Double quantum dot Andreev molecules: Phase diagrams and critical evaluation of effective models

We investigate the phase diagram of a parallel double-quantum-dot Andreev molecule, where the two quantum dots are coupled to a common superconducting lead. Using the numerical renormalization group method, we study the ground state evolution over a broad parameter range, examining variations in level detuning, superconducting gap size, coupling to the lead, and interdot coupling strength. The resulting phase diagrams reveal a range of ground states, including singlet, doublet, and a notably rare triplet state, the latter highlighting significant lead-mediated interactions between the quantum dots. We assess the accuracy of simplified effective models, such as the atomic limit and zero-bandwidth approximations, in replicating the complex behavior of this setup. Our findings point to substantial limitations in these models, emphasizing the need for caution when applying them beyond their established scope. Notably, all but the extended zero-bandwidth approximation failed to capture the triplet ground state and led to incorrect predictions. This study offers valuable insights for interpreting experimental data and for the development of superconducting quantum-dot devices.