Knotted Solitons Research Articles

Emergence of zero-field non-synthetic single and interchained antiferromagnetic skyrmions in thin films

Antiferromagnetic (AFM) skyrmions are envisioned as ideal localized topological magnetic bits in future information technologies. In contrast to ferromagnetic (FM) skyrmions, they are immune to the skyrmion Hall effect, might offer potential terahertz dynamics while being insensitive to external magnetic fields and dipolar interactions. Although observed in synthetic AFM structures and as complex meronic textures in intrinsic AFM bulk materials, their realization in non-synthetic AFM films, of crucial importance in racetrack concepts, has been elusive. Here, we unveil their presence in a row-wise AFM Cr film deposited on PdFe bilayer grown on fcc Ir(111) surface. Using first principles, we demonstrate the emergence of single and strikingly interpenetrating chains of AFM skyrmions, which can co-exist with the rich inhomogeneous exchange field, including that of FM skyrmions, hosted by PdFe. Besides the identification of an ideal platform of materials for intrinsic AFM skyrmions, we anticipate the uncovered knotted solitons to be promising building blocks in AFM spintronics.

Optical excitations of Skyrmions, knotted solitons, and defects in atoms

Analogies between non-trivial topologies of matter and light have inspired numerous studies, including defect formation in structured light and topological photonic band structures. Three-dimensional topological objects of localised particle-like nature attract broad interest across discipline boundaries from elementary particle physics and cosmology to condensed matter physics. Here we propose how simple structured light beams can be transformed into optical excitations of atoms with considerably more complex topologies representing three-dimensional particle-like Skyrmions. This construction can also be described in terms of linked Hopf maps, analogous to knotted solitons of the Skyrme-Faddeev model. We identify the transverse polarisation density current as the effective magnetic gauge potential for the Chern-Simons helicity term. While we prepare simpler two-dimensional baby-Skyrmions and singular defects using the traditional Stokes vectors on the Poincaré sphere for light, particle-like topologies can only be achieved in the full optical hypersphere description that no longer discards the variation of the total electromagnetic phase of vibration.

Hopfions emerge in ferroelectrics

Paradigmatic knotted solitons, Hopfions, that are characterized by topological Hopf invariant, attract an intense attention in the diverse areas of physics ranging from high-energy physics, cosmology and astrophysics to biology, magneto- and hydrodynamics and condensed matter physics. Yet, while being of broad interest, they remain elusive and under-explored. Here we demonstrate that Hopfions emerge as a basic configuration of polarization field in confined ferroelectric nanoparticles. Our findings establish that Hopfions are of fundamental importance for the electromagnetic behavior of the nanocomposits and can result in advanced functionalities of these materials.

Stable knotted structure in spin-1 Bose-Einstein condensates with spin-orbit coupling

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.

Topological energy bounds for frustrated magnets

Frustrated magnets are known to support two-dimensional topological solitons, called skyrmions. A continuum model for frustrated magnets has recently been shown to support both two-dimensional skyrmions and three-dimensional knotted solitons (hopfions). In this note we derive lower bounds for the energies of these solitons expressed in terms of their topological invariants. The bounds are linear in the degree in the case of skyrmions and scale as the Hopf degree to the power 3/4 in the case of hopfions.

Knotted Solitons in an Interacting Mixture of a Charged and a Neutral Superfluid for Neutron Stars

By making use of the decomposition of U(1) gauge potential theory and the φ-mapping method we discuss a mixture of interacting neutral and charged Bose condensates, which is supposed being realized in the interior of neutron stars in the form of a coexistent neutron superfluid and protonic superconductor. We propose that this system possesses vortex lines and two classes of knotted solitons. The topological charge of the vortex lines are characterized by the Hopf indices and the Brower degrees of φ-mapping, and the knotted solitons are described by nontrivial Hopf invariant and the BF action respectively.

Knot solitons in the AFZ model

This paper studies the topological properties of knotted solitons in the (3 + 1)-dimensional Aratyn–Ferreira–Zimerman (AFZ) model. Topologically, these solitons are characterized by the Hopf invariant I, which is an integral class in the homotopy group π3(S3) = Z. By making use of the decomposition of U(1) gauge potential theory and Duan's topological current theory, it is shown that the invariant is just the total sum of all the self-linking and linking numbers of the knot family while only linking numbers are considered in other papers. Furthermore, it is pointed out that this invariant is preserved in the branch processes (splitting, merging and intersection) of these knot vortex lines.

Non-Meissner electrodynamics and knotted solitons in two-component superconductors

I consider electrodynamics and the problem of knotted solitons in two-component superconductors. Possible existence of knotted solitons in multicomponent superconductors was predicted several years ...

Vortex Lines and Monopoles in Electrically Conducting Plasmas

Based on the ϕ-mapping topological current theory and the decomposition of gauge potential theory, the vortex lines and the monopoles in electrically conducting plasmas 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 topological term n⃗ · (∂in⃗ × ∂jn⃗), and both these topological structures are characterized by the ϕ-mapping topological numbers–Hopf indices and Brouwer degrees. Furthermore, the spatial bifurcation of vortex lines and the generation and annihilation of monopoles are also discussed. At last, we point out that the Hopf invariant is a proper topological invariant to describe the knotted solitons.

Knotted solitons in a charged two-condensate Bose system

By making use of the decomposition of U(1) gauge potential theory and the \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 \phi-mapping, and the knotted solitons are described by the nontrivial Hopf invariant and the BF action, respectively.

Static knot energy, Hopf charge, and universal growth law

We present a family of static knotted soliton energy functionals governing the configuration maps from the Euclidean space R 4 n − 1 into the unit sphere S 2 n so that the knot charges are naturally represented by the Hopf invariants in the homotopy group π 4 n − 1 ( S 2 n ) and the special case n = 1 recovers the classical Faddeev knot energy. We establish the general result that the minimum energy or the knot mass E N of knotted solitons of the Hopf charge N satisfies the universal fractional-exponent growth law E N ∼ | N | ( 4 n − 1 ) / 4 n , in which the fractional exponent depends only on the dimensions of the domain and range spaces of the configuration maps but does not depend on the detailed structure of the knot energy.

Andreev-Bashkin effect and knot solitons in an interacting mixture of a charged and a neutral superfluid with possible relevance for neutron stars

We discuss a mixture of interacting neutral and charged Bose condensates, which is supposed being realized in the interior of neutron stars in the form of a coexistent neutron superfluid and protonic superconductor. We show that in this system, besides ordinary vortices of the ${S}^{1}\ensuremath{\rightarrow}{S}^{1}$ map, the neutron condensate also allows for (meta)stable finite-length knotted solitons, which are characterized by a nontrivial Hopf invariant and in some circumstances may be stabilized by a Faddeev-Skyrme term induced by the drag effect. We also consider a helical protonic flux tube in this system and show that, in contrast, it does not induce a Faddeev-Skyrme term.

Existence of Energy Minimizers as Stable Knotted Solitons in the Faddeev Model

In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q, which is an integral class in the homotopy group π3(S 2 )= . We prove in the full space situation that there exists an infinite subset of such that for any m∈ , the Faddeev energy, E, has a minimizer among the topological class Q=m. Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for = . As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality E≤C|Q|3/4, where C>0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.

Vortices carrying an arbitrary fraction of magnetic flux quantum, neutral superfluidity and knotted solitons in two-gap Ginzburg–Landau model

We show that in two-gap superconductors there exist vortices which carry an arbitrary fraction of magnetic flux quantum and in two dimensions under certain conditions these vortices undergo a BKT transition which marks onset of quasi-long-range order only in a difference of phases of the two order parameters. In the second part of the talk we show that an U(1)× U(1) Ginzburg–Landau model or a GL model where U(1)× U(1) symmetry is weakly broken to U(1) is exactly equivalent to a version of the Faddeev's the nonlinear O(3) sigma model. This implies in particular that such a system possesses a hidden O(3) symmetry and besides that allows for the formation of knotted solitons. The second part of the talk is based on a joint work with L.D. Faddeev and A. Niem: Phys. Rev. B 65 (2002) 100512.

EFFECTIVE STRING THEORIES AND FIELD THEORIES IN FOUR DIMENSIONS

We argue that in four dimensions bosonic strings can be bundled together in a manner which resembles the way how individual filaments become bundled together to form a cable. The energy of such a bundle of strings is described by its extrinsic geometry, and it involves both torsion and curvature contributions. This leads to ordinary four-dimensional field theories, that describe the bundles of strings in terms of closed knotted solitons. Examples of field theories that can be constructed in this manner include the Faddeev model and the Skyrme model.

Hidden symmetry and knot solitons in a charged two-condensate Bose system

We show that a charged two-condensate Ginzburg-Landau model or equivalently a Gross-Pitaevskii functional for two charged Bose condensates, can be mapped onto a version of the nonlinear O(3) \ensuremath{\sigma} model. This implies in particular that such a system possesses a hidden O(3) symmetry and allows for the formation of stable knotted solitons.

Aspects of electric and magnetic variables in SU(2) Yang–Mills theory

We introduce a novel decomposition of the four dimensional SU(2) gauge field. This decomposition realizes explicitely a symmetry between electric and magnetic variables, suggesting a duality picture between the corresponding phases. It also indicates that at large distances the Yang-Mills theory involves a three component unit vector field, a massive Lorentz vector field, and a neutral scalar field that condenses which yields the mass scale. Our results are consistent with the proposal that the physical spectrum of the theory contains confining strings which are tied into stable knotted solitons.

Knots in interaction

We study the geometry of interacting knotted solitons. The interaction is local and advances either as a three-body or as a four-body process, depending on the relative orientation and a degeneracy of the solitons involved. The splitting and adjoining is governed by a four-point vertex in combination with duality transformations. The total linking number is preserved during the interaction. It receives contributions both from the twist and the writhe, which are variable. Therefore solitons can twine and coil and links can be formed.

KNOT SOLITONS

The existence of ring-like and knotted solitons in O(3) nonlinear σ-model is analyzed. The role of isotopy of knots/links in classifying such solitons is pointed out. Appearance of torus knot solitons is seen.

Stable knot-like structures in classical field theory

Using methods of high performance computing, we have found indications that knotlike structures appear as stable finite energy solitons in a realistic 3+1 dimensional model. We have explicitly simulated the unknot and trefoil configurations, and our results suggest that all torus knots appear as solitons. Our observations open new theoretical possibilities in scenarios where stringlike structures appear, including physics of fundamental interactions and early universe cosmology. In nematic liquid crystals and 3He superfluids such knotted solitons might actually be observed.