Consider a bulk uniform system with a boundary (edge, surface, etc.) and we want to study some transport effects mainly due to some boundary states. One possible experimental scenario might be to attach thin contacts to the top surface of a thick sample.
The bare boundary Green's function is $g=(\omega-h_\mathrm{b})^{-1}$, assuming the Hamiltonian $h_\mathrm{b}$ only for boundary degrees of freedom (e.g., sites in the top layer of a system) is decoupled from the bulk. And one can also derive a full Green's function $G=(g^{-1}-\Sigma)^{-1}$ for the boundary with $\Sigma$ accounting for the coupling to the other parts.
The question is what current operator should one use? $j_1=-\partial g^{-1}/\partial k=\partial h_\mathrm{b}/\partial k$, which is purely from the bare boundary, or $j_2=-\partial G^{-1}/\partial k$, which includes also some bulk electromagnetic coupling effect?
