Dc conductivity as a geometric phase

The zero-frequency conductivity (Dc), the criterion to distinguish between conductors and insulators, is expressed in terms of a geometric phase. Dc is also expressed using the formalism of the modern theory of polarization. The tenet of Kohn, namely that insulation is due to localization in the many-body space, is refined as follows. Wave functions, which are eigenfunctions of the total current operator, give rise to a finite Dc and are therefore metallic. They are also delocalized. Based on the value of Dc it is also possible to distinguish purely metallic states from states in which the metallic and insulating phases coexist. Several examples which corroborate the results are presented, as well as a numerical implementation. The formalism is also applied to the Hall conductance, and the quantization condition for zero Hall conductance is derived to be eΦBNhc=QM, with Q and M as integers. © 2013 American Physical Society.