Scalable effective models for superconducting nanostructures: Applications to double, triple, and quadruple quantum dots

We introduce a versatile and scalable framework for constructing effective models of superconducting (SC) nanostructures described by the generalized SC Anderson impurity model with multiple quantum dots and leads. Our Chain Expansion (ChE) method maps each SC lead onto a finite tight-binding chain with parameters obtained from \emph{Padé} approximants of the tunneling self-energy. We provide an explicit algorithm for the general case as well as simple analytical expressions for the chain parameters in the wide-band and infinite-chain limits. This mapping preserves low-energy physics while enabling efficient simulations: short chains are tractable using exact diagonalization, and longer ones are handled with density matrix renormalization group methods. The approach remains reliable and computationally efficient across diverse geometries, both in and out of equilibrium. We use ChE to map the ground-state phase diagrams of double, triple, and quadruple quantum dots coupled to a single SC lead. While half-filled symmetric systems show similar overall diagrams, the particular phases differ substantially with the dot number. Here, large parameter regions are entirely missed by the widely used zero-bandwidth approximation but are captured by ChE. Away from half-filling, additional dots markedly increase diagram complexity, producing a rich variety of stable phases. These results demonstrate ChE as a fast, accurate, and systematically improvable tool for exploring complex SC nanostructures.