Dear Xiao-Xiao Zhang, To compute the conductivity tensor of systems with PBC you do need periodic velocity operators. However, if you compute the conductivity locally in the bulk, away from the edges you will get a correct result.
You can obtain the local value of the conductivity or any other spectral density, by restricting the extent of the vectors used in the KPM expansion to a finite region of space. The need for periodic velocity operators comes from the fact that those are computed using the distance in real space between sites that are connected with a hopping. When a site at the boundary connects the with the opposite boundary through the translational symmetry, then the distance and velocity has a sudden jump in magnitude, which creates spurious boundary effects. To overcome this issue, I've created a module to construct periodic velocity operators and other utilities in general Bloch Hamiltonians (that is, Kwant systems with all translational symmetries "wrapped-around"). Please check this repository, and the example therein: https://gitlab.com/kpm-tools/bloch Best regards, Pablo
