`Interaction annealing' to determine effective quantized valence and orbital structure: an illustration with ferro-orbital order in WTe$_2$

Correlated materials are known to display qualitatively distinct emergent behaviors at low energy. Conveniently, upon absorbing rapid quantum fluctuations, these rich low-energy behaviors can always be effectively described by dressed particles with fully quantized charge, spin, and orbital structure. Such a powerful and simple description is, however, difficult to access through bare particles used in most many-body computations, especially when fluctuations are strong such as in $4d$ and $5d$ compounds. To decipher the dominant quantized structure, we propose an easy-to-implement `interaction annealing' approach that utilizes suppressed charge fluctuation through enhancing ionic charging energy. We establish its theoretical foundation using an exactly treated two-site Hubbard model as a generic example. We then demonstrate its applications with more affordable density functional calculations to a representative $3d$ Mott insulator La${_2}$CuO${_4}$ and a highly fluctuating $5d$ semi-metal WTe${_2}$. In the latter, it reveals an emergent local electronic structure that makes possible an unprecedented explanation of several experimental observations. Finally, we demonstrate the effectiveness of this approach in studying competing local electronic structures in functional materials.