Bernevig & Hughes' book "Topological Insulators and Topological Superconductors" have a derivation of the TKNN invariant in terms of the finite-temperature Green's function. From the correlation function
$$R_{ij}(q,\,\tau-\tau')=\langle T[j_i(q,\,\tau),\,j_j(-q,\,\tau')]\rangle$$
where $j$ is the current and $\tau$ is imaginary time, the authors are able to represent the above in terms of the Matsubara Green's function, and from there perform a flat-band analysis which allows them to write the Green's functions in terms of projectors in the occupied and empty bands.
This derivation seems to be a rigorous linear response approach to the TKNN derivation, but I would like to read other sources which discuss this projection method they use at greater length. Are there any books, papers, articles, etc. which approach the TKNN derivation in a similar fashion? If not the TKNN proper, then any resources that discuss the derivation of the retarded correlation function in such a fashion?