Polarization and localization in insulators: generating function approach

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  • 2 November 1999
Abstract
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 wavefunction. The central quantity is a cumulant generating function which yields, upon successive differentiation, all the cumulants and moments of the probability distribution of an appropriately defined center of mass X/N of the electrons in an extended system with N electrons in a periodic volume V. In the case of the average polarization q_e/V we recover the well-known Berry's phase expression. The mean-square fluctuation can be used to define a localization length xi_i for the electrons along each cartesian direction. Using the fluctuation-dissipation theorem we show that this a well-defined measurable quantity, which in the thermodynamic limit is finite for insulators and divergent for conductors. The present formalism can be recast in terms of Kohn's theory of the insulating state, in which the many-electron insulating wavefunction breaks up into a sum of functions which are localized in disconnected regions of the high-dimensional configuration space. It is in this sense that electrons are localized in an insulator, and the desired distribution of the center of mass is that for the N electrons calculated from the partial wavefunction in one disconnected region in configuration space. The width of this distribution is sqrt(N)xi_i and is related to the observable polarization fluctuations; for non-interacting electrons, xi_i^2 is a lower bound to the average quadratic spread of the occupied Wannier functions. The quadratic fluctuations have a geometrical interpretation in terms of a metric (distance between nearby states) on a manifold of quantum states parametrized by the twisted boundary conditions.

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