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

Abstract

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.

Figures (31)

• Figure 1: Illustration of $N_d$ QDs coupled to $N_l$ SC leads. Here, only nearest neighbor QDs interact with each other through hopping $t_d$ and capacitance coupling $W$. Dots' interactions with leads are described by parameters $V_{ij\mathbf{k}}$.
• Figure 2: Mapping of an SC lead onto a finite tight-binding chain with site-dependent hopping amplitudes $\widetilde{h}_\ell$ and local SC pairing. In the derivation of the ChE models, it is convenient to introduce the substitutions $\gamma=\sqrt{\Gamma\Delta h_0}$ and $\widetilde{h}_\ell=\sqrt{h_\ell}\Delta$ with dimensionless coefficients $h_\ell$.
• Figure 3: Square roots of first thirteen coefficients ${h_\ell}$ for ChE models with chain lengths ranging for (from left to right) $L = 2$, $3$, $4$, $6$, $10$, $14$, $20$, and $22$. Blue squares represent the ChE($L$) model in WBL, orange circles correspond to ChE($L$, $D_{\Delta} = 100$), and green crosses to ChE($L$, $D_\Delta = 10$). Lines are included as guides to the eye. Note that we plot $\sqrt{h_\ell}$ as this is proportional to the hopping terms in the chain. Coefficients for $\ell>12$ are practically aligned for all three cases.
• Figure 4: Comparison of the approximate function ${\cal P}_L(\omega_n,{h_\ell})$ with the full hybridization function $\widetilde{\Gamma}(\omega_n;D)$ for $\Delta = 1$. (a) WBL, where the coefficients ${h_\ell}$ take a simple analytic form \ref{['eq:coefWBL']}. (b) Finite-bandwidth case with $D_\Delta = 10$, where ${h_\ell}$ are obtained using the method described in the main text and Appendix \ref{['app:CF']}, based on matching the Padé approximant to the full tunneling self-energy. (c) Truncated expansion of the infinite-chain model, where the last coefficient is evaluated using Eq. \ref{['eq:trancation']}, with $D_\Delta = 10$. See also Fig. \ref{['fig:rozvoj_app']}.
• Figure 5: Illustration of systems used to benchmark the ChE models against previous results. (a) Single QD coupled to one lead. (b) Junction bridged by a QD. (c) Single QD coupled to three leads. (d) DQD in serial configuration. (e) DQD in parallel configuration.