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Chern number of a two-level system

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The bulk of my question relates to a two-level system, but I have some questions about the Chern number in general as well.

The Chern number of a gapped periodic system (free fermions or mean field) is given by \begin{equation} \mathcal C = \frac{1}{2\pi} \sum_n \int_{\textrm{BZ}} d^2\mathbf k \, \mathcal F_n(\mathbf k), \end{equation} where the sum runs over the occupied bands. Here $\mathcal F_n(\mathbf k)$ is the Berry curvature of the $n$-th Bloch band which can be calculated from the corresponding Bloch eigenstates.

  • Why is the Chern number only defined for gapped systems? Or is it just that it is an integer only for gapped systems?
  • Why is the Chern number of a gapped continuum model a half integer? Or does this only apply to massive Dirac fermions?

As an example, consider the Berry curvature of the ground state of a two-level system with Bloch Hamiltonian $H(\mathbf k)=d_i(\mathbf k) \sigma_i$:\begin{equation} \mathcal F(\mathbf k) = \frac{1}{2} \hat{\mathbf d} \cdot \left( \partial_{k_x} \hat{\mathbf d} \times \partial_{k_y} \hat{\mathbf d} \right). \end{equation} Up to a constant, this is the Jacobian of the map $X \rightarrow Y : \mathbf k \mapsto \hat{\mathbf d}(\mathbf k)$ which is a parameterization of the unit 2-sphere. Therefore, the flux through the image $Y$ in $S^2$ is given by \begin{equation} \int_Y \hat{\mathbf d} \cdot d\mathbf S = 2 \int_X d^2\mathbf k \, \mathcal F(\mathbf k). \end{equation} If the mapping covers the sphere exactly ones, i.e. it wraps around the sphere one time, then the integral equals $4\pi$ and therefore the Chern number (or wrapping number) is one.

  • Is this correct? Also, the unit vector $\hat{\mathbf d}(\mathbf k)$ is not defined if the gap closes ($\frac{0}{0}$). But this is just a single point, why does the integral make no sense in this case?

  • Why does the mapping wrap around the 2-sphere an integer amount of times if the domain $X$ is compact (e.g. Brillouin zone torus)? Or is this statement incorrect?

What if a band is only partially filled? In this case, the domain of the mapping is compact (or not?), but the integral is not quantized?

Thanks!

This post imported from StackExchange Physics at 2015-07-15 14:47 (UTC), posted by SE-user Praan
asked Jul 10, 2015 in Theoretical Physics by Praan (5 points) [ no revision ]
Typically we prefer one question per post on this site, though sometimes two closely related questions can be asked in one post. Whether these questions are sufficiently distinct to warrant separate questions is beyond my ken, hopefully someone could enlighten us on whether they should be posted separately or kept here. Either way, Welcome to Physics!

This post imported from StackExchange Physics at 2015-07-15 14:47 (UTC), posted by SE-user Kyle Kanos
One way to see that the integral does not make sense with gapless points is that one needs to regularize it near the point, and the Chern number depends on how you regularize it. Physically this is just saying that the system is sitting at a critical point between phases with different Chern numbers.

This post imported from StackExchange Physics at 2015-07-15 14:47 (UTC), posted by SE-user Meng Cheng
@Meng Cheng What do you mean with "regularize"? This is a word that pops up everywhere in physics and seems to have different meanings depending on the context. The integrand becomes singular at the $k$-value where the gap closes. In numerical calculations (at least for the models I have tried) one finds the average of the Chern numbers at the critical point.

This post imported from StackExchange Physics at 2015-07-15 14:47 (UTC), posted by SE-user Praan

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