Quantum Fisher information (QFI) quantifies the infinitesimal distance between two quantum states and characterizes the ultimate precision limit for the estimation of a parameter. While critical enhancement of QFI is well understood in gapped systems, its behavior in gapless topological phases remains less explored. In this work, we establish a general scaling criterion governing the emergence of singular many-body QFI in gapless fermionic systems. We show that, although single-particle QFI generically diverges near band-closing points when a control parameter shifts the position of the node, whether this divergence survives at the many-body level is determined by the codimension of the gapless manifold. Specifically, we demonstrate that in three-dimensional systems, the many-body QFI density diverges only when the codimension is less than or equal to two. This result leads to a sharp metrological distinction between Weyl semimetals and nodal-line semimetals. Furthermore, we construct explicit measurement protocols that saturate the quantum Cramér-Rao bound, showing that the predicted enhancement is, in principle, accessible via mode-resolved measurements. These results establish a direct connection among band topology, quantum geometry, and quantum metrology.

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• Geometric & topological phases
• Quantum metrology