Magnetic Hopfions in the Faddeev-Skyrme-Maxwell model

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 $\phi_3=\pm 1$.


I. INTRODUCTION
Topological solitons appear as classical solutions in various non-linear models, they have been intensively studied over last decades. These regular localized field configurations with finite energy attracted a lot of attention, they emerge in variety of physical, chemical and biological systems.
Interesting examples of stable topological solitons exist in the family of Skyrme-type scalar theories, which can be considered as deformations of the non-linear sigma model. It includes so-called baby Skyrmions in (2+1)-dimensional O(3) model [1,3], Skyrmions in the conventional (3+1)-dimensional Skyrme model [4] and its modifications [5,6], and the Hopfions in the Faddeev-Skyrme model [7,8]. A unifying feature of all these models is that the have the same structure, the corresponding Lagrangians always include the usual σ-model term, the Skyrme term, which is quartic in derivatives of the field, and a potential term, which does not contain the derivatives. According to the Derrick's theorem [9] the potential is optional in 3 + 1 dimensions, however it is obligatory to stabilise the soliton solutions of the planar baby-Skyrme model. Notably, all the models of the Skyrme family on non-compact domain do not saturate the topological bound. In order to attain the bound which yields a relation between the static energy of the solitons and their topological charges Q, one has to modify the model preserving its topological properties, for example, truncate the Faddeev-Skyrme model [11], or oppositely, extend the Skyrme model via coupling it to an infinite tower of vector mesons [12], or completely change the original theory to the form, which supports self-dual equations [5,6,14,15]. Thus, the energy of interaction between the solitons is relatively large, they may attract each other forming various multisoliton configurations, see e.g. [19,20].
Intuitively, the Hopfions can be constructed by considering baby Skyrmions restricted to the plane, which is orthogonal to the direction of the position curve of the string-like configuration [10]. The topological charge of such a soliton corresponds to the product of the winding number of the planar Skyrmions and the number of the twists of the entire configuration in the extra spatial direction. Physically, solitons of that type can be considered as a vortex, which is bending and twisting. The identification of the end points of the vortex yields the loop, which can transform itself into a knot to minimise its energy.
A peculiarity of the interaction potential, both in the case of Skyrmions and Hopfions, is that the asymptotic decay of the fields, which defines the character of interaction, strongly depends to the explicit form of the potential [5,6,[21][22][23][24]29]. Further, various symmetry breaking potentials were considered to construct half-Skyrmions [26-28, 30, 31] or fractional Hopfions [32].
There is another possibility to make alterations to the structure of multisoliton solutions.
In the Faddeev-Skyrme model the vacuum boundary condition should be imposed in such a way that all the points on the boundary are identified. It yields the compactification of the domain space from R 3 to S 3 . Hence the Hopfions are invariant with respect to the global SO(2) symmetry of the vacuum. This allows us to construct the U (1) gauged Faddeev-Skyrme-Maxwell theory by analogy with the extension of the gauged planar baby Skyrme model [34,41,42]. Clearly, electromagnetic interaction will strongly affect the usual pattern of interaction in the system of Hopfions.
Unfortunately, the task of explicit construction of the solutions of the Faddeev-Skyrme-Maxwell theory has been hampered by numerous technical obstacles. Since there is no analytical solutions of the corresponding field equations, the minimizers of the corresponding energy functional can only be obtained numerically. However, it is known that the Hopfions of lowest degrees Q = 1, 2 are axially symmetric [16][17][18], thus in the paper [35] the consideration was restricted to the case of the static axially symmetric gauged unlinked Hopfions A 1,1 and A 2,1 . Assumption of axial symmetry simplifies the consideration significantly since the problem then can be reduced to the numerical solution of system of coupled ordinary differential equations. However, this symmetry is not a general property of general solutions of the Faddeev-Skyrme model supplemented by the Maxwell term, thus this problem should be revisiting.
In this paper we investigate the structure of multisoliton solutions of full coupled Faddeev-Skyrme-Maxwell system. Usually there is an ambiguity in the choice of the topological boundary conditions on the scalar field, however, in the U (1) gauged Faddeev-Skyrme model it becomes dependent on the definition of the electromagnetic group. We consider two choices of the vacuum boundary conditions φ ∞ = (0, 0, 1) and φ ∞ = (1, 0, 0). In both cases we perform full 3d numerical computations to find the corresponding magnetic Hopf solitons in the sectors of degrees up to Q = 8. We study numerically the dependence of masses of the Hopfions and the corresponding magnetic fluxes on the gauge coupling constant. We confirm, that in the strong coupling limit the magnetic fluxes of the Hopfion become quantized in units of 2π.
We found that in a general case the magnetic fluxes of gauged Hopfions are defined by the preimages of the vectors φ = (0, 0, ±1), there is an intrinsic interplay between the topology of the Hopf map and the structure of the magnetic field of the configuration.
The rest of the paper is structured as follows. In the next section we briefly describe the Faddeev-Skyrme-Maxwell model. In the Section 3, for the sake of completeness, we review the rational map approximation used as input in our numerical simulations. Numerical results are presented in Section 4, where we describe various magnetic Hopfion solutions.
For the sake of compactness, we restrict the analysis to the solitons with topological charges up to eight, as a particular example we present a more detailed discussion of evolution of the Q = 5 Hopfions. Conclusions and remarks are formulated in the last Section.

II. U (1) GAUGED FADDEEV-SKYRME MODEL
We consider the Faddeev-Skyrme theory coupled to the Abelian gauge field in (3 + 1) dimensions. The model is defined by the rescaled Lagrangian where the real scalar triplet φ = (φ 1 , φ 2 , φ 3 ) is constrained to the surface of unit sphere, | φ · φ| = 1, so the target space is the sphere S 2 . Since the potential term is optional, we do not consider it. However, the global SO(3) symmetry will be broken as we impose the topological vacuum boundary conditions, like φ ∞ = (0, 0, 1), which yield a one-point compactification of the domain space R 3 to S 3 . Note that this common choice is not unique, below we will also consider another case, φ ∞ = (1, 0, 0).
Thus, the field of the Hopfion is a map φ : R 3 → S 2 which belongs to an equivalence class characterized by the homotopy group π 3 (S 2 ) = Z. Explicitly, the Hopf invariant is defined non-locally as where F ij = φ · (∂ i φ × ∂ j φ) and one-form A = A k dx k is defined via F = dA, i.e the two-form F is closed, dF = 0.
The model (1) includes also the usual Maxwell term with the field strength tensor F µν = Note that under a spacial rescaling x → λx, this term in the action scales as λ −1 , i.e. it has the same scaling properties as the quartic in derivatives Skyrme term. The flat metric is g µν = diag(1, −1, −1, −1) and the coupling of the scalar triplet to the gauge field is given by the covariant derivative [34,35,40,41] where the unit vector n = (0, 0, 1) defines the direction of the electromagnetic subgroup. Explicitly, where φ ⊥ = φ 1 +iφ 2 are planar components of the scalar field. Thus, the third component φ 3 remains decoupled from the gauge potential. However, since the scalar triplet is restricted to the surface of the unit sphere, coupling of the planar components φ ⊥ to the gauge sector still affects the component φ 3 indirectly.
The Abelian gauge transformations act on the fields as thus, we can make use of this symmetry to set A 0 = 0. Further, restricting our analysis to static configurations, we consider purely magnetic field The static energy functional of the model (1) is Here we are using normalized units of energy, rescaling it as E → E/(32π 2 √ 2). The Hopfions correspond to the stationary points of this functional. Note that the condition of finiteness In other words, on the spacial asymptotic the field of the gauged Hopfion must lie in an orbit of the gauge group, it is not necessarily a constant there.
The complete set of the field equations, which follows from the variation of the action of the model (1), is Here the scalar current is and a source in the corresponding Abelian Maxwell equations is j µ = n · J µ . This system is similar to the corresponding equations of the planar Skyrme-Maxwell theory [40][41][42], however, the topological properties of the fields are different.
Unlike other solitons, the location of the Hopfions does not correspond to the maximum of the topological charge density, the Hopfions are extended string-like configurations in 3 dimensional space. The maxima of the energy density distribution can be identified as the curve of positions of the preimage of the point φ 0 = (0, 0, −1), which is antipodal to the vacuum [18]. This curve is usually referred to as the position curve [18]. In the gauged Faddeev-Skyrme model this curve has also another meaning.
Note that we can make use of the trigonometrical parametrization of the scalar field φ = (sin ψ cos σ , sin ψ sin σ , cos ψ) , where two functions ψ(x, y, z), σ(x, y, z) satisfy the boundary conditions on the Hopfion configuration in a given topological sector. Although this parametrization is not the most convenient from the point of view of numerical simulations [44], it automatically takes into account restriction of the scalar field to S 2 . This in particular, allows for a more transparent understanding of many peculiarities of the gauged Faddeed-Skyrme model.
In a simple case of the axially symmetric gauged Hopfions A 1,1 and A 2,1 [16], the function σ can be explicitly written in spherical coordinates (r, θ, ϕ) as σ = nϕ − mG(r, θ), here two winding numbers n, m ∈ Z correspond to the planar winding and the twisting of the configuration, respectively. The phase function of the axially symmetric configuration G(r, θ) increases by 2π after one revolution around the core, thus the Hopf number of the soliton is just a product of two windings, Q = mn and the axially symmetric configuration of the type A m,n can be thought of as composed from planar baby Skyrmion of charge n twisted m times along the circle [16].
In a general case, by analogy with the similar situation in the gauged planar Skyrme model [43], the Abelian current can be written as We can assume that the gauge potential A i slowly varies in space. Then, from the second equation in (7), we can see that, in the limit of infinitely large gauge coupling g → ∞ this equation is satisfied only if the Abelian current (10) becomes zero. Evidently, if φ 3 = ±1, the current is vanishing when A i = ∂ i σ, i.e. the magnetic potential becomes a pure gauge everywhere in 3d space apart the curves C ± = φ −1 (0, 0, ±1).
Considering the magnetic flux through the area, transverse to the direction of the φ 3 , we can see that where Γ is a closed contour encircling the points, where φ = (0, 0, ±1).
The consideration above explains the effective quantization of the magnetic fluxes of the gauged axially symmetric Hopfions of degrees Q = 1, 2 in the strong coupling limit [35]. It was observed that the configuration is associated with two magnetic fluxes, one of which represents a circular vortex, and the second one is orthogonal to the position curve [48]. In the strong coupling limit the former flux is quantized in units of the winding number n while the latter flux is quantized in units of m.
Indeed, the position curve of the Hopfion is defines as the preimage of the point φ = (0, 0, −1) on the target space. On the other hand, for the axially symmetric Hopfions the component φ 3 is approaching the vacuum on the symmetry axis, so φ = (0, 0, 1) as r = 0.
Therefore there are two associated magnetic fluxes, both becomes quantized in the strong coupling limit. More generally, the curves of φ 3 = ±1 define the structure of the magnetic fluxes of a gauged magnetic Hopfion.

III. INITIAL APPROXIMATION
The task of finding of multi-soliton solutions of the Faddeev-Skyrme model in a given sector of degree Q is very complicated, it can be performed only numerically. Moreover, it involves rather sophisticated numerical technique see, e.g., [18].
As usual, the energy minimization scheme needs an appropriate initial configuration in a given sector. The most effective approach here is related with the rational map approximation, suggested by Sutcliffe [18]. One can consider two complex variables which parameterize where f (r) is a monotonically decreasing function with the boundary values f (0) = π and f (∞) = 0. The coordinates Z 1 , Z 0 are restricted to the unit sphere S 3 , i.e. |Z 1 | 2 + |Z 2 | 2 = 1.
This allows us to construct a map R 3 → S 3 ∈ C 2 .
The components of the scalar field φ, which are coordinates on the target space S 2 , are given by the rational map W : S 3 ∈ C 2 → CP 1 : where the polynomials P (Z 1 , Z 0 ) and Q(Z 1 , Z 0 ) have no common roots on the two-sphere S 2 . The rational map ansatz (13) produces a curve in R 3 , therefore the first Hopf map There are three different types of input configurations. The axially symmetric Hopfions A mn are produced by the rational map [18] This Hopfion has a single position curve C − = φ −1 (0, 0, −1).
More generally, we can consider initial configurations, which are given by maps of the where α is a positive integer and β is a non-negative integer. These maps have Hopf degree Q = αb + βa, the corresponding configuration is a torus knot K a,b . In a particular case when a and b are not coprime integers, the rational map (15) is degenerated producing a link with two or more interlinked and disconnected position curves. Configurations of that type are labeled as L n,m a,b , here the subscripts label the Hopf indexes of the unknots and the superscript above each subscript counts the secondary linking number, which appears due to inter-linking with the other components.
Since these loops are linked Q times, the definition of the linking number can be related with the positions of the preimages of these points: Q = link(C − , C 1 ). Other choice of the preimages are also possible [32].
The input for the magnetic potential A i at a finite value of the gauge coupling g can be taken as a generalization of the limiting form of the pure gauge condition above, where the function σ(φ ⊥ ) appeared in the trigonometric parametrization (9) of the scalar fields. The smooth function A(φ 3 ) must satisfy the restrictions A(±1) = 0 and A(0) 1, it agrees with the parametrization used previously to construct axially symmetric solutions of the gauged Faddeev-Skyrme model [35]. Thus, we can take A(φ 3 ) = 1−φ 2 3 as an appropriate choice.

IV. NUMERICAL RESULTS
For our numerical computations we used algorithm of minimization of the energy functional (6) described in [34,45]. The fields are discretized on the grid with 100 3 or 150 3 points with grid spacing ∆x = 0.1. The initial configurations were produced via the rational map approximation as above. As a consistency check, we verify that our algorithm correctly reproduces the known results for the Hopfion configurations of the usual decoupled Faddeev-Skyrme model at g = 0 and for the gauged axially symmetric configurations A 11 , A 21 previously discussed in [35].
The solutions of that type, A 11 and A 21 are global minima in the sectors of degrees Q = 1, 2, respectively. They represent axially symmetric unknots with the position curve Note that with the usual choice of the vacuum boundary conditions φ ∞ = (0, 0, 1), the vector n, which appears in the definition of the covariant derivative (3), is parallel to φ ∞ .
Below we will also consider another situation, when on the spacial boundary φ ∞ = (1, 0, 0), and n is transverse to φ ∞ .
As the gauge coupling gradually increases from g = 0, the energy of the configuration decreases since the magnetic flux is formed and core of the Hopfions shrinks. The magnetic energy is initially increasing, however its contribution starts to decrease as g becomes larger than g = 1 [35]. The structure of the magnetic field follows the pattern above, in the weak coupling regime there is a toroidal magnetic field, which encircles the position curve of the Hopfion. As the gauge coupling constant increases, the curves of the preimages of the points C ± = φ −1 (0, 0, ±1) paves the way for magnetic flux tubes. Indeed, we can clearly identify two fluxes along these curves, the first flux is directed alon the symmetry axis of the configuration, the second circular magnetic flux is orthogonal to the x − z plane, see Fig 1. Both fluxes become quantized in units of 2π in the strong coupling limit [35].
Further, increase of the gauge coupling leads to shrinkage of the Hopfion, magnetic field effectively squeezes the configuration. This effect is opposite to the isorotations of the Hopfions, which also affect the structure of the solutions [36,37].
Note that there is another axially symmetric Hopfion configuration in the sector of degree two, 2A 12 [16,18,38,39]. It can be thought as two Q = 1 Hopfions stacked one above the other. In the limit g = 0 this solution is a saddle point configuration, which has higher energy than the A 21 Hopfion. As g increases, it still remains as a saddle point, see Fig. 3 Interestingly, there is a certain similarity between the structure of the magnetic field of Q = 1, 2 axially symmetric Hopfions and toroidal magnetic fields which are well known in solar and plasma physics, see e.g. [46]. In the latter case the magnetic field appears as a solution of so called force free equation for a plasma current j × B = 0.
In order to see the difference between the cases of magnetic field of the gauged Hopfions and the magnetic field in a plasma device, such as a stellerator or tokamak, let us assume that the scalar current j is a plasma current. However, the field free equation leads to B = α j, thus for a constant α we obtain the Helmholtz equation: On the other hand, the magnetic field of the Hopfions is generated by the scalar current (8), which can be written as where K = ∇σ − A. Thus, the corresponding Maxwell equation becomes Since in the strong coupling regime the magnetic flux tubes follow the directions of the curves of preimages of the vectors φ = (0, 0, ±1), we can assume that outside of these curves Since B = ∇ × A = − ∇ × K, we can see that the counterpart of the force free equation for the magnetic field of the Hopfions can be written as Notably, this is a London type equation with the penetration depths parameter 1 √ 2g , the mass term here has a sign opposite to the one in the force free equation (17). Thus, in the strong coupling limit the magnetic field of a Hopfion exhibit a sort of Meissner effect. For Hopf degree Q = 4 there are possibilities to construct initial configurations of types 4A 22 , 4A 41 and 4L 11 11 . In the usual Faddeev-Skyrme model without the magnetic field, the axially symmetric Hopfion A 22 , which may be thought of as two adjacent 2A 21 solitons in the maximally attractive channel of interaction, represents the global minimum [18]. Numerical relaxation of the initial 4A 41 configuration yields a buckled Hopfion 4 A 41 , however in the limit g = 0 its energy is about 2% above the global minimum. Situation changes as the gauge coupling increases, the interaction with magnetic field tends to straighten out the position curve, thus the axially symmetric Hopfion 4A 41 has lower energy than 4A 22 .
In the usual Faddeev-Skyrme model at g = 0, in the sector of degree four the link 4L 1,1 1,1 does not exist as a local minimum. However, this type of solution, 5L 1,1 2,1 is a minimizer for Q = 5 Hopfions. We observe that increase of the gauge coupling transforms it into configuration of a different type. As g ∼ 1, the magnetic attraction between the fluxes, associated with the collection of loops C − = φ −1 (0, 0, −1), deforms the position curve, it corresponds to two adjacent loops which are not linked to each other, see Fig. 1. Further, such a configuration is not a global minimum in this sector, as g 0.2, the axially symmetric configuration A 51 has lower energy, see Fig. 4.
Note that the magnetic field of all axially symmetric gauged Hopfions A n1 represents two magnetic fluxes, one flux encircles the position curve of the Hopfion and the second one is directed along the symmetry axis, this pattern is is illustrated in Fig. 1. As we discussed above, in the strong coupling limit both fluxes are quantized in units of 2π and 2πn, respectively.
Considering the structure of the magnetic field of a A n1 Hopfion with axial symmetry weakly broken, we observe that one of the magnetic fluxes follows the position curve C − = φ −1 (0, 0, −1). Another flux is associated with direction of the vector n = (0, 0, 1) in (3). This is also the case of the Q = 5 Hopfion, as seen in the 9th row of Fig. 1 the magnetic fluxes follow the loops, which are preimages of C − = φ −1 (0, 0, −1) and C + = φ −1 (0, 0, 1).
The axially symmetric configurations A n1 represent global minima in the strong coupling regime up to Q ≤ 5. When Q = 6 there are two different links 6L 1,1 2,2 and 6L 1,1 3,1 , and the axially symmetric configurations of two types, A 32 and A 61 , respectively. As g = 0 the link 6L 1,1 2,2 represent the global minimum, it has a lower energy than other configurations in that sector for all values of g. Increase of the coupling constant and related magnetic interaction deforms this initial configuration into an axially symmetric soliton 6A 32 , as g = 1 it has energy about 16% lower than another axially symmetric Hopfion 6A 61 .
The trefoil knot 7K 3,2 is the only minimiser in the sector of degree seven as g = 0.
However, as the gauge coupling grows, the magnetic fluxes, associated with the loops C − = φ −1 (0, 0, −1) tends to merge because of attraction between them. At g = 1 the knot becomes deformed into configuration, whose position curve represents two contacting loops, which are not interlinked, see Fig. 1. This structure is similar with the corresponding solution in the sector Q = 5.
At degree Q = 8 there are three energy minima, which represent a link 8L 1,1 3,3 , a knot 8K 3,2 and axially symmetric Hopfions 8A 42 , respectively. As g = 0 the link L 1, 1 3,3 has the energy a little less than the knot 8K 3,2 . However, as g increase, the axial symmetry is recovered and the 8A 4,2 becomes global minimizer in that sector as g = 1. This Hopfion is composed of two 4A 41 Hopfions stacked one above the other with the orientation in the maximally attractive channel. Another axially symmetric configuration 8A 81 in the strong coupling limit has a bit higher energy. Again, we observe that the structure of the magnetic fluxes is completely determined by the preimages of C − = φ −1 (0, 0, −1) and C + = φ −1 (0, 0, 1), see Unlike gauged planar baby Skyrme model [34,41,42], in the 3 dimensional Faddeev-Skyrme-Maxwell model (1) the U (1) gauging prescription (3) is not necessarily correlated with the vacuum boundary condition imposed on the scalar field. The topological restriction on the scalar field is that on the spacial boundary φ must approach the same vacuum value regardless of direction, then R 3 → S 3 .
On the other hand, the condition of finiteness of the energy of the system for any choice of the vacuum requires the same restrictions on the spacial infinity Choosing an appropriate gauge, we can just impose φ ∞ = const.
In our consideration above we suppose that the vector n = (0, 0, 1) and the vacuum φ ∞ are parallel, let us consider another possibility imposing the boundary condition [49] φ The results of energy minimization simulations are summarized in Fig. 2. We confirm that the structure of the magnetic field of the gauged Hopfion is always determined by the collection of loops C ± = φ −1 (0, 0, ±1), for any value of the parameter β.
We observe that as the parameter β is increasing, the energy of the Q = 1 Hopfion is decreasing, however, the static energy of the configurations of higher degrees is increasing. C. Dependence on gauge coupling for Q = 5 As a particular example of the parametric dependency of the gauged Hopfions on the coupling constant g, we considered solitons in the sector of degree Q = 5, both in the case of the vacuum φ ∞ = (0, 0, 1) and φ ∞ = (1, 0, 0).
In Fig. 4 we have plotted the graphs of total energy of the gauged 5A 5,1 and 5L 1,1 1,2 Hopfions, defined by the functional (6), and magnetic energy as function of the gauge coupling g. As g = 0 both choices of the vacuum are equivalent, the energy of the link 5L 1,1 1,2 is lower than the axially symmetric Hopfion 5A 5,1 . However, as the gauge coupling increases from zero, the latter configuration becomes a global minimum in the vacuum φ ∞ = (0, 0, 1), while the link 5L 1,1 1,2 still remains the minimal energy solution in the vacuum φ ∞ = (1, 0, 0), as seen in the left plot, Fig. 4.
General observation is that, as the gauge coupling increases, the energy of the gauged Hopfion monotonically decreases. On the other hand, the magnetic energy initially increases from zero, it attains its maximum at g 0.7. Further increase of the coupling leads to decrease of the magnetic energy, as shown in the right plot, Fig. 4. As expected, the size of the Hopfions decreases as the coupling g increases.
In Figs. 5,6 we display the pattern of evolution of the initial 5A 5,1 and 5L 1,1 1,2 configurations, as the gauge coupling g is growing from zero. In the model (1) with the usual choice of the vacuum φ ∞ = (0, 0, 1), coupling to the magnetic field, directed along the vector n φ ∞ , recovers the axial symmetry of the 5A 5,1 Hopfion, which is violated as g 0. 25. Within that range of values of g the bent axial solution 5 A 5,1 is a local energy minimum, the minimal energy configuration in this sector remains the link 5L 1,1 1,2 . However, as g 0.25 the lowest energy solution is the axially symmetric Hopfion 5A 5,1 . The magnetic fluxes of this configuration are directed through the center of the Hopfion and around the symmetry axis, as shown in Fig. 5, upper panel. In the strong coupling limit the link 5L 1,1 1,2 becomes strongly deformed, see The work here should be taken further by considering the electrically charged configurations, another interesting direction is to investigate the soliton solutions of the SO (3) gauged Faddeev-Skyrme model. It might be also interesting to consider gauged Hopfions in frustrated magnets, which combine nearest-neighbour ferromagnetic and higher-neighbour anti-ferromagnetic interactions [47]. We hope we can address these issues in our future work.