Localization and fluctuations of local spectral density on treelike structures with large connectivity: Application to the quasiparticle line shape in quantum dots

We study fluctuations of the local density of states (LDOS) on a treelike lattice with large branching number m. 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 α~1/m, where α=(V/W)², V is the typical value of the hopping matrix element, and W is the width of the distribution of random site energies. For α>1/m² the LDOS fluctuations (with respect to this average form) are weak. In the opposite case α<1/m², the fluctuations become strong and the average LDOS ceases to be representative, which is related to the existence of the Anderson transition at αc~1/m²log²m. On the localized side of the transition the spectrum is discrete and the LDOS is given by a set of δ-like peaks. The effective number of components in this regime is given by $1/P$, with P being the inverse participation ratio. It is shown that P has in the transition point a limiting value Pc close to unity, 1-Pc~1/logm, 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 ~exp-constlogm[(α-αc)/αc]^-1/2 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)].