Pole expansion of self-energy and interaction effect for topological insulators

We study the effect of interactions on time-reversal-invariant topological insulators. Their topological indices are expressed by interacting Green's functions. Under the local self-energy approximation, we connect topological index and surface states of an interacting system to an auxiliary noninteracting system, whose Hamiltonian is related to the pole expansions of the local self-energy. This finding greatly simplifies the calculation of interacting topological indices and gives a noninteracting pictorial description of interaction driven topological phase transitions. Our results also bridge studies of the correlated topological insulating materials with the practical dynamical-mean-field-theory calculations.

Figure 1

$Z_2$ index (red line) and gap (blue line) of noninteracting model [Eq. (5)] as a function of $m$. The phase boundaries between different $Z_2$ indices are in accordance with the gap closing points.

Figure 2

Left panel: Paths of the mapping $\omega \mapsto G_{\mathrm{imp}}^{-1}=i\omega -\frac{V^2}{i\omega -P}$ with $V=1$, $P=2$ (blue line), and $P=0.15$ (red line). Gray line indicates the large circle we appended to close the contour. Crosses on real axis indicate the eigenenergy continuum of the noninteracting Hamiltonian, whose lower (upper) bound equals $\Delta$ ($\Gamma$). Topological transition between TI and metal occurs when $\left|P\right|\Delta =V^2$, where the blue line touches the lower bound. The gap reopens for $\left|P\right|\Gamma <V^2$, and results in a normal insulating phase. Right panel: Phase diagram determined by calculating the $Z_2$ index of the pseudo-Hamiltonian (see the PE coefficients in Table ). Shaded and white region have $Z_2=1$ and 0, respectively, the yellow region denotes the metallic state. The phase boundaries $\left|P\right|\Delta =V^2$ and $\left|P\right|\Gamma =V^2$ are in accordance with the FDWN analysis.

Figure 3

Surface spectral functions for self-energy case (a) with $V=1$. Upper panel: $P=2$ is in TI phase. Lower panel: $P=0.15$ is normal insulator. The chemical potential lies at zero.