Astrophysical Quantum Matter Revisited: Flat-Band Topological States on a Zero-Flux Dipole Sphere

Abstract: We study strongly correlated fractional topological phases on a two-sphere threaded by a magnetic dipole field with globally vanishing flux. Solving the Dirac equation in this background produces spheroidal wavefunctions forming a highly degenerate manifold of normalizable zero modes, with degeneracy proportional to the total absolute flux. We introduce a non-Abelian spin gauge field near the equator to hybridize the north and south domain-confined modes, forming a global flat band. Projecting interactions into this band yields Laughlin-type correlated states. The entanglement spectrum shows a chiral tower consistent with a virtual edge, demonstrating bulk-edge correspondence in a closed geometry. This generalizes the zero-flux flat-band construction of \cite{Parhizkar:2024som} to curved backgrounds, with potential applications to synthetic and astrophysical systems.