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Journal of Physics A Mathematical and General

Mermin Ho vortices and monopoles in three-component spinor BEC

Yi-Shi Duan, Xin Liu, Pengming Zhang

Published Dec 17, 2002• 18 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/36/2/318

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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.
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.
A rich variety of order parameter manifolds of multicomponent Bose-Einstein condensates (BECs) admit various kinds of topological excitations, such as fractional vortices, monopoles, skyrmions, and knots. In this paper, we discuss two topological excitations in spinor BECs: non-Abelian vortices and knots. Unlike conventional vortices, non-Abelian vortices neither reconnect themselves nor pass through each other, but create a rung between them in a topologically stable manner. We discuss the collision dynamics of non-Abelian vortices in the cyclic phase of a spin-2 BEC. In the latter part, we show that a knot, which is a unique topological object characterized by a linking number or a Hopf invariant [π3 (S2) = Z], can be created using a conventional quadrupole magnetic field in a cold atomic system.
Magnetic monopoles, which are particle-like field configurations with which one can associate a topological charge, widely exist in various three-dimensional condensate systems. In this paper, by making use of Duan 's topological current theory, we obtain the charge density of magnetic monopoles and their bifurcation theory in a charged two-condensate Bose–Einstein system. The evolution of magnetic monopoles is studied from the topological properties of a three-dimensional vector field. The magnetic monopoles are found generating or annihilating at the limit points and encountering, splitting or merging at the bifurcation points.
The structure and stability of various vortices in F=1F=1F=1 spinor Bose-Einstein condensates are investigated by solving the extended Gross-Pitaevskii equation under rotation. We perform an extensive search for stable vortices, considering both axisymmetric and nonaxisymmetric vortices and covering a wide range of ferromagnetic and antiferromagnetic interactions. The topological defect called the Mermin-Ho (Anderson-Toulouse) vortex is shown to be stable for the ferromagnetic case. The phase diagram is established in a plane of external rotation $\ensuremath{\Omega}$ versus total magnetization M by comparing the free energies of possible vortices. It is shown that there are qualitative differences between axisymmetric and nonaxisymmetric vortices which are manifested in the $\ensuremath{\Omega}$ and M dependences.