Abe homotopy classification of topological excitations under the topological influence of vortices
Abe homotopy classification of topological excitations under the topological influence of vortices
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DOI：
10.1016/j.nuclphysb.2011.11.003
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发表时间：
2011-10
期刊：
影响因子：
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通讯作者：
S. Kobayashi;Michikazu Kobayashi;Y. Kawaguchi;M. Nitta;Masahito Ueda
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文献类型：
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作者：
S. Kobayashi;Michikazu Kobayashi;Y. Kawaguchi;M. Nitta;Masahito Ueda
拓扑激励通常由第n个同伦群πn来分类。然而，对于与涡旋共存的拓扑激发，由于涡旋的影响，存在π n的元素不能正确描述拓扑激发的电荷的情况。这是因为对应于拓扑激发的电荷的π n的元素在拓扑激发环绕涡旋时可能改变。这种现象被称为π 1对πn的作用。在本文中，我们表明，拓扑激励共存的旋涡分类的阿部同伦群κn。第n个Abe同伦群κ被定义为π 1和πn的半直积。在这个框架中，π 1对π的作用被理解为源于π 1和πn之间的非对易性。我们表明，物理电荷的拓扑激发可以描述的共轭类的安倍同伦群。此外，阿部同伦群自然地描述涡对的创建和湮灭过程，这也影响拓扑激发。我们计算了序参数流形为Sn/K的情况下涡对拓扑激励的影响，其中Sn是n维球面，K是SO（n+1）的离散子群.我们证明了只有当n是偶数且K包含O（n）/SO（n）的非平凡元素时，涡对拓扑激发的影响才存在.
Topological excitations are usually classified by the nth homotopy group πn. However, for topological excitations that coexist with vortices, there are cases in which an element of πncannot properly describe the charge of a topological excitation due to the influence of the vortices. This is because an element of πncorresponding 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 π1on πn. In this paper, we show that topological excitations coexisting with vortices are classified by the Abe homotopy group κn. The nth Abe homotopy group κnis defined as a semi-direct product of π1and πn. In this framework, the action of π1on πnis understood as originating from noncommutativity between π1and πn. 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 Sn/K, where Snis 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(n)/SO(n).