Chiral numerical renormalization group

TL;DR

This paper extends Wilson's numerical renormalization group method to handle systems with unidirectional channels, such as chiral edge modes, by introducing a Wilson ladder, enabling the study of two impurities coupled via these channels.

Contribution

The authors develop a novel Wilson ladder construction to adapt NRG for unidirectional channels, allowing accurate analysis of multi-impurity systems in topological materials.

Findings

Local properties are independent of impurity separation in chiral channels.

The method accurately captures the physics of two impurities coupled to a single chiral channel.

Extensions to more impurities and helical channels are feasible.

Abstract

The interplay between the Kondo screening of quantum impurities (by the electronic channels to which they couple) and the interimpurity RKKY interactions (mediated by the same channels) has been extensively studied. However, the effect of unidirectional channels (e.g., chiral or helical edge modes of 2D topological materials) which greatly restrict the mediated interimpurity interactions, has only more recently come under scrutiny, and it can drastically alter the physics. Here we take Wilson's numerical renormalization group (NRG), the most established numerical method for treating quantum impurity models, and extend it to systems consisting of two impurities coupled at different locations to unidirectional channel(s). This is challenging due to the incompatibility of unidirectionality with one of the main ingredients in NRG -- the mapping of the channel(s) to a Wilson chain -- a tight-binding chain with the impurity at one end and hopping amplitudes which decay exponentially with the distance. We bridge this gap by introducing a "Wilson ladder" consisting of two coupled Wilson chains, and demonstrate that this construction successfully captures the unidirectionality of the channel(s), as well as the distance between the two impurities. We use this mapping in order to study two Kondo impurities coupled to a single chiral channel, showing that all local properties and thermodynamic quantities are indifferent to the interimpurity distance, and correspond to two separate single-impurity models. Extensions to more impurities and/or helical channels are possible.