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(Color online) (a) The preimages of two points on the S² form two linked loops. The torus corresponds to the preimage of . (b) The isosurfaces of (the torus) and (the inner ring). The variations of color on the torus represent the angle between Sx and Sy which reflects the twist and chirality of the knot.
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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 a...
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Citations
... In the axially asymmetric waveguide, VRs are strongly deformed and can either break up into vortex lines or shrink and annihilate for sufficiently strong barriers [55]. Further, great efforts have been made to study knots in many different physical areas, such as classical fluids [56,57], nonlinear FitzHugh-Nagumo excitable media [58][59][60][61][62], and quantum knots in BEC systems [63][64][65][66][67][68]. Stable knots examples which might preserve their topology have been reported [57][58][59][60][61][62][63][64][65]69]; however, there exist many knot structures which would decay or finally untie due to reconnections [56,62,65,66]. ...
... Further, great efforts have been made to study knots in many different physical areas, such as classical fluids [56,57], nonlinear FitzHugh-Nagumo excitable media [58][59][60][61][62], and quantum knots in BEC systems [63][64][65][66][67][68]. Stable knots examples which might preserve their topology have been reported [57][58][59][60][61][62][63][64][65]69]; however, there exist many knot structures which would decay or finally untie due to reconnections [56,62,65,66]. ...
We investigate dynamics of a vortex ring encountering a coaxial spherical obstacle in Bose-Einstein condensates. For a perfect spherical obstacle, the vortex ring would first expand and then shrink when passing over the obstacle. For the case of a periodically modified spherical obstacle, the circular vortex ring is deformed to be gear shaped during the passing-over process. This gear-shaped vortex ring can evolve into a helical Kelvin-wave state, or split into several smaller vortex rings. In the latter situation, these split vortex rings would travel some distances forwards and then reconnect with each other, forming one new forward-moving Kelvin wave, or two Kelvin waves of different sizes, with the bigger one moving forwards and the smaller one moving backwards.
... Three-dimensional (3D) skyrmions and knots which are topological solitons classified by the third homotopy group have been a fascinating subject for decades [7][8][9]. The 3D skyrmions, which are identified by counting the number covering the 3D sphere surface SU(2) S 3 , have been studied widely [10][11][12][13][14][15][16][17]. Knots are identified by mapping from a three-dimensional sphere S 3 to S 2 and are classified by the homotopy classes with π 3 (S 2 ) Z [18][19][20][21][22][23]. ...
... In the two-component BECs, the conventional 3D skyrmion is composed of a ring component and a line component with boundary values ξ † = (1,0) for r → ∞ [12,13]. It is usually expressed as [11] ...
... Based on this analogy, we name the 3D texture in Fig. 2 with the terminology of 3D dimeron. We mention that in the conventional 3D skyrmion there is a 2D skyrmion in each vertical half-plane [13]. ...
Searching for novel topological objects is always an intriguing task for scientists in various fields. We study
a three-dimensional (3D) topological structure called a 3D dimeron in trapped two-component Bose-Einstein
condensates. The 3D dimeron differs from the conventional 3D skyrmion for the condensates hosting two
interlocked vortex rings. We demonstrate that the vortex rings are connected by a singular string and the
complexity constitutes a vortex molecule. The stability of the 3D dimeron is examined in two different models
using the imaginary time evolution method. We find that the stable 3D dimeron can be naturally generated from
a vortex-free Gaussian wave packet incorporating a synthetic non-Abelian gauge potential into the condensates.
The oxygen reduction reaction (ORR), a pivotal process in hydrogen fuel cells crucial for enhancing fuel cell performance through suitable catalysts, remains a challenging aspect of development. This study explores the catalytic potential of germanene on Al (111), taking advantage of the successful preparation of stable reconstructed germanene layers on Al (111) and the excellent catalytic performance exhibited by germanium-based nanomaterials. Through first-principles calculations, we demonstrate that the O2 molecule can be effectively activated on both freestanding and supported germanene nanosheets, featuring kinetic barriers of 0.40 and 0.04 eV, respectively. The presence of the Al substrate not only significantly enhances the stability of the reconstructed germanene but also preserves its exceptional ORR catalytic performance. These theoretical findings offer crucial insights into the substrate-mediated modulation of germanene stability and catalytic efficiency, paving the way for the design of stable and efficient ORR catalysts for future applications.
We show that the realization of synthetic magnetic fields via light-matter coupling in the Λ scheme implements a natural geometrical construction of magnetic fields, namely, as the pullback of the area element of the sphere to Euclidean space via certain maps. For suitable maps, this construction generates linked and knotted magnetic fields, and the synthetic realization amounts to the identification of the map with the ratio of two Rabi frequencies which represent the coupling of the internal energy levels of an ultracold atom. We consider examples of maps which can be physically realized in terms of Rabi frequencies and which lead to linked and knotted synthetic magnetic fields acting on the neutral atomic gas. We also show that the ground state of the Bose-Einstein condensate may inherit the topological properties of the synthetic gauge field, with linked and knotted vortex lines appearing in some cases.
The dynamics of a pair of three-dimensional matter-wave harmonic oscillators (HOs) coupled by a repulsive cubic nonlinearity is investigated through direct simulations of the respective GrossPitaevskii equations (GPEs) and with the help of the finite-mode Galerkin approximation (GA),which represents the two interacting wave functions by a superposition of 3 + 3 HO p -wave eigenfunctions with orbital and magnetic quantum numbers l = 1 and m = 1; 0; 1. First, the GA very accurately predicts a broadly degenerate set of the system's ground states in the p -wave manifold, in the form of complexes built of a dipole coaxial with another dipole or vortex, as well as complexes built of mutually orthogonal dipoles. Next, pairs of non-coaxial vortices and/or dipoles, including pairs of mutually perpendicular vortices, develop remarkably stable dynamical regimes, which feature periodic exchange of the angular momentum and periodic switching between dipoles and vortices. For a moderately strong nonlinearity, simulations of the coupled GPEs agree very well with results produced by the GA, demonstrating that the dynamics is accurately spanned by the set of six modes limited to l = 1.
Toroidal modes in the form of so-called Hopfions, with two independent
winding numbers, a hidden one (twist, s), which characterizes a circular vortex
thread embedded into a three-dimensional soliton, and the vorticity around the
vertical axis m, appear in many fields, including the field theory,
ferromagnetics, and semi- and superconductors. Such topological states are
normally generated in multi-component systems, or as trapped quasi-linear modes
in toroidal potentials. We uncover that stable solitons with this structure can
be created, without any linear potential, in the single-component setting with
the strength of repulsive nonlinearity growing fast enough from the center to
the periphery, for both steep and smooth modulation profiles. Toroidal modes
with s=1 and vorticity m=0,1,2 are produced. They are stable for m<=1, and do
not exist for s>1. An approximate analytical solution is obtained for the
twisted ring with s=1, m=0. Under the application of an external torque, it
rotates like a solid ring. The setting can be implemented in BEC, by means of
the Feshbach resonance controlled by inhomogene-ous magnetic fields.
