Nonextensive effects in tight-binding systems with long-range hopping

Abstract

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 .

I. Nonextensive Statistical Mechanics and Thermodynamics: Historical Background and Present Status

“…The T = 0 electron diffusive properties corresponding to this Hamiltonian exhibit a variety of anomalies intimately related toÑ , as preliminary shown by Nazareno and Brito [199]. More details have been recently exhibited by Borland, Menchero and myself [200]: see Fig. 24.…”

“…It is clear, however, that similar nonextensivity is expected to emerge in quantum systems if longrange interactions are present. One such Hamiltonian is the tight-binding-like which follows [199,200]:…”

Discrete nonlinear Schrödinger equation dynamics in complex networks

“…Only when the particle is spread initially equally on all sites it may stay in this delocalized state corresponding to the Perron-Frobenius eigenstate of the lattice. We thus find that in the linear case while the NN limit favors delocalization the MF limit promotes localization [16,17]. This feature will dominate the behavior of DNLS in the large-B limit.…”

Magnetic phase diagram for a nonextensive system: Experimental connection with manganites

“…[13][14][15][16][17] For condensed matter problems, applications of Eq. ͑1͒ include Ising ferromagnets, 18 -20 molecular field approximation, 21,22 Landau diamagnetism, 23 electron-phonon systems and tight-binding-like Hamiltonians, 24,25 metallic 26 and superconductor 27 systems, etc. The first evidence that the magnetic properties of manganites could be described within the framework of Tsallis statistics was presented by Reis et al, 28 followed by an analysis 29 of the unusual paramagnetic susceptibility of La 0.67 Ca 0.33 MnO 3 , measured by Amaral et al 30 Maximization of Eq.…”