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arXiv:cond-mat/0106103

Novel Theory for Topological Structure of Vortices in BEC

Yi-Shi Duan, Xin Liu, Pengming Zhang

Published Jun 6, 2001• 0 citations
arXivHTMLPDF

Abstract

By making use of the $ϕ$-mapping topological current theory, a novel expression of $\nabla \times \vec{V}$ in BEC is obtained, which reveals the inner topological structure of vortex lines characterized by Hopf indices and Brouwer degrees. This expression is just that formula Landau and Feynman expected to find out long time ago. In the case of superconductivity, the decomposition theory of U(1) gauge potential in terms of the condensate wave function gives a rigorous proof of London assumption, and shows that each vortex line should carry a quantized flux. The $ϕ$-mapping topological current theory of $\nabla \times \vec{V}$ can also gives a precise bifurcation theory of vortex lines in BEC.

DOI
10.48550/arxiv.cond-mat/0106103

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By making use of the ϕϕϕ-mapping topological current theory, a novel expression of ∇×V⃗\nabla \times \vec{V}∇×V in BEC is obtained, which reveals the inner topological structure of vortex lines characterized by Hopf indices and Brouwer degrees. This expression is just that formula Landau and Feynman expected to find out long time ago. In the case of superconductivity, the decomposition theory of U(1) gauge potential in terms of the condensate wave function gives a rigorous proof of London assumption, and shows that each vortex line should carry a quantized flux. The ϕϕϕ-mapping topological current theory of ∇×V⃗\nabla \times \vec{V}∇×V can also gives a precise bifurcation theory of vortex lines in BEC.
Through the application of ϕ-mapping topological theory, the properties of vortices in quantum Röntgen effect are thoroughly studied. The explicit expression of the vorticity is obtained, in which the δ function indicates that the vortices can only stem from the zero points of ϕ, and the magnetic flux of the consequent monopoles is quantized in terms of the Hopf indices and Brouwer degrees. The evolution of vortex lines is discussed. The reduced dynamic equation and a conserved dynamic quantity on stable vortex lines are obtained.
Using the phi-mapping topological theory, we study the topological structure of vortex lines in a two-dimensional generalized Gross-Pitaevskii theory in (3+1)-dimensional space-time. We obtain the reduced dynamic equation in the framework of the two-dimensional Gross-Pitaevskii theory, from which a conserved dynamic quantity is derived on the stable vortex lines. Such equations can also be used to discuss Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We obtain an exact dynamic equation with a topological term, which is ignored in traditional hydrodynamic equations. The explicit expression of vorticity as a function of the order parameter is derived, where the delta function indicates that the vortices can only be generated from the zero points of Phi and are quantized in terms of the Hopf indices and Brouwer degrees. The phi-mapping topological current theory also provides a reasonable way to study the bifurcation theory of vortex lines in the two-dimensional Gross-Pitaevskii theory.
In the light of the ϕ-mapping topological current theory, the Mermin–Ho vortices and the monopoles in three-component spinor BEC are studied. It is pointed out that these two topological structures respectively inhere in two-dimensional and three-dimensional topological currents which can be derived from the same field tensor k μν , and both these topological structures are characterized by the ϕ-mapping topological numbers—Hopf indices and Brouwer degrees. Furthermore, the spatial bifurcation of Mermin–Ho vortices and the generation and annihilation of monopoles are also discussed.
The topological solutions of the De Broglie–Bohm quantum mechanics are presented. Starting from the Schrödinger equation for one particle system and ϕ-mapping topological current theory, the trajectory of the particle is derived explicitly, and can be used as the world line of the particle. The world line is just at the zero point of the wave function and it is shown that the vorticity of the world line can be expressed by Hopf index and Brouwer degree. The evolution of the world line at the bifurcation point is given.