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