Abstract
We study an effective model of regular quantum dot clusters coupled to a common superconductor. By applying a canonical transformation, we map the system onto a particle-number-conserving representation, making it directly accessible to standard fermionic neural-network quantum-state variational Monte Carlo methods. We show that the superconducting gap closes at a particular high-symmetry point, which, in finite non-interacting systems, corresponds to crossings between singlet ground states of different character. Combining exact methods, density matrix renormalization group, and neural quantum-state variational Monte Carlo calculations, we identify three distinct interacting regimes: a trivial superconducting singlet phase, a strongly correlated regime connected to an effective Heisenberg model, and a critical intermediate regime with qualitatively different behavior in one and two dimensions. In one-dimensional systems, the intermediate regime exhibits a sequence of singlet-doublet transitions and becomes gapless in the thermodynamic limit even for finite Coulomb interaction. In two-dimensional clusters, we find robust triplet ground states. Furthermore, our results demonstrate that relatively standard fermionic neural quantum states provide an efficient approach for correlated superconducting nanostructures.