Topological invariants for interacting topological insulators with inversion symmetry

For interacting $Z_2$ topological insulators with inversion symmetry, we propose a simple topological invariant expressed in terms of the parity eigenvalues of the interacting Green's function at time-reversal invariant momenta and zero frequency. We derive this result from our previous formula involving the integral over the frequency-momenta space. This formula greatly simplifies the explicit calculation of $Z_2$ topological invariants in inversion-symmetric insulators with strong interactions.

Figure 1

Topological invariance of R/L-curves on the ${\mu }_{\alpha }$ complex plane. (a) $A$ and $B$ is L-zero and R-zero, respectively, and the corresponding curve is an L/R-curve, respectively. (b) Compactified complex plane, on which an R-curve $l$ is shown. Here the only crossing point of $l$ with real axis is $D$. The point $\infty$ is not reached by $l$, although $l$ approaches $\infty$ as $\omega \to \pm \infty$. (c) The R-curve $l_1$ can be deformed into another R-curve $l_2$ following the path indicated by the arrow; however, it cannot be deformed into an L-curve if 0 and $\infty$ are avoided, i.e., the topological class is preserved during smooth deformations.

Figure 2

Deformation of integral contours. The $p$ axis refers to $p=(p_0,p_1,p_2,p_2,p_4)$. The two contours in (a) correspond to $\lambda >0$ and $\lambda <0$, respectively. The opposite orientations come from the negative sign in $-W(G,\lambda =0^{-})$. We can deform these two contours to those illustrated in (b), which does not change $W\left(G\right)$ due to topological invariance. In the calculation of the jump of $W\left(G\right)$, the “dashed lines”(which are in fact 5d manifolds) in (b) cancel each other, and we are left with the “solid circle”, which is the 5d sphere $S^5$.