Single-particle Green’s functions and interacting topological insulators

We study topological insulators characterized by the integer topological invariant $\mathbb{Z}$, in even and odd spatial dimensions. These are well understood in the case when there are no interactions. We extend the earlier work on this subject to construct their topological invariants in terms of their Green’s functions. In this form, they can be used even if there are interactions. Specializing to one and two spatial dimensions, we further show that if two topologically distinct topological insulators border each other, the difference of their topological invariants is equal to the difference between the number of zero-energy boundary excitations and the number of zeros of the Green’s function at the boundary. In the absence of interactions Green’s functions have no zeros; thus, there are always edge states at the boundary, as is well known. In the presence of interactions, in principle, Green’s functions could have zeros. In that case, there could be no edge states at the boundary of two topological insulators with different topological invariants. This may provide an alternative explanation to the recent results on one-dimensional interacting topological insulators.

Figure 1

The contour which $\Omega$ is taken around in order to prove that the eigenvalues of the Green’s functions cannot vanish in the upper half plane.

Figure 2

A possible scenario outlining the change in the position of the poles (solid lines) and zeros (dashed lines) as a function of some parameter $\lambda$ as the interaction strength is varied. (a) An energy level crossing zero as a function of $\lambda$ for a noninteracing system. (b) Two zeros and and two nearby poles appear in the determinant of the Green’s function [Eq. (37)] as the interactions are increased. (c) Zeros move closer to the positions of the original poles of the noninteracting problem. (d) Zeros “annihilate” the poles, resulting in a system with exactly the same change of the topological invariant, but without any energy levels crossing zero energy.