Elementary band representations for the single-particle Green's function of interacting topological insulators

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 noninteracting system Hₓ (k) =-G^-1 (0, k), known as the topological Hamiltonian, whose bands can be labeled by EBRs. This labeling is robust 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 phases is discussed, but remains an open question.