Charged Nielsen-Olesen vortices from a Generalized Abelian Chern-Simons-Higgs Theory
aa r X i v : . [ h e p - t h ] J a n Charged Nielsen-Olesen vortices from aGeneralized Abelian Chern-Simons-HiggsTheory
Lucas Sourrouille
Departamento de F´ısica, Universidade Federal do Maranh˜ao,65085-580, S˜ao Lu´ıs, Maranh˜ao, Brazil [email protected]
July 23, 2018
Abstract
We consider a generalization of abelian Chern-Simons-Higgs model by introducinga nonstandard kinetic term. In particular we show that the Bogomolnyi equations ofthe abelian Higgs theory may be obtained, being its solutions Nielsen-Olesen vorticeswith electric charge. In addition we study the self-duality equations for a generalizednon-relativistic Maxwell-Chern-Simons model.
Keywords : Chern-Simons gauge theory, Topological solitons
PACS numbers :11.10.Kk, 11.10.Lm
The two dimensional matter field interacting with gauge fields whose dynamics is gov-erned by a Chern-Simons term support soliton solutions[1], [2]. These models have theparticularity to became auto-dual when the self-interactions are suitably chosen[3, 4].When this occur the model presents particular mathematical and physics properties,such as the supersymmetric extension of the model[5], and the reduction of the motionequation to first order derivative equation[6]. The Chern-Simons gauge field inher-its its dynamics from the matter fields to which it is coupled, so it may be eitherrelativistic[3] or non-relativistic[4]. In addition the soliton solutions are of topologicaland non-topological nature[7].The addition of non-linear terms to the kinetic part of the Lagrangian has interest-ing consequences for topological defects, making it possible for defects to arise without symmetry-breaking potential term [8]. In the recent years, theories with nonstan-dard kinetic term, named k -field models, have received much attention. The k -fieldmodels are mainly in connection with effective cosmological models[9] as well as thetachyon matter[10] and the ghost condensates [11]. The strong gravitational waves[12]and dark matter[13], are also examples of non-canonical fields in cosmology. One in-teresting aspect to analyze in these models concern to its topological structure. In thiscontext several studies have been conducted, showing that the k -theories can supporttopological soliton solutions both in models of matter as in gauged models[14, 15].These solitons have certain features such as their characteristic size, which are not nec-essarily those of the standard models. From the theoretical point of view, it has beenrecently shown, in Ref.[16], that the introduction of a nonstandard kinetic term in thenon-relativistic Jackiw-Pi model Lagrangian[4], leads to a topological model in whichthe self-duality equations are the same of the relativistic ablelian Chern-Simons-Higgsmodel.In this paper we study a Chern-Simon-Higgs model with a generalized dynamics.This nonstandard dynamics is introduced by a function ω , which depend on the Higgsfield. We study the Bogomolnyi limit for such system. In particular we will show thatchoosing a suitable ω , the Bologmolnyi equations of Maxwell-Higgs theory may beobtained. The soliton solutions of these equations are identical in form to the Nielsen-Olesen vortices. The difference lies in the fact that, unlike the usual abelian Higgsmodel, our vortex solutions have electric charge. Finally, we propose a generalizationof a non-relativistic Maxwell-Chern-Simons model introduced by Manton[17], whoseself-duality equation are the Bogomolnyi equations of the Higgs model, and analyzethe Bogomolnyi framework, obtaining as a solution the Chern-Simons-Higgs vortices. Let us start by considering briefly the relativistic Chern-Simons-Higgs model and itssoliton solutions[3, 7]. The dynamics of this model is descried by the action S = Z d x (cid:16) κ ǫ µνρ A µ ∂ ν A ρ + | D µ φ | − V ( | φ | ) (cid:17) (1)This is (2 + 1)-dimensional model with Chern-Simons gauge field coupled to complexscalar field φ ( x ). Here, the covariant derivative is defined as D µ = ∂ µ + ieA µ ( µ =0 , , g µν = (1 , − , −
1) and ǫ µνλ is the totally antisymmetrictensor such that ǫ = 1.The corresponding field equations are D µ D µ φ = − ∂V∂φ ∗ F µν = eκ ǫ µνρ j ρ (2) here j i = i (cid:16) φ ∗ D ρ φ − ( D ρ φ ) ∗ φ (cid:17) . The time component of the second equation in (2)is B = eκ j , (3)which is the Chern-Simons version of Gauss’s law. Integrating this equation, overthe entire plane, we obtain the important consequence that any object with charge Q = e R d xρ also carries magnetic flux Φ = R Bd x [18]:Φ = − κ Q (4)The energy may be found from the energy momentum tensor T µν = D µ φ ∗ D ν φ + D µ φ ∗ D ν φ − g µν [ | D α φ | − V ( | φ | )] , (5)Integration on the time-time component yields E = Z d x (cid:16) | D φ | + | D i φ | + V ( | φ | ) (cid:17) (6)In order to find the minimum of the energy, the expression (6) can be rewritten as E = Z d x (cid:16) | D φ ∓ iκ ( | φ | − υ ) φ | + | ( D ± iD ) φ | − κ ( | φ | − υ ) | φ | + V ( | φ | ) ∓ υ B (cid:17) (7)where we have used the Chern-Simons Gauss law and the identities | D i φ | = | ( D ± iD ) φ | ∓ eB | φ | ± ǫ ij ∂ i J j (8)and | D φ ∓ iκ ( | φ | − υ ) φ | = | D φ | ± iκ ( | φ | − υ )[ φ ∗ D φ − ( D φ ) ∗ φ ] +1 κ ( | φ | − υ ) | φ | (9)Thus, if the potential is chosen to take the self-dual form V ( | φ | ) = 1 κ ( | φ | − υ ) | φ | , (10)the expression (7) is reduced to a sum of two squares plus a topological term E = Z d x (cid:16) | D φ ∓ iκ ( | φ | − υ ) φ | + | ( D ± iD ) φ | ∓ υ B (cid:17) (11)Then the energy is bounded below by a multiple of the magnitude of the magnetic flux(for positive flux we choose the lower signs, and for negative flux we choose the uppersigns): E ≥ υ | Φ | (12) he bound is saturated by solutions to the first-order equations( D ± iD ) φ = 0 D φ = ± iκ ( | φ | − υ ) φ (13)which, when combined with the Gauss law constraint (3) become the self-duality equa-tions: ( D ± iD ) φ = 0 B = ± κ ( | φ | − υ ) | φ | (14)These equations are clearly very similar to the self-duality equations of Nielsen Olesenvortices[19]. However, their solutions are magnetic vortices that carry electric chargeand may be topological as well as no topological. We will consider, here, a generalization of the Chern-Simons-Higgs model (1). Wemodify this model by changing the canonical kinetic term of the scalar field, S = Z d x (cid:16) κ ǫ µνρ A µ ∂ ν A ρ + ω ( ρ ) | D φ | − | D i φ | − V ( ρ ) (cid:17) (15)The function ω ( ρ ) is, in principle, an arbitrary function of the complex scalar field φ and V ( ρ ) is the scalar field potential to be determined below. Here, we have abbreviatedthe notation, naming, ρ = | φ | .Variation of this action yields the field equations ∂ω ( ρ ) ∂φ ∗ | D φ | − ω ( ρ ) D D φ + D i D i φ − ∂V∂φ ∗ = 0 B = eω ( ρ ) κ j F µν = eκ ǫ µνρ j ρ (16)The second equation of (16) is the Gauss’s law of Chern-Simons dynamics, modified,here, by the function ω ( ρ ). Notice that R d xeω ( ρ ) j is the conserved charge associatedto the U (1) global symmetry δφ = iαφ , (17)Indeed, by the Nother theorem J = ∂ L ∂ ( ∂ φ ) δφ + ∂ L ∂ ( ∂ φ ∗ ) δφ ∗ = e i ω ( ρ ) (cid:16) φ ∗ D φ − ( D φ ) ∗ φ (cid:17) = eω ( ρ ) j (18) here we have chosen α = e .Here, we are interested in time-independent soliton solutions that ensure the finite-ness of the action (15). These are the stationary points of the energy which for thestatic field configuration reads E = Z d x (cid:16) − κA B − e ω ( ρ ) A ρ + | D i φ | + V ( ρ ) (cid:17) (19)By varying with respect to A , we obtain the relation A = − κ e Bρω ( ρ ) (20)Substitution of (20) into (19) leads to E = Z d x (cid:16) κ e B ρω ( ρ ) + | D i φ | + V ( ρ ) (cid:17) (21)Consider, now, the following choice for the ω ( ρ ) function ω ( ρ ) = ρ − κ e (22)Then, by using the identity (8), the energy may be written as E = Z d x (cid:16) B + | ( D ± iD ) φ | ∓ eBρ + V ( ρ ) (cid:17) , (23)where we have dropped a surface term. So, we have obtained an expression of theenergy which is the same of the Abelian Higgs model. The form of the potential V ( ρ )that we choose is motivated by the desire to find self-dual soliton solution. Thus, if wechoose the potential as V ( ρ ) = λ ρ − , (24)the energy may be rewritten as follows E = Z d x (cid:16) [ B ∓ e ( ρ − + | D ± φ | + ( ρ − ( λ − e ) ∓ eB (cid:17) (25)Notice that the potential (24) is the symmetry breaking potential of the Abelian Higgsmodel. Also, when the symmetry breaking coupling constant λ is such that λ = 2 e , (26)i.e. when the self-dual point of the Abelian Higgs model is satisfied, the energy (25) re-duce to a sum of square terms which are bounded below by a multiple of the magnitudeof the magnetic flux: E ≥ e | Φ | (27) n order to the energy be finite the covariant derivative must vanish asymptotically.This fixes the behavior of the gauge field A i and implies a non-vanishing magnetic flux:Φ = Z d xB = I | x | = ∞ A i dx i = 2 πN (28)where N is a topological invariant which takes only integer values. The bound issaturated by fields satisfying the first-order self-duality equations: D ± φ = ( D ± iD ) φ = 0 (29) B = ± e ( ρ −
1) (30)These are just the Bogomolnyi equations of the abelian Higgs model[6]. The differencelies in the fact that, here, our vortices not only carry magnetic flux, as in the Higgsmodel, but also U (1) charge. This is a consequence that in our theory the dynamics ofgauge field is dictated by a Chern-Simons term instead of a Maxwell term as in Higgstheory. Therefore, as consequence of the Gauss law of (16), if there is magnetic fluxthere is also electric charge: Q = e Z d x ω ( ρ ) j = κ Z d x B = κ Φ (31)Since, the fields A i and φ satisfy the same self-duality equations as in abelian Maxwell-Higgs theory the solutions will be the same for the same boundary conditions. Anotherinteresting aspect is that, although this is a Chern-Simons-Higgs model, we expectto find only topological solitons in contrast to the usual abelian Chern-Simons-Higgstheory which support both, topological and non-topological solitons. Of course theself-duality equations of ordinary abelian Chern-Simons-Higgs model, and therefore itstopological and non-topological solitons, may be achieved by taking ω ( ρ ) = 1.It is also interesting to compare our study with the results obtained in Ref.[20].In this last work the authors considered a class of generalization of the abelian Higgsmodel described by S = Z d x (cid:16) − ω ( ρ ) F µν F µν + | D µ φ | − V ( | φ | ) (cid:17) (32)Then, they chose ω ( ρ ) = ρ − κ e (33)and as potential term V ( | φ | ) = e κ ρ ( ρ − , (34)which is the self-dual potential of the Chern-Simons-Higgs theory. With this conditions,they were able to obtain the Bogomolnyi equations of the Chern-Simons-Higgs theory( D ± iD ) φ = 0 B = ± κ ( | φ | − υ ) | φ | (35) he vortex solutions of this equations are identical in form to the Chern-Simons-Higgsvortex solutions. However the difference lies in that these solutions have no electriccharge, in contrast with the charged vortices of Chern-Simons-Higgs model. In someway, we have performed an inverse procedure of the developed in Ref.[20], since wehave started from a generalization of abelian Chern-Simons-Higgs theory and we havearrived to the Bogomolnyi equation of the abelian Higgs model, whose solutions areNielsen-Olesen vortices which possess electric charge. Also it is interesting to note thatthe dielectric function ω ( ρ ) = ρ − κ e are very similar to our ω ( ρ ) = ρ − κ e . We conclude this note by analyzing a generalized dynamics of a non-relativistic Maxwell-Chern-Simons model proposed by Manton[17]. This model is governed by a (2+1)-dimensional action consisting on a mixture from the standard Landau-Ginzburg andthe Chern-Simons model, S = Z d x (cid:16) − B + iγ ( φ † ∂ t φ + iA | φ | ) −
12 ( D i φ ) † D i φ + κ ( A B + A ∂ A ) + γA + λ ( | φ | − − A i J Ti (cid:17) (36)Here, γ , κ and λ are real constants and the term γA is related to the possibility ofa condensate in the ground state[21]. In order to hold the Galilean invariance of themodel, the transport current J Ti should transform as J Ti → J Ti + γv i under a boost[17].Choosing a frame where J Ti = 0, the field equations takes the form iγD φ = − D i D i φ − λ ( | φ | − φǫ ij ∂ j B = eJ i + κǫ ij E j κB = eγ ( | φ | −
1) (37)where E i = ∂ i A − ∂ A i is the electric field. Using the identity (8) and the Gauss lawof (37) the energy of the model for static field configuration may be written as[22]: E = Z d x (cid:16) | ( D ± iD ) φ | + ( ∓ e γ κm + e γ κ − λ )( | φ | − ∓ m B (cid:17) (38)For λ = ∓ e γ κm + e γ κ the potential terms cancel, and the energy takes the minimumby the fields obeying the first order self-duality equations( D ± iD ) φ = 0 κB = eγ ( | φ | −
1) (39) otice that for κ = ± γ this set of equations become the Bogomolnyi equations of theHiggs model[22].Let us then propose the following generalization of the Manton model (36) S = Z d x (cid:16) − ω ( ρ ) − B + iγω ( ρ )( φ † ∂ t φ + iA | φ | ) −
12 ( D i φ ) † D i φ + κ ( A B + A ∂ A ) + γω ( ρ ) A − V ( ρ ) (cid:17) (40)Here, V ( ρ ) is again the scalar field potential to be determined below.By varying with respect to A we obtain the Gauss law for this system B = eγκ ω ( ρ )( ρ −
1) (41)After using the identity (8) and the Gauss law, the energy for the static field configu-ration may be written as follows: E = Z d x (cid:16) e γ κ ω ( ρ )( ρ − + 12 | ( D ± iD ) φ | ∓ e γ κm ω ( ρ )( ρ − ρ + V ( ρ ) (cid:17) (42)We will show that the solutions of this theory are related to those present in the abelianChern-Simons-Higgs model, if we choose a suitable ω ( ρ ). That choice is ω ( ρ ) = ± e γκ ρ (43)Then, the Gauss law takes the form B = ± e κ ρ ( ρ −
1) (44)and the energy functional (42) is written as E = Z d x (cid:16) ( ρ − ρ [ ρ ( ± e γκ − e κ + λ ) + ( ∓ e γκ − λ )] + 12 | ( D ± iD ) φ | (cid:17) (45)where we have chosen a specific form of potential motivated by the desire to findself-dual topological soliton solutions. That form is V ( ρ ) = λρ ( ρ − (46)In order to the write the expression (45) as sum of a square term plus a topologicalterm, we choose γ = κ and λ = ∓ e κ + e κ . Then, the energy (45) becomes E = Z d x (cid:16) | ( D ± iD ) φ | ∓ e B (cid:17) (47)Therefore, the energy is bounded below by a multiple of the magnitude of the magneticflux, which is saturated by the field satisfying( D ± iD ) φ = 0 B = ± e κ ρ ( ρ −
1) (48) he equations (48) are just the set of Bogomolnyi self-duality equations of the abelianChern-Simons-Higgs model. Thus, starting from a Manton model,in which the self-duality equations are identical to those of the abelian Higgs theory, we were able toconstruct a generalized model with the same Bogomolnyi equations as those of theabelian Chern-Simons-Higgs model. Another interesting aspect is that in our general-ization we have proposed a potential which is six order in the field just like the scalarpotential introduced in the Chern-Simons-Higgs system. It is also interesting to notethat the generalized model support also non-topological solitons which are not presentin the Manton model.In summary we have discussed the Bogomolnyi framework for two generalized mod-els obtained by the introduction of a nonstandard kinetic terms in the Lagrangian. Inthe first case we proposed a Chern-Simons-Higgs generalized system and showed thatthe self-duality equations are those of the abelian Higgs model. The second case con-sist on a generalizacion of a non-relativistic Maxwell-Chern-Simons model. We showed,here, that the Bogomolnyi equations of this model are the Chern-Simons Higgs self-duality equations. Acknowledgements
I would like to thank Peter Horvathy for many helpful comments and discussions. AlsoI would like to thank Rodolfo Casana and Manoel Messias Ferreira for suggestionsand for hospitality during the realization of this work. This work is supported byCAPES(PNPD/2011).
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