Bulk boundary correspondence = difference in Chern numbers?

There seems to be no general but simple derivation to this fact. However there are some papers concerning the issue that involve mathematical techniques too advanced for me still. The works that I found are

• Bulk-Edge Correspondence for Chern Topological Phases: A Viewpoint from a Generalized Index Theorem. T. Fukui, K. Shiozaki, T. Fujiwara, S. Fujimoto. J. Phys. Soc. Jpn. 81 114602, (2012); arXiv:1206.4410.

• Edge states and the bulk-boundary correspondence in Dirac Hamiltonians. Roger S. K. Mong, Vasudha Shivamoggi. Phys. Rev. B 83, 125109 (2011); arXiv link

I also found some lecture notes on the topic also giving some more detailed background information at arXiv:1501.02874. Maybe it helps somebody.

There is this paper of Fidkowski, Jackson, and Klich. They have the simplest description I've found so far. The idea is that no matter the model, you can form a spectrum-flattened version

$$H = 1 - 2P,$$

where $P$ projects onto the filled states. Then, an edge may be added by forming the Hamiltonian

$$H' = 1 - P - PVP,$$

where $V$ is a boundary profile function, which is $-1$ for $x < - \Delta$, $+1$ for $x > \Delta$, and $X/\Delta$ in some width $\Delta$. Thus, for $x \ll 0$ it is $H$ and for $x \gg 0$ it is the trivial Hamiltonian $1$.

The point that relates this all to the Chern number is that

$$PXP$$

is the Berry connection, so the Berry holonomy (in the direction perpendicular to the boundary) contributes a shift to the spectrum in the presence of a boundary.

In particular, in 2d, as we move around the Brillouin zone in the parallel direction, the holonomy winds $n$ times, where $n$ is the Chern number. It's easy to show the spectrum of this simple system is $m + nk_{||}/2\pi$ where $m \in \mathbb{Z}$ and $k_{||} \in [0,2\pi)$. Thus, at any value of the chemical potential, the bands cross zero energy $n$ times in the same direction, meaning there are $n$ modes with chiral $k_{||}$. One can identify these with the edge modes because the original $H$ has integer spectrum.

NOTE: I don't know if anyone showed that you can continuously connect the regimes where the domain wall looks like a step function (and the edge mode profile is like $e^{-x}$) to where the domain wall is a linear profile (and the edge mode is like $e^{-x^2}$). If somebody did this rigorously I would be very happy to look at it.