Polarization and localization in insulators: Generating function approach

We develop the theory and practical expressions for the full quantum-mechanical distribution of the intrinsic macroscopic polarization of an insulator in terms of the ground state wave function. The central quantity is a cumulant generating function, which yields, upon successive differentiation, all the cumulants and moments of the probability distribution of the center of mass X/N of the electrons, defined appropriately to remain valid for extended systems obeying twisted boundary conditions. The first moment is the average polarization, where we recover the well-known Berry phase expression. The second cumulant gives the mean-square fluctuation of the polarization, which defines an electronic localization length ξᵢ along each direction i: ξᵢ²=(〈Xᵢ²〉-〈Xᵢ〉²)/N. It follows from the fluctuation-dissipation theorem that in the thermodynamic limit ξᵢ diverges for metals and is a finite, measurable quantity for insulators. In noninteracting systems ξᵢ² is related to the spread of the Wannier functions. It is possible to define for correlated insulators maximally localized ``many-body Wannier functions,'' which for large N become localized in disconnected regions of the high-dimensional configuration space, establishing a direct connection with Kohn's theory of the insulating state. Interestingly, the expression for ξᵢ², which involves the second derivative of the wave function with respect to the boundary conditions, is directly analogous to Kohn's formula for the ``Drude weight'' as the second derivative of the energy.