English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Topological excitations are usually classified by the $n$th homotopy group$\pi_n$. However, for topological excitations that coexist with vortices, thereare case in which an element of $\pi_n$ cannot properly describe the charge ofa topological excitation due to the influence of the vortices. This is becausean element of $\pi_n$ corresponding to the charge of a topological excitationmay change when the topological excitation circumnavigates a vortex. Thisphenomenon is referred to as the action of $\pi_1$ on $\pi_n$. In this paper,we show that topological excitations coexisting with vortices are classified bythe Abe homotopy group $\kappa_n$. The $n$th Abe homotopy group $\kappa_n$ isdefined as a semi-direct product of $\pi_1$ and $\pi_n$. In this framework, theaction of $\pi_1$ on $\pi_n$ is understood as originating from noncommutativitybetween $\pi_1$ and $\pi_n$. We show that a physical charge of a topologicalexcitation can be described in terms of the conjugacy class of the Abe homotopygroup. Moreover, the Abe homotopy group naturally describes vortex-paircreation and annihilation processes, which also influence topologicalexcitations. We calculate the influence of vortices on topological excitationsfor the case in which the order parameter manifold is $S^n/K$, where $S^n$ isan $n$-dimensional sphere and $K$ is a discrete subgroup of $SO(n+1)$. We showthat the influence of vortices on a topological excitation exists only if $n$is even and $K$ includes a nontrivial element of $O(n)/SO(n)$.

Abe homotopy classification of topological excitations under the topological influence of vortices