Bulk boundary correspondence = difference in Chern numbers?

Question

In topological insulators the bulk boundary correspondence is frequently stated as the principle that the number of edge modes equals the difference in Chern numbers at that edge. I found this e.g. in "Topological Band Theory and the Z2 Invariant" by C.L. Kane pp.18 . However in there is no "derivation" to this in the text by C.L. Kane.

Can anyone present to me an argument (maybe even a proof) that the number of edge states corresponds to the change in Chern numbers? A source where I can find such an argument would also be great!

Answer 1

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.

Answer 2

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.

Answer 3

This paper about the mathematical foundation of bulk boundary correspondence may be a good reference:

Emil Prodan, Hermann Schulz-Baldes. Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics. https://arxiv.org/abs/1510.08744

Answer 4

I would formulate the argument as follows:

1. Take a Thouless pump that adiabatically pumps charge, 1 negative energy state by 1 unit cell to the right per cycle. Thus the Chern number of the pump cycle is 1.

2. Because of unitarity, the net result of one cycle must be: 2A) 1 negative energy state must be converted to a positive energy state on the right end 2B) 1 positive energy state must be converted to a negative energy state on the left end 2C) 1 positive energy state must be pumped by 1 unit cell to the left.

3. Now promote time "t" to momentum "ky". You thus obtain a 2-dimensional system, which is infinite along y, but finite along x, with a left edge and a right edge. From 2A), you find that in the bulk gap, there is 1 edge state on the right edge moving up (more precisely, one more moving up than moving down) From 2B), you find an edge state on the left edge, moving down.

This is bulk--boundary correspondence (I just used Chern number 1 for simplicity, but it holds for any Chern number).

We try to spell this out in our lecture notes: https://arxiv.org/abs/1509.02295