# Effective hamiltonian for the second-order degenerate perturbation theory

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

• Would you accept a derivation here or do you really need an external reference? – DanielSank Aug 7 '15 at 21:52
• @DanielSank, I would accept the deriviation here, thanks in advance. – SpinningCat Aug 8 '15 at 6:15

## 1 Answer

I found an answer myself and I would like to share it via this answer. The process of arriving to this Hamiltonian is described in details in the following book:

G.L. Bir, G.E. Pikus "Symmetry and strain-induced effects in semiconductors"

The process is described in chapter 15 below the topic "Perturbation theory for the degenerate case"

The approach the authors use is making an infinitesimal basis transformation of the following form: $$\hat{H}_{new}=e^{-\hat{S}}\hat{H}e^{\hat{S}}$$ that reduces the Hamiltonian to block form. They examine not only the perturbation theory of the second order, but also of the third order as well as describing the process of achieving higher orders.

However, I'm not quite sure if it is possible to find the electronic version of this book in English.

• It looks like Schwinger-Wolff transformation, isn't it? – FangXie Jan 21 '19 at 23:54