# Theory of non-collinear interactions beyond Heisenberg exchange; applications to bcc Fe

## TL;DR

This paper investigates the limitations of the Heisenberg model in describing non-collinear magnetic interactions, showing that non-Heisenberg terms appear generally, but certain exchange parameters in bcc Fe remain effectively Heisenberg-like.

## Contribution

It demonstrates that while non-Heisenberg interactions generally arise in non-collinear spin configurations, specific exchange parameters in bcc Fe can still be accurately modeled by a Heisenberg Hamiltonian.

## Key findings

- Non-Heisenberg terms appear in general non-collinear configurations.
- Nearest neighbor T2g orbital exchange parameters in bcc Fe are Heisenberg-like.
- Certain exchange interactions can be mapped onto a Heisenberg model despite non-collinearity.

## Abstract

We show for a simple non-collinear configuration of the atomistic spins (in particular, where one spin is rotated by a finite angle in a ferromagnetic background) that the pairwise energy variation computed in terms of multiple scattering formalism cannot be fully mapped onto a bilinear Heisen- berg spin model even in the lack of spin-orbit coupling. The non-Heisenberg terms induced by the spin-polarized host appear in leading orders in the expansion of the infinitesimal angle variations. However, an Eg-T2g symmetry analysis based on the orbital decomposition of the exchange param- eters in bcc Fe leads to the conclusion that the nearest neighbor exchange parameters related to the T2g orbitals are essentially Heisenberg-like: they do not depend on the spin configuration, and can in this case be mapped onto a Heisenberg spin model even in extreme non-collinear cases.