# Chiral ferromagnetism beyond Lifshitz invariants

## TL;DR

This paper investigates a chiral contribution to micromagnetic energy in 2D Rashba ferromagnets, revealing new terms beyond traditional Lifshitz invariants caused by spin-orbit coupling effects.

## Contribution

It demonstrates that certain chiral energy terms cannot be expressed through Lifshitz invariants beyond linear SOC, highlighting the role of spin rotation symmetry breaking.

## Key findings

- Identification of non-Lifshitz invariant terms in micromagnetic energy
- Analysis of effects on magnetic phase diagrams and spin-wave dispersion
- Classification of symmetry-allowed terms for crystallographic groups

## Abstract

We consider a contribution $w_{\text{ch}}$ to the micromagnetic energy density that is linear with respect to the first spatial derivatives of the local magnetization direction. For a generalized 2D Rashba ferromagnet, we present a microscopic analysis of this contribution and, in particular, demonstrate that it cannot be expressed through Lifshitz invariants beyond the linear order in the spin-orbit coupling (SOC) strength. Terms in $w_{\text{ch}}$ beyond Lifshitz invariants emerge as a result of spin rotation symmetry breaking caused by SOC. Effects of these terms on the phase diagram of magnetic states and spin-wave dispersion are discussed. Finally, we present a classification of terms in $w_{\text{ch}}$, allowed by symmetry, for each crystallographic point group.