Single-particle Green's function zeroes cannot reliably characterize interacting topological phases because they miss excitations in Fock sectors tied to topological degeneracy.
Rosch, Breakdown of Luttinger’s theorem in two-orbital Mott insulators, The European Physical Journal B59, 495 (2007)
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Fundamental limitations of single-particle Green's-function zeroes as probes of many-body topology
Single-particle Green's function zeroes cannot reliably characterize interacting topological phases because they miss excitations in Fock sectors tied to topological degeneracy.