Dual Haldane sphere and quantized band geometry in chiral multifold fermions

We show that the chiral multifold fermions present a dual Haldane sphere problem in momentum space. Owing to Berry monopole at degenerate point, Landau level emerges trace of quantum metric, with which quantized geometric invariant is defined through surface integration. further demonstrate potential manifestations measurable, physical observables. With lower bound derived for finite spread Wannier functions, anomalous phase coherence identified accordingly flat band superconductivity. briefly comment on stability these results under perturbations. Potential experimental probes metric are also discussed.