Generic reduction theory for Fermi sea topology in metallic systems

The Fermi sea of a metal can host exotic quantum topology, which governs its conductance quantization and is characterized by the Euler characteristic (${\chi }_F$). In contrast to the well-known band topology, which is determined by the global features of wave functions, the topology of such metallic systems is intrinsically linked to the geometry of the Fermi sea. As a result, probing and identifying ${\chi }_F$ in high-dimensional systems presents a challenge. Here, we propose a generic dimensional reduction theory for the Fermi sea topology in $d$-dimensional metallic systems, showing that ${\chi }_F$ can be determined by the features of so-called reduced critical points on Fermi surfaces. Moreover, we reveal that ${\chi }_F$ can be interpreted as a topological invariant of band topology by mapping a metallic system to a gapped system. Building on this nontrivial result, we identify a broad class of topological superconductors (SCs) whose topological numbers are precisely determined by the ${\chi }_F$ of their normally filled bands. This provides an indirect method to capture ${\chi }_F$ by measuring the (pseudo)spin polarizations of these topological SCs. Our findings are expected to significantly advance research into Fermi sea topology.