[论文解读] Generalized Skyrmions

An Wang, Zimo Zhao+11 more|arXiv (Cornell University)|Sep 25, 2024

Advanced Topics in Algebra被引用 6

一句话总结

该论文定义了一个新的 coproduct 拓扑不变量，the generalized Skyrmion number，取值于一个可数的 coproduct of Z，适用于非紧致化场，并证明其在传播过程中对光学偏振场的鲁棒性。

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.

研究动机与目标

• Motivate the need for a topological count for fields without compact boundary conditions.
• Introduce a generalized Skyrmion number based on De Rham cohomology of compactly supported forms.
• Develop a framework that assigns an integer tuple to a field via boundary partitions.
• Demonstrate topological robustness of the generalized Skyrmion number under propagation and media variations.
• Show experimental evidence of generalized Skyrmion robustness in optical polarization fields.

提出的方法

• Define generalized Y-valued Skyrmions using psi-compactifiability and the degree of extended maps.
• Introduce omega-Skyrmion numbers for non-compactifiable fields via forms in Lambda^n(Y).
• Use (psi,F)-extensions to interpret omega-Skyrmion numbers topologically and relate them to degrees.
• Prove topological protection results (Theorems 1 and 2) ensuring invariance under smooth deformations.
• Apply the theory to optical polarization fields and discuss propagation in free space and media.
• Present experimental measurements of Stokes fields to extract standard and generalized Skyrmion numbers.

实验结果

研究问题

• RQ1 Can a topological invariant be defined for fields that are not boundary-compactifiable?
• RQ2 How can one assign integer topological numbers to each connected component carved by the field boundary?
• RQ3 Is there a propagation-preserving invariant for optical Skyrmions beyond the usual Skyrmion number?
• RQ4 How do singularities and boundary inadequacies affect topological information storage in fields?
• RQ5 Can the generalized Skyrmion number be demonstrated experimentally in optical beams?

主要发现

• A new coprod_Z^i-valued topological number is defined for fields with target manifold Y.
• The omega-Skyrmion number provides a way to assign integer values even when compactification fails.
• (psi,F)-extensions connect non-compactifiable fields to a topological degree via extended maps.
• Experiments show the generalized Skyrmion number remains stable during propagation where the usual Skyrmion number does not.
• Generalized Skyrmion numbers offer higher data density by allowing multiple independent topological components.
• The framework is abstract and applicable across various vector and abstract fields beyond optics.