Ground State and Excitations of Quantum Dots with "Magnetic Impurities"

TL;DR

This paper investigates the many-body spectrum of a quantum impurity coupled to a finite electron reservoir, revealing how spectral features depend on coupling strength, impurity nature, and system parity, with implications for experimental detection.

Contribution

It provides a detailed analysis of the finite-size spectrum of quantum dots with magnetic impurities, including strong and weak coupling limits, and explores observable effects in tunneling spectroscopy and magnetic susceptibility.

Findings

Spectrum evolution reflects many-body correlations.

Finite-size effects alter magnetic susceptibility from bulk behavior.

Scaling functions are derived for different system parities and ensembles.

Abstract

We consider an "impurity" with a spin degree of freedom coupled to a finite reservoir of non-interacting electrons, a system which may be realized by either a true impurity in a metallic nano-particle or a small quantum dot coupled to a large one. We show how the physics of such a spin impurity is revealed in the many-body spectrum of the entire finite-size system; in particular, the evolution of the spectrum with the strength of the impurity-reservoir coupling reflects the fundamental many-body correlations present. Explicit calculation in the strong and weak coupling limits shows that the spectrum and its evolution are sensitive to the nature of the impurity and the parity of electrons in the reservoir. The effect of the finite size spectrum on two experimental observables is considered. First, we propose an experimental setup in which the spectrum may be conveniently measured using tunneling spectroscopy. A rate equation calculation of the differential conductance suggests how the many-body spectral features may be observed. Second, the finite-temperature magnetic susceptibility is presented, both the impurity susceptibility and the local susceptibility. Extensive quantum Monte-Carlo calculations show that the local susceptibility deviates from its bulk scaling form. Nevertheless, for special assumptions about the reservoir -- the "clean Kondo box" model -- we demonstrate that finite-size scaling is recovered. Explicit numerical evaluations of these scaling functions are given, both for even and odd parity and for the canonical and grand-canonical ensembles.