Helical Domain-Wall-Ring Networks Reshape Superconducting Correlations

Abstract

Extended domain-wall networks that emerge in moir\'e materials provide a distinct platform for quasi-one-dimensional electronic states. However, the interaction-driven orders in confined networks remain largely unexplored. Here, we discuss superconducting (SC) correlations in interacting helical domain-wall-ring networks in the closed topological domains formed within the moir\'e patterns of an underlying twisted bilayer honeycomb lattice. We first analyze the system within the framework of an infinite-size theory and show that inter-ring SC-pair tunneling is renormalization-group relevant and thus enhances SC correlations through inter-ring phase locking. To address finite-size effects resulting from the ring-network geometry, we present a self-consistent variational approach. Our analysis shows that even in the regime where the infinite-size theory predicts strongly coupled pair tunneling, the induced phase-locking scale remains strongly suppressed. In contrast, the SC scaling dimension continues to decrease with stronger inter-ring density-density interaction and a decreasing twist angle, while remaining insensitive to the pair-tunneling strength. This discrepancy demonstrates that ring networks do not simply approach their infinite-size counterparts but can exhibit qualitatively distinct collective behavior. Our study highlights how the interplay of confinement effects and ring-network geometry can reshape SC correlations.