MMultimagnetic Monopoles
D. Bazeia, M. A. Liao, and M. A. Marques
2, 1 Departamento de F´ısica, Universidade Federal da Para´ıba, 58051-970 Jo˜ao Pessoa, PB, Brazil Departamento de Biotecnologia, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa, PB, Brazil (Dated: January 29, 2021)In this work we investigate the presence of magnetic monopoles that engender multimagneticstructures, which arise from an appropriate extension of the SU(2) gauge group. The investigationis based on a modified relativistic theory that contain several gauge and matter fields, leading to aBogomol’nyi bound and thus to a first order framework, from which stable multimagnetic solutionscan be constructed. We illustrate our findings with several examples of stable magnetic monopoleswith multimagnetic properties.
I. INTRODUCTION
Magnetic monopole has been an important object ofinvestigation in theoretical physics ever since Dirac firstshowed that it can be incorporated into the framework ofelectromagnetism in a simple and elegant way [1, 2]. Ofparticular importance is the demonstration, also givenby Dirac, that their mere existence could, when com-bined with the postulates of quantum mechanics, providea natural explanation for the quantization of electric (andmagnetic) charge.Despite this theoretical triumph, lack of experimentalevidence for the existence of monopoles led Dirac’s the-ory to be regarded as a mathematical curiosity. Almostfifty years later, ’t Hooft [3] and Polyakov [4] indepen-dently found a topological monopole solution for a Yang-Mills-Higgs theory; see also Refs. [5–7]. The ’t Hooft-Polyakov monopole presents a magnetic charge which isdirectly proportional to a topological invariant called thewinding number, or topological degree [8, 9]. Perhapseven more important than the solution itself was the re-alization that magnetic monopoles were an unavoidableconsequence of spontaneous symmetry breaking in grandunification theories [3]. The so called monopole problem,regarding the apparent absence of these particles in ouruniverse despite the theoretical predictions of their pro-duction in the early universe [10] was one of the mainmotivations for the development of inflationary universemodels [11, 12].The study of magnetic monopoles has further evolvedby the introduction of models with enhanced gauge sym-metry [13, 14]. This enhancement allows for more com-plex monopoles, which now emerge from a much richertopology and present internal structure. Works [15–17]have been developed along these lines, and we have alsoconducted a similar investigation in (2,1) dimensions,which led to vortices possessing a novel internal struc-ture, which is reflected by the magnetic field associatedwith these solutions [18]. In this sense, we believe thatthe present study, in which we deal with models wherethe SU(2) symmetry is enhanced to describe products ofseveral SU(2) factors will also induce interest in planarsystems, following the lines of [18]. In the case of vortices,enhancement of the basic U(1) symmetry to a product of N factors, U(1) × U(1) × · · · ×
U(1), can be directly con-nected to the addition of several order parameters in thesystem. The interest here can be extended to vortices insuperconductors described by the Ginzburg-Landau the-ory and is directly connected with the study of two- andthree-component systems that appeared in Refs. [19–21].The subject is also of interest to the case of the multi-component Gross-Pitaevskii equations that describe themean-field dynamics of spin-1 and spin-2 Bose-Einsteincondensates; see, e.g., Ref. [22] and references thereinfor more information on these issues. Moreover, the ideacan also be very naturally extended to kinks in the realline, in direct connection with the work [23], in which theZ symmetry is enhanced to Z × Z , and can be used inapplications similar to the very recent one, described in[24], where geometrically constrained kinks contribute tomodify the behavior of fermions, leading to situations ofpractical interest for the constructions of electronic de-vices at the nanometric scale. There are other more in-volved situations, in which the gauge group can appear asa product of the form SU(N ) × SU(N ) × · · · × SU(N p );this is the case, for instance, in the Atiyah, Drinfeld,Hitchin, and Manin construction of multi-instantons withproduct group gauge theories that arise from D3-branes;see, e.g., [25] and references therein.In this work we will build upon the investigation con-ducted in [17] to construct multimagnetic structures,which emerge from the superposition of magnetic shells,each of which is the result of spontaneous symmetrybreaking in a SU(2) subgroup of the full gauge group,under whose transformations our models will be takento be invariant. Research from the last decade hasbrought to light a great variety of interesting applica-tions for bimagnetic structures, ranging from engineer-ing to biomedicine [26–29]. One particularly interest-ing application lies in the possibility of using the ex-change bias present on multilayers of magnetic materialfor beating the superparamagnetic limit, which normallyconstrains the miniaturization of magnetic recording de-vices [30, 31]. Such multilayered or onyon-like struc-tures have been investigated in relation to nanoparti-cles [32, 33] and, as multimagnetic structures expandupon the bimagnetic models studied earlier, so we hopethat the systems investigated here may be useful to ex- a r X i v : . [ h e p - t h ] J a n pand on the aforementioned applications. Another in-teresting property of our models lie in the presence of aBogomol’nyi-Prasad-Sommerfield (BPS) limit, in whichthe problem is reduced to the solution of a system of firstorder differential equations. Solutions of this type areglobal minima of the energy, and are therefore expectedto be stable under small fluctuations.Magnetic monopoles have also been found in spin icesystems [34, 35] and more recently, the direct observationof static and dynamics of emergent magnetic monopolesin a chiral magnet has been reported in Ref. [36]. Theyare also object of theoretical and experimental study inheavy-ion collisions and in neutron stars [37], and also inthe international research collaboration MoEDAL run-ning at CERN with the prime goal of searching for themagnetic monopole [38]. These studies motivate that weinvestigate new models of magnetic monopoles in high en-ergy physics, hoping to contribute with the constructionof models that allow the addition of internal structure,bringing novel properties and/or features at the funda-mental level.In order to disclose the investigation, we organize thepresent work as follows: In Sec. II we introduce the gen-eralized Lagrangian that defines our models, and derivethe second order equations that follow from it. We theninvestigate the conditions under which a Bogomol’nyibound is attainable, and examine the first order equa-tions which must be satisfied by BPS solutions. We thenproceed, in Sec. III and IV, to solve some examples whichengender the sought-after multimagnetic structures, andinvestigate their properties. Finally, in Sec. V, we discussour results and add some perspectives of future works inthe subject. II. GENERAL PROCEDURE
We work in four dimensional Minkowski spacetimewith metric ( − , + , + , +), and consider a class of non-abelian models with the Lagrangian density of the form L = − N (cid:88) n =1 P ( n ) ( {| φ |} )4 F ( n ) aµν F ( n ) aµν − N (cid:88) n =1 M ( n ) ( {| φ |} )2 D ( n ) µ φ ( n ) a D ( n ) µ φ ( n ) a − V ( {| φ |} ) , (1)where N is a positive integer, ( {| φ |} ) stands for theset ( | φ (1) | , | φ (2) | , ..., | φ ( N ) | ), D ( k ) µ φ ( k ) a = ∂ µ φ ( k ) a + g ( k ) ε abc A ( k ) bµ φ ( k ) c is the action of covariant derivative inthe k -th sector with coupling g ( k ) and F ( k ) aµν = ∂ µ A ( k ) aν − ∂ ν A ( k ) aµ + g ( k ) ε abc A ( k ) bµ A ( k ) cν is the k -th field strength.The potential is denoted by V ( {| φ |} ).This Lagrangian is invariant under the action of theproduct of N SU (k) (2) groups, in the form SU (1) (2) × ... × SU (N) (2). Each scalar field is a SU(2)-valued quan-tity, given by φ ( n ) = φ ( n ) a T a , where T a = − iσ a / σ a . Each φ ( n ) is charged only in therespective SU (n) (2) subgroup, and is thus coupled to agauge field A ( n ) µ = A ( n ) aµ T a . The condition (cid:104) T a , T b (cid:105) = − a T b ) = δ ab may be used to define an innerproduct in SU(2), under which one obtains the norm | φ ( n ) | = (cid:112) φ ( n ) a φ ( n ) a . Throughout this work, the Ein-stein summation convention is always implied for Lorentzand SU(2) internal indices, but not for the remainingones, which simply label the SU(2) subgroups and arealways enclosed in parentheses.The functions P ( n ) and M ( n ) are nonnegative and may,in principle, depend on all the scalar fields {| φ |} , respect-ing the gauge symmetry, but we shall shortly restricttheir forms, aimed to implement some possibilities capa-ble of giving interesting stable minimum energy configu-rations. We generically require that these functions takefinite values when the field configurations approach thevacuum manifold, which will be topologically non trivial,in order to allow for finite energy topological solutions.The solutions can be classified according to a set of N integers, which, in analogy to the standard monopole,we may relate to quantized “magnetic” charges emerg-ing from the unbroken U(1) symmetry from each SU(2)submanifold.The field equations derived from (1) for the k -th sectorare D ( k ) µ (cid:16) M ( k ) D ( k ) µ φ ( k ) a (cid:17) = 14 N (cid:88) n =1 ∂P ( n ) ∂φ ( k ) a F ( n ) bµν F ( n ) bµν + 12 N (cid:88) n =1 ∂M ( n ) ∂φ ( k ) a D ( n ) µ φ ( n ) b D ( n ) µ φ ( n ) b + ∂V∂φ ( k ) a , (2a) D ( k ) µ (cid:16) P ( k ) F ( k ) aµν (cid:17) = g ( k ) M ( k ) ε abc φ ( k ) b D ( k ) ν φ ( k ) c , (2b)where the covariant derivatives also act in the followingform D ( n ) µ F ( n ) aµν = ∂ µ F ( n ) aµν + g ( n ) ε abc A ( n ) bµ F ( n ) cµν .Let us now consider static solutions, which may beobtained as time independent fields which obey equationsof motion in a gauge such that A ( n )0 = 0 (leading tothe non-abelian analogues of Gauss’ law to be triviallysolved). The energy functional in this case is given by E = N (cid:88) n =1 (cid:90) d x (cid:20) M ( n ) | D ( n ) k φ ( n ) | + 12 P ( n ) | B ( n ) k | + V ( {| φ |} )] , (3)where B ( n ) k ≡ ε ijk F ij ( n ) / M ( n ) = 1 /P ( n ) for each n and taking the limit V ( {| φ |} ) → E = N (cid:88) n =1 (cid:90) d x P ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B ( n ) k ± D ( n ) k φ ( n ) P ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 4 π N (cid:88) n =1 v ( n ) g ( n ) | Q ( n ) t | , (4)where Q ( n ) t ∈ Z is the topological charge associated withthe ( φ ( n ) , A ( n ) µ ) solutions, defined in the same way as inthe standard SU(2) Yang-Mills-Higgs theory [9]. The sec-ond term in the right hand side is recognized as the sum-mation of the fluxes arising from the magnetic fields ineach sector. The observation that the first integral in (4)is non-negative leads to the Bogomol’nyi bound E ≥ π N (cid:88) n =1 v ( n ) g ( n ) | Q ( n ) t | . (5)The equality in (5) is attained if and only if the first orderequations B ( n ) k = ∓ D ( n ) k φ ( n ) P ( n ) (6)are satisfied for each n . By direct derivation it is straight-forward to show that solutions of (6) always solve thesecond order equations. Deviations from the static casegive rise to quadratic terms in F ( n ) j and ∂ φ ( n ) whichmust be added to the integral in (3). Such terms mayonly increase the energy, and thus configurations thatsolve (6) are global minima of this functional.We shall henceforth concern ourselves with sphericallysymmetric static solutions. Since we are choosing A ( n )0 =0, as is well known, these restrictions allow us to writethe fields in the form [3, 4] φ ( n ) a = x a r H ( n ) ( r ) , (7a) A ( n ) ai = ε aib x b g ( n ) r (1 − K ( n ) ( r )) , (7b)subject to the conditions H ( n ) (0) = 0 , K ( n ) (0) = 1 ,H ( n ) ( ∞ ) → ± v ( n ) , K ( n ) ( ∞ ) → . (8)These requirements ensure that the ensuing energyis finite, while also enforcing symmetry breaking andthus resulting in topological solutions. The form of thisAnsatz implies | Q ( n ) t | = 1 for all n , so that the magneticcharge in each subsystem achieves the smallest non-zerovalue allowed by the quantization condition. In the stan-dard magnetic monopole the vacuum manifold is the 2-sphere S , but in the present model it is the product of N S × S × · · · × S . As one knows, the second homotopy group of S is the group of theintegers, Z ; however, we also know that the second ho-motopy group of the product of N S is theproduct of N second homotopy group of the 2-sphere S ,so we have Z × Z × · · · × Z , that is, a set of N integers.In this sense, the classification by a set of N integers ex-hausts the finite energy solutions, including in particularthe spherically symmetric solutions that we will describebelow. Under the above assumptions, the second orderequations take the form1 r (cid:16) r M ( n ) H ( n ) (cid:48) (cid:17) (cid:48) − M ( n ) H ( n ) K ( n )2 r − N (cid:88) k =1 ∂P ( k ) ∂H ( n ) (cid:32) K ( k ) (cid:48) g ( k )2 r − (1 − K ( k )2 ) g ( k )2 r (cid:33) − N (cid:88) k =1 ∂M ( k ) ∂H ( n ) (cid:32) H ( k ) (cid:48) + 2 H ( k )2 K ( k )2 r (cid:33) − ∂V∂H ( n ) = 0 , (9a)and r (cid:16) P ( n ) K ( n ) (cid:48) (cid:17) (cid:48) − K ( n ) (cid:110)(cid:16) M ( n ) g ( n )2 r H ( n )2 (cid:17) + (cid:16) P ( n ) (1 − K ( n )2 ) (cid:17)(cid:111) = 0 , (9b)where the prime denotes differentiation with respect tothe radial coordinate. These equations constitute a setof 2 N coupled and nonlinear ordinary differential equa-tions. This system is very hard to solve, even when wechoose simple couplings among the N sectors with SU(2)symmetry to complete the model. For this reason, letus substitute the Ansatz into (6) and take the same as-sumptions used in that derivation to find the first orderequations in the presence of spherical symmetry. For each n , there result two independent equations, obtained byprojecting (6) into directions parallel and orthogonal to φ ( n ) , so that the energy is minimized by configurationsthat solve the N pairs of first order equations H ( n ) (cid:48) = ± P ( n ) − K ( n )2 g ( n ) r , (10a) K ( n ) (cid:48) = ∓ g ( n ) H ( n ) K ( n ) P ( n ) , (10b)for n = 1 , ..., N . These configurations saturate the Bogo-mol’nyi bound, with energy E = 4 π N (cid:88) k =1 v ( k ) g ( k ) , (11)so that each SU(2) subsystem contributes with an ad-ditive factor 4 πv ( k ) /g ( k ) to the total energy. This, ofcourse, is a reflection of the fact that these solutions haveunity topological charges in each sector. We also notethat the energy does not depend on any of the P k , sothe Bogomol’nyi bound only depends on the asymptoticvalues of the fields, which always approach the trivialvacuum solutions when the boundary conditions (8) aresatisfied.We may choose the functions P k in a way that makesit easier to solve the first order equations while still keep-ing the subsystems coupled in a non-trivial way. In orderto achieve this, we can consider several distinct possibil-ities, some of them will be considered below to illustratethe general situation. Before doing this, however, let usnotice that the energy density for solutions of the firstorder equations can be written in the form ρ ( r ) = N (cid:88) k =1 ρ ( n ) ( r ) , (12)where ρ ( n ) ( r ) = 2 P ( n ) K ( n ) (cid:48) ( g ( n ) r ) + H ( n ) (cid:48) P ( n ) , = P ( n ) (1 − K ( n )2 ) ( g ( n ) r ) + 2( H ( n ) K ( n ) ) r P ( n ) , (13)which represents the energy density for the n -th sector. III. MODELS WITH N=3
Let us first illustrate the aforementioned procedurewith a few examples, in which the Lagrangian is of theform (1), with gauge group SU(2) × SU(2) × SU(2).For simplicity, let us take g (1) = g (2) = g (3) = 1 and v (1) = v (2) = v (3) = 1 in our calculations and, sincethe upper and lower signs in equations (10) are relatedby changing H → − H , we then only consider the uppersigns in the investigation that follows. A. First Model
As a first example, we shall consider a model with astandard monopole core, meaning P (1) = 1. The k = 1equations in (10) are thus H (1) (cid:48) = 1 − K (1) r , (14a) K (1) (cid:48) = − H (1) K (1) , (14b)which are simply the Bogomol’nyi equations for the ’tHooft-Polyakov monopole, for which an analytic solutionhas long been known [5]: H (1) ( r ) = coth( r ) − r , K (1) ( r ) = r csch(r) . (15)The energy density for this solution can be foundfrom (13) to be ρ ( r ) = [ r − sinh ( r )] r sinh ( r ) + 2( r coth( r ) − r sinh ( r ) . (16) We may now proceed as in [17] and introduce a shellstructure by the choices P (2) = | φ (1) | − α , P (3) = | φ (2) | − β ,where α and β are positive integers. With these choices,we are led to the equations H (2) (cid:48) = (cid:18) r coth( r ) − (cid:19) α − K (2)2 r − α , (17a) K (2) (cid:48) = − H (2) K (2) (cid:18) coth( r ) − r (cid:19) α . (17b)and H (3) (cid:48) = (cid:18) H (2) (cid:19) β − K (3)2 r , (18a) K (3) (cid:48) = − H (3) K (3) (cid:16) H (2) (cid:17) β . (18b)Note that we have inserted the solution (15) for H (1) ( r )in equations (17), resulting in a system of two first or-der equations which can now be solved for the two un-known functions H (2) and K (2) . We have not been ableto obtain H (2) and K (2) in closed form, but the problemis amenable to numerical analysis. At large distances, P ( n ) ( r ) → H (2) = H (2)0 , K (2) = 1 − K (2)0 , with H (2) ∝ r ( − − α + √ α +2 α +9) / and K (2) ∝ r (1+ α + √ α +2 α +9) / . These expressions tellus how different choices of the parameter α may affectthe short-distance behavior of the solution. A similaranalysis can be conducted for the other shells and for ev-ery example we present here, but the explicit dependenceon the radial coordinate becomes increasingly more com-plicated for the outermost shells, and shall therefore beomitted.Once the full equations are solved numerically, we mayplug the result H (2) in (18) in order to solve these equa-tions. We have done this for the case α = 3 , β = 10, andpresent the results in FIGs 1, 2 and 3. One sees that in allcases these functions present a behavior that is qualita-tively similar to that of the standard ’t Hooft-PolyakovMonopole. The contributions ρ , ρ and ρ to the en-ergy density can be calculated from (13). Their behavioris represented in Figs. 1, 2 and 3. The last two contribu-tions are strikingly different from the standard monopolesolution (which has energy density equal to what we herehave called ρ ), presenting a maximum for r (cid:54) = 0.The total energy could be calculated by direct inte-gration of ρ = ρ + ρ + ρ , but these solutions solve thefirst order equations, so we may simply apply (11) to find E = 12 π .As a visual aid that may be useful for the interpretationof our results, we depict, in Fig. 4, a planar section of theenergy density, in which the the contributions ρ , ρ and ρ have been plotted simultaneously. We see how a novelstructure, comprised by a standard monopole core andtwo shells, arises from the superposition of the different FIG. 1. In the left panel, we show the solutions H (1) ( r )(ascending line) and K (1) ( r ) (descending line) of (14) for α = 3 , β = 10. Their analytical form is given by (15). Inthe right panel, we show the contribution ρ to the energydensity, given by (16).FIG. 2. In the left panel, we show the solutions H (2) ( r )(ascending line) and K (2) ( r ) (descending line) of (17) for α = 3 , β = 10. In the right panel, we show the contribu-tion ρ , calculated from (13), relative to this sector. subsystems. The size of each shell is controlled by thechoice of the parameters α and β . The maxima of ρ ( r ), ρ ( r ) and ρ ( r ) are attained at r = 0, r ≈ . r ≈ . B. Second Model
As our second example, we abandon the standard coreof the previous one, and take P (1) = | φ (1) | − α , implyinga self-interaction, as well as P (2) = | φ (1) | − β and P (3) = | φ (2) | − γ . This combination of parameters will result ina triple shell structure, in which all three contributionsto the energy density go to zero at the origin. In thisexample, the system of first order equations takes theform H (1) (cid:48) = (cid:18) H (1) (cid:19) α − K (1)2 r , (19a) K (1) (cid:48) = − H (1) K (1) (cid:16) H (1) (cid:17) α , (19b) FIG. 3. In the left panel, we show the solutions H (3) ( r )(ascending line) and K (3) ( r ) (descending line) of (18) for α = 3 , β = 10. In the right panel, we show the contribu-tion ρ , calculated from (13), relative to this sector. Here, (cid:15) = 0 . ρ (red), ρ (blue)and ρ (green) calculated in subsection III A are shown, givingrise to a multimagnetic structure. We depict the case α =3 , β = 10. and H (2) (cid:48) = (cid:18) H (1) (cid:19) β − K (2)2 r , (20a) K (2) (cid:48) = − H (2) K (2) (cid:16) H (1) (cid:17) β , (20b)and H (3) (cid:48) = (cid:18) H (2) (cid:19) γ − K (3)2 r , (21a) K (3) (cid:48) = − H (3) K (3) (cid:16) H (2) (cid:17) γ . (21b)Closed form solutions have not been discovered for anyof the functions in this example, but we may solve theseequations numerically. Once again, we find that the H ( n ) and K ( n ) present profiles that are qualitatively similarto those of the standard monopole solution. These solu-tions, as well as the individual contributions to the en-ergy density, are represented in Figs. 5, 6 and 7. Thetotal energy is again equal to 12 π .In Fig. 8, we show the densities ρ , ρ and ρ simulta-neously, which allows for visualization of the triple shell FIG. 5. In the left panel, we show the solutions H (1) ( r )(ascending line) and K (1) ( r ) (descending line) of (19) for α = 1 , β = 10 and γ = 30. In the right panel, we show thecontribution ρ , calculated from (13), relative to this sector.FIG. 6. In the left panel, we show the solutions H (2) ( r )(ascending line) and K (2) ( r ) (descending line) of (20) for α = 1 , β = 10 and γ = 30. In the right panel, we show thecontribution ρ , calculated from (13), relative to this sector.Here we have (cid:15) = 0 . structure of this solution. FIG. 7. In the left panel, we show the solutions H (3) ( r )(ascending line) and K (3) ( r ) (descending line) of (21) for α = 1 , β = 10 and γ = 30. In the right panel, we show thecontribution ρ , calculated from (13), relative to this sector.Here, (cid:15) = 0 . ρ (red), ρ (blue)and ρ (green) calculated in subsection III B are shown, givingrise to a multimagnetic structure. We depict the case α = 1, β = 10, γ = 30. C. Third Model
We may also consider another possibility, which canbe achieved by taking P (2) and P (3) as functions of | φ (1) | alone. As an example, let us consider P (1) = 1, P (2) = | φ (1) | − α and P (3) = | φ (1) | − β . This leads to equations (14)and (17), coupled to the additional pair of equations H (3) (cid:48) = (cid:18) H (1) (cid:19) β − K (3)2 r , (22a) K (3) (cid:48) = − H (3) K (3) (cid:16) H (1) (cid:17) β . (22b)This modification doesn’t change the four other equa-tions, the functions H (1) , K (1) , H (2) and K (2) are thesame discussed in subsection III A. The substitution H (1) ( r ) = coth( r ) − /r in (22) gives rise to a systemof ODEs that can be solved for H (3) and K (3) . Theirfeatures are shown in Fig. 9, for β = 12. The multi-magnetic structure that arises from the solution of theequations of motion is depicted in Fig. 10. FIG. 9. In the left panel, we show the solutions H (3) ( r ) (as-cending line) and K (3) ( r ) (descending line) of (22) for β = 12.In the right panel, we show the contribution ρ , calculatedfrom (13), relative to this sector. Here, (cid:15) = 0 . FIG. 10. Planar section, passing through the center, of theenergy density, in which the contributions ρ (red), ρ (blue)and ρ (green) calculated in subsection III C are shown, givingrise to a multimagnetic structure. We depict the case α =3 , β = 12. D. Fourth Model
As a final example of the N = 3 class of models,let us consider another example, in which P (1) = 1, P (2) = | φ (1) | − α and P (3) = ( | φ (1) || φ (2) | ) − β . This modelis different from the previous examples in that both the φ (1) and φ (2) fields appear explicitly in the third sector,with first order equations that are given by H (3) (cid:48) = (cid:18) H (1) H (2) (cid:19) β − K (3)2 r , (23a) K (3) (cid:48) = − H (3) K (3) (cid:16) H (1) H (2) (cid:17) β . (23b)These equations are coupled to (14) and (17), and wecan plug in the results for H (1) and H (2) to solve (23).This has been done for the case α = 3, β = 10. Thesolutions of (23) are displayed in Fig 11 and the mul-timagnetic structure associated to this case is shown inFig 12. IV. MODELS WITH N=4
Let us now investigate another system, resulting in thegroup SU(2) × SU(2) × SU(2) × SU(2). This requires thatwe have four pairs of first order differential equations.The simplest way to achieve this is by adding a fourthpair of fields to the systems described in the previous sec-tion, and we investigate some distinct possibilities below.
A. First Model
As a first example, we take P (1) , P (2) and P (3) asin our very first model of the previous Section and, ad-ditionally, choose P (4) = | φ (3) | − γ . We are then led toequations (14), (17) and (18), with the additional pair of FIG. 11. In the left panel, we show the solutions H (3) ( r )(ascending line) and K (3) ( r ) (descending line) of (23) for α =3, β = 10. In the right panel, we show the contribution ρ ,calculated from (13), relative to this sector. Here, (cid:15) = 0 . ρ (red), ρ (blue)and ρ (green) calculated in subsection III D are shown, givingrise to a multimagnetic structure. We depict the case α =3 , β = 10. first order equations H (4) (cid:48) = (cid:18) H (3) (cid:19) γ − K (4)2 r , (24a) K (4) (cid:48) = − H (4) K (4) (cid:16) H (3) (cid:17) γ . (24b)Since a solution for H (3) has already been found nu-merically in subsection III A, we may plug this result inthe equations above, and then solve them numerically.The result for the case α = 3, β = 10, γ = 20, along withthe contribution ρ to the energy density, is shown inFig. 13. We see that ρ attains its maximum at r (cid:39) . π factor,raising its value to 16 π .Moreover, we depict in Fig. 14 the multimagneticstructure which results from the superposition of the foursubsystems considered. FIG. 13. In the left panel, we show the graph of the func-tions H (4) (ascending line) and K (4) (descending line) calcu-lated from (24) which, together with the functions obtained insubsection III A, form a solution of the equations of motion.In the right panel, we show the contribution ρ , calculatedfrom (13), relative to this sector. Both plots correspond tothe choices α = 3, β = 10, γ = 20 of the model’s parameters.Here, (cid:15) = 0 . ρ (red), ρ (blue), ρ (green) and ρ (black) relative to the model presented in subsection IV Afor α = 3, β = 10, γ = 20. B. Second Model
As another example, we present a solution with fourshells. To achieve this, we take P , P and P as insubsection IV A, as well as P (4) = | φ (3) | − ζ . This caseresults in a system of equations comprised by (19), (20)and (21), which are now coupled with the additional pairof equations H (4) (cid:48) = (cid:18) H (3) (cid:19) ζ − K (4)2 r , (25a) K (4) (cid:48) = − H (4) K (3) (cid:16) H (3) (cid:17) ζ . (25b)The equations have again been solved through numericalprocedures, and the results can be seen in Fig. 15 bellow.The energy density ρ ( r ) has a peak at r ≈ .
2. We thenpresent in Fig. 16 the multimagnetic structure that arisesfrom the superposition of the four subsystems considered.In the case with N = 4 there are several other pos-sibilities, similar to the case of N = 3 which we haveinvestigated above. FIG. 15. In the left panel, we show the graph of the func-tions H (4) and K (4) which solve (25) and, together with thefunctions calculated in subsection III B, form a solution of theequations of motion. In the right panel, we show the contri-bution ρ to the energy density. Both plots correspond tothe choices α = 1, β = 10, γ = 30, ζ = 8 of the model’sparameters. Here, (cid:15) = 0 . ρ (red), ρ (blue), ρ (yellow) and ρ (green) to the energy density relative to the model presentedin subsection IV B for α = 1, β = 10, γ = 30, ζ = 8. V. DISCUSSION
In this work we studied the presence of magneticmonopoles in a relativistic model with a modified La-grangian that is invariant under N copies of the SU(2)symmetry, that is, SU (1) (2) × SU (2) (2) × · · · × SU (N) (2).We developed a general investigation leading to the pres-ence of first order differential equations that solve theequations of motion and ensure the presence of minimumenergy configurations. We then discussed some exampleswith N = 3 and N = 4, with shells on top of a standardmonopole core and in other cases, giving rise to novelonionlike structures. We believe that these multimag-netic structures may find applications in several areas ofnonlinear science. In particular, the solutions generalizethe bimagnetic structures considered before in [17], andthey may find applications in several different scenariosin condensed matter [26, 27, 36, 39].It is worthwhile to notice that instead of the replicationof the SU(2) symmetry, we could also chose other mod-els with gauge groups of the form G × ... × G N , wherethe G k are themselves non-abelian gauge groups. Oneparticularly interesting example is found for the gaugegroup SO(10). This particular grand unification the-ory has been studied for quite some time [40, 41] andis considered a relatively simple and promising unifica-tion model [42, 43]. This gauge group may undertakesymmetry breaking to SU(3) × SU(2) × SU(2) × U(1) asan intermediary step before reduction to the gauge groupof the standard model [44]. Thus, we could take, for ex-ample, an SU(3) core with two shells provided by thesymmetry breaking from the SU(2) × SU(2) subgroup.Although mathematically more challenging, this case is,in principle, a straightforward extension of the examplesprovided here. In the same way, we could also considerthe SU(3) and SU(5) gauge groups, since they also sup-port monopoles [45, 46] and may be used with distinctmotivations. Another line of current interest concernsthe study of the magnetic monopoles found in the presentwork interacting with gravity, to see how they change thestandard scenario of black holes in magnetic monopoles[47]. As we can see, the underlying principle leading tothe solutions studied in this work may lead to a widerange of multimagnetic monopoles, differing by the gaugegroup and by the many possible choices for the func-tions that modify the standard model. We hope that theabove results will foster new investigation on magneticmonopoles.The main results of the present work motivate us tothink of diminishing the spatial dimension, investigat-ing, for instance, vortices in planar systems. In this case,the basic U(1) symmetry can be enhanced to becomeU (1) × U (1) × · · · × U N (1), bringing into action a sys-tem with N order parameters. This study can be devel-oped in direct connection with the works [18–21], and ispresently under consideration. It is also of interest tothe case of multi-component Gross-Pitaevskii equationsto describe solutions in the case of spin-1 and spin-2 Bose-Einstein condensates; see, e.g., Ref. [22] and referencestherein. We can also go further down and consider lineal systems, and study kinks in one spatial dimension. Herewe can follow the lines of [23], where we studied kinks ina model with Z × Z symmetry. Extension to the caseof the Z × Z × · · · × Z symmetry can follow the presentinvestigation, and is also under consideration. We hopeto report on some of these issues in the near future.Another line of investigation is related to the recentstudy in Ref. [48], where the authors found magneticmonopole solution in a multivector boson theory in whicha replication of the basic SU(2) symmetry is also consid-ered. The construction of the magnetic monopole de-scribed in [48] is based on the dimensional deconstruc-tion mechanism [49] and the Higgsless theory [50], andthis is different from the procedure which we described inthe present work, where we focus on stable minimum en-ergy configurations with multimagnetic properties. Wewould also like to add that the multimagnetic structureswhich appeared in this work may be constructed underthe guidance of the recent advances on the study of mul-tishelled hollow micro-nanostructures with triple or moreshells; see, e.g., Refs. [51, 52] and references therein. Inthese investigations, the main motivation included poten-tial applications mainly in the fields of energy conversionand storage, sensors, photocatalysis, and drug delivery,but the above results may foster investigations concern-ing the presence of shells with distinct magnetic proper-ties. ACKNOWLEDGMENTS
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