# Exact solutions for a ferromagnet with Dzyaloshinskii-Moriya interaction

## TL;DR

This paper derives exact static solutions for a 2D ferromagnet model with Dzyaloshinskii-Moriya interaction, revealing ground and excited states, and uncovering potential symmetries at specific magnetic field values.

## Contribution

It constructs three families of exact solutions for the non-linear sigma model with Dzyaloshinskii-Moriya interaction, including ground, excited, and special discrete states.

## Key findings

- Ground state is a clockwise helix.
- Excited state is a counterclockwise helix.
- Additional solutions may emerge at specific magnetic fields.

## Abstract

On the two-dimensional non-linear $\Sigma$-model describing a ferromagnet with Dzyaloshinskii-Moriya interaction, we build three families of exact static solutions depending on a single Cartesian variable. One of them describes a clockwise helix configuration, that characterizes the ground state of the system. A second one corresponds to a counterclockwise helix, representing an excited state. These two families of solutions are parameterized by a continuous parameter that depends on the magnetic field and the Dzyaloshinskii-Moriya coupling. Finally, the third family exists only for isolated values of the same parameter, corresponding to a discrete family of Viviani curves on the target sphere. The degeneracy of the resulting spectrum suggests that an approximate symmetry may emerge at specific values of the magnetic field, at which additional solutions could then exist.