Chern number of a two-level system

+ 1 like - 0 dislike
873 views

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 $$\mathcal C = \frac{1}{2\pi} \sum_n \int_{\textrm{BZ}} d^2\mathbf k \, \mathcal F_n(\mathbf k),$$ 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 $H(\mathbf k)=d_i(\mathbf k) \sigma_i$:$$\mathcal F(\mathbf k) = \frac{1}{2} \hat{\mathbf d} \cdot \left( \partial_{k_x} \hat{\mathbf d} \times \partial_{k_y} \hat{\mathbf d} \right).$$ 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 $$\int_Y \hat{\mathbf d} \cdot d\mathbf S = 2 \int_X d^2\mathbf k \, \mathcal F(\mathbf k).$$ 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
@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.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOver$\varnothing$lowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.