# Hopfions emerge in ferroelectrics

## TL;DR

This paper demonstrates that Hopfions, knotted topological solitons characterized by the Hopf invariant, can naturally form in confined ferroelectric nanoparticles, revealing new functionalities in their electromagnetic responses.

## Contribution

It introduces the emergence of Hopfions as fundamental polarization configurations in ferroelectric nanoparticles, linking topological physics with nanomaterial functionalities.

## Key findings

- Hopfions are found in confined ferroelectric nanoparticles.
- They influence the electromagnetic response of nanomaterials.
- Potential for novel technological applications.

## Abstract

Paradigmatic knotted solitons, Hopfions, that are characterized by topological Hopf invariant, are widely investigated in the diverse areas ranging from high energy physics, cosmology and astrophysics to biology, magneto- and hydrodynamics and condensed matter physics. Yet, while holding high promise for applications, they remain elusive and under-explored. Here we demonstrate that Hopfions emerge as a basic configuration of polarization field in confined ferroelectric nanoparticles. Our findings establish that Hopfions govern a wealth of novel functionalities in the electromagnetic response of composite nanomaterials opening route to unprecedented technological applications.