I have a simple question regarding sign conventions pertaining to the Chern number and Hall conductance (and what seems to be inconsistencies in the literature).

In a 2D band insulator, the Chern number $C$ is given by \begin{align} C = \frac{1}{2\pi } \int_{BZ} d^2 k\,\, f_{12}(k) \end{align} where $f_{12}(k)$ is the Berry curvature and is given by $f_{12}(k) = \epsilon_{ij} \partial_i a_j(k)$. Here $a_j(k)$ is the Berry connection.

Now, some papers in the literature define the Berry connection as \begin{align} a^-_j (k) = -i \langle \psi_k | \partial_{k_j} | \psi_k \rangle \end{align} (note the minus sign). But some papers define it as \begin{align} a^+_j (k) = i \langle \psi_k | \partial_{k_j} | \psi_k \rangle . \end{align}

The Hall conductance and Chern number are related via $\sigma_{xy} = C \frac{e^2}{h}$. My issue is that papers state that $\sigma_{xy} = C \frac{e^2}{h}$ regardless of the sign convention they use for the Berry connection.

My question is which convention for the sign of the Berry connection should be chosen to ensure $\sigma_{xy} = C \frac{e^2}{h}$?

I've calculated the Hall conductance using simple perturbation theory and I'm pretty sure the correct choice is $a^-_j(k)$. The paper by Qi, Hughes, and Zhang (arXiv:0802.3537) uses the convention $a_j^-(k)$ but I haven't been able to get through the full path integral derivation of the Hall conductance yet.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.