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

作者：Kobayashi Shingo^*; Kobayashi Michikazu; Kawaguchi Yuki; Nitta Muneto; Ueda Masahito

来源：Nuclear Physics B, 2012, 856(2): 577-606.

DOI：10.1016/j.nuclphysb.2011.11.003

• Topological defect
• Homotopy theory
• Gaseous BECs
• Superfluid helium 3
• Liquid crystal

Topological excitations are usually classified by the nth homotopy group pi(n). However, for topological excitations that coexist with vortices, there are cases in which an element of pi(n) cannot properly describe the charge of a topological excitation due to the influence of the vortices. This is because an element of pi(n) corresponding to the charge of a topological excitation may change when the topological excitation circumnavigates a vortex. This phenomenon 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 by the Abe homotopy group kappa(n). The nth Abe homotopy group kappa(n) is defined as a semi-direct product of pi(1) and pi(n). In this framework. the action of pi(1) on pi(n) is understood as originating from noncommutativity between pi(1) and pi(n). We show that a physical charge of a topological excitation can be described in terms of the conjugacy class of the Abe homotopy group. Moreover, the Abe homotopy group naturally describes vortex-pair creation and annihilation processes, which also influence topological excitations. We calculate the influence of vortices on topological excitations for the case in which the order parameter manifold is S-n/K, where S-n is an n-dimensional sphere and K is a discrete subgroup of SO(n + 1). We show that the influence of vortices on a topological excitation exists only if n is even and K includes a nontrivial element of O(t)/SO(n).

• 出版日期2012-3-11