$Z_2$ invariant and Wannier states switching partner

Question

I have been reading about $Z_2$ topological invariant recently. However, after some literature survey, I still cannot understand $Z_2$ invariant in language of time reversal polarization.

Basically, my struggle includes the following two questions:

1. As the ref paper says(see the picture below): On the edge, however, partner switching results in the occupied Wannier orbital at t = 0 switching partners and having as a partner at t = T/2, a formerly unoccupied Wannier orbital. If we consider a two band system for simplicity(one occupied valence band and one empty conduction band), all wannier states of valence band are occupied and where is this formerly unoccupied Wannier orbital? Is this a Wannier orbital from conduction band?

2. Still, why such a partner switching will result in an edge state connecting valence and conduction band in quantum spin hall effect(with $k_y$ acting as $t$)?

(ref http://link.aps.org/doi/10.1103/PhysRevB.74.195312)

Answer 1

I am new to this and just happened to have similar questions. Here are some of my thoughts -- not sure they are correct and I would certainly welcome some discussion.

I think it is important to remember what is drawn in your attached figure are Wannier orbitals rather than band states.

To answer your first question -- it is not shown. The figure you attached only shows the occupied $2N$ Wannier orbitals. Take a look at the left edge Wannier orbital at $t=T/2$. It appears to not have a TR partner. However, TR dictates that it must have one, then it must come from a formerly unoccupied orbital. Note that "is this a Wannier orbital from conduction band?" in your question is inaccurate as Wannier orbitals are defined in real space and band is defined in momentum ($k_x$) space. I think the more accurate answer is that it comes from a formally unoccupied edge Wannier orbital.

Regarding your second question, if the pumping cycle keeps going on, the switching keeps going on too. An edge Wannier orbital occupant can go all the way to occupy the highest energy Wannier orbital. In $k_y$ language, this is just an edge energy band traversing the bulk gap. See the figure in the Fu-Kane 2007 paper (or page 137 of Bernevig's book).