# Chern number of projection-Topological magic in physics

I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $$-1$$) and the full paper can be found here click me. The Landau level is the first eigenstate of a Schrödinger operator with magnetic field.

The nice physics behind this computation is that it shows that the quantum Hall effect (which is the transverse current created as a response to an external current in a conductor that is placed in a magnetic field) is quantized and given in terms of a topological quantity, the Chern number.

Yet, I have difficulties understanding what happened in this computation.

The projection is given by the integral kernel

$$\Pi(x,y) = \frac{qB}{h} e^{-(qB/4\hbar)(x-y)^2-i(qB/2\hbar)x\wedge y}.$$

The authors compute a "derivative" of this expression and get an integral expression for the Chern character and claim it is equal to $$-1$$. It does not seem like a very sophisticated computation, but it is just very hard to understand so I would like to ask here about this since it is very likely that I am missing something.

• It seems like you are asking for someone to write an exposition of that paper. I think that an exposition would be too long for the sort of short and specific answers that this web site was designed to encourage. – Ben McKay Jan 17 at 9:05
• @BenMcKay Mhmm- I think my question: How is the Chern number of this Landau level computed refers to roughly half a page of this paper and I would like to ask for more details on that. Why do you say I ask for an exposition of the paper? – Ben Curnow Jan 17 at 15:53
• It's tough to tell from your question where your sticking point is. Maybe you could give some more details about which parts you're having trouble understanding? – d_b Jan 17 at 20:46
• my question is just: Given the Landau Hamiltonian, how do they end up with this integral expression for the chern number. So I wonder particularly about this integral – Ben Curnow Jan 17 at 20:50