Normal Modes of a Spin Cycloid or Helix

TL;DR

This paper numerically investigates the normal modes of finite-length spin cycloids and helices on a lattice, revealing how interactions and exchange types influence mode structure and comparing results with continuum models.

Contribution

It provides the first detailed numerical analysis of normal modes in finite spin cycloids and helices with different interactions, highlighting the dependence on interaction type and exchange.

Findings

AF/DM and FM/DM cases have a single Goldstone mode.

AF/CE and FM/CE cases have three normal modes.

Non-Goldstone modes show mixed tangential and transverse oscillations in FM exchange.

Abstract

Although spin cycloids and helices are quite common, remarkably little is known about the normal modes of a spin cycloid or helix with finite length on a discrete lattice. Based on simple one-dimensional lattice models, we numerically evaluate the normal modes of a spin cycloid or helix produced by either Dzyaloshinskii-Moriya (DM) or competing exchange (CE) interactions. The normal modes depend on the type of interaction and on whether the nearest-neighbor exchange is antiferromagnetic (AF) or ferromagnetic (FM). In the AF/DM and FM/DM cases, there is only a single Goldstone mode; in the AF/CE and FM/CE cases, there are three. For FM exchange, the spin oscillations produced by non-Goldstone modes contain a mixture of tangential and transverse components. For the DM cases, we compare our numerical results with analytic results in the continuum limit. Examples are given of materials that fall into all four cases.