  We discuss the applicability of elementary band representations (EBRs) to
diagnose spatial- and time-reversal-symmetry protected topological phases in
interacting insulators in terms of their single-particle Green's functions. We
do so by considering an auxiliary non-interacting system
$H_{\textrm{T}}(\mathbf{k}) = -G^{-1}(0, \mathbf{k})$, known as the topological
Hamiltonian, whose bands can be labeled by EBRs. This labeling is stable if
neither (i) the gap in the spectral function at zero frequency closes, (ii) the
Green's function has a zero at zero frequency or (iii) the Green's function
breaks a protecting symmetry. We demonstrate the use of EBRs applied to the
Green's function on the one-dimensional Su-Schrieffer-Heeger model with Hubbard
interactions, which we solve by exact diagonalization for a finite number of
unit cells. Finally, the use of EBRs for the Green's function to diagnose
so-called symmetry protected topological (SPT) phases is discussed, but remains
an open question.
