Generalized Skyrmions

TL;DR

This paper introduces a generalized topological invariant called the generalized Skyrmion number, applicable to fields with arbitrary boundary conditions, enabling richer data encoding and robustness in various physical contexts.

Contribution

It develops a new abstract formalism for Skyrmions based on De Rham cohomology, extending topological classification to boundary-free fields and broadening potential applications.

Findings

Defines a new $igoplus_{i=1}^ Z^i$-valued topological number

Demonstrates increased data capacity in optical polarization fields

Shows robustness of the generalized Skyrmion number

Abstract

Skyrmions are important topologically non-trivial fields characteristic of models spanning scales from the microscopic to the cosmological. However, the Skyrmion number can only be defined for fields with specific boundary conditions, limiting its use in broader contexts. Here, we address this issue through a generalized notion of the Skyrmion derived from the De Rham cohomology of compactly supported forms. This allows for the definition of an entirely new $\coprod_{i=1}^\infty \mathbb{Z}^i$-valued topological number that assigns a tuple of integers $(a_1, \ldots, a_k)\in \mathbb{Z}^k$ to a field instead of a single number, with no restrictions to its boundary. The notion of the generalized Skyrmion presented in this paper is completely abstract and can be applied to vector fields in any discipline, not unlike index theory within dynamical systems. To demonstrate the power of our new formalism, we focus on the propagation of optical polarization fields and show that our newly defined generalized Skyrmion number significantly increases the dimension of data that can be stored within the field while also demonstrating strong robustness. Our work represents a fundamental paradigm shift away from the study of fields with natural topological character to engineered fields that can be artificially embedded with topological structures.