I'm currently trying to figure out the way we arrive to the Hamiltonian of a topological insulator. In an article by Xiao-Liang Qi (arXiv: http://arxiv.org/abs/1005.1682) in a process of arriving to the Hamiltonian using $k \cdot p$ - theory he refers to degenerate perturbation theory formalism (in the article it is briefly reviewed in appendix C (from (C18) to (C21) ) ). The author's approach is to deal with an effective Hamiltonian that acts only in the degenerate subspace. This Hamiltonian (up to the terms of the second order) looks the following way: $$H^{eff}_{mm'}=E_m\delta_{mm'}+H'_{mm'}+\frac{1}{2}\sum_{l}H'_{ml}H'_{lm'}\left( \frac{1}{E_m-E_l} + \frac{1}{E_{m'}-E_{l}}\right).$$ Here $m$ is taken from degenerate subspace and $l$ is taken from all other states, exept for those degenerate ones. $H'$ denotes the perturbation to the initial Hamiltonian and $E_j$ stands for unperturbed energies.
The question: I wonder if you could supply me with a reference to a book or a paper where the process of arriving to this effective Hamiltonian is described in details. An answer with the deriviation is also a suitable variant.
Addition: the same effective Hamiltonian can be found on Wikipedia. However, there is no actual reference to the way of obtaining it: http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)#Effective_Hamiltonian
