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arXiv:cond-mat/0112325

Knots in Condensed Matters

Y. M. Cho

Published Dec 18, 2001• 2 citations
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Abstract

We propose two types of topologically stable knot solitons in condensed matters, one in two-component Bose-Einstein condensates and one in two-gap superconductors. We identify the knot in Bose-Einstein condensates as a twisted vorticity flux ring and the knot in two-gap superconductors as a twisted magnetic flux ring. In both cases we show that there is a remarkable interplay between topology and dynamics which transforms the topologcal stability to the dynamical stability, and vise versa. We discuss how these knots can be constructed in the spin-1/2 condensate of $^{87}{\rm Rb}$ atoms and in two-gap superconductor of $MgB_2$.

DOI
10.48550/arxiv.cond-mat/0112325

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We propose two types of topologically stable knot solitons in condensed matters, one in two-component Bose-Einstein condensates and one in two-gap superconductors. We identify the knot in Bose-Einstein condensates as a twisted vorticity flux ring and the knot in two-gap superconductors as a twisted magnetic flux ring. In both cases we show that there is a remarkable interplay between topology and dynamics which transforms the topologcal stability to the dynamical stability, and vise versa. We discuss how these knots can be constructed in the spin-1/2 condensate of 87Rb^{87}{\rm Rb}87Rb atoms and in two-gap superconductor of MgB2MgB_2MgB2​.
We demonstrate the existence of the helical vortex solution in two-component Bose-Einstein condensates which can be identified as a twisted vorticity flux. Based on this we argue that the recently proposed knot in two-component Bose-Einstein condensates can be interpreted as a vorticity knot, a vortex ring made of the helical vortex. This picture shows that the knot is made of two quantized vorticity fluxes linked together, whose topology π3(S2)π_3(S^2)π3​(S2) is fixed by the linking number of two vorticity fluxes. Due to the helical structure the knot has both topological and dynamical stability. We estimate the energy of the lightest knot to be about 3×10−3eV3\times 10^{-3} eV3×10−3eV.
The recent experimental creation of knotted solitons in spin-1 Bose-Einstein condensates opens an interesting avenue for exploring the stability and dynamics of knot solitons. Knotted and helical vortices can be accommodated in spin-orbit-coupled antiferromagnetic spin-1 Bose-Einstein condensates. We demonstrate the stability of these topological structures via numerical simulations of the corresponding Gross-Pitaevskii equation. By changing the strength of the spin-orbit coupling, we find that the ground state undergoes a change from a knotted spin texture to a helical spin texture. Our results reveal that these types of topological object are experimentally realizable in spin-1 Bose-Einstein condensates.
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.
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.
The knot of spin texture is studied within the two-component Bose-Einstein condensates which are described by the nonlinear Gross-Pitaevskii equations. We start from the non-interacting equations including an axisymmetric harmonic trap to obtain an exact solution, which exhibits a non-trivial topological structure. The spin-texture is a knot with an integral Hopf invariant. The stability of the knot is verified by numerically evolving the nonlinear Gross-Pitaevskii equations along imaginary time.
Knots are familiar entities that appear at a captivating nexus of art, technology, mathematics, and science. As topologically stable objects within field theories, they have been speculatively proposed as explanations for diverse persistent phenomena, from atoms and molecules to ball lightning and cosmic textures in the universe. Recent experiments have observed knots in a variety of classical contexts, including nematic liquid crystals, DNA, optical beams, and water. However, no experimental observations of knots have yet been reported in quantum matter. We demonstrate here the controlled creation and detection of knot solitons in the order parameter of a spinor Bose-Einstein condensate. The experimentally obtained images of the superfluid directly reveal the circular shape of the soliton core and its accompanying linked rings. Importantly, the observed texture corresponds to a topologically non-trivial element of the third homotopy group and demonstrates the celebrated Hopf fibration, which unites many seemingly unrelated physical contexts. Our observations of the knot soliton establish an experimental foundation for future studies of their stability and dynamics within quantum systems.