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