Universal Quantized Berry-Dipole Flat Bands

Perfectly flat bands with nontrivial quantum geometry have emerged as a frontier for exotic topological phenomena and superconductors. Here, we unveil a universal family of quantized Berry-dipole flat bands in chiral-symmetric (2n+1)-band systems, where the central perfectly flat band carries a Berry-dipole moment d=n, with n an arbitrary integer, while preserving zero Chern number. We construct explicit lattice models to showcase three topological phenomena characterized by the Berry-dipole moment: a flat-band returning pump featuring bidirectional, soliton-like displacement of Wannier centers by exactly n unit cells per half cycle, a dipolar Haldane phase diagram arising from the competition between time-reversal and parity symmetries, and n pairs of bulk helical zero modes whose existence depends on the orientation of pseudomagnetic field. Our findings establish a universal framework for the topology beyond Chern class in perfectly flat bands and provide a tunable platform for exploring quantum geometry and interaction-driven phases.