Effective Josephson Junction Networks and π-Flux Trapping in Single-Crystal Superconductors with Frustrated Magnetic Textures

We demonstrate that anisotropic Josephson couplings can promote the emergence of a network of Josephson junctions within a single-crystal superconductor.  Although structurally homogeneous, the system self-organizes into many coupled superconducting grains and can allow for π-flux trapping for unconventional pairing orders, indicating spontaneous time-reversal symmetry breaking.  Such Josephson couplings can arise, for example, in a Kondo model, in which itinerant electrons are strongly coupled to a frustrated local exchange field [arXiv:2506.15661].  We provide a self-consistent solution for a model on a triangular lattice, demonstrating that a frustrated magnetic texture can give rise to many weakly coupled superconducting grains.  Our results are pertinent to superconducting 4Hb-TaS₂ and reveal new routes to engineer Josephson networks from the interplay of frustrated magnetism and unconventional superconducting order.