Title: Effect of Off-Diagonal Elements in Wannier Hamiltonian on DFT+DMFT for low-symmetry material: Study of Li$_2$MnO$_3$

Abstract: We study the effect of the off-diagonal elements of the Wannier Hamiltonian on the electronic structure of low-symmetry material Li$_2$MnO$_3$ ($C2/m$), using dynamical mean field theory calculations with continuous-time Quantum Monte Carlo impurity solver. Presence of significant off-diagonal elements leads to a pronounced suppression of the energy gap. The off-diagonal elements are largest when the Wannier projection is used based on the global coordinate, and they remain substantial even with the projection using the local coordinate close to the direction of Mn-O bonds. We show that the energy gap is enhanced by the diagonalization of the Mn $d$ block in the full $p$-$d$ Hamiltonian, with applying unitary rotation matrix. Additionally, the inclusion of a small double counting energy is crucial for achieving the experimental gap by reducing $p$-$d$ hybridization. Furthermore, we establish the efficiency of a low-energy ($d$-only basis) model for studying the electronic structure of Li$_2$MnO$3$, as the Wannier basis represents a hybridized state of Mn $d$ and O $p$ orbitals. These findings suggest an appropriate new approach for investigating low-symmetry materials using the DFT+DMFT method. To the best of our knowledge, no systematic study of the effect of off-diagonal terms has been conducted thus far. We also find that the antiferromagnetic ground state $\Gamma_{2u}$ is stable with $U \leq 2$ eV within density functional theory+$U$ calculations, which is much smaller than widely used $U$=5 eV.