Our method is based on the combination of a Wilson chain construction for the leads and a mean-field BCS description for the superconducting scatterers. We propose a study of their transport based on the analysis of the currents after quantum quenches connecting such devices with external leads. Here we present a matrix product state approach to simulate the real-time dynamics of these systems. The study of their transport properties is non-trivial and some of their non-perturbative aspects are hard to capture with the most ordinary techniques. Superconducting devices subject to strong charging energy interactions and Coulomb blockade are one of the key elements for the development of nanoelectronics and constitute common building blocks of quantum computation platforms and topological superconducting setups. Extensions to more impurities and/or helical channels are possible. We use this mapping in order to study two Kondo impurities coupled to a single chiral channel, demonstrating that all local properties and thermodynamic quantities are indifferent to the impurity distance, and correspond to two separate single-impurity models.
We bridge this gap by introducing a "Wilson ladder" consisting of two coupled Wilson chains, and demonstrate that this construction successfully captures the uni-directionality of the channel(s), as well as the distance between the two impurities. This is challenging due to the incompatibility of uni-directionality 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. 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 uni-directional channel(s). However, the effect of uni-directional channels (e.g., chiral or helical edge modes of 2D topological materials) which greatly restrict the mediated inter-impurity interactions, has only more recently come under scrutiny, and it can drastically alter the physics. The interplay between the Kondo-screening of quantum impurities (by the electronic channels to which they couple) and the inter-impurity RKKY interactions (mediated by the same channels) has been extensively studied. The 4CK critical point was located using calculations in which Ns = 65000 states were. Results with z = 0 and 0.5 were averaged for 3CK and 4CK to reduce discretization artifacts.
1 were used, and ρ1J1 = 0.075 with ρ2J2 ≈ ρ1J1 tuned to the MCK criti- cal point in each case.
We stress that the fixed points and crossovers are described accurately, as is the universal critical physics which depends only on proximity to the critical point Comparison of entropy Simp(T ) vs T /D1 for 1CK, 2CK, 3CK and 4CK models, calculated using inter- leaved NRG. The increase in d RMS (T ) around T * FL is simply due to the fact that the relevant perturbation generating the crossover grows under RG at slightly different rates for slightly different T * FL. This is indeed found to be the case, with the characteristic error for Λ = 3, 5 and 7 being ∼ 10 −4, 10 −5 and 10 −6 respectively. Since the truncation is justified by energy-scale separation controlled by Λ, one naturally expects the er- ror to decrease as Λ increases.
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