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 $H_{\text{T}}\left(\mathbf{k}\right)=-G^{-1}(0,\mathbf{k})$, 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.

Figure 1

Eigenvalues of the topological Hamiltonian at high symmetry $k$-points can be labeled by irreps of the little group of the $k$-point. The gap in the spectral function at $\omega =0$ causes the eigenvalues to be real. Depending on the sign of the eigenvalue, it is either an R-zero or an L-zero. As the parameters in the many-body Hamiltonian are varied, the eigenvalues move on the real axis as long as the gap is maintained. The gap further prevents the eigenvalues from crossing zero. By assumption, we excluded a zero eigenvalue of $G(0,\mathit{k})$, so all eigenvalues of the topological Hamiltonian are finite. If also the Green's function does not break a symmetry while the parameters of the many-body Hamiltonian are varied, the multiplicity of irreps of R-zeros cannot change.

Figure 2

(a) The SSH model with the unit cell, sites, and hoppings shown. (b) Unit cell of the SSH model with Wyckoff positions $1a$, $1b$, and $2c$. The centers of the inversion symmetry $\mathcal{I}$ are located in $1a$ and $1b$. See the text for the description of the model.

Figure 3

Phase diagram obtained from an EBR analysis for the Green's function of the $\text{SSH}+\text{U}$ model. The phases are labeled by the EBR lowest in energy, derived from the inversion eigenvalues (see Table 1). The respective value of $N_1$ is also shown. The Green's function EBRs are independent of the value of the Hubbard interaction $U$, hence yielding the same phase diagram for any value of $U$.

Figure 4

Exact diagonalization results for the topological Hamiltonian as a function of $U$. Each figure corresponds to a representative choice of the hopping parameters $t_1$ and $t_2$ for each topological phase. Labeled by the EBR of the lower band, the phases shown are (a) ${\mathrm{p}}_{1\mathrm{a}}$, (b) ${\mathrm{p}}_{1\mathrm{b}}$. The respective left plot shows the winding around the origin of the Pauli matrix expansion coefficients $q_1\left(k\right)$ and $q_2\left(k\right)$ [see Eq. (15)] when $k$ sweeps the Brillouin zone. The respective right plot shows the eigenvalues of the topological Hamiltonian together with the inversion eigenvalues of the respective eigenstates of the topological Hamiltonian at the high-symmetry $k$-points denoted by $+$ or $-$. The inversion eigenvalues fully determine the irreps. Increasing the Hubbard interaction $U$ enlarges the expansion coefficients $q_1\left(k\right)$ and $q_2\left(k\right)$ without changing the topology.

Figure 5

Exact diagonalization results for the topological Hamiltonian near the line $t_1=t_2$. In each case, $t_1=1$ and $U=1$ are fixed. The respective left figure shows the winding around the origin of the Pauli matrix expansion coefficients $q_1\left(k\right)$ and $q_2\left(k\right)$ [see Eq. (15)] when $k$ sweeps the Brillouin zone. The respective right plot shows the eigenvalues of the topological Hamiltonian together with the inversion eigenvalues of the respective eigenstates of the topological Hamiltonian at the high-symmetry $k$-points denoted by $+$ or $-$. The inversion eigenvalues fully determine the irreps. The results indicate that the transition happens by a swap over infinity of the eigenvalues of the topological Hamiltonian labeled by the inversion eigenvalues.

Figure 6

Illustration of the topological invariant $N_1$ for the (noninteracting) SSH model with $t_1=1$ fixed. The respective left plot shows the winding of the Pauli matrix expansion coefficients $q_1\left(k\right)$ and $q_2\left(k\right)$ around the origin when $k$ sweeps the Brillouin zone [see Eq. (15)]. The respective right plot shows the band structure with inversion eigenvalues of the eigenstates at the high-symmetry $k$-points, denoted by $+$ or $-$. The inversion eigenvalues fully determine the irreps. The value of $N_1$ can already be read from looking at the inversion eigenvalues. In the $N_1=0$ phase, the inversion values within the same band have the same sign, in the $N_1=2$ phase they have opposite signs. In (a), for $t_1>t_2$ one finds $N_1=0$ except for the point of the transition at $t_1=t_2=1$ where the gap closes. In (b), for $t_2>t_1$ one finds $N_1=2$.