Towards ideal band insulation: Modulating the localization tensor using optimal auxiliary fields

When applied to band insulators, electric fields cause transitions between (filled) valence bands and (empty) conduction bands. This gives rise to macroscopic electron transport. Such transport properties are closely related to the localization tensor [reviewed, e.g., in R. Resta, J. Phys.: Condens. Matter 22, 123201 (2010)], which diverges for metals but remains finite (though nonvanishing) for insulators in the thermodynamic limit. By invoking ideas related to Berry's transitionless quantum driving [M. V. Berry, J. Phys. A: Math. Theor. 42, 365303 (2009)], we seek auxiliary fields that aim to reduce the localization tensor and thus improve the quality of the insulator. By considering certain classes of auxiliary fields (e.g., ones that are feasible to produce experimentally), we optimize a certain figure of merit that directly characterizes the effects of transitions. For given classes of auxiliary fields, this strategy yields fields that optimally reduce the localization tensor, taking it as close as possible to the case of an ideal insulator, for which it would vanish.