# Geometrical contributions to the exchange constants: Free electrons with spin-orbit interaction

## TL;DR

This paper derives a new expression for exchange constants incorporating geometrical band structure effects, highlighting their significance especially with spin-orbit interaction, and validates the approach with the Rashba model.

## Contribution

It introduces a formalism linking exchange constants to geometrical properties of band structures and demonstrates its application with free electrons and spin-orbit interaction.

## Key findings

- Geometrical contributions significantly affect exchange constants.
- The formalism aligns with gauge-field approaches in the Rashba model.
- Differences are observed between helical and cycloidal spin spirals due to SOI.

## Abstract

Using thermal quantum field theory we derive an expression for the exchange constant that resembles Fukuyama's formula for the orbital magnetic susceptibility (OMS). Guided by this formal analogy between the exchange constant and OMS we identify a contribution to the exchange constant that arises from the geometrical properties of the band structure in mixed phase space. We compute the exchange constants for free electrons and show that the geometrical contribution is generally important. Our formalism allows us to study the exchange constants in the presence of spin-orbit interaction (SOI). Thereby, we find sizable differences between the exchange constants of helical and cycloidal spin spirals. Furthermore, we discuss how to calculate the exchange constants based on a gauge-field approach in the case of the Rashba model with an additional exchange splitting and show that the exchange constants obtained from this gauge-field approach are in perfect agreement with those obtained from the quantum field theoretical method.