Thermodynamic perturbation theory for non-interacting quantum particles with application to spin-spin interactions in solids

TL;DR

This paper develops a perturbation theory for non-interacting fermionic systems to calculate the Landau free energy and applies it to describe various spin-spin interactions in solids, including exchange and RKKY interactions.

Contribution

It introduces a higher-order perturbation approach for fermionic systems and explicitly derives formulas for multiple spin interaction mechanisms at finite temperature.

Findings

Derived explicit formulas for exchange interactions up to fourth order.

Unified treatment of RKKY, superexchange, and two-electron exchange interactions.

Applicable to metals, semiconductors, and insulators.

Abstract

The determination of the Landau free energy (the grand thermodynamic potential) by a perturbation theory is advanced to arbitrary order for the specific case of non-interacting fermionic systems perturbed by a one-particle potential. Peculiar features of the formalism are highlighted, and its applicability for bosons is indicated. The results are employed to develop a more explicit approach describing exchange interactions between spins of Anderson's magnetic impurities in metals, semiconductors, and insulators. Within the fourth order our theory provides on the equal footing formulae for the Ruderman-Kittel-Kasuya-Yosida, Bloembergen-Rowland, superexchange, and two-electron exchange integrals at non-zero temperature.