dc conductivity as a geometric phase

The zero-frequency conductivity (D-c), the criterion to distinguish between conductors and insulators, is expressed in terms of a geometric phase. D-c is also expressed using the formalism of the modern theory of polarization. The tenet of Kohn [Phys. Rev. 133, A171 (1964)], 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 D-c and are therefore metallic. They are also delocalized. Based on the value of D-c 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 e Phi B/Nhc = Q/M, with Q and M as integers.