Unambiguous and robust formulation for Wannier localization

We provide a new variational definition for the spread of an orbital under periodic boundary conditions (PBCs) that is continuous with respect to the gauge, consistent in the thermodynamic limit, well suited to diffuse orbitals, and systematically adaptable to schemes computing localized Wannier functions. Existing definitions do not satisfy all these desiderata, partly because they depend on an ``orbital center''---an ill-defined concept under PBCs. Based on this theoretical development, we showcase a robust and efficient ($10\ifmmode\times\else\texttimes\fi{}$--$70\ifmmode\times\else\texttimes\fi{}$ fewer iterations) localization scheme across a range of materials.