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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.

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