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

Alexander D. Mirlin; Yan V. Fyodorov

doi:10.1103/PhysRevB.56.13393

paper

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

arXiv:cond-mat/9702120 · doi:10.1103/PhysRevB.56.13393

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 $α_{c} \sim 1/ (m^{2} lo g^{2} m)$ . 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, $1 - P_{c} \sim 1/ lo g m$ , 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 $\sim exp {- co n s t lo g m [(α - α_{c}) / α_{c}]^{- 1/2}}$ 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.

12 pages, 1 figure. Misprints in eqs.(21) and (28) corrected, section VII added. Accepted for publication in Phys. Rev. B

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

Abstract

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