Good articles on quantum doubles

Very nice review on 3-cocycles, history and connection to many topics in math and physics:

Representation Theory of Twisted Group Double
D. Altschulera, A. Costeb, J-M. Maillard,

Click to access Coste_aflb294m159.pdf

An important paper on Finite group modular data:

Click to access 0001158.pdf

Orbifold approach to crystallography

Click to access 2104.09561.pdf

https://www.sciencedirect.com/science/article/pii/S0393044019300075

And the famous 2001 paper by Convay et al. https://arxiv.org/abs/math/9911185

No. 227: notation is $\latex 2^+\colon 2$:

Key word here is “classifying space of space group”. Note that the classifying space of 3d point groups are all known; see the 1999 Conway paper

https://link.springer.com/article/10.1023/A:1015851621002

Absolute majority convention

Berry connection:

A_\mu = i \langle u| \partial_\mu |u\rangle \rightarrow i u^\dag \partial_\mu u

Berry curvature:

F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu

F_{\mu\nu} = -2\text{Im} (\partial_\mu u^\dag \partial_\nu u )

Chern number:

C = \frac{1}{2\pi} \int k_x k_y F_{xy}

Hall conductivity:

\sigma_{xy} = \frac{e^2}{h} C

E.g. for the Qi-Wu-Zhang model

H = \left(\begin{array}{cc} \cos k_x + \cos k_y - m & \sin k_x - i \sin k_y \\ \sin k_x + i \sin k_y & m - \cos k_x - \cos k_y\end{array}\right)

For the lower band we should get

C_{\text{lower}} = \left\{\begin{array}{ll} 0 & m<-2\\ -1 &-2<m<0 \\ 1 & 0<m<2 \\ 0 & m>2\end{array}\right.

If we flip the sign of $m$ in the above Qi-Wu-Zhang model, of course the Chern number will be changed.

In the Fukui-Hatsugai method of numerically computing the Berry curvature and Chern number, we have to use

In the Fukui-Hatsugai paper https://arxiv.org/pdf/cond-mat/0503172.pdf, their definition of the Chern number differs from the above by a minus sign. This can be seen in their first page, where they defined \sigma_{xy} as the negative of the sum of Chern numbers. As a consequence, when using their Eq. (9) to numerically compute the Chern number, one has to add a minus sign to get the conventional defition of Chern number above. (see my chern_insulator.py.)

The reason to care about these convention is that: we really want a filled band with Chern number C=1 to have a counterclockwise edge charge current, so that a right-hand rule is satisfied. One can carefully work out the signs and show this is TRUE.

A few more usual conventions: classical Hall effect: set up is B_z >0 along +z, and E_x>0 along +x which directs current. Then this current is deflected by the magnetic field: either positive or negative charge carriers are deflected towards the small y edge; but the Hall voltage is exactly opposite: V_H = V(\text{large }y)-V(\text{small }y) is negative (positive) for positive (negative) charge carriers. Then, the Hall resistence R_{xy} (not to be confused with Hall coefficient R_H)! is defined as R_{xy} := V_H/E_x, note that by our setup we have E_x>0, so positive (negative) charge carriers gives negative (positive) R_{xy}, hence

\sigma_H =\frac{1}{2} (\sigma_{xy} - \sigma_{yx}) \approx \sigma_{xy} >0 \text{ for negative charge carriers (like electrons)}

Now, restricting to electrons, i.e. negative charge carriers. Again assuming the above setup (i.e. positive magnetic field along z and positive electric field along x). Then, using the skipping orbit picture we see that that electrons will travel clockwisely on the edge, which means that there will be an edge current travelling counterclockwisely along the edge. Therefore, the Chern number convention above is specially designed so that the right-hand rule for current for the negatively charged carriers (i.e. electrons) is satisfied.

Furthermore, by putting the QWZ model on a slab with the above setup (i.e. k_x still good quantum number; edges parallel to y) we will find that the filled band edge states have the correct rule: i.e. the edge dispersion \sin k_x, whose filled part is for $k_x<0$, is localized on the large y edge, so the upper edge has a electron density whose group velocity is positive, i.e. the electron density moves clockwisely, so the electron charge current moves counterclockwisely. This is fully consistent with the skipping orbit picture.