Electrically and magnetically charged vortices in the Chern–Simons–Higgs theory

Robin Ming Chen, Yujin Guo, Daniel Spirn, Yisong Yang

Abstract

In this paper, we prove the existence of finite-energy electrically and magnetically charged vortex solutions in the full Chern–Simons–Higgs theory, for which both the Maxwell term and the Chern–Simons term are present in the Lagrangian density. We consider both Abelian and non-Abelian cases. The solutions are smooth and satisfy natural boundary conditions. Existence is established via a constrained minimization procedure applied on indefinite action functionals. This work settles a long-standing open problem concerning the existence of dually charged vortices in the classical gauge field Higgs model minimally extended to contain a Chern–Simons term.

1. Introduction

Dirac (1931), in his celebrated work, showed that the existence of a magnetic monopole solution to the Maxwell equations has the profound implication that electric charges in the universe are all quantized. Later, Schwinger (1969) further explored the idea of Dirac and proposed the existence of both electrically and magnetically charged particle-like solutions, called dyons, and used them to model quarks. In particular, Schwinger (1969) generalized the electric charge quantization condition of Dirac (1931) to a quantization condition relating electric and magnetic charges of a dyon. In modern theoretical physics, dyons are considered as excited states of magnetic monopoles. Both magnetic monopoles and dyons and their abundance are predicted by grand unified theories (Lykken & Strominger 1980; l’Yi et al. 1982; Grossman 1983; Nelson 1983; Preskill 1984; Affleck 1986; Vachaspati 1996). The well-known finite-energy singularity-free magnetic monopole and dyon solutions in the Yang–Mills–Higgs theory include the monopole solutions due to Polyakov (1974), ’t Hooft (1974), Bogomol’nyi (1976), Prasad & Sommerfeld (1975), Jaffe & Taubes (1980) and Taubes (1982) and the dyon solutions due to Julia & Zee (1975), Bogomol’nyi (1976) and Prasad & Sommerfeld (1975). See also Cho & Maison (1997) and Yang (1998) for the construction of dyon solutions in the Weinberg–Salam electroweak theory. These are all static solutions of the governing gauge field equations in three-space dimensions.

Vortices arise as static solutions to gauge field equations in two-space dimensions. Unlike monopoles, magnetic vortices not only arise as theoretical constructs but also play important roles in areas such as superconductivity (Abrikosov 1957; Ginzburg & Landau 1965; Jaffe & Taubes 1980), electroweak theory (Ambjorn & Olesen 1988, 1989a,b, 1990) and cosmology (Vilenkin & Shellard 1994). The mathematical existence and properties of such vortices have been well studied (Jaffe & Taubes 1980; Berger & Chen 1989; Neu 1990; Du et al. 1992; Spruck & Yang 1992a,b; Bethuel et al. 1994; Weinan 1994; Bethuel & Rivière 1995; Lin 1995, 1998; Ovchinnikov & Sigal 1997; Bauman et al. 1998; Serfaty 1999; Pacard & Rivière 2000; Yang 2001; Montero et al. 2004; Tarantello 2008; B. J. Plohr 1980, unpublished data). Naturally, it will be interesting and important to establish the existence of dyon-like vortices, simply called electrically charged vortices, carrying both electric and magnetic charges. Such dually charged vortices have applications in a wide range of areas including high-temperature superconductivity (Khomskii & Freimuth 1995; Matsuda et al. 2002), optics (Bezryadina et al. 2006), the Bose–Einstein condensates (Inouye et al. 2001; Kawaguchi & Ohmi 2004), the quantum Hall effect (Sokoloff 1985) and superfluids.

Surprisingly, unlike static gauge field theory in three-space dimensions, it is recognized that there can be no finite-energy electrically charged vortex solutions in two-space dimensions for the classical Yang–Mills–Higgs equations, Abelian or non-Abelian. The impossibility of finite-energy electrically charged solutions is known as the Julia–Zee theorem (Julia & Zee 1975; Spruck & Yang 2009). Owing to the pioneering studies of Jackiw & Templeton (1981), Schonfeld (1981), Deser et al.(1982a,b), Paul & Khare (1986), de Vega & Schaposnik (1986a,b) and Kumar & Khare (1986), it has become accepted that, in order to accommodate electrically charged vortices, one needs to introduce into the action Lagrangian a Chern–Simons topological term (Chern & Simons 1971, 1974), which has become a central structure in anyon physics (Wilczek 1982, 1990; Fröhlich & Marchetti 1989). Therefore, an imperative problem one encounters is to develop an existence theory for the solutions of the full Chern–Simons–Higgs equations (de Vega & Schaposnik 1986a,b; Paul & Khare 1986) governing such electrically charged vortices.

This basic existence problem, however, has not yet been tackled in the literature, despite some successful numerical solutions reported (Jacobs et al. 1991). In fact, the lack of understanding of the solutions of the full system of equations has led to some dramatic trade-wind changes in the research on the Chern–Simons vortices, starting from the seminal papers of Hong et al. (1990) and Jackiw & Weinberg (1990), in which the Maxwell term is removed from the Lagrangian density while the Chern–Simons term stands out alone to govern the dynamics of electromagnetism. Physically, this procedure recognizes the dominance of the Chern–Simons term over the Maxwell term over large distances; mathematically, it allows one to pursue a Bogomol’nyi reduction (Bogomol’nyi 1976) when the Higgs potential takes a critical form as that in the classical Abelian Higgs model (Bogomol’nyi 1976; Jaffe & Taubes 1980). Such an approach triggered a wide range of explorations on the reduction of numerous Chern–Simons models, Abelian and non-Abelian, relativistic and non-relativistic (see Dunne (1995) for a review), and a rich spectrum of mathematical existence results for the Bogomol’nyi-type Chern–Simons vortex equations has been obtained (Spruck & Yang 1992c, 1995; Caffarelli & Yang 1995; Tarantello 1996, 2008; Chae & Kim 1997; Yang 1997; Nolasco & Tarantello 1999; Chae & Yu 2000; Ricciardi & Tarantello 2000; Chan et al. 2002; Nolasco 2003; Lin et al. 2007). We note that the existence of planar Abelian Chern–Simons models with no Maxwell term for non-Bogomol’nyi regimes has been recently established in Chen & Spirn (2009) and Spirn & Yan (2009). Although these contributions lead to considerable understanding of the properties of charged vortices when interaction between vortices is absent, the original problem of the existence of charged vortices, which are necessarily subject to interaction owing to the lack of a Bogomol’nyi structure, in the Chern–Simons–Higgs theory containing a Maxwell term (de Vega & Schapesnik 1986a,b; Kumar & Khare 1986; Paul & Khare 1986) remains unsolved.

In the present paper, we will establish the existence of charged vortices in the full Chern–Simons–Higgs theory with the Maxwell term (de Vega & Schapesnik 1986a,b; Kumar & Khare 1986; Paul & Khare 1986) in both Abelian and non-Abelian cases.

The rest of the paper is organized as follows. In §2, we review the Abelian Chern–Simons–Higgs theory, discuss some basic properties of charged vortices and their governing equations and state our main existence theorem. Then, we discuss the methods used in our proofs. In §3, we describe the basic set-up of our problem and introduce our constraint space. In §4 to §6, we prove the existence of weak solutions. In §7, we show that our weak solutions are, in fact, classical solutions. In §8, we establish the quantization formulas (2.16) and (2.17) expected for the magnetic and electric charges. Finally, in §9, we apply our methods to solve the non-Abelian Chern–Simons–Higgs equations.

2. Abelian Chern–Simons–Higgs equations and main existence theorem

After adding a Chern–Simons term to the classical Abelian Higgs Lagrangian density (Nielsen & Olesen 1973; Jaffe & Taubes 1980) and taking normalized units, the minimally extended action density, or the Chern–Simons–Higgs Lagrangian density introduced in Paul & Khare (1986) and de Vega & Schaposnik (1986a), defined over the Minkowski space–time Embedded Image with metric ημν=diag(1,−1,−1), may be written in the form Embedded Image 2.1 where ϕ is a complex scalar function, the Higgs field, Aμ (μ=0,1,2), is a real-valued vector field, the Abelian gauge field, Fμν=∂μAν−∂νAμ, is the induced electromagnetic field, Dμ=∂μ+iAμ is the gauge-covariant derivative, κ>0 is a constant referred to as the Chern–Simons coupling parameter, εμνγ is the Kronecker skew-symmetric tensor with ε012=1 and summation convention over repeated indices is observed. The extremals of the Lagrangian density (2.1) formally satisfy its Euler–Lagrange equations or the Abelian Chern–Simons–Higgs equations (Paul & Khare 1986), Embedded Image 2.2 and Embedded Image 2.3 in which equation (2.3) expresses the modified Maxwell equations so that the current density Jμ is given by Embedded Image 2.4 Recall that we may rewrite Jμ into a decomposed form Jμ=(ρ,J), such that ρ represents electric charge density and J=Jk represents electric current density. Here, and in the sequel, we use the Latin letters j,k=1,2 to denote the indices of spatial components.

Therefore, since we will consider static configurations only so that all the fields are independent of the temporal coordinate, t=x0, we have Embedded Image 2.5 which indicates that a non-trivial temporal component, A0, of the gauge field Aμ is essential for the presence of electric charge. Besides, also recall that the electric field E=Ej (in the spatial plane) and magnetic fields H (perpendicular to the spatial plane) induced from the gauge field Aμ are Embedded Image 2.6 respectively. The static version of the Chern–Simons–Higgs equations (2.2) and (2.3) takes the explicit form Embedded Image 2.7Embedded Image 2.8 and Embedded Image 2.9 On the other hand, since the Chern–Simons term gives rise to a topological invariant, it makes no contribution to the energy–momentum tensor Tμν of the action density (2.1), which may be calculated as Embedded Image 2.10 where Embedded Image is obtained from the Lagrangian (2.1) by setting κ=0. Hence, it follows that the Hamiltonian Embedded Image or the energy density of the theory is given by Embedded Image 2.11 which is positive definite and the terms in equation (2.11) not containing A0 are exactly those appearing in the classical Abelian Higgs model (Nielsen & Olesen 1973; Jaffe & Taubes 1980). Thus, the finite-energy condition Embedded Image 2.12 leads us to arrive at the following natural asymptotic behaviour of the fields A0, Aj and ϕ: Embedded Image 2.13Embedded Image 2.14 and Embedded Image 2.15 as Embedded Image. In analogue to the Abelian Higgs model (Nielsen & Olesen 1973; Jaffe & Taubes 1980), we see that a finite-energy solution of the Chern–Simons–Higgs equations (2.7)–(2.9) should be classified by the winding number, say Embedded Image, of the complex scalar field ϕ near infinity, which is expected to give rise to the total quantized magnetic charge (or magnetic flux).

The resolution of the aforementioned open problem for the existence of charged vortices in the full Chern–Simons–Higgs theory amounts to prove that, for any integer N, the coupled nonlinear elliptic equations (2.7)–(2.9) over Embedded Image possess a smooth solution (A0,Aj,ϕ) satisfying the finite-energy condition (2.12) and natural boundary conditions (2.13)–(2.15) so that the winding number of ϕ near infinity is N.

Here is our main existence theorem, which solves the above problem.

Theorem 2.1

For any given integer N, the Chern–Simons–Higgs equations (2.7)–(2.9) overEmbedded Imagehave a smooth finite-energy solution (A0,Aj,ϕ) satisfying the asymptotic properties (2.13)–(2.15) asEmbedded Image, such that the winding number of ϕ near infinity is N, which is also the algebraic multiplicity of zeros of ϕ inEmbedded Image, and the total magnetic charge or flux Φ and electric charge Q are given by the quantization formulasEmbedded Image 2.16andEmbedded Image 2.17Such a solution represents an N-vortex soliton, which is indeed both magnetically and electrically charged.

The proof of theorem 2.1 is contained in the proofs of propositions 5.2, 6.1 and 8.1. In the subsequent sections, we shall establish this theorem.

Methodology. We use the following standard ansatz to represent a radially symmetric N-vortex solution of the Abelian Chern–Simons–Higgs equations so that the N vortices are clustered at the origin: Embedded Image 2.18Embedded Image 2.19 and Embedded Image 2.20 As derived by Paul & Khare (1986) (and also de Vega & Schaposnik 1986a), the equations of motion (2.7)–(2.9) become Embedded Image 2.21Embedded Image 2.22and Embedded Image 2.23 Regularity and finite-energy condition prompt us to impose the boundary conditions Embedded Image 2.24Embedded Image 2.25 and Embedded Image 2.26 Here w0 is some finite constant, depending on N, λ and κ, that should arise from our constrained minimization procedure.

In order to establish existence, we note that equations (2.21)–(2.23) are the Euler–Lagrange equations of the indefinite action functional Embedded Image Here G(u,v) is the standard Ginzburg–Landau functional for radially symmetric vortices, studied by B. J. Plohr (1980, unpublished data) and Berger & Chen (1989). The functional Ju,v(w) is indefinite and a source of difficulty in our existence problem.

Note that, in view of the radially symmetric ansatz (2.18)–(2.20), the total energy calculated from the Hamiltonian density (2.11) is Embedded Image 2.27

In §3, we discuss some general notation and definitions used throughout the paper and set up our constrained minimization space. In particular, we will minimize I(u,v,w) over the space Embedded Image, consisting of triples (u,v,w) such that w is a weak solution to equation (2.23) with u and v given. This approach is similar to those of Schechter & Weder (1981) and Yang (2001) for the dyon problem in three spaces.

In §4, we assume bounded G(u,v) energy and we show that Ju,v(w) has a minimizer, say wu,v, among Embedded Image functions, and this minimizer is the unique critical point of Ju,v(w). Here we first show that we have a uniform control of the radius R, such that Embedded Image (say) outside the ball BR, which implies both the boundedness of Ju,v(w), CJu,v(w)≥−C and the control of the H1 norm of w. Such boundedness and H1 control give us the existence of a minimizer for Ju,v.

We prove the existence of weak solutions of equations (2.21)–(2.23) in §5 and §6. To do so we show that I(u,v,w)≥G(u,v) for Embedded Image, which implies the coercivity of I(u,v,w). Once we have this coercivity behaviour, we can take a minimizing sequence in Embedded Image and obtain a constrained minimizer. Such a minimizer can be shown to solve the equations (2.21)–(2.23) at least in a weak sense. Here some extra attention will be given to proving the existence of a Fréchet derivative.

In §7, we establish the boundary conditions and expected full regularity of our solutions. In §8, we obtain the quantization formulas for the magnetic and electric charges. In §9, we construct non-Abelian Chern–Simons–Higgs vortex solutions using our methods presented in the previous sections.

3. Radial equations action principle and the constrained admissible space

Recall that a radially symmetric solution of the Chern–Simons–Higgs theory with N vortices clustered at the origin satisfies equations (2.21)–(2.23), which can be derived from the indefinite action functional Embedded Image 3.1 Let Embedded Image 3.2 and Embedded Image 3.3 Then I(u,v,w)=G(u,v)−Ju,v(w). Note that G(u,v) does not depend on w and has the form of the Ginzburg–Landau energy, whereas Ju,v(w) contains an indefinite part Embedded Image.

The natural admissible space Embedded Image is defined by Embedded Image 3.4 Note that here we leave out the boundary condition (2.26) in the admissible set because it cannot be simply recovered from a finite energy requirement. However, condition (2.26) will be obtained when we construct a constrained admissible space.

Our goal is to find a critical point of the functional (3.1) in the admissible space Embedded Image. However, the difficulty comes from both the negative definite energy part and the indefinite energy part, which is an obstacle to getting the coerciveness of I(u,v,w). Motivated by the idea of the constrained minimization methods by Schechter & Weder (1981) and Yang (2001), we look for a suitable set of constraints to restrict the consideration of (3.1) over a smaller admissible space, say Embedded Image. With this choice of Embedded Image, I(u,v,w) becomes coercive on Embedded Image and the minimizer of I(u,v,w) over Embedded Image can be shown to be a critical point over the original admissible space Embedded Image, and thus is a solution of equations (2.21)–(2.23).

In order to make I(u,v,w) coercive over a properly constrained admissible space Embedded Image, we need to control Ju,v(w). To do so, we need to ‘freeze’ the unknown w, which certainly cannot be done arbitrarily since we are looking for a solution of equations (2.21)–(2.23) eventually. Hence, we naturally require w to satisfy equation (2.23) in a suitable weak sense for given u and v. In this way, we are led to considering seeking, for each fixed pair (u,v), a critical point of the functional Ju,v(w). In order to get a good convergence result, we restrict further to considering Embedded Image, where Embedded Image 3.5

We often use f(r) to unambiguously denote the radial dependence of the function f over Embedded Image, which is symmetric about the origin of Embedded Image.

Note that Embedded Image implies Embedded Image (Strauss 1977). In fact, it is easily seen that the set of all Embedded Image so that Embedded Image is an affine linear space. Besides, since Ju,v is strictly convex with respect to w for each given pair (u,v), Embedded Image 3.6 If w is a critical point, then Embedded Image 3.7 for all Embedded Image such that Embedded Image In this way, we may define the constrained admissible space Embedded Image 3.8

We need to make sure that Embedded Image is not empty. A natural way is to use the variational approach, that is, to consider minimizing Ju,v(w) over Embedded Image for certain fixed (u,v). The major difficulty is that, when it comes to minimizing I(u,v,w), one is looking at a class of (u,v). Moreover, Ju,v(w) contains an indefinite part, which, after applying Cauchy–Schwartz, introduces a term Embedded Image that cannot be controlled by Embedded Image only. Therefore, we have to enlist the second term Embedded Image in Ju,v(w) to help control the H1 norm of w.

4. Minimization of Ju,v(w)

Since u may vanish in a finite-energy setting, we need to control the size of the set in which Embedded Image.

Proposition 4.1

Suppose that (u,v) satisfies thatEmbedded Image. Then there exists an R independent of u such thatEmbedded Image, where BRis a ball inEmbedded Imageof radius R centred at the origin.

Proof.

Consider a pair (u,v) such that Embedded Image. Then using the result in Ginzburg–Landau theory (Berger & Chen 1989), we know that Embedded Image. We also know that ||u|′|≤|u′| a.e. (Gilbarg & Trudinger 1983). Hence, we have Embedded Image In this way, we may choose Embedded Image 4.1 so that Embedded Image for |x|≥R. ▪

We are now ready to study the minimization problem for Ju,v(w) over Embedded Image, for a fixed pair (u,v) such that Embedded Image, u(0)=v(0)=0 and Embedded Image.

Lemma 4.2

For each (u,v) withEmbedded Image, the following minimization problemEmbedded Image 4.2has a unique solution. HenceEmbedded Image.

Proof.

The uniqueness of the minimizer can be seen from the fact that the functional Ju,v(w) is strictly convex.

In order to prove the existence of the minimizer, we need to first show that Ju,v(w) is bounded from below, provided that Embedded Image. Using Cauchy–Schwartz, we have Embedded Image where R in the last inequality is defined by equation (4.1).

Take a smooth function η(x) on Embedded Image such that Embedded Image, 0≤η(x)≤1 and η≡1 on BR. Let Embedded Image. Then Embedded Image. Hence using Poincaré’s inequality, we have Embedded Image However, Embedded Image Therefore, Embedded Image 4.3 Hence, we obtain Embedded Image From equation (4.3) and the above inequality, we can also derive the following control of H1 norm of w in terms of Ju,v(w): Embedded Image 4.4

Now we can take a minimizing sequence {wn} in Embedded Image. Then, by equation (4.4), Embedded Image is uniformly bounded. Hence (up to a subsequence) Embedded Image Then by the compactness lemma in Strauss (1977), we know that Embedded Image From Berger & Chen (1989), we know that Embedded Image implies that Embedded Image and Embedded Image. Thus, the weak lower semicontinuity of the L2 norm, the Fatou’s lemma and the weak convergence of wn imply that Embedded Image Therefore, w solves equation (4.2).

Critical points of Ju,v(w), of course, satisfy equation (3.7), Embedded Image, and the lemma is proved. ▪

Remark

From the above proposition, we understand the structure of Embedded Image explicitly: for any pair (u,v) satisfying Embedded Image, equations (2.4) and (2.5), then Embedded Image is the unique triplet such that w is the unique solution to equation (2.5) and in fact minimizing Ju,v(w). Thus each pair u,v unambiguously defines w=w(u,v) and Embedded Image looks like the image of the map (u,v)↦w(u,v) in Embedded Image.

5. Minimization of I(u,v,w)

In this section, we try to solve the minimization problem of the full energy I(u,v,w) over the constrained admissible space Embedded Image. We first show that I(u,v,w) is positive definite and coercive with respect to u and v on Embedded Image.

Proposition 5.1

ForEmbedded Image, Embedded Image 5.1

Proof.

Considering equation (3.7) for (u,v,w) and taking Embedded Image, we get Embedded Image Therefore, Embedded Image 5.2 Hence, we have equation (5.1). ▪

Proposition 5.2

The minimization problemEmbedded Image 5.3has a solution.

Proof.

By lemma 4.2, we can take a minimizing sequence {(un,vn,wn)} of equation (5.3). Since all terms involving function u appear in a quadratic form, we may take all un≥0. From equation (5.1), we know that {G(un,vn)} is uniformly bounded. Therefore, from Berger & Chen (1989), we know that Embedded Image and Embedded Image are uniformly bounded, where Embedded Image 5.4

Hence, Embedded Image

From equation (5.2), we know that Jun,vn(wn)≤0. So by equation (4.4), Embedded Image is uniformly bounded. Therefore, Embedded Image Moreover, we have Embedded Image

Next we check that Embedded Image, that is, equation (3.7) is satisfied for all Embedded Image with Embedded Image.

Weak convergence of {wn} in L2 and {vn} in CS implies that Embedded Image As for the second term in equation (3.7), Embedded Image Using the compact embedding of Embedded Image for any p>2 (Chabrowski 1992), Embedded Image Similarly, Embedded Image and Embedded Image Therefore, we have proved that Embedded Image. To show that the limiting configuration (u,v,w) is a minimizer of equation (5.3), we consider equation (3.7) for (u,v,w)=(un,vn,wn) and take Embedded Image. Then Embedded Image In the same way, considering equation (3.7) for (u,v,w) and taking Embedded Image, we get Embedded Image Therefore, Embedded Image Thus, using weak lower semicontinuity and Fatou’s lemma, we have Embedded Image Hence, we conclude that such a limit (u,v,w) satisfies equation (5.3). ▪

6. Weak solutions of the governing equations

We use the idea developed in Yang (2001) to establish the existence of weak solutions.

Proposition 6.1

The action minimizing solution (u,v,w) of problem (5.3) is a weak solution of equations (2.21)–(2.23), subject to the partial boundary conditions (2.24) and (2.25).

Proof.

As discussed earlier, with Embedded Image 6.1 the constrained set Embedded Image may be viewed as the image of the map Embedded Image, (u,v)↦(u,v,w(u,v)) with w=w(u,v) being determined by equation (3.7), which is the weak form of equation (2.23). Consequently, χ is a differentiable map in an obvious sense. Besides, the minimizer (u,v,w) of the constrained problem (5.3) obtained in proposition 5.2 may simply be viewed as the image under χ of an absolute minimizer (u,v) of the functional I(u,v,w(u,v)) over the unconstrained class Embedded Image.

Let h be a real parameter confined in a small interval, say, |h|<1 and Embedded Image (functions with compact supports). Set Embedded Image. We use the following notations: Embedded Image Then we use Δw as a test function in equation (3.7) to get Embedded Image We also have Embedded Image Subtracting the first equality from the second, we get Embedded Image Using Cauchy–Schwartz, we obtain Embedded Image Hence, Embedded Image 6.2 From equations (4.4) and (5.2), we know that Embedded Image where C depends on Embedded Image. By assumption, we know |h|<1. Then Embedded Image and Embedded Image Hence, Embedded Image where C depends on Embedded Image, not on h.

Similarly, we obtain that Embedded Image where C is independent of h. Thus, Embedded Image 6.3 where C is independent of h. Taking Embedded Image in equation (6.3), we have Embedded Image The third term in Ju,v(Dw) can be bounded as follows by using proposition 9.1 and equation (4.4): Embedded Image where we have used equation (4.4) from the second inequality to the third and proposition 9.1 from the third to the fourth. Therefore, we get Embedded Image. Hence, Embedded Image Furthermore from the above estimates, we also obtain that Embedded Image. Therefore, equation (3.7) is satisfied with Embedded Image.

Since (u,v) minimizes I(u,v,w(u,v)), we have Embedded Image which gives Embedded Image 6.4 The left-hand side of the above leads to the validity of a weak form of equation (2.21).

Similarly, we fix a compactly supported test function Embedded Image and consider Embedded Image as before. We can show in a similar way as we did for equation (6.4) that Embedded Image 6.5 which is the weak form of equation (2.22). Therefore, the proof of the proposition is complete. ▪

7. Full set of boundary conditions and regularity

In this section, we show that the remaining boundary condition (2.26) also holds for the solution (u,v,w) obtained in the last section and then prove that the solution (u,v,w) is indeed a classical solution to equations (2.21)–(2.23).

Lemma 7.1

Let (u,v,w) be the solution of equations (2.21)–(2.23) obtained in the last section. Then equation (2.26) holds for a certain suitable w0.

Proof.

From the finite-energy configuration, we know Embedded Image and we have Embedded Image 7.1

We rewrite equation (2.22) as Embedded Image 7.2

Integrating equation (7.2) and using equation (7.1), we have Embedded Image 7.3

On the other hand, the condition Embedded Image implies that Embedded Image 7.4 Using equation (7.4) to integrate equation (2.23), we obtain, in view of equation (7.2), that Embedded Image 7.5

For I1(r), the Schwartz inequality gives us Embedded Image 7.6 where C may depend on the upper bound of |u|. Similarly, for I2(r) and I3(r), we have Embedded Image 7.7 and Embedded Image 7.8 Note that each of the right-hand sides of equations (7.6)–(7.8) appears in the energy functionals.

Integrating equation (7.5) and using equations (7.6)–(7.8), we see that the limit Embedded Image exists as hoped. ▪

Lemma 7.2

Through the ansatz (2.18)–(2.20), the solution (u,v,w) of the radial equations (2.21)–(2.23) obtained in the last section gives rise to a classical (smooth) solution (ϕ,Aj,A0) of the static Chern–Simons–Higgs equations (2.7)–(2.9) overEmbedded Image.

Proof.

We first prove the interior regularity of solutions. From the minimization procedure, we obtain that the weak solution lives in the space: Embedded Image, vCS, where CS is defined in equation (5.4), and Embedded Image. For any 0<δ<R, let Ω=BR\Bδ, then (u,v,w) is a generalized solution of the system Embedded Image on Ω. The right-hand side of the third equation is in L2(Ω). Hence, wH2(Ω). In the second equation, Embedded Image Hence, we also have vH2(Ω). In the first equation, Embedded Image where we have used the fact (Berger & Chen 1989) Embedded Image In this way, uH2(Ω). Therefore, by standard regularity theory of elliptic equations and using the iterative bootstrap argument, we conclude that (u,v,w) is a classical solution of equations (2.21)–(2.23) on Ω.

Since both u and w satisfy the property that Embedded Image 7.9 by the removable singularity theorem, the regularity of u and w extends to the origin, so does the regularity of ϕ(x) and A0(x) as in equations (2.18) and (2.20).

As for v, we first look at Aj(x). Since ∂jAj(x)=0 (divergence free) away from the origin, we know that, in Ω, Aj(x) satisfies Embedded Image 7.10 for some hjL2(Ω). Since Aj(x) is an H1(Ω) solution, from the previous interior regularity argument, we know that it is also an H2(Ω) solution. Hence, we may apply the same removable singularity theorem to extend the regularity of Aj(x) to the origin.

Bootstrap then shows that ϕ,Aj and A0 are all smooth across the origin. For example, for ϕ, we notice that equation (2.7) may be rewritten as Embedded Image 7.11 Therefore, we know that (u,v,w) gives rise to a classical solution. ▪

8. Quantization of magnetic flux and electrostatic charge

We finish with the proof of (2.16) and (2.17).

Proposition 8.1

The solution satisfies the quantization relationshipEmbedded Image where Q is the electrostatic charge and Φ is the magnetic flux.

Proof.

In the static case, the μ=0 component of equation (2.3) is the Gauss law, Embedded Image 8.1

On the other hand, within the radial ansatz (2.19), we know that the magnetic field is represented by Embedded Image Therefore, equation (2.23) is exactly the radial form of the Gauss law (8.1), which correctly relates the magnetic field F12 to the electric charge density J0 and implies that electricity and magnetism must coexist when the Chern–Simons coupling parameter is non-trivial, κ≠0. Thus, the total magnetic charge (flux) is given by Embedded Image 8.2

Since Embedded Image, then Embedded Image 8.3 Multiplying equation (2.23) by r, integrating and using equation (8.3), we get Embedded Image In particular, Embedded Image which explicitly shows how electric charge is proportional to magnetic flux. ▪

9. Application to non-Abelian Chern–Simons–Higgs equations

We start from the simplest non-Abelian case (de Vega & Schaposnik 1986a), where the gauge group is SU(2) and the scalar fields are two scalar fields represented adjointly. For convenience, use isovectors. The Chern–Simons–Higgs field-theoretical Lagrangian density reads (de Vega & Schaposnik 1986a) Embedded Image 9.1 where Embedded Image (μ=0,1,2), ϕ=(ϕ1,ϕ2,ϕ3) and ψ=(ψ123) are isovectors, Embedded Image and Embedded Image and the Higgs potential density is chosen to be Embedded Image 9.2 The equations of motion of (9.1) are Embedded Image 9.3 and Embedded Image 9.4 Following de Vega & Schaposnik (1986a), we take the following radially symmetric ansatz for an electrically charged static vortex solution so that ϕ and ψ are orthogonal in isospace, Embedded Image 9.5 and Embedded Image 9.6 where u, v and w are real-valued functions. Then the governing equations (9.3) and (9.4) become (2.21)–(2.23) when N=1, Embedded Image 9.7Embedded Image 9.8Embedded Image 9.9 subject to the boundary conditions (2.24)–(2.26). (Note that, in de Vega & Schaposnik (1986a), equation (2.26) is stated in a stronger form that the constant w0 assumes zero value. However, we have seen in our present study that w0 cannot be determined by the structure of the governing equations. This undeterminedness does affect the regularity, finiteness of energy and quantization of electric and magnetic charges of solutions.) Thus, the existence of electrically and magnetically charged static vortex solutions as described in de Vega & Schaposnik (1986a) follows.

We next describe how to apply our work to the study of the dually charged vortex solutions in the general non-Abelian Chern–Simons–Higgs gauge field theory. To be specific, we consider the SU(n) (n≥3) theory formulated in de Vega & Schaposnik (1986b). We use su(n) to denote the Lie algebra of SU(n) consisting of n by n Hermitian matrices with vanishing trace. The inner product over su(n) is then defined by (A,B)=Tr(AB)=Tr(AB) (A,Bsu(n)). Recall that the dimension of the Cartan subalgebra, or the rank, of SU(n) is n−1. Following de Vega & Schaposnik (1986b), we consider the Chern–Simons–Higgs field theory housing 2(n−1) Higgs scalar particles ϕa and ψa (a=1,2,…,n−1) in the adjoint representation of SU(n) given by the Lagrangian density Embedded Image 9.10 where Aμsu(n),Fμν=∂μAν−∂νAμ+[Aμ,Aν] and Dμ=∂μ+[Aμ,], and the potential density may be chosen to take the typical form Embedded Image 9.11 in which Vab’s are some functions satisfying Vab≥0 and Vab(0)=0 (1≤a,bn−1) and λa,ηa,μa,γa (1≤an−1) are positive coupling constants.

Recall that we can use the Cartan–Chevalley–Weyl basis {Ha,ER} to decompose su(n), where {Ha|a=1,2,…,n−1} is a basis of the (Abelian) Cartan subalgebra and R=(R1,…,Rn−1) are root vectors, so that the spaces H and E, spanned by {Ha} and {ER}, respectively, satisfy Embedded Image. With these facts, it is consistent to impose the condition that the gauge field Aμ lies in H and ϕa and ψa stay in E and H, respectively, for which ψa takes a constant value in H (a=1,2,…,n−1). Therefore, the equations of motion of (9.10) contain Aμ and Φα only, which are rewritten as (de Vega & Schaposnik 1986b) Embedded Image 9.12 and Embedded Image 9.13 where Embedded Image is the matter current generated from the Higgs particles.

To proceed, we follow de Vega & Schaposnik (1986b) to write down the group element Embedded Image 9.14m=0,1,…,n−1, which lies in the Cartan subgroup and is responsible for the degeneracy of vacuum space. Then set Embedded Image 9.15

The radially symmetric static vortex solutions of the SU(n) Chern–Simons–Higgs theory formulated in de Vega & Schaposnik (1986b) are given by the ansatz Embedded Image 9.16 and Embedded Image 9.17 realizing a solution asymptotically associated with the mth non-trivial vacuum state represented by an integral class in the fundamental group of the coset space of centre Zn of SU(n), that is, by Embedded Image, where the ladder generators {ERa} are chosen to assume the normalized forms (de Vega & Schaposnik 1986b) Embedded Image 9.18

Substituting equations (9.16) and (9.17) into equations (9.12) and (9.13) and using equation (9.18), we arrive at the radial version of the equations of motion (de Vega & Schaposnik 1986b) Embedded Image 9.19Embedded Image 9.20Embedded Image 9.21 subject to the boundary condition consisting of Embedded Image 9.22Embedded Image 9.23Embedded Image 9.24 The associated action functional to the above equations is Embedded Image 9.25 which is again indefinite of course. Here and in the sequel, we use the vector notation u=(ua)=(u1,…,un−1). Let Embedded Image Then it is clear that Embedded Image.

The total energy is Embedded Image 9.26 Thus the natural admissible space Embedded Image is Embedded Image 9.27

We will first minimize Embedded Image for (u,v) such that Embedded Image in order to construct our constraint set. From the argument before, we need to control the size of the set where |ua(x)|≤ηa/2 for each a=1,2,…,n−1.

Proposition 9.1

Suppose that (u,v) satisfies thatEmbedded Image. Then there exists an R independent of uasuch thatEmbedded Imagefor all a=1,2,…,n−1, where BRis a ball inEmbedded Imageof radius R centred at the origin.

Proof.

Consider (u,v) such that Embedded Image. From the result on Ginzburg–Landau theory, we know that Embedded Image. Hence, we have Embedded Image In this way, we may choose Embedded Image 9.28 Then |ua(x)|>ηa/2 for |x|≥R, a=1,…,n−1. ▪

In particular, denote Embedded Image Hence from the above proposition |ua|>η/2 for |x|≥R. Then using the same argument as in lemma 4.2, we have the following.

Lemma 9.2

For each (u,v) withEmbedded Image, the following minimization problemEmbedded Image 9.29has a unique solution.

Proof.

The uniqueness of the minimizer can be seen from the fact that the functional Embedded Image is strictly convex in w.

First, we derive the lower bound for Embedded Image. Using Cauchy–Schwartz, we have Embedded Image where R in the last inequality is defined by equation (9.28) and we have used equation (4.3) to obtain the last inequality. Choosing ε to satisfy Embedded Image 9.30 we obtain Embedded Image From this and equation (4.3), we also obtain the control of Embedded ImageEmbedded Image 9.31 where ε satisfies equation (9.30). The rest of the proof will be the same as in lemma 4.2. ▪

In this way, we can construct the constraint set to the minimization problem of Embedded Image to be Embedded Image 9.32 Therefore, from lemma 9.2, we know that Embedded Image, and for each minimizer w, we have Embedded Image 9.33

Therefore, the minimization of Embedded Image can be done via the similar method as before.

Proposition 9.3

The minimization problemEmbedded Image 9.34has a solution.

Furthermore, the existence and regularity of solutions and the verification of the boundary conditions can all be achieved by the same argument as in §§6 and 7. As for the quantization relations, following de Vega & Schaposnik (1986b), we introduce an electromagnetic tensor Embedded Image 9.35 Then the quantization of the magnetic flux and the electric charge can also be obtained by the same method as in §8.

Summing up all of the above, we have the following.

Theorem 9.4

For any given integer m∈{1,…,n−1}, the non-Abelian Chern–Simons–Higgs equations expressed in equations (9.12) and (9.13) overEmbedded Imagehave a smooth finite-energy solution (A0,A,ϕ), where ϕ=(ϕa) represents a multiplet of n−1 Higgs fields each lying in the Cartan subalgebra of su(n), satisfying the asymptotic propertiesEmbedded Image asEmbedded Image. Moreover, the total magnetic flux Φ and electric charge Q are given, respectively, by the quantization formulasEmbedded Image 9.36andEmbedded Image 9.37Such a magnetically and electrically charged solution realizes an SU(n) vortex configuration asymptotically and topologically represented by the mth integral class in the classification space of the vortex vacuum manifold SU(n)/Zn, that is, byEmbedded Imagefor m=1,…,n−1.

To conclude, in this paper, we have developed an existence theory for the electrically and magnetically charged vortex solutions arising in the classical Abelian and non-Abelian Chern–Simons–Higgs models using a constrained variational approach. Such a construction is of a general nature and does not rely on exploring the self-dual or BPS formulation of the problem.

Acknowledgements

The first three authors would like to thank Ellen Shi Ting Bao for many fruitful discussions. R.M.C. was supported in part by NSF grant no. DMS-0908663. D.S. was supported in part by NSF grant no. DMS-0707714. Y.Y. was supported in part by NSF grant no. DMS-0406446.

Footnotes

  • Address from 1 September 2009: Department of Mathematics, Polytechnic Institute of New York University, Brooklyn, NY 11201, USA.

    • Received April 14, 2009.
    • Accepted July 16, 2009.

References

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