Nonextensive E ects in Tight - Binding Systems with Long - Range Hopping

Consequences of long-range hopping in one-dimensional tight-binding models are studied. A hopping term proportional to 1=r ij is used, where rij denotes the distance between atoms i and j and determines the range of the interactions within the system. Calculations of the di usion of an electron along the lattice yield interesting e ects of nonextensivity. In particular, we nd that the mean square displacement scales anomalously as Dt in the following way: For 0 < < 1, we nd D / NN , where N is the number of atoms on the lattice and N = N 1 1 1 is related to the number of elements interacting at a given . In this regime the behaviour is subdi usive (:5 < 1) but approaches normal di usion ( = 1) for = 1. There exists a transition region between 1 < < 2, where the di usion coe cient loses its system size dependency and becomes size independent for all 2. In addition, we nd 1 < 2 (superdi usion) for > 1. Ballistic motion ( = 2) is recovered for all 1:5 and is maintained in the nearest neighbour limit. Speci c heat and internal energy as a function of temperature and system size are also analyzed. They appear extensive on the macroscopic level for all values of .