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