Loewdin calculus for multiband Hamiltonians

TL;DR

This paper details the derivation and numerical implementation of the Loewdin calculus for multiband Hamiltonians, incorporating p-d and s-d exchange couplings, and explains the Bogoliubov transform for systems with inversion asymmetry.

Contribution

It provides a comprehensive derivation and implementation guide for the Loewdin calculus in multiband Hamiltonians with exchange interactions and inversion asymmetry.

Findings

Explicit formulas for the Loewdin transformation in multiband systems.

Methodology for implementing sp-d exchange in tight-binding models.

Complete procedure for the Bogoliubov transform in asymmetric systems.

Abstract

This appendix to the paper by Werpachowska and Dietl "Theory of spin waves in ferromagnetic (Ga,Mn)As" (DOI: 10.1103/PhysRevB.82.085204) is a response to inquiries about the derivation of the Loewdin calculus and its numerical implementation. It presents the detailed version of the Loewdin calculus for the multiband Hamiltonian taking into account both the p-d and s-d exchange couplings. This should explain how to implement the sp-d exchange interaction in the tight-binding computational scheme. I also include the complete procedure of the Bogoliubov transform for systems with spatial inversion asymmetry.