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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 [{\it Phys. Rev.} {\bf 133} A171 (1964)], namely, that insulation is due to localization in the many-body space, is refined as follows. Wavefunctions which are eigenfunctions of the total current operator give rise to a finite $D_c$ and are therefore metallic. They are also delocalized. 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 $\frac{eΦ_B}{N h c} = \frac{Q}{M}$, with $Q$ and $M$ integers.