# Topological energy bounds for frustrated magnets

## TL;DR

This paper derives mathematical lower bounds for the energies of topological solitons, such as skyrmions and hopfions, in frustrated magnets, linking their energy to their topological invariants.

## Contribution

It provides the first analytical energy bounds for both two-dimensional skyrmions and three-dimensional hopfions in frustrated magnet models.

## Key findings

- Energy bounds are linear in skyrmion degree.
- Hopfion energy bounds scale as degree to the 3/4 power.
- Results connect topological invariants with energetic stability.

## Abstract

Frustrated magnets are known to support two-dimensional topological solitons, called skyrmions. A continuum model for frustrated magnets has recently been shown to support both two-dimensional skyrmions and three-dimensional knotted solitons (hopfions). In this note we derive lower bounds for the energies of these solitons expressed in terms of their topological invariants. The bounds are linear in the degree in the case of skyrmions and scale as the Hopf degree to the power 3/4 in the case of hopfions.