  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.

Mesoscale and Nanoscale Physics;Quantum Gases;Superconductivity