  We introduce a topological theory to study quasiparticles in interacting
and/or disordered many-body systems, which have a finite lifetime due to
inelastic and/or elastic scattering. The one-body quasiparticle Hamiltonian
includes both the Bloch Hamiltonian of band theory and the self-energy due to
interactions, which is non-Hermitian when quasiparticle lifetime is finite. We
study the topology of non-Hermitian quasiparticle Hamiltonians in momentum
space, whose energy spectrum is complex. The interplay of band structure and
quasiparticle lifetime is found to have remarkable consequences in zero- and
small-gap systems. In particular, we predict the existence of topological
exceptional point and bulk Fermi arc in Dirac materials with two distinct
quasiparticle lifetimes.
