The effect of long-range interactions and its connection with nonextensivity was analyzed by determining the propagating properties of wave packets in a one-dimensional ordered structure. It was assumed that a power-law decaying hopping term with an exponent $\alpha$ determined the range of the interaction. One can clearly notice different regimes of propagation according to the different $\alpha$ values. When $\alpha =0,$ a situation in which every site in the lattice sees the others with the same intensity, the wave packet gets localized around the starting position, i.e., self-trapping takes place. By increasing $\alpha$ the localization is lost, the $X_{\mathrm{MSD}}\equiv 〈n^2〉$ grows with time but shows oscillations that disappear as soon as $\alpha$ reaches the value $1.$ In this case, and for very short times $(t\lesssim 2\hspace{0.5em}\mathrm{ps}),$ the packet diffuses with a diffusion coefficient that increases with the number of sites in the lattice, that is $〈n^2〉{=D}_1\mathrm{}\left(N\right)t.\mathrm{}$ This size effect reported here is absent in the familiar nearest-neighbor (NN) interaction case. For later times and $\alpha =1,$ the particle propagates subdiffusively. As for bigger $\alpha$ values, the propagation can be characterized by the following $X_{\mathrm{MSD}}{=Ct}^{\mu }$ where the exponent $\mu$ approaches rapidly the value $2$ as long as $\alpha$ is greater than $2.$ In other words, for great $\alpha$’s we have ballistic propagation, as was the case for the NN interaction. When a dc electric field is applied we get a localized wave packet, i.e., it is the phenomenom of dynamic localization.

• Received 18 December 1998

DOI:https://doi.org/10.1103/PhysRevB.60.4629