class of materials, such as natural orbitals from CI, or for the inclusion of multi-reference effects that capture the static correlations for an accurate prediction of magnetic properties. The necessity for multi-reference treatment is motivated by extrapolating the dynamic correlations to the exact limit. DFT$+U$ with Mn magnetic moments adjusted to experiment predict the Dirac crossing in bulk to be close to the Fermi level, within $\sim 120$ meV, in agreement with the experiments. Our non-stoichiometric slab calculations show that the Dirac crossing approaches even closer to the Fermi level, suggesting the possible realization of Chern magnetism in this limit.