Faddeev Model Research Articles

Hopf solitons in the AFZ model

The Aratyn–Ferreira–Zimerman (AFZ) model is a conformal field theory in three-dimensional space. It has solutions that are topological solitons classified by an integer-valued Hopf index. There exist infinitely many axial solutions which have been found analytically. Static axial, knot and linked solitons are found numerically using a modified volume preserving flow for Hopf index one to eight, allowing for comparison with other Hopf soliton models. Solutions include a static trefoil knot at Hopf index five. A one-parameter family of conformal Skyrme–Faddeev models, consisting of linear combinations of the Nicole and AFZ models, are also investigated numerically. The transition of solutions for Hopf index four is mapped across these models. A topological change between linked and axial solutions occurs, with fewer models (or a limited range of parameter values) permitting axial solitons than linked solitons at Hopf index four.

Soliton configurations in generalized Mie electrodynamics

The generalization of the Mie electrodynamics within the scope of the effective 8-spinor field model is suggested, with the Lagrangian including Higgs-like potential and higher degrees of the invariant AµAµ. Using special Brioschi 8-spinor identity, we show that the model includes the Skyrme and the Faddeev models as particular cases. We investigate the large-distance asymptotic of static solutions and estimate the electromagnetic contribution to the energy of the localized charged configuration.

Global solutions of the evolutionary Faddeev model with small initial data

We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space ℝ1+n to the unit sphere $$ \mathbb{S} $$ 2, which obey a system of non-linear wave equations. The nonlinearity enjoys the null structure and contains semi-linear terms, quasi-linear terms and unknowns themselves. We prove that the Cauchy problem is globally well-posed for sufficiently small initial data in Sobolev space.

Hopf solitons in the Nicole model

The Nicole model is a conformal field theory in a three-dimensional space. It has topological soliton solutions classified by the integer-valued Hopf charge, and all currently known solitons are axially symmetric. A volume-preserving flow is used to construct soliton solutions numerically for all Hopf charges from 1 to 8. It is found that the known axially symmetric solutions are unstable for Hopf charges greater than 2 and new lower energy solutions are obtained that include knots and links. A comparison with the Skyrme–Faddeev model suggests many universal features, though there are some differences in the link types obtained in the two theories.

Axially symmetric soliton solutions in a Skyrme–Faddeev-type model with Gies's extension

We construct static soliton solutions with non-zero Hopf topological charges to a theory which is the extended Skyrme–Faddeev model with a further quartic term in derivatives. We use an axially symmetric ansatz based on toroidal coordinates, and solve the resulting two coupled nonlinear partial differential equations in two variables by a successive over-relaxation method. We construct numerical solutions with the Hopf charge up to 4. The solutions present an interesting behavior under the changes of a special combination of the coupling constants of the quartic terms.

A new approximation of relaxed energies for harmonic maps and the Faddeev model

We propose a new approximation for the relaxed energy E of the Dirichlet energy and prove that the minimizers of the approximating functionals converge to a minimizer u of the relaxed energy, and that u is partially regular without using the concept of Cartesian currents. We also use the same approximation method to study the variational problem of the relaxed energy for the Faddeev model and prove the existence of minimizers for the relaxed energy \({\tilde{E}_F}\) in the class of maps with Hopf degree ±1.

On the geometrized Skyrme and Faddeev models

The higher power derivative terms involved in both Faddeev and Skyrme energy functionals correspond to σ 2 -energy, introduced by Eells and Sampson (1964) [1]. This paper provides a detailed study of the first and second variation formulae associated to this energy. Some classes of (stable) critical points are outlined.

Toroidal spin ordering in BiFeO3, GaFeO3, and Cr2O3 in the Faddeev model with a magnetic field

The Fadeev model is used for describing the recently discovered toroidal spin ordering in piezoelectric and ferrimagnetic GaFeO3 and piezo- and magnetoelectric Cr2O3 and BiFeO3. A stable toroidal solution of the Faddeev model with the topological charge Q= 1 in an external homogeneous magnetic field was obtained using the trial function method. The energy of a toroid as a function of its radius (R) was determined at various values of the external magnetic field (H). It was shown that the energy minimum is shifted toward smaller R’s with an increase in H. At a critical field value, the torus collapses so that the local spin structure disappears. It is suggested to use magnetic field for controlling the torus size in multiferroics, promising materials of spintronics.

APPROXIMATE VORTEX SOLUTION OF FADDEEV MODEL

Through an ansatz specifying the azimuthal-angle dependence of the solution, the static field equation for vortex of the Faddeev model is converted to an algebraic ordinary differential equation. An approximate analytic expression of the vortex solution is explored so that the energy per unit vortex length becomes as small as possible. It is observed that the minimum energy of vortex is approximately proportional to the integer which specifies the solution.

DECOMPOSITION OF SU(2) CONNECTION AND KNOT STRUCTURE IN SKYRME–FADDEEV MODEL

In this paper, spinor and vector decomposition of SU(2) gauge potential are presented and their equivalence is constructed using a simple proposal. We also obtain the action of Skyrme–Faddeev model from the SU(2) massive gauge field theory which is proposed according to the gauge invariant principle. Then, the knot structure in Skyrme–Faddeev model is discussed in terms of the so-called ϕ-mapping topological current theory. The topological charge of the knot is characterized by the Hopf indices and the Brouwer degrees of ϕ-mapping, naturally. At last, we briefly discussed the topological invariant–Hopf invariant which describes the topology of these knots. It is shown that Hopf invariant is the total number of all the linking numbers and self-linking numbers of these knots.

Geometric Property of the Faddeev Model

Two functions u and v are used in expressing the solutions of the Faddeev model. The geometric property of the surface S determined by u and v is discussed and the shape of the surface is demonstrated as an example. The Gaussian curvature of the surface S is negative.

Comment on: “Reduction of static field equation of Faddeev model to first order PDE” [Phys. Lett. B 652 (2007) 384

The authors of the article [M. Hirayama, C.-G. Shi, Phys. Lett. B 652 (2007) 384, arXiv: 0707.2207] propose an interesting method to solve the Faddeev model by reducing it to a set of first order PDEs. They first construct a vectorial quantity α, depending on the original field and its first derivatives, in terms of which the field equations reduce to a linear first order equation. Then they find vectors α1 and α2 which identically obey this linear first order equation. The last step consists in the identification of the αi with the original α as a function of the original field. Unfortunately, the authors overlook a constraint implied by their construction, which invalidates most of their subsequent results.

Pipe Shape Solution of Faddeev Model

Utilizing the results that the Faddeev model is equivalent to the mesonic sector of the SU(2) Skyrme model, where the baryon number current vanishes everywhere, some exact solutions including the vortex solutions of the Faddeev model are discussed. The solutions are classified by the first Chern number. When the Chern number equals 2, the new multisoliton solutions are obtained and the pipe shape distribution of the energy density of the solutions are found.

Knots in the Skyrme–Faddeev model

The Skyrme–Faddeev model is a modified sigma model in three-dimensional space, which has string-like topological solitons classified by the integer-valued Hopf charge. Numerical simulations are performed to compute soliton solutions for Hopf charges up to 16, with initial conditions provided by families of rational maps from the three-sphere into the complex projective line. A large number of new solutions are presented, including a variety of torus knots for a range of Hopf charges. Often these knots are only local energy minima, with the global minimum being a linked solution, but for some values of the Hopf charge they are good candidates for the global minimum energy solution. The computed energies are in agreement with Ward's conjectured energy bound.

Reduction of static field equation of Faddeev model to first order PDE

A method to solve the static field equation of the Faddeev model is presented. For a special combination of the concerned field, we adopt a form which is compatible with the field equation and involves two arbitrary complex functions. As a result, the static field equation is reduced to a set of first order partial differential equations.

Spin-charge separation and the Pauli electron

The separation between the spin and the charge converts the quantum mechanical Pauli Hamiltonian into the Hamiltonian of the non-Abelian Georgi-Glashow model, which is notorious for its magnetic monopoles and confinement. The independent spin and charge fluctuations both lead to the Faddeev model, suggesting the existence of a deep duality structure and indicating that the fundamental carriers of the spin and charge are knotted solitons.

Toroidal solitons in magnetic oxides, Bose-Einstein condensates, and other media

We consider toroidal soliton solutions in a number of nonlinear models of field theory (Skyrme, Faddeev, and Higgs models). Such models are used in various areas of solid-state chemistry and physics, chemical physics, astrophysics, and plasma chemistry and physics. The stability of toroidal solitons is shown to have a topological nature. We examine different methods of generating a nonlinear contribution to the Lagrangian in order to insure stability of the torus to collapse. Using the Faddeev and Skyrme models, we analyze the toroidal ordering in Bose-Einstein condensates of 23Na, 39K, and 87Rb alkali-metal atoms, the toroidal structures recently found in the magnetic oxides BiFeO3 and GaFeO3, and potential applications of such structures in spintronics. In addition, we address several critical issues in the chemistry and physics of solitons with high topological charges.

Glueball mass from quantized knot solitons and gauge-invariant gluon mass

We propose an approach which enables one to obtain simultaneously the glueball mass and the gluon mass in the gauge-invariant way to shed new light on the mass gap problem in Yang-Mills theory. First, we point out that the Faddeev (Skyrme--Faddeev-Niemi) model can be induced through the gauge-invariant vacuum condensate of mass dimension two from SU(2) Yang-Mills theory. Second, we obtain the glueball mass spectrum by performing the collective coordinate quantization of the topological knot soliton in the Faddeev model. Third, we demonstrate that a relationship between the glueball mass and the gluon mass is obtained, since the gauge-invariant gluon mass is also induced from the relevant vacuum condensate. Finally, we determine physical values of two parameters in the Faddeev model and give an estimate of the relevant vacuum condensation in Yang-Mills theory. Our results indicate that the Faddeev model can play the role of a low-energy effective theory of the quantum SU(2) Yang-Mills theory.

Energy Splitting, Substantial Inequality, and Minimization for the Faddeev and Skyrme Models

In this paper, we prove that the Faddeev energy E1 at the unit Hopf charge is attainable. The proof is based on utilizing an important inequality called the substantial inequality in our previous paper which describes how the Faddeev energy splits into its sublevels in terms of energy and topology when compactness fails. With the help of an optimal Sobolev estimate of the Faddeev energy lower bound and an upper bound of E1, we show that E1 is attainable. For the two-dimensional Skyrme model, we prove that the substantial inequality is also valid, which allows us to greatly improve the range of the coupling parameters for the existence of unit-charge solitons previously guaranteed in a smaller range of the coupling parameters by the validity of the concentration-compactness method.

Estimation of the Lin-Yang Bound of the Least Static Energy of the Faddeev Model

Lin and Yang's upper bound EQ ≦ c Q¾ of the least static energy EQ of the Faddeev model in a sector with a fixed Hopf index Q is investigated. By constructing an explicit trial configuration for the Faddeev field n, a possible value of the coefficient c is obtained numerically, which is much smaller than the value obtained quite recently by analytic discussions.