Inner topological structure of Hopf invariant

Ren, Ji‐Rong; Li, Ran; Duan, Yi‐Shi

doi:10.1063/1.2747614

Cited by 21 publications

(30 citation statements)

References 40 publications

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“…It is well known that the Hopf invariant is an important topological invariant to describe the topological characteristics of the knot family. In a closed three-manifold M , the Hopf invariant is defined as [13,24]…”

Section: The Hopf Invariant Of the Knotted Rs Vorticesmentioning

confidence: 99%

“…i.e. the RS vortices with the Hopf invariant which usually can be used to describe the linkage of the knotted family in mathematics [24], and reveal the inner relationship between the Hopf invariant and the topological knotted characteristic numbers of knotted RS vortices. Furthermore, the conservation of the Hopf invariant in the splitting, the mergence and the intersection processes is also discussed in details.…”

Section: Introductionmentioning

confidence: 99%

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Knotted Topological Phase Singularities of Electromagnetic Field

Ren,

Zhu,

2007

Preprint

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In this paper, knotted objects (RS vortices) in the theory of topological phase singularity in electromagnetic field have been investigated in details. By using the φ-mapping topological current theory proposed by Prof. Duan, we rewrite the topological current form of RS vortices and use this topological current we reveal that the Hopf invariant of RS vortices is just the sum of the linking and self-linking numbers of the knotted RS vortices. Furthermore, the conservation of the Hopf invariant in the splitting, the mergence and the intersection processes of knotted RS vortices is also discussed.

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Section: The Hopf Invariant Of the Knotted Rs Vorticesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Knotted Topological Phase Singularities of Electromagnetic Field

Ren,

Zhu,

2007

Preprint

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“…, entails the Hopf curvature two-form. We can cast this Hopf invariant into a more practical form [51]…”

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confidence: 99%

Hopf characterization of two-dimensional Floquet topological insulators

Ünal

Eckardt

Slager

2019

Phys. Rev. Research

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We present a topological characterization of time-periodically driven two-band models in 2 + 1 dimensions as Hopf insulators. The intrinsic periodicity of the Floquet system with respect to both time and the underlying two-dimensional momentum space constitutes a map from a three dimensional torus to the Bloch sphere. As a result, we find that the driven system can be understood by appealing to a Hopf map that is directly constructed from the micromotion of the drive. Previously found winding numbers are shown to correspond to Hopf invariants, which are associated with linking numbers describing the topology of knots in three dimensions. Moreover, after being cast as a Hopf insulator, not only the Chern numbers, but also the winding numbers of the Floquet topological insulator become accessible in experiments as linking numbers. We exploit this description to propose a feasible scheme for measuring the complete set of their Floquet topological invariants in optical lattices. arXiv:1904.03202v2 [cond-mat.quant-gas]

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“…In our topological theory of knotted vortex filaments, the Hopf invariant relates to the topological characteristic numbers of the knotted vortex filaments family. In a closed three-manifold M the Hopf invariant is defined as [16,24] …”

Section: Hopf Invariant Constraint On Scroll Wavementioning

confidence: 99%

“…Recently, Duan's topological current theory [15,16,23,24,25,26] has been applied to study the topological properties of spiral waves and scroll waves. Zhang et al [15] presented a rigorous topological description of spiral waves and scroll waves.…”

Section: Introductionmentioning

confidence: 99%

Branch Processes of Vortex Filaments and Hopf Invariant Constraint on Scroll Wave

Zhu¹,

Ren²,

Mo³

2009

Commun. Theor. Phys.

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In this paper, by making use of Duan's topological current theory, the evolution of the vortex filaments in excitable media is discussed in detail. The vortex filaments are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points of a complex function Z( x, t). It is also shown that the Hopf invariant of knotted scroll wave filaments is preserved in the branch processes (splitting, merging, or encountering) during the evolution of these knotted scroll wave filaments. Furthermore, it also revealed that the "exclusion principle" in some chemical media is just the special case of the Hopf invariant constraint, and during the branch processes the "exclusion principle" is also protected by topology.

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Inner topological structure of Hopf invariant

Cited by 21 publications

References 40 publications

Knotted Topological Phase Singularities of Electromagnetic Field

Knotted Topological Phase Singularities of Electromagnetic Field

Hopf characterization of two-dimensional Floquet topological insulators

Branch Processes of Vortex Filaments and Hopf Invariant Constraint on Scroll Wave

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