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

TL;DR

This paper explores how elementary band representations (EBRs) can be used to identify topological phases in interacting insulators through their single-particle Green's functions, extending tools from non-interacting systems.

Contribution

It introduces a method to apply EBRs to the Green's function via the topological Hamiltonian, enabling diagnosis of topological phases in interacting systems.

Findings

EBRs can label bands of the topological Hamiltonian in interacting insulators.

Stability of EBR labeling requires no gap closing, zero in Green's function, or symmetry breaking.

Application demonstrated on the 1D Su-Schrieffer-Heeger model with Hubbard interactions.

Abstract

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 phases is discussed, but remains an open question.