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 $\mathbf{X}/N$ 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 ${\ensuremath{\xi}}_{i}$ along each direction i: ${\ensuremath{\xi}}_{i}^{2}=(〈{X}_{i}^{2}〉\ensuremath{-}〈{X}_{i}{〉}^{2})/N.$ It follows from the fluctuation-dissipation theorem that in the thermodynamic limit ${\ensuremath{\xi}}_{i}$ diverges for metals and is a finite, measurable quantity for insulators. In noninteracting systems ${\ensuremath{\xi}}_{i}^{2}$ 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 ${\ensuremath{\xi}}_{i}^{2},$ 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.