Generalized Peierls substitution for Wannier obstructions: response to disorder and interactions

We study the interplay between quantum geometry, interactions, and external fields in complex band systems. When Wannier obstructions preclude a description based solely on atomic-like orbitals, this complicates the prediction of electromagnetic responses particularly in the presence of disorder and interactions. In this work, we introduce a generalized Peierls substitution framework based on Lagrange multipliers to enforce the constraints of the Wannier obstruction in the band of interest. Thus we obtain effective descriptions of interactions and disorder in the presence of non-trivial quantum geometry of that band. We apply our approach to examples including the diamagnetic response in flat-band superconductors and delocalization effects in flat-band metals caused by interactions and disorder.