Majorana modes in quantum dots coupled via a floating superconducting island

Majorana modes can be engineered in arrays where quantum dots (QDs) are coupled via grounded superconductors, effectively realizing an artificial Kitaev chain. Minimal Kitaev chains, composed by two QDs, can host fully localized Majorana modes at discrete points in parameter space, known as Majorana sweet spots. Unlike grounded superconductors, where the superconducting phase is a conserved quantum number, floating islands have a well-defined number of charges. The coexistence of charging effects and Majorana physics allows us to explore novel phenomenology, including teleportation and topological Kondo effect. Here, we extend previous works by theoretically investigating a setup with two QDs coupled via a floating superconducting island. We study the effects of the charging energy of the island and the properties of the resulting minimal Kitaev chain. We initially employ a minimal perturbative model, valid in the weak QD-island coupling regime, to derive analytic expressions for the Majorana sweet spots and the splitting of the ground-state degeneracy as a function of tunable physical parameters. The conclusions from this perturbative approximation are then benchmarked using a microscopic model that explicitly describes the internal degrees of freedom of the island. Our work shows the existence of Majorana sweet spots, even when the island is not tuned at a charge-degeneracy point. In contrast to the Kitaev chains in grounded superconductors, these sweet spots involve a degeneracy between states with a well-defined number of particles and allow us to explore the interplay between Majorana states and charging effects.