Failure of Topological Invariants in Strongly Correlated Matter

We show exactly that standard "invariants" advocated to define topology for noninteracting systems deviate strongly from the Hall conductance whenever the excitation spectrum contains zeros of the single-particle Green's function, G, as in general strongly correlated systems. Namely, we show that if the chemical potential sits atop the valence band, the "invariant" changes without even accessing the conduction band but by simply traversing the band of zeros that might lie between the two bands. Since such a process does not change the many-body ground state, the Hall conductance remains fixed. This disconnect with the Hall conductance arises from the replacement of the Hamiltonian, h(k), with G^{-1} in the current operator, thereby laying plain why perturbative arguments fail.