Frequency domain winding number and interaction effect on topological insulators

We study the effect of interactions on the time-reversal invariant topological insulators in four and three spatial dimensions. Their topological indices are expressed by the interacting Green's functions. Under the local self-energy approximation, we find that interaction could induce nontrivial frequency-domain winding numbers and change the topological classes of the system. Our results suggest that the topological phases could be destroyed without developing long-range orders. Practical issues on the accurate frequency-momentum integration combined with dynamical-mean-field theory and diagrammatic calculations of the interacting Green's functions are also addressed.

Figure 1

(a)–(c) The mapping $\omega \mapsto G_{\mathrm{imp}}^{-1}\left(\omega \right)$ appended with appropriate large circles defines a contour in the complex plane. The result of the contour integration gives FDWN. Crosses denote the singular points of the integrand $\frac{1}{{(z^2-h_{\mathbf{k}}^2)}^3}$. Double lines mean the contour wraps around twice. (d)–(g) The imaginary part of the PHS: (d) $G_{\mathrm{imp}}^{-1}=$ $i\omega$, (e) $i\omega -\frac{1}{i\omega }$, (f) $i\omega -\frac{1}{{\left(i\omega \right)}^3}$, and (g) $i\omega -\frac{1}{i\omega -1}-\frac{1}{i\omega +1}$. They have FDWNs of 1, 0, 2, and 1 respectively. The result can be read from the number of right- and left-moving intersections of the curve with a horizontal line: $\gamma$ equals the difference between them. One can also get the results from the complex plane contour integrations with contours drawn in (a), (b), (c), and(a), respectively.