Electrically charged localized structures
EElectrically charged localized structures
D. Bazeia, M.A. Marques, and R. Menezes
3, 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 Departamento de Ciˆencias Exatas, Universidade Federal da Para´ıba, 58297-000 Rio Tinto, PB, Brazil (Dated: December 18, 2020)This work deals with an Abelian gauge field in the presence of an electric charge immersed in amedium controlled by neutral scalar fields, which interact with the gauge field through a generalizeddielectric function. We develop an interesting procedure to solve the equations of motion, whichis based on the minimization of the energy, leading us to a first order framework where minimumenergy solutions of first order differential equations solve the equations of motion. We investigatetwo distinct models in two and three spatial dimensions and illustrate the general results withsome examples of current interest, implementing a simple way to solve the problem with analyticalsolutions that engender internal structure.
I. INTRODUCTION
Localized finite energy structures play an importantrole in nonlinear science in general. In high energyphysics, in particular, localized structures may attaintopological profile and may appear as kinks in the realline, vortices in the plane and monopoles in the threedimensional space [1, 2]. Kinks can be immersed in theplane as domain ribbons and in space as domain walls,and vortices can behave as stringlike objects when im-mersed in the space. These structures are well-knownobjects and have been studied in several distinct contextsin high energy physics, and in applications in several ar-eas of nonlinear science.In this work we will study localized structures, butfollowing another route. The study will deal with sys-tems composed of a single charge immersed in a mediumwith electric permittivity controlled by real scalar fieldsthat regularize the energy of the system. Since the scalarfield is electrically neutral, the coupling with the Abeliangauge field is via the Maxwell term, as it appears in theLagrange density (1), for instance. This is not a standardcoupling, but it has been used in several distinct situa-tions, in particular in the so called Friedberg-Lee model[3, 4], where a dielectric function is used to describe abag similar to the MIT [5] and SLAC [6] bag models.This coupling has also appeared in [7–9], in connectionwith the presence of vortices in the plane. It was also ex-plored in connection with the AdS/CFT correspondence,to describe insulators and metals within the holographicsetup [10, 11]. The holographic scenario has been furtherexplored in [12] and, in the context of the gauge/gravityduality, it was also used to investigate hydrodynamic be-havior in a hot and dense strongly coupled relativisticfluid in the framework of the Einstein-Maxwell-dilatontheory [13].The use of a field-dependent function coupled with thegauge field dynamical term has also been recently con-sidered in [14] in the electric context, that is, in the pres-ence of an electric charge fixed at the origin; there, weshowed that the electric field has a behavior that cap- tures the basic feature of asymptotic freedom, an effectthat is usually associated to quarks and gluons. It wasalso considered, for instance, in [15–19] to describe vor-tex configurations with internal structures in the planein the magnetic context. Moreover, very recently, in [20]the authors studied the dielectric Skyrme model, that is,the Skyrme model where both the kinetic and Skyrmeterms are multiplied by field-dependent functions, lead-ing to new results of current interest. As one knows,the Skyrme model has a direct connection with mesonsand baryons [21] and the binding energies of nuclei [22]and, in the same line, in [23] another investigation hasbeen implemented, with focus on lowering the bindingenergies of the Skyrme solutions towards more realisticvalues. See also [24] for the investigation of exact self-dual skyrmions, and [25] for the construction of Lorentzinvariant compactlike structures.The field-dependent function can be used to modifythe magnetic properties of the medium to study vortices,as in [7–9, 15–19], but it can also be considered to mod-ify electrical properties of the medium, as in [3, 4] andin [14], for instance. Inspired by the results of [14], inthe present work we first consider the field-dependentfunction as a dielectric function to be governed by a sin-gle scalar field. In the sequence, we add an extra scalarfield, that modifies both the dielectric function and thedynamical term of the first scalar field in the Lagrangedensity. We develop a first order framework that helpsus to describe the system via first order differential equa-tions, and show that it is valid for the two models. Also,in the case of two scalar fields the first order frameworkhelps us to describe novel configurations that supportelectrically charged multilayered ringlike structures. Animportant feature of the first order framework is thatit allows the presence of minimum energy field configu-rations that are solved analytically. The present studywill focus on the basic aspects of the problem, whichconcerns the construction of electrically charged local-ized structures described by scalars and the electric field.However, the electrically charged structures which we re-port in the present work may also be of interest to otherareas of physics, in particular, to optical fibers [26] when a r X i v : . [ phy s i c s . g e n - ph ] D ec the system supports axial symmetry, and to the studyof memory and other devices based on ferroelectricity[27, 28].To implement the investigation, we organize the workas follows. In Sec. II we first describe and illustrate thecase with a single scalar field, and then introduce an-other model, described by two scalar fields, with is alsoillustrated with some distinct examples. Since we will bedealing with electrically charged localized structures, andsince the electric field is a vector, the passage from two tothree space dimensions is somehow smooth and requiresno extra degrees of freedom, so we will study the caseof two and three spatial dimensions in the present work,which are of potential applications with dielectric andferroelectric materials. The electric case is different fromthe case of magnetic structures, which may describe vor-tices in the plane or magnetic monopoles in space; sincethe magnetic field is pseudo vector, the passage from twoto three spatial dimensions requires the inclusion of ex-tra degrees of freedom, suggesting that we change fromthe Abelian U (1) symmetry in the plane in the case ofvortices [29], to the non Abelian SU (2) symmetry, forinstance, when one moves on to the case of monopolesin three spatial dimensions [30, 31]. We then close thework in Sec. III, reviewing the main results and addingcomments on some new lines of investigation of currentinterest. II. THE MODELS
The investigation will focus on the presence of electri-cally charged localized structures in two and three spatialdimensions, so we split this Section in two distinct parts,the first dealing with the case of two spatial dimensions,and the second one with the case of three space dimen-sions. We will study two models, which are describedby two distinct Lagrange densities that do not changewhen we go from two to three spatial dimensions. Forthis reason, we start describing the two models on gen-eral grounds, before specializing in the cases of two andthree spatial dimensions.We first consider the model with Lagrange density L = − ε ( φ )4 F µν F µν + 12 ∂ µ φ∂ µ φ − A µ j µ , (1)where A µ describes the Abelian gauge field, F µν = ∂ µ A ν − ∂ ν A µ stands for the electromagnetic strength ten-sor, φ is a real scalar field and j µ is an external currentdensity. ε ( φ ) is a real nonnegative function which onlydepends on φ . It represents a dielectric function thatcouples the neutral real scalar field to the Abelian gaugefield. Here we consider natural units, with (cid:126) = c = 1 andmore, we take time, space, fields and coupling constantsdimensionless, for simplicity. Also, we will consider aunity electric charge, that is, e = 1.If the scalar field is constant and uniform, the modelreduces to the case of an Abelian gauge field generated by the external current j µ . To circumvent this possibility,we use the scalar field to get to models of current inter-est. Moreover, we consider the case where the currentdensity is a timelike vector of the form j µ = ( j ,(cid:126)j = 0)with j time-independent. In the present investigationwe will then focus on how the scalar field may contributeto introduce modifications in the standard scenario.The equations of motion associated to the Lagrangedensity (1) are ∂ µ ∂ µ φ + 14 ε φ F µν F µν = 0 , (2a) ∂ µ ( εF µν ) = j ν . (2b)where ε φ = ∂ε/∂φ . We will search for static field con-figurations, and since there are no spatial components ofthe current density, there is no magnetic field present inthe system. We then define the components of electricfield E as E i = F i to show that the above equationsbecome ∇ φ + 12 ε φ | E | = 0 , (3a) ∇ · ( ε E ) = j . (3b)The last equation is the Gauss’ law of the model. Theenergy density associated to the solutions of the aboveequations is calculated standardly; it has the form ρ = 12 ( ∇ φ ) + 12 ε ( φ ) | E | + A j . (4)The energy is found by integrating the above energy den-sity.The second model enlarges the above model, introduc-ing an additional scalar field, χ , to help control the dy-namics of the gauge and the scalar field φ . It is definedas L = − ε ( φ, χ )4 F µν F µν + 12 f ( χ ) ∂ µ φ∂ µ φ + ∂ µ χ∂ µ χ − A µ j µ , (5)where ε ( φ, χ ) and f ( χ ) are real non negative functions.Notice that the dielectric function now depends on both φ and χ in this new model. The presence of the secondscalar field χ is inspired on the recent work [32], in whichthe function f ( χ ) is introduced to modify the kinematicsof the φ field. As we will see, this new model will bringnovelties, modifying the internal struture of the field con-figurations.In this new model, the equations of motion are ∂ µ ( f ∂ µ φ ) + 14 ε φ F µν F µν = 0 , (6a) ∂ µ ∂ µ χ − f χ ∂ µ φ∂ µ φ + 14 ε χ F µν F µν = 0 , (6b) ∂ µ ( εF µν ) = j ν . (6c)In the case of static solutions we get ∇ · ( f ∇ φ ) + 12 ε φ | E | = 0 , (7a) ∇ χ − f χ ( ∇ φ ) + 12 ε χ | E | = 0 , (7b) ∇ · ( ε E ) = j . (7c)We can also write the energy density in the form ρ = 12 f ( χ ) ( ∇ φ ) + 12 ( ∇ χ ) + 12 ε ( φ, χ ) | E | + A j . (8)The above results are general and can be used in twoand three spatial dimensions, with the coordinate sys-tems which we consider below. A. Two Spatial dimensions
In the plane, we then consider polar coordinates ( r, θ ),for convenience, since we are interested in studying a sin-gle charge immersed in a medium with modified dielectricfunction. In two spatial dimensions, the charge is repre-sented by j = δ ( r ) /r .
1. First model
From Eq. (3b), we obtain the electric field E = 1 ε ( φ ) ˆ rr . (9)Also, knowing that the scalar electric fields do not dependon θ , Eq. (3a) can be rewritten as1 r ddr (cid:18) r dφdr (cid:19) + 12 ε φ | E | = 0 . (10)In this case, both E and φ do not depend on θ , and sothe above equation becomes, after using the expressionfor the electric field in Eq. (9), r ddr (cid:18) r dφdr (cid:19) = ddφ (cid:18) ε (cid:19) . (11)This is a second order differential equation with nonlin-earities introduced by the dielectric function ε ( φ ).We now follow the lines of Ref. [33] to find a first-orderformalism for the above model. One can use Eq. (4) towrite the energy density as ρ = ρ f + ρ c , where ρ f = 12 (cid:18) dφdr (cid:19) + 12 r ε ( φ ) , (12a) ρ c = A δ ( r ) r . (12b)with the indices f and c standing for field and charge,respectively. The total energy can be written in the form E = E f + E c , as the integral of the above energy densities,respectively.The key point here is that one can introduce an auxil-iary function W = W ( φ ) to write ρ f in the form ρ f = 12 (cid:18) dφdr ∓ W φ r (cid:19) + 12 r ε ( φ ) − W φ r ± r dWdr . (13)Notice the presence of the factor 1 /r in the last term; it isimportant since under integration to get the energy, theline element dx dy = r dr dθ makes dW/dr be a surfaceterm after integration. When the dielectric function isgiven by ε ( φ ) = 1 W φ , (14)one can show that the energy E f is bounded, i.e., E f ≥ E B , where E B = 2 π | W ( φ ( r → ∞ )) − W ( φ ( r = 0)) | . (15)For solutions obeying the first-order equations dφdr = ± W φ r , (16)we get to the minimum energy case, with E f = E B . Thetotal energy is E = E B + E c , where E c = 2 πA ( r = 0)is the energy due to the electric charge. One can showthat the above equation is compatible with the equationof motion (11). We further notice that equations of theabove form, with the 1 /r factor, were considered before inRefs. [14, 34], and they engender scale invariance. More-over, the above first-order equations have two signs; theyare related by the change r → /r , so we only deal withthe positive sign from now on. We further notice thatequations similar to the above first order equations (16)appeared before in [15, 17], for instance, in the study ofvortices with internal structure.The first order equation allows that we write the energydensity in Eq. (12a) in the form ρ f = (cid:18) dφdr (cid:19) = W φ r (17)Since E = −∇ A , we can use Eqs. (9), (14) and (16)with positive sign to obtain A ( r ) = − W ( φ ( r )) , (18)such that the energy of the charge, E c , is written as E c = 2 π | W ( φ (0)) | , (19)and it can be calculated straightforwardly, by just know-ing the value of φ ( r ) at r = 0.Before illustrating the general results, let us notice thatif we consider ε ( φ ) such that ε (0) = 1, the model (1)becomes the standard model, with the electric field cor-rectly given by (9) with ε (0) = 1 in the standard case.However, when one implements the first order frameworkto get minimum energy configurations, the first orderequations (16) are mandatory, and they impose that theconstant field φ = ¯ φ has to obey W φ ( ¯ φ ) = 0, meaningthat a constant scalar field configuration cannot be cho-sen at will anymore. A direct consequence of this appearsin Eq. (14): any constant field ¯ φ that obeys the first or-der equations (16) induces a divergence in the dielectricfunction ε ( φ ), and this imposes that the electric field inEq. (9) has to vanish in this case to regulate the energyof the system.Another interesting issue arises from the first orderequations (16); as it was shown in Ref. [34], if onechanges r → e x the equations (16) become dφdx = ± W φ , (20)which are first order equations that describe static con-figurations of a (1 ,
1) dimensional scalar field theory withpotential V ( φ ) = 12 W φ . (21)In this sense, if one chooses W ( φ ) = λ a φ − λ φ , (22)where λ and a are real parameters, we get the scalar fieldmodel L = 12 ∂ µ φ∂ µ φ − V ( φ ) , (23)with the potential V ( φ ) = 12 λ ( a − φ ) , (24)and the first order equations dφdx = ± λ ( a − φ ) , (25)with the standard solutions φ ( x ) = ± a tanh[ λ ( x − x )].The above (1 ,
1) dimensional scalar field model is theprototype of the Higgs field; it is driven by a double-wellpotential that engenders spontaneous symmetry break-ing and has important consequences in nonlinear science.In this sense, the scalar field which drives the symme-try breaking may be seen as an order parameter similarto the spontaneous polarization which is studied, for in-stance, in ferroelectric materials; see, e.g., Refs. [35–37].We notice that the parameters λ and a which we addedin (22) control the energy barrier in the double-well po-tential of the scalar field, so they are of direct interestin applications in ferroelectrics [35–37], for instance. Forsimplicity, however, we will take λ = a = 1 in this work.Motivated by the above results, let us now illustratethe planar model with W ( φ ) as in Eq. (22) (with λ = a = 1). Here the dielectric function in Eq. (14) is thenwritten as ε ( φ ) = (cid:0) − φ (cid:1) − , (26)and Eq. (16) gives dφdr = 1 − φ r . (27)It is solved by φ ( r ) = r − r r + r , (28)which has energy density ρ f = 16 r r ( r + r ) . (29)In Fig. 1, we display the solution of the above first-orderequation and its energy density. It engenders scale in-variance, so we use the condition φ (1) = 0, for simplic-ity. From Eqs. (15) and (19), we get that the energyis E = E B + E c , where E B = 8 π/ E c = 4 π/ r, θ ) plane. We further noticethat it is controlled by the dielectric function ε ( φ ) thatcouples the gauge field to the scalar field.As we have already commented on, the scalar fieldmodel described by (22) develops spontaneous symmetrybreaking and can be related to the potential based on thephenomenological Landau-Ginzburg-Devonshire or LGDtheory of ferroelectrics [35–37]. The recent measurements[37] of the intrinsic double-well energy landscape in athin layer of ferroelectric material integrated into a het-erostructure with a second dielectric layer, suggest thatthe negative capacitance [38] of the dielectric materialhas its origin in the energy barrier of the double-wellpotential. The negative capacitance can be used to pro-vide, for instance, voltage amplification for low powernanoscale devices; this can be done, for instance, aftertrading a standard insulator with a ferroelectric insula-tor of appropriate thickness [39].
2. Second model
We now focus on the second model. Here, consider-ing the presence of the same electric charge at the ori-gin, the electric field has the same form of Eq. (9) with ε ( φ ) → ε ( φ, χ ), so the equations of motion that describethe scalar fields are r ddr (cid:18) r dχdr (cid:19) − r f χ (cid:18) dχdr (cid:19) = ddχ (cid:18) ε (cid:19) , (30a) r ddr (cid:18) rf dφdr (cid:19) = ddφ (cid:18) ε (cid:19) . (30b) FIG. 1: The solution φ ( r ) that appears in (28) (top) and itsenergy density ρ f ( r ) in (29) (bottom), displayed in terms ofthe radial coordinate.FIG. 2: The energy density of the solution (29), displayedwith the intensity of the blue color increasing with the in-creasing values of the energy density. The energy density for the solutions of the above equa-tions can be written as ρ = ρ f + ρ c , where ρ f = 12 (cid:18) dχdr (cid:19) + 12 f ( χ ) (cid:18) dφdr (cid:19) + 12 r ε ( φ, χ ) , (31a) ρ c = A δ ( r ) r . (31b) As before, here we also introduce an auxiliary function W = W ( φ, χ ) to write ρ f as ρ f = 12 (cid:18) dχdr ∓ W χ r (cid:19) + 12 f ( χ ) (cid:18) dφdr ∓ W φ rf ( χ ) (cid:19) + 12 r ε ( φ, χ ) − W χ r − W φ r f ( χ ) ± r dWdr . (32)We choose the dielectric function in the form ε ( φ, χ ) = (cid:32) W φ f ( χ ) + W χ (cid:33) − , (33)to make the energy bounded, i.e., E f ≥ E B , where E B = 2 π | ∆ W | , (34)such that ∆ W = W ( φ ( ∞ ) , χ ( ∞ )) − W ( φ (0) , χ (0)). Onecan show that the bound is saturated to E f = E B if thefollowing first-order equations are satisfied dχdr = ± W χ r , (35a) dφdr = ± W φ rf ( χ ) . (35b)It is not hard to show that solutions of the above firstorder equations also solve the equations of motion (30).In this situation, the total energy is E = E B + E c , where E c = 2 πA ( r = 0). The upper and lower signs are relatedby the change r → /r , so we only consider the positivesign from now on. By using the above equations, onecan write the energy density (31a) as the sum of twocontributions, in the form ρ f = ρ + ρ , where ρ = f ( χ ) (cid:18) dφdr (cid:19) (36a) ρ = (cid:18) dχdr (cid:19) . (36b)The first-order equations (35) with positive sign may becombined with Eqs. (9) under the change ε ( φ ) → ε ( φ, χ )and (33) to give A ( r ) = − W ( φ ( r ) , χ ( r )) . (37)The energy associated to the electric charge is E c = | W ( φ (0) , χ (0)) | . (38)We remark that the energy contributions presented in(34) and (38) do not depend on f ( χ ). Also, the function W = W ( φ, χ ) depends on both φ and χ , so the first orderequations (35) are coupled and must be solved simultane-ously, in general. However, an interesting case arises for W ( φ, χ ) = g ( φ )+ h ( χ ). In this specific situation, the firstorder equation (35a) can be solved independently, andthis simplifies the calculation importantly. This allowsthat we add another integration constant in the problem,leading to a more general situation. For simplicity, how-ever, we will not consider this possibility in the presentwork. As in the previous model, here we also have to becareful with the choice of constant fields, ¯ χ and ¯ φ , sincein the first order framework they have to obey first or-der differential equations. This issue is similar to the onediscussed before in the paragraph below Eq. (19), so wedo not comment on it anymore.To illustrate the new possibility, we take h ( χ ) = αχ − αχ , (39)where α is a positive real parameter. From Eqs. (35a) and(36b), we get the solution, χ ( r ), and the energy density ρ , in the form χ ( r ) = r α − r α + 1 and ρ ( r ) = 16 α r α − ( r α + 1) . (40)Their profiles for α = 1 can be seen in Fig. 1, where theyappear as the dotted lines. Next, we use the above solu-tion to feed the function f ( χ ), which we firstly consideras f ( χ ) = 1 /χ . This is perhaps the simplest choice,which is not negative, is unity for ¯ χ = ±
1, as requiredby the choice of h ( χ ) in Eq. (39), and engenders the ap-propriate profile. To find the behavior of φ ( r ), we choose g ( φ ) in the form g ( φ ) = φ − φ . (41)The dielectric function becomes ε ( φ, χ ) = ( χ (1 − φ ) + α (1 − χ ) ) − , (42)and the first order equation (55b) now changes to dφdr = (cid:0) r α − (cid:1) ( r α + 1) r (cid:0) − φ (cid:1) . (43)It supports the analytical solution φ ( r ) = tanh (cid:18) ln r − r α − α ( r α + 1) (cid:19) . (44)The energy density (36a) takes the form ρ ( r ) = (cid:0) r α − (cid:1) ( r α + 1) r sech (cid:18) ln r − r α − α ( r α + 1) (cid:19) . (45)The energy of this model is E = E B + E c , where E B = 8 π (1 + α ) / E c = 4 π (1 + α ) /
3, matchingwith Eqs. (34) and (38). The solution (44) and the aboveenergy density are displayed in Fig. 3. Notice that the so-lution presents a plateau around r = 1 that gets wider as α decreases. This introduces a hole in the energy densityat this point, which becomes more evident as α decreases.In order to highlight this feature, we plot the above en-ergy density in the plane in Fig. 4. The structure presents FIG. 3: The solution φ ( r ) in Eq. (44) (top) and its energy den-sity ρ ( r ) in Eq. (45) (bottom), depicted for α = 2 (dashedline) and 10 (solid line).FIG. 4: The energy density in Eq. (45) in the plane for α =2 (top) and 10 (bottom). The intensity of the blue colorincreases with the increasing of the of the energy densities. a hole at the center and a ring around it, which is morevisible as α increases. This behavior is controlled by thedielectric function ε ( φ ).The model described by Eq. (41) with f ( χ ) = χ − supports rings around the central hole in the energy den-sity. We keep the same g ( φ ) in Eq. (41) with the χ fieldas in Eq. (40), but now we consider f ( χ ) = sec ( nπχ ),where n is a natural number. This is another choice,which is also not negative, unity for ¯ χ = ±
1, as requiredby the choice of h ( χ ) in Eq. (39) above, and engendersthe interesting wavelike profile which will bring internalmodification in the electric structure. We also remarkthat, since W ( φ, χ ) does not change, the energy remainsthe same, i.e., E = E B + E c , where E B = 8 π (1 + α ) / E c = 4 π (1 + α ) /
3. In this case, we have to change χ → cos ( nπχ ) in the factor that multiplies (1 − φ ) inthe dielectric function shown in Eq. (42). Moreover, thefirst order equation (55b) takes the form dφdr = 1 r cos (cid:18) nπ r α − r α + 1 (cid:19) (cid:0) − φ (cid:1) . (46)It supports the solution φ ( r ) = tanh ϑ ( r ) , (47a) ϑ ( r ) = 12 ln r + 14 α (cid:18) Ci (cid:18) nπr α r α + 1 (cid:19) − Ci (cid:18) nπr α + 1 (cid:19)(cid:19) , (47b)where Ci( z ) denotes the cosine integral function. Theenergy density in Eq. (36a) takes the form ρ ( r ) = 1 r cos (cid:18) nπ r α − r α + 1 (cid:19) sech ϑ ( r ) . (48)Since we have already seen how the parameter α modi-fies the configurations, we fix α = 3 and plot the abovesolution and energy density for n = 1 and 2 in Fig. 5.We see that, as n increases, the solution exhibits moreand more plateaux which appear in the energy densityas valleys, points in which ρ = 0. Including the centralvalley, one gets 2 n + 1 valley in the solutions. To illus-trate this feature, we plot the above energy density inthe plane for the very same values of α and n in Fig. 6.The localized configuration develops an interesting inter-nal multilayered ringlike structure. B. Three spatial dimensions
In the case of three spatial dimensions, we have twodistinct possibilities to work. The first one is the case ofcylindrical symmetry, with the spatial position describedby the vector ( r, θ, z ). This can be achieved when onesupposes that the electric charge is described by a uni-formly charged wire in the z axis. In this case the z dimension in unimportant, and the planar ( r, θ ) dimen-sions describe the planar system studied above. Thismeans that the planar results that we described in Sec. FIG. 5: The solution in Eq. (47) and its energy density (48)for α = 3 and for n = 1 (top and middle top) and 2 (middlebottom and bottom), respectively. II A are also valid in the three dimensional space withaxial symmetry.The other possibility is to consider spherical symmetry.In this case, we take the current density in the specificform j µ = ( δ ( r ) /r , , ,
0) and study the two modelsdescribed above, in this scenario in the presence of rota-tional symmetry in three spatial dimensions.
FIG. 6: The energy density in Eq. (48) in the plane for α = 3and n = 1 (top) and 2 (bottom). The intensity of the bluecolor increases with the increasing of the energy densities.
1. First model
Let us now focus on the presence of electrically chargedlocalized structures in the model (1) in (3 ,
1) dimensions.Here we first notice that Eqs. (2)-(4) are also valid in thiscase, and more, there is no magnetic field. However, inthe spatial case the electric field changes to E = 1 ε ( φ ) ˆ rr . (49)In the presence of spherical symmetry, the equation ofmotion (3a) for the scalar field becomes1 r ddr (cid:18) r dφdr (cid:19) + 12 ε φ | E | = 0 . (50)Combining this with the above electric field, we obtain r ddr (cid:18) r dφdr (cid:19) = ddφ (cid:18) ε (cid:19) . (51)We use the Eq. (4) in order to write the energy densityas ρ = ρ f + ρ c , where ρ f = 12 (cid:18) dφdr (cid:19) + 12 r ε ( φ ) , (52a) ρ c = A δ ( r ) r . (52b) FIG. 7: The top and bottom panels show the solution andenergy density in Eq. (54), respectively.
As before, the indices f and c stand for field and charge,respectively. Following similar steps of the previous Sec.II A, one can write the dielectric function as in Eq. (14)and show that E f = 4 π | W ( φ ( r → ∞ )) − W ( φ ( r = 0)) | isthe minimum energy if the solutions obey the first orderequation dφdr = ± W φ r . (53)We notice that first order equations similar to the aboveones appeared before in [40, 41], in the study of mag-netic monopoles with internal structure in three spatialdimensions.Regarding the function A , it has the very same ex-pression displayed in Eq. (18), so E c = 4 π | W ( φ ( r = 0)) | .Considering the function in Eq. (22), one can show thatthe above equation with negative sign, which we choosefor convenience, supports the solution and energy density φ ( r ) = tanh (cid:18) r (cid:19) and ρ f ( r ) = 1 r sech (cid:18) r (cid:19) . (54)We plot them in Fig. 7. The planar section of the energydensity passing through the center of the structure is sim-ilar to the configuration displayed in Fig. 2, so we do notdisplay it here. One can show that the total energy is E = E f + E c , where E f = E c = 8 π/
2. Second model
We now turn attention to the second model, which isdefined in Eq. (5). We can also develop a first orderframework in (3 ,
1) dimensions. In this case, one gets thevery same dielectric function in Eq. (33) and, also, thefirst order equations (35) with the change r → r in theright hand side of the equations, that is, dχdr = ± W χ r , (55a) dφdr = ± W φ r f ( χ ) . (55b)Considering the χ field described by the same h ( χ ) inEq. (39), we obtain the following solution and associatedenergy density χ ( r ) = tanh (cid:16) αr (cid:17) and ρ ( r ) = α r sech (cid:16) αr (cid:17) . (56)They can be seen in Fig. 7 for α = 1. By using theabove solution, one can use the g ( φ ) in Eq. (41) with f ( φ ) = sec ( nπχ ) to obtain φ ( r ) = tanh ϑ ( r ) , (57a) ϑ ( r ) = 12 r + 14 α (Ci ( ξ + ( r )) − Ci ( ξ − ( r ))) , (57b) ξ ± ( r ) = 2 πn (cid:16) ± tanh (cid:16) αr (cid:17)(cid:17) , (57c)such that its corresponding contribution in the energydensity is ρ ( r ) = 1 r cos (cid:16) nπ tanh (cid:16) αr (cid:17)(cid:17) sech ϑ ( r ) . (58)By using the Bogomol’nyi bound, one gets the energy E = E B + E c , where E B = E c = 8 π (1 + α ) /
3. In Fig. 8,we display the solution (57) and the above energy density.We see that the most external ring is almost invisible. Ingeneral, the energy density engenders n + 1 rings, whoseintensities are controlled by α . To highlight the ringlikeprofile of the energy density, we plot the planar sectionof the energy density passing through the center of thestructure in Fig. 9. As both Figs. 7 and 8 show, the shellstructure in the spatial case is different from the ringlikestructure in the planar case. III. CONCLUSION
In this paper, we have studied the effects of a staticelectric charge immersed in a medium with generalizeddielectric function. We have investigated models with asingle scalar field, and also with two scalar fields in twoand three spatial dimensions. We implemented a firstorder framework, in which the equations of motion aresolved by solutions of first order differential equations,
FIG. 8: The top and bottom panels show the solution (57)and the the energy density in Eq. (58) for n = 2 and α = 0 . n = 2 and α = 0 . which describe field configurations that minimize the en-ergy of the localized structures. In the case of two scalarfields, we found solutions that allow for the presence ofconfigurations with several internal structures. In bothsituations, the first order equations simplify the problemand we could find exact analytical solutions in two andthree spatial dimensions.The models investigated in this work are based on anAbelian gauge field with U (1) symmetry, so it would beof current interest to investigate the possibility to extendthe present study to other cases, as in the Born-Infeld[42, 43] modification of the Maxwell term that leads tononlinear electromagnetism, and also in the case of nonAbelian gauge fields. The case of an SU(2) gauge the-0ory appears of current interest here, since this is close tothe case of magnetic monopoles [30, 31, 40, 41]. Anotherpossibility refers to the case of Q-balls in the presence of U (1) gauged two-component model similar to the mod-els recently investigated in Refs. [44, 45]. The dielectricmodification that we included in this work may suggestfurther research in the subject.We notice that first order equations similar to the equa-tions (16) appeared before in [15, 17] in the study of vor-tices with internal structure in the planar case. Moreover,in three spatial dimensions, equations of the first ordertype similar to the equations (53) also appeared before in[40, 41] in the investigation of magnetic monopoles. It isinteresting to see that these first order equations changeas one changes from two to three spatial dimensions, butthe change is directly related to the study developed in[34], in which we investigated the presence of topologicalstructures constructed by a single real scalar field in arbi-trary dimensions, circumventing the scaling theorem dueto Derrick and Hobart [46, 47], which shows that there isno topological structure in dimensions greater than onewhen one deals with standard scalar field theory. We re-call here that in Ref. [34] one changed the potential ofthe scalar field, adding specific spatial dependence whichallowed to construct topological solutions in arbitrary di-mensions. In this sense, the investigation in [34] seems toprovide a unifying approach to deal with the presence oflocalized structures described by real scalar fields in twoand three spatial dimensions.The effect responsible for the presence of internal struc-ture appears from the requirement of finite energy of thelocalized structure. Indeed, for higher and higher valuesof the dielectric function, the electric field has to dimin-ish toward zero to keep the energy finite. In this sense, ifone thinks of applications of the present study to otherareas of nonlinear science, one would require materials with high dielectric constants, which can be found, forinstance, in ceramic elements like the ones investigated inRefs. [48–50]. There are other possibilities as the ones in-vestigated, for instance, in [51, 52]; in [51] the study dealtwith dielectric film with a high dielectric constant usingchemical vapor deposition-grown graphene interlayer andin [52] the authors investigated the construction of dielec-tric gels with a new type of polymer-based dielectric ma-terial in order to design gels that achieve ultra-high val-ues for the dielectric constant. In the case of three spatialdimensions with axial symmetry, the results of the pla-nar case may suggest the study of optical fibers with anelectrically charged ultra thin wire encapsulated at thecore of the fibers and other possible realizations. More-over, in the case of three spatial dimensions, the local-ized structure can be seem as core and shell nanoparticlesin the form of polymer based nanocomposite dielectricsthat create hierarchically structured composites in whicheach sublayer may contribute a distinct function to yielda multilayered multifunctional material [53, 54] similar tothe core and shell magnetic structures described in Ref.[55], which appears as the magnetic counterpart of theelectrically charged structures described in the presentwork. Acknowledgments
This work is supported by the Brazilian agen-cies Conselho Nacional de Desenvolvimento Cient´ıficoe Tecnol´ogico (CNPq), grants Nos. 404913/2018-0(DB), 303469/2019-6 (DB) and 306504/2018-9 (RM),and Paraiba State Research Foundation (FAPESQ-PB)grants Nos. 0003/2019 (RM) and 0015/2019 (DB andMAM). [1] A. Vilenkin, E.P.S. Shellard,
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