RKKY to Kondo crossover in Helical Edge of a Topological Insulator

TL;DR

This paper investigates the crossover between RKKY interaction and Kondo screening in helical edges of topological insulators, using numerical renormalization group methods to analyze how impurity distance, anisotropy, and temperature influence the dominant magnetic interactions.

Contribution

It provides a detailed numerical analysis of the Kondo-RKKY competition on helical edges, including a pedagogical introduction and a clear physical picture of the crossover behavior.

Findings

Kondo screening dominates at large impurity distances.

A crossover from RKKY to Kondo occurs with increasing impurity distance.

Finite bandwidth effects are discussed.

Abstract

Two spatially separated magnetic impurities coupled to itinerant electrons give rise to a dynamically generated exchange (RKKY) inter-impurity interaction that competes with the individual Kondo screening of the impurities. It has been recently shown by Yevtushenko and Yudson (2018), that the RKKY interaction and the RKKY vs. Kondo competition become nontrivial on helical edges of two-dimensional topological insulators where there is lock-in relation between the electron spin and its direction of motion. Kondo screening always takes over and dominates at large inter-impurity distances and it can also dominate all the way to short distances if the Kondo coupling is sufficiently large and anisotropic. In the present paper, we study the Kondo-RKKY competition in detail on a qualitative and quantitative level. For this we employ the numerically exact numerical renormalization group (NRG) for a broad parameter scan of two Kondo coupled impurities vs. magnetic anisotropy, impurity distance and temperature, and comment on the role of finite bandwidth. We give a pedagogical introduction on the the setup of the two-impurity setting within the NRG in the helical context. Overall we establish a plain crossover from RKKY to Kondo with increasing impurity distance which permits an intuitive physical picture by simply comparing length scales set by the Kondo screening cloud, the thermal length scale vs. the impurity distance.