标题： 可扩展的有效模型用于超导纳米结构：双、三和四量子点的应用 显示英文标题

标题： Scalable Effective Models for Superconducting Nanostructures: Applications to Double, Triple, and Quadruple Quantum Dots

摘要： 我们引入了一个通用且可扩展的框架，用于构建由广义超导（SC）安德森杂质模型描述的超导纳米结构的有效模型，该模型包含多个量子点和电极。 我们的链展开（ChE）方法将每个SC电极映射到一个有限的紧束缚链，其参数通过隧穿自能的Padé逼近获得。 我们提供了通用情况的显式算法，以及在宽频带和无限链极限下链参数的简单解析表达式。 这种映射保留了低能物理特性，同时实现了高效的模拟：短链可通过精确对角化处理，而更长的链则可通过密度矩阵重整化群方法处理。 该方法在各种几何结构中保持可靠且计算效率高，无论是在平衡还是非平衡状态下。 我们使用ChE来映射耦合到单个SC电极的双量子点、三量子点和四量子点的基态相图。 虽然填充对称系统显示出相似的整体相图，但随着量子点数量的不同，特定相位有显著差异。 在这里，大量参数区域被广泛使用的零带宽近似完全遗漏，但被ChE捕捉到了。 远离填充时，额外的量子点显著增加了相图的复杂性，产生了丰富的稳定相。 这些结果证明ChE是一种快速、准确且可系统改进的工具，用于探索复杂的SC纳米结构。 显示英文摘要

摘要： 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 Pad\'e 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 with exact diagonalization, and longer ones 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 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.