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We present a theory unifying the topological responses and anomalies ofvarious gapless fermion systems exhibiting Fermi surfaces, including those withBerry phases, and nodal structures, which applies beyond non-interacting limit.As our key finding, we obtain a general approach to directly relate gaplessfermions and topological insulators in phase space, including first- andhigher-order insulators. Using this relation we show that the low-energyproperties and response theories for gapless fermionic systems can be directlyobtained without resorting to microscopic details. Our results provide aunified framework for describing such systems using well-developed theoriesfrom the study of topological phases of matter.

Gapless Fermionic Systems as Phase-space Topological Insulators: Non-perturbative Results from Anomalies