Nonequilibrium Transport Through Kondo Quantum Dots: A Real-Time Renormalization Group Analysis

Abstract

Kondo physics first surfaced into the scientific world in 1934 via an unexpected observations made by a group of experimentalists, who measured the temperature-dependent resistance of gold wires in their laboratory at the University of Leiden. When decreasing the temperature below 4 Kelvin, they observed an unexpected increase of the resistance, which resulted in a resistance minimum that was not accounted for by the theoretical understanding at that time. Only 30 years later, Jun Kondo, a Japanese physicist from Tokyo, was the first to explain this curiosity, known as the Kondo effect, under the assistance of the Kondo model. Although his findings explained the increased resistance, a more fundamental problem was raised: the Kondo problem, an unphysical divergence of the resistance at very low temperatures. To solve this problem required the development of more advanced methods beyond the scope of pure perturbative methods, namely the use of renormalization group (RG) methods. More than twenty years after Kondo’s first steps towards the solution of the resistance minimum, the Kondo effect experienced a scientific revival due to the emergence of quantum dot systems in solid state physics. A quantum dot is a very localized and spatially restricted area within a metal, where only a few electrons fit in the area. The limited spatial spread constrains the possible positions for the electrons on the dot, and forces them into strong interactions with each other–they are strongly correlated. In quantum dot systems, not only the Kondo effect in equilibrium situations between the dot and its surrounding reservoirs can be studied, but also the effects of nonequilibrium situations, where a finite bias voltage drives a current through the quantum dot. However, the nonequilibrium Kondo effect is far less explored compared to its equilibrium counterpart and it requires the development of new methods beyond established equilibrium methods. To fill this gap, we use the recently developed real-time renormalization group (RTRG) method in the weak-coupling regime and its extension to the crossover regime, from weak coupling to strong coupling. Furthermore, different combinations between the number of electrons on the dot and the number of reservoirs of electrons in the surrounding metal allow for distinct physical situations, which can be studied within the framework of the RTRG method. We subsequently characterize the properties of the the overscreened, the underscreened, and the fully screened Kondo model by analyzing their physical observables and, in parts, their time evolution. The perturbative foundation of the RTRG method gives us the opportunity to gather new analytic insights for higher spin Kondo systems, on the one hand, and provides for a qualitative comparison of obtained results to various other methods and analytic results, on the other hand.