We consider a four dimensional field theory with target space being CP^N which constitutes a generalization of the usual Skyrme-Faddeev model defined on CP^1. We show that it possesses an integrable sector presenting an infinite number of local conservation laws, which are associated to the hidden symmetries of the zero curvature representation of the theory in loop space. We construct an infinite class of exact solutions for that integrable submodel where the fields are meromorphic functions of the combinations (x^1+i x^2) and (x^3+x^0) of the Cartesian coordinates of four dimensional Minkowski space-time. Among those solutions we have static vortices and also vortices with waves traveling along them with the speed of light. The energy per unity of length of the vortices show an interesting and intricate interaction among the vortices and waves.
We look at properties of vortex solutions of the extended CP^N Skyrme-Faddeev model. We show that only holomorphic solutions of the CP^N model are also solutions of the Skyrme-Faddeev model. As the total energy of these solutions is infinite these solutions should be interpreted as describing time dependent vortices. We describe their dynamics and, in partcular, point out that one of the terms in the energy density is related to the Noether charge of the model.
Solitons in the Skyrme-Faddeev model on R^2xS^1 are shown to undergo buckling transitions as the circumference of the S^1 is varied. These results support a recent conjecture that solitons in this field theory are well-described by a much simpler model of elastic rods.
The extended Skyrme-Faddeev model possesses vortex solutions in a (3+1) dimensional Minkowski space-time with target space $CP^N$. They have finite energy per unit of length and contain waves propagating along vortices with the speed of light. We introduce various types of the potentials which correspond with holomorphic solutions of the integrable sector and also with several numerical solutions outside of this sector. The presented solutions constitute a strong indication that the current model contains large class of solutions with much wider range of coupling constants than the previously …
The strongly coupled limit of the Skyrme-Faddeev-Niemi model (i.e., without quadratic kinetic term) with a potential is considered on the spacetime S^3 x R. For one-vacuum potentials two types of exact Hopf solitons are obtained. Depending on the value of the Hopf index, we find compact or non-compact hopfions. The compact hopfions saturate a Bogomolny bound and lead to a fractional energy-charge formula E \sim |Q|^{1/2}, whereas the non-compact solitons do not saturate the bound and give E \sim |Q|. In the case of potentials with two vacua compact shell-like hopfions are derived. Some remarks…
A topological lower bound on the Skyrme energy which depends explicity on the pion mass is derived. This bound coincides with the previously best known bound when the pion mass vanishes, and improves on it whenever the pion mass is non-zero. The new bound can in particular circumstances be saturated. New energy bounds are also derived for the Skyrme model on a compact manifold, for the Faddeev-Skyrme model with a potential term, and for the Aratyn-Ferreira-Zimerman and Nicole models.
We point out a duality between steady incompressible Euler flows and solutions of the strongly coupled Faddeev-Skyrme sigma model with potential (mass) term. We supplement this result with various applications and several explicit examples.
The higher-power derivative terms involved in both Faddeev and Skyrme energy functionals correspond to $σ_2$-energy, introduced by Eells and Sampson. The paper provides a detailed study of the first and second variation formulae associated to this energy. Some classes of (stable) critical maps are outlined.
In the present article we show that the Skyrme--Faddeev model possesses nonlinear wave solutions, which can be expressed in terms of elliptic functions. The Whitham averaging method has been exploited in order to describe slow deformation of periodic wave states, leading to a quasi-linear system. The reduction to general hydrodynamic systems have been considered and it is compared with other integrable reductions of the system.
The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity. For the Skyrme model, the codomain is any Lie group, while for the Faddeev-Hopf model it is $S^2$. It is determined when the topology of configuration space permits fermionic and isospinorial quantization of the solitons of the model within generalizations of the frameworks of Finkelstein-Rubinstein and Sorkin. Fermionic quantization of Sky…
We construct analytical and numerical vortex solutions for an extended Skyrme-Faddeev model in a $(3+1)$ dimensional Minkowski space-time. The extension is obtained by adding to the Lagrangian a quartic term, which is the square of the kinetic term, and a potential which breaks the SO(3) symmetry down to SO(2). The construction makes use of an ansatz, invariant under the joint action of the internal SO(2) and three commuting U(1) subgroups of the Poincaré group, and which reduces the equations of motion to an ODE for a profile function depending on the distance to the $x^3$-axis. The vortices …
We construct exact vortex solutions in 3+1 dimensions to a theory which is an extension, due to Gies, of the Skyrme-Faddeev model, and that is believed to describe some aspects of the low energy limit of the pure SU(2) Yang-Mills theory. Despite the efforts in the last decades those are the first exact analytical solutions to be constructed for such type of theory. The exact vortices appear in a very particular sector of the theory characterized by special values of the coupling constants, and by a constraint that leads to an infinite number of conserved charges. The theory is scale invariant …
The Faddeev-Skyrme model, a modified O(3) nonlinear sigma model in three space dimensions, is known to admit topological solitons that are stabilized by the Hopf charge. The Faddeev-Skyrme model is also related to the low-energy limits of SU(2) Yang-Mills theory. Here, the model is reformulated into its gauge-equivalent expression, which turns out to be Skyrme-like. The solitonic solutions of this Skyrme-like model are analyzed by the rational map ansatz. The energy function and the Bogomolny-type lower bound of the energy are established. The generalized Faddeev-Skyrme model that originates f…
We construct numerical vortex solutions in a (3+1) dimensional Minkowski space-time for the extended version of the Skyrme-Faddeev model with target space $CP^N$. The solutions are essentially composed of $N$-th single vortex which does not belong to the integrable sector. They have finite energy per unit length and contain waves propagating along vortices with the speed of light. In this Letter we report on the case N=2 and show existence of the solutions with the charges $(n_1,n_2)=(1,2)$.
Firstly, we consider Yang-Mills theory on ${\mathbb R}^{3,1}$ with an adjoint Higgs field spontaneously breaking a compact gauge group $G$ to a subgroup $H$, so that the Higgs vacuum manifold forms the coset $G/H$. It is shown that in the low-energy limit, when the Higgs vacuum value is large, the 4d Yang-Mills-Higgs theory reduces to the Faddeev sigma model on ${\mathbb R}^{3,1}$ with $G/H$ as target. Its action contains the standard two-derivative sigma-model term as well as the four-derivative Skyrme-type term, which stabilizes solutions against scaling. Secondly, we put the Higgs field in …
We observe that the Faddeev-Skyrme model emerges as a low-energy limit of scalar QED with two charged scalar fields and a selfinteraction potential of a special form (inspired by supersymmetric QCD). Then we discuss possible Hopf solitons of the "twisted-toroidal" type.
We discuss the existence of knot solitons (Hopfions) in a Skryme-Faddeev-Niemi-type model on the target space $SU(3)/U(1)^2$, which can be viewed as an effective theory of both the $SU(3)$ Yang-Mills theory and the $SU(3)$ anti-ferromagnetic Heisenberg model. We derive the knot solitons with two different types of ansatz: the first is a trivial embedding configuration of $SU(2)$ into $SU(3)$, and the second is a non-embedding configuration that can be generated through the Bäcklund transformation. The resulting Euler-Lagrange equations for both ansatz reduce exactly to those of the $CP^1$ Skyr…
We construct non-axially symmetric static soliton solutions, with non-zero topological charges, of an extension of the Skyrme-Faddeev model. The model has an extra quartic-derivative term and we choose its coupling to the Skyrme-term to be negative. We solve the full equations of motion to find numerical solutions with topological charge up to seven and find that the model favours large ring-like solutions.
Recently we proposed a decomposition for fields of the Georgi-Glashow model and interpreted Cho's decomposition as a result of some constraints on Georgi-Glashow's fields. Now, using the decomposition form that Faddeev and Niemi proposed, we introduce a simple and novel method to derive the Skyrme-Faddeev Lagrangian from the reformulation of the Georgi-Glashow model with an extra constraint. As we showed before, this extra constraint leads to appearance of both vortices and monopoles.
Motivated by the sigma model limit of multicomponent Ginzburg-Landau theory, a version of the Faddeev-Skyrme model is considered in which the scalar field is coupled dynamically to a one-form field called the supercurrent. This coupled model is investigated in the general setting where physical space is an oriented Riemannian manifold and the target space is a Kaehler manifold. It is shown that supercurrent coupling destroys the topological stability enjoyed by the usual Faddeev-Skyrme model, so that there can be no globally stable knot solitons in this model. Nonetheless, local energy minimiz…
We study vortex knotting in the Faddeev-Skyrme model. Starting with a straight vortex line twisted around its axis we follow its evolution under dissipative energy minimization dynamics. With low twist per unit length the vortex forms a helical coil, but with higher twist numbers the vortex becomes knotted or a ring is formed around the vortex.
We discuss the physical impacts of the ``Cho decomposition'' (or the ``Cho-Faddeev-Niemi-Shabanov decomposition'') of the non-Abelian gauge potential on QCD. We show how the decomposition makes a subtle but important modification on the non-Abelian dynamics, and present three physically equivalent quantization schemes of QCD which are consistent with the decomposition. In particular, we show that the decomposition enlarges the dynamical degrees of QCD by making the topological degrees of the non-Abelian gauge symmetry dynamical. Furthermore with the decomposition we show that the Skyrme-Faddee…
Skyrme theory on S^2 (Faddeev coset proposal), is analyzed with a generalization of 0-curvature integrability, based on gauge techniques. New expressions valid for models in the sphere are given. The relation of the minimum energy configurations to gauge vacua is clarified. Consequences of adding a potential term to break the SO(3) symmetry are discussed.
We consider 5d Yang-Mills-Higgs theory with a compact ADE-type gauge group $G$ and one adjoint scalar field on $\mathbb{R}^{3,1}\times\mathbb{R}_+$, where $\mathbb{R}_+=[0,\infty)$ is the half-line. The maximally supersymmetric extension of this model, with five adjoint scalars, appears after a reduction of 6d ${\cal N}{=}\,(2,0)$ superconformal field theory on $\mathbb{R}^{3,1}\times\mathbb{R}_+\times S^1$ along the circle $S^1$. We show that in the low-energy limit, when momenta along $\mathbb{R}^{3,1}$ are much smaller than along $\mathbb{R}_+$, the 5d Yang-Mills-Higgs theory reduces to a n…
We construct static soliton solutions with non-zero Hopf topological charges to a theory which is an extension of the Skyrme-Faddeev model by the addition of a further quartic term in derivatives. We use an axially symmetric ansatz based on toroidal coordinates, and solve the resulting two coupled non-linear partial differential equations in two variables by a successive over-relaxation (SOR) method. We construct numerical solutions with Hopf charge up to four, and calculate their analytical behavior in some limiting cases. The solutions present an interesting behavior under the changes of a s…
We construct new solutions of the Faddeev-Skyrme-Maxwell model, which represent Hopf solitons coupled to magnetic fluxes. It turns out that coupling to the magnetic field allows for transmutations of the solitons, however, the results depend both on the type of the vacuum boundary condition and on the strength of the gauge coupling. It is shown that the structure of the magnetic fluxes of a gauged Hopfion is governed by the preimages of the points $ϕ_3=\pm 1$.
Class of exact solutions of the Skyrme and the Faddeev model are presented. In contrast to previously found solutions, they are produced by the interplay of the two terms in the Lagrangians of the models. They are not solitonic but of wave character. With an appropriate choice of field variables, the field equations of the two models are written in exactly the same form. The solutions supply us with examples of the superposition of two plane waves in nonlinear field theories.
This paper describes a natural one-parameter family of generalized Skyrme systems, which includes the usual SU(2) Skyrme model and the Skyrme-Faddeev system. Ordinary Skyrmions resemble polyhedral shells, whereas the Hopf-type solutions of the Skyrme-Faddeev model look like closed loops, possibly linked or knotted. By looking at the minimal-energy solutions in various topological classes, and for various values of the parameter, we see how the polyhedral Skyrmions deform into loop-like Hopf Skyrmions.
The Skyrme-Faddeev model has planar soliton solutions with target space $\mathbb{C}P^N$. An Abelian Chern-Simons term (the Hopf term) in the Lagrangian of the model plays a crucial role for the statistical properties of the solutions. Because $Π_3(\mathbb{C}P^1)=\mathbb{Z}$, the term becomes an integer for $N=1$. On the other hand, for $N>1$, it becomes perturbative because $Π_3(\mathbb{C}P^N)$ is trivial. The prefactor $Θ$ of the Hopf term is not quantized, and its value depends on the physical system. We study the spectral flow of the normalizable fermions coupled with the baby-Skyrme model …
We analytically construct vortex solutions in the integrable sector of the extended Skyrme-Faddeev model. The solutions are holomorphic type which satisfy the zero curvature condition. For the model parameter $βe^2=1$ there is a lump solution, and for $βe^2 \neq 1$ new potentials are introduced for the several molecular-shaped solutions with half-integer or integer charges. They necessarily have infinite number of conserved currents and some of the examples are shown. By performing an annealing simulation with our potentials, we verify the existence of the solutions of the integrable sector.
The $\mathbb{C}P^N$ extended Skyrme-Faddeev model possesses planar soliton solutions. We consider quantum aspects of the solutions applying collective coordinate quantization in regime of rigid body approximation. In order to discuss statistical properties of the solutions we include an Abelian Chern-Simons term (the Hopf term) in the Lagrangian. Since $Π_3(\mathbb{C}P^1)=\mathbb{Z}$ then for $N=1$ the term becomes an integer. On the other hand for $N>1$ it became perturbative because $Π_3(\mathbb{C}P^N)$ is trivial. The prefactor of the Hopf term (anyon angle) $Θ$ is not quantized and its val…
The Skyrme-Faddeev model is a three-dimensional non-linear field theory that has topological soliton solutions, called hopfions, which are novel string-like solutions taking the form of knots and links. Solutions found thus far take the form of torus knots and links of these, however torus knots form only a small family of known knots. It is an open question whether any non-torus knot hopfions exist. In this paper we present a construction of knotted fields with the form of cable knots to which an energy minimisation scheme can be applied. We find the first known hopfions which do not have the…
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 sixteen, 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 s…
Several materials, such as ferromagnets, spinor Bose-Einstein condensates, and some topological insulators, are now believed to support knotted structures. One of the most successful base-models having stable knots is the Faddeev-Skyrme model and it is expected to be contained in some of these experimentally relevant models. The taxonomy of knotted topological solitons (Hopfions) of this model is known. In this paper we describe some aspects of the dynamics of Hopfions and show that they do indeed behave like particles: during scattering the Hopf charge is conserved and bound states are formed…
We propose a generalized Skyrme-Faddeev type theory with an additional scalar field. In a special case of model parameters one has a theory which admits exact knot solutions given by a class of exact toroidal solitons from Aratyn-Ferreira-Zimerman (AFZ) integrable CP1 model. In a general case the theory admits an exact knot solution for a unit Hopf charge. For higher Hopf charges we perform numeric analysis of the solutions and obtain estimates for the knot energies using energy minimization procedure based on ansatz with AFZ field configurations and with rational functions. We show that AFZ c…
Analytical and numerical vortex solutions for the extended Skyrme-Faddeev model in a (3+1) dimensional Minkowski space-time are investigated. The extension is obtained by adding to the Lagrangian a quartic term, which is the square of the kinetic term, and a potential which breaks the SO(3) symmetry down to SO(2). The construction of the solutions has been done in twofold: one makes use of an axially symmetric ansatz and solves the resulting ODE by an analytical and a numerical way. The analytical vortices are obtained for special form of the potentials, and the numerical ones are computed usi…
We study geometric variational problems for a class of effective models in quantum field theory known as Faddeev-Skyrme models. Mathematically one considers minimizing an energy functional on homotopy classes of maps from closed 3-manifolds into homogeneous spaces of compact Lie groups. The energy minimizers known as Hopfions describe stable configurations of subatomic particles such as protons and their strong interactions. In particular, we introduce novel Sobolev spaces suitable for our variational problem and develop the notion of homotopy type for maps in such spaces that generalizes homo…
A class of exact solutions of the Faddeev model, that is, the modified SO(3) nonlinear sigma model with the Skyrme term, is obtained in the four dimensional Minkowskian spacetime. The solutions are interpreted as the isothermal coordinates of a Riemannian surface. One special solution of the static vortex type is investigated numerically. It is also shown that the Faddeev model is equivalent to the mesonic sector of the SU(2) Skyrme model where the baryon number current vanishes.
We construct new solutions of the Faddeev-Skyrme model with a symmetry breaking potential admitting $S^1$ vacuum. It includes, as a limiting case, the usual $SO(3)$ symmetry breaking mass term, another limit corresponds to the potential $m^2 ϕ_1^2$, which gives a mass to the corresponding component of the scalar field. However we find that the spacial distribution of the energy density of these solutions has more complicated structure, than in the case of the usual Hopfions, typically it represents two separate linked tubes with different thicknesses and positions. In order to classify these c…
The Skyrme-Faddeev model is a (3+1)-dimensional model which has knotted, string-like, soliton solutions. In this paper we investigate a Skyrme-Faddeev model with an SO(3) symmetry breaking potential. We then rescale this model and take the mass to infinity. This infinite mass model is found to have compact knotted solutions. In all of the investigated massive models we find similar charged solutions as in the usual, m=0, model. We also find that their energies follow a similar power growth as the m=0 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.
A regular method is suggested for constructing vortex-like solutions with cylindrical symmetry in the Skyrme-Einstein chiral model. The method is based on the expansion of metric and field functions in power series with respect to the two small parameters entering the model. The length mass density of the vortex is estimated.
We examine non-linear sigma (plus Skyrme term) model in flat space, in particular with a twist, which comprises a twisted baby Skyrmion vortex solution with an added dependence on a twist term $mkz$, where $z$ is the vertical coordinate. We find that the solutions should have asymptotic form. We also find that mass per unit length of string, $μ$, increases roughly linearly with $ζ$, where $ζ=4λ^2K_s(mk)^2$, $λ$ is scaling constant, $K_s$ is Skyrme term, $m$ is integer and $k$ is wave number.
We construct a molecule of fractional vortices with fractional topological lump charges as a baby Skyrmion with the unit topological lump charge in the anti-ferromagnetic (or XY) baby Skyrme model, that is, an O(3) sigma model with a four derivative term and an anti-ferromagnetic or XY-type potential term quadratic in fields. We further construct configurations with topological lump charges Q <= 7 and find that bound states of vortex molecules constitute regular polygons with 2Q vertices as vortices, where the rotational symmetry SO(2) in real space is spontaneously broken into a discrete subg…
Hopf solitons in the Skyrme-Faddeev system on $R^3$ typically have a complicated structure, in particular when the Hopf number Q is large. By contrast, if we work on a compact 3-manifold M, and the energy functional consists only of the Skyrme term (the strong-coupling limit), then the picture simplifies. There is a topological lower bound $E\geq Q$ on the energy, and the local minima of E can look simple even for large Q. The aim here is to describe and investigate some of these solutions, when M is $S^3$, $T^3$ or $S^2 \times S^1$. In addition, we review the more elementary baby-Skyrme syste…
We propose a new Skyrme-like model with fields taking values on the sphere S^3 or, equivalently, on the group SU(2). The action of the model contains a quadratic kinetic term plus a quartic term which is the same as that of the Skyrme-Faddeev model. The novelty of the model is that it possess a first order Bogomolny type equation whose solutions automatically satisfy the second order Euler-Lagrange equations. It also possesses a lower bound on the static energy which is saturated by the Bogomolny solutions. Such Bogomolny equation is equivalent to the so-called force free equation used in plas…
The BPS baby Skyrme models are submodels of baby Skyrme models, where the nonlinear sigma model term is suppressed. They have skyrmion solutions saturating a BPS bound, and the corresponding static energy functional is invariant under area-preserving diffeomorphisms (APDs). Here we show that the solitons in the BPS baby Skyrme model, which carry a nontrivial topological charge $Q_{b} \in π_2(S^2)$ (a winding number), are dual to vortices in a BPS vortex model with a topological charge $Q_{v}\in π_1(S^1)$ (a vortex number), in the sense that there is a map between the BPS solutions of the two m…
We derive a generalized Skyrme-Faddeev action as the effective action of QCD in the low energy limit. Our result demonstrates the existence of a mass gap in QCD which triggers the confinement of color.
Based on the proposal that the Skyrme theory is a theory of monopole we provide a new interpretation of Skyrme theory, that the theory can also be viewed as an effective theory of strong interaction which is dual to QCD, where the monopoles (not the quarks) are confined through the Meissner effect. This dual picture leads us to predict the existence of a topological glueball in QCD, a chromoelectric knot which is dual to the chromomagnetic Faddeev-Niemi knot in Skyrme theory, whose mass and decay width are estimated to be around $60 GeV$ and $8 GeV$. As importantly, the existence of the magnet…
Hopf solitons in the Skyrme-Faddeev model are string-like topological solitons classified by the integer-valued Hopf charge. In this paper we introduce an approximate description of Hopf solitons in terms of elastic rods. The general form of the elastic rod energy is derived from the field theory energy and is found to be an extension of the classical Kirchhoff rod energy. Using a minimal extension of the Kirchhoff energy, it is shown that a simple elastic rod model can reproduce many of the qualitative features of Hopf solitons in the Skyrme-Faddeev model. Features that are captured by the mo…
In the Gribov-Zwanziger scenario the confinement of gluons is attributed to an enhancement of the spectrum of the Faddeev-Popov operator near eigenvalue zero. This has been observed in functional and also in lattice calculations. The linear rise of the quark-anti-quark potential and thus quark confinement on the other hand seems to be connected to topological excitations. To investigate whether a connection exists between both aspects of confinement, the spectrum of the Faddeev-Popov operator in two topological background fields is determined analytically in SU(2) Yang-Mills theory. It is foun…
We argue that the Skyrme theory describes the chromomagnetic (not chromoelectric) dynamics of QCD. This shows that the Skyrme theory could more properly be interpreted as an effective theory which is dual to QCD, rather than an effective theory of QCD itself. This leads us to predict the existence of a new type of topological knot, a twisted chromoelectric flux ring, in QCD which is dual to the chromomagnetic Faddeev-Niemi knot in Skyrme theory. We estimate the mass and the decay width of the lightest chromoelectric knot to be around $50 GeV$ and $117 MeV$.
Applying Cho-Faddeev-Niemi decomposition for SU(2) Yang-Mills field, we obtain the Abelian-Higgs Lagrangian by some approximation. Abelian-Higgs Lagrangian with a spontaneous symmetry breaking potential has vortex solutions known as Nielsen-Olesen solutions. We conclude that vortices as well as magnetic monopoles can exist in Cho-Faddeev-Niemi decomposition of SU(2) Yang-Mills field.
A regular method for constructing vortex-like solutions with cylindrical symmetry to the equations of the SU(2) Skyrme chiral model is proposed. A numerical estimate for the length density of mass is given.
We consider a modification of QCD in which conventional fundamental quarks are replaced by Weyl fermions in the adjoint representation of the color SU(N). In the case of two flavors the low-energy chiral Lagrangian is that of the Skyrme-Faddeev model. The latter supports topologically stable solitons with mass scaling as N^2. Topological stability is due to the existence of a nontrivial Hopf invariant in the Skyrme-Faddeev model. Our task is to identify, at the level of the fundamental theory, adjoint QCD, an underlying reason responsible for the stability of the corresponding hadrons. We argu…
A supersymmetric extension of the Skyrme model was obtained recently, which consists of only the Skyrme term in the Nambu-Goldstone (pion) sector complemented by the same number of quasi-Nambu-Goldstone bosons. Scherk-Schwarz dimensional reduction yields a kinetic term in three or lower dimensions and a potential term in two dimensions, preserving supersymmetry. Euclidean solitons (instantons) are constructed in the supersymmetric Skyrme model. In four dimensions, the soliton is an instanton first found by Speight. Scherk-Schwarz dimensional reduction is then performed once to get a 3-dimensio…
We discuss a mixture of interacting a neutral and a charged Bose condensates, which is supposed being realized in interior of neutron stars in form of coexistent neutron superfluid and protonic superconductor. We show that in this system, besides ordinary vortices of $S^1\to S^1$ map, the neutron condensate also allows for (meta)stable finite-length knotted solitons, which are characterized by a nontrivial Hopf invariant and in some circumsatnces are stablized by Faddeev-Skyrme term induced by drag effect. We also consider a helical protonic fluxtube in this system and show that, in contrast, …
The Skyrme crystal, a solution of the Skyrme model, is the lowest energy-per-charge configuration of skyrmions seen so far. Our numerical investigations show that, as the period in various space directions is changed, one obtains various other configurations, such as a double square wall, and parallel vortex-like solutions. We also show that there is a sudden "phase transition" between a Skyrme crystal and the charge 4 skyrmion with cubic symmetry as the period is gradually increased in all three space directions.
