Spin-1/2 molecules on superconductors represent a promising platform for advanced quantum devices. Recent experiments have shown that the ground-state phase and subgap states of molecular-superconductor hybrids, such as TBTAP [1] can be effectively tuned. A quantum phase transition can be induced by changing the distance between the STM tip and the molecule, or by adding another molecule and varying their mutual positions. These processes are faithfully described by the Superconducting Impurity Anderson Model (SCIAM) [2], which accommodates one or several impurities connected to one or multiple leads [3]. We investigate the phase diagrams and the evolution of subgap states for one, two, and three molecules coupled to the same lead. To achieve this goal, we have derived a Chain Expansion (ChE) method [4] which maps superconducting leads into finite chains. We show that ChE-based effective models closely match the Numerical Renormalization Group (NRG) solutions of the full SCIAM across a broad parameter range, already for short chains solvable via Exact Diagonalization (ED). In more challenging regimes, the agreement between NRG and ChE calculations systematically improves with increasing chain length. The one-dimensional nature of ChE enables the usage of effective models with longer chain, inaccessible to ED, via the Density Matrix Renormalization Group.  Interestingly, simpler systems, such as single quantum dot on a superconductor, require longer chains for converged ground-state expectation values in certain experimentally relevant regimes. Conversely, for more complex configurations, such as three coupled dots (trimers), shorter chains often suffice even there. Our findings demonstrate that ChE is a powerful tool for studying intricate superconducting systems, including those inaccessible to NRG, thus enabling investigations of superconducting structures relevant for quantum technologies and fundamental physics.

 

 

[1] C.Li, V. Pokorný, M. Žonda, et. al., Individual Assembly of Radical Molecules on Superconductors: Demonstrating Quantum Spin Behavior and Bistable Charge Rearrangement, ACS Nano 19 (3), (2025)

[2] V. Meden, The Anderson-Josephson quantum dot- a theory perspective, J. Phys. Condens. Matter 31, 163001 (2019)

[3] P. Zalom, M. Žonda, T. Novotný, Hidden Symmetry in Interacting-Quantum-Dot Based Multiterminal Josephson Junctions, Phys. Rev. Lett.132, 126505(2024)

[4] D. Bobok, L. Frk, V. Pokorný, and M. Žonda, Scalable effective models for superconducting nanostructures: Applications to double, triple, and quadruple quantum dots, Phys. Rev. B 112, 205418 (2025)