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arXiv:cond-mat/0703475 · Physical Review B

Knotted solitons in a charged two-condensate Bose system

Yi-Shi Duan, Xinhui Zhang, Yu-Xiao Liu, Li Zhao

Published Oct 10, 2006• 10 citations
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Abstract

By making use of the decomposition of U(1) gauge potential theory and the $\ensuremath{\phi}$-mapping method, we propose that a charged two-condensate Bose system possesses vortex lines and two classes of knotted solitons. The topological charges of the vortex lines are characterized by the Hopf indices and the Brower degrees of $\ensuremath{\phi}$-mapping, and the knotted solitons are described by the nontrivial Hopf invariant and the BF action, respectively.

DOI
10.1103/physrevb.74.144508

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By making use of the decomposition of U(1) gauge potential theory and the \ensuremathϕ\ensuremath{\phi}\ensuremathϕ-mapping method, we propose that a charged two-condensate Bose system possesses vortex lines and two classes of knotted solitons. The topological charges of the vortex lines are characterized by the Hopf indices and the Brower degrees of \ensuremathϕ\ensuremath{\phi}\ensuremathϕ-mapping, and the knotted solitons are described by the nontrivial Hopf invariant and the BF action, respectively.
We study the topological objects in two-component Bose-Einstein condensates. We compare two competing theories of two-component Bose-Einstein condensates, the popular Gross-Pitaevskii theory, and the recently proposed gauge theory of two-component Bose-Einstein condensate which has an induced vorticity interaction. We show that two theories produce very similar topological objects, in spite of the obvious differences in dynamics. Furthermore we show that the gauge theory of two-component Bose-Einstein condensates, with the U(1)U(1)U(1) gauge symmetry, is remarkably similar to the Skyrme theory. Just like the Skyrme theory this theory admits the non-Abelian vortex, the helical vortex, and the vorticity knot. We construct the lightest knot solution in two-component Bose-Einstein condensates numerically, and discuss how the knot can be constructed in the spin-12\frac{1}{2}21​ condensate of 87Rb^{87}\mathrm{Rb}87Rb atoms.
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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.
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.