Universal classification scheme for the spatial-localization properties of one-particle states in finite, d-dimensional systems.

Abstract

The spatial-localization properties of general lattice distributions (e.g., charge distribution of one-particle eigenstates) are studied. We have introduced a localization quantity, the structural entropy characteristic of the decay form (shape) of the distribution. We have found that there exists a nontrivial relation between the structural entropy and the well-known delocalization index (participation ratio) where the latter parameter measures the spatial extension of the distribution. This relation is universal in the sense that it is independent of the geometrical arrangement of the atomic network, of the lattice constant, and of the size of the system, and is fully determined by the decay form of the eigenstates and the dimension d of the lattice. Based on this relation we have developed a classification scheme for the characterization of the shape and extension of the one-particle states, which makes possible the identification of various decay forms (Gaussian, exponential, power law, etc.) in numerical calculations even on finite systems. The implementation of our method in already existing program systems is particularly easy. Besides presenting the theoretical background we demonstrate its applicability on some numerical calculations.