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

Abstract

We study fluctuations of the local density of states (LDOS) on a tree-like lattice with large branching number $m$. The average form of the local spectral function (at given value of the random potential in the observation point) shows a crossover from the Lorentzian to semicircular form at $α\sim 1/m$, 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 $α>1/m^2$ the LDOS fluctuations (with respect to this average form) are weak. In the opposite case, $α<1/m^2$, the fluctuations get strong and the average LDOS ceases to be representative, which is related to the existence of the Anderson transition at $α_c\sim 1/(m^2\log^2m)$. On the localized side of the transition the spectrum is discrete, and 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 $P_c$ close to unity, $1-P_c\sim 1/\log m$, so that the system undergoes a transition directly from the deeply localized to extended phase. On the side of delocalized states, the peaks in LDOS get broadened, with a width $\sim\exp\{-{const}\log m[(α-α_c)/α_c]^{-1/2}\}$ being exponentially small near the transition point. We discuss 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.