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 $\alpha \sim 1/m$, where $\alpha {=(V/W)\mathrm{}}^2$, $V$ is the typical value of the hopping matrix element, and $W$ is the width of the distribution of random site energies. For $\alpha >{1/m}^2$ the LDOS fluctuations (with respect to this average form) are weak. In the opposite case $\alpha <{1/m}^2$, the fluctuations become strong and the average LDOS ceases to be representative, which is related to the existence of the Anderson transition at ${\alpha }_c\sim {1/m}^2{\log }^{2}m$. On the localized side of the transition the spectrum is discrete and the LDOS is given by a set of $\delta$-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 $P_c$ close to unity, $1-P_c\sim 1/\log m$, 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 $\sim \exp \{-\mathrm{const}\log m\left[\right(\alpha -{\alpha }_c)/{\alpha }_c{]}^{-1/2}\}$ 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)].