Abstract
We study fluctuations of the local density of states (LDOS) on a treelike lattice with large branching number . The average form of the local spectral function (at a given value of the random potential in the observation point) shows a crossover from the Lorentzian to a semicircular form at , where , is the typical value of the hopping matrix element, and is the width of the distribution of random site energies. For the LDOS fluctuations (with respect to this average form) are weak. In the opposite case , the fluctuations become strong and the average LDOS ceases to be representative, which is related to the existence of the Anderson transition at . On the localized side of the transition the spectrum is discrete and the LDOS is given by a set of -like peaks. The effective number of components in this regime is given by , with being the inverse participation ratio. It is shown that has in the transition point a limiting value close to unity, , so that the system undergoes a transition directly from the deeply localized phase to the extended phase. On the side of delocalized states, the peaks in the LDOS become broadened, with a width being exponentially small near the transition point. We discuss the application of our results to the problem of the quasiparticle line shape in a finite Fermi system, as suggested recently by Altshuler, Gefen, Kamenev, and Levitov [Phys. Rev. Lett. 78, 2803 (1997)].
- Received 12 February 1997
DOI:https://doi.org/10.1103/PhysRevB.56.13393
©1997 American Physical Society