3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Ben McKay Jan 17 at 9:05
  • $\begingroup$ @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? $\endgroup$ – Ben Curnow Jan 17 at 15:53
  • $\begingroup$ 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? $\endgroup$ – d_b Jan 17 at 20:46
  • $\begingroup$ 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 $\endgroup$ – Ben Curnow Jan 17 at 20:50

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.