Skyrmion topology of scalar waves in real and momentum space

Abstract

Generalizations of magnetic skyrmions to the context of other areas of physics, such as plasmonics, optics (including microcavity polaritons) and even water waves have become an important research topic. However, the generalization of the skyrmion number in such systems is still the subject of active research. While magnetic skyrmions textures are inherently formed by vectors (magnetic spin vectors), recent developments in other areas of physics have generalized the concept of skyrmions and skyrmionic patterns to the case in which scalar fields are used to define 3-component pseudo-spin vectors. This gives rise to a wide variety of skyrmion-like textures, depending on the detailed functional forms of the scalar fields. For example, steady-state outgoing waves in driven-dissipative polariton systems fail to have a well-defined global skyrmion number in real (configuration) space but can have one in momentum space. We classify the skyrmion integral for various scalar fields with different asymptotic behaviors in real space as well as momentum space, and in both cases give conditions for the skyrmion number not to exist, or be integer, half-integer or non-integer. Examples of physical systems to which our analysis can be applied include polariton condensates, water waves and gain-guided lasers.