Vortex solutions in the Abelian Higgs Model with a neutral scalar
VVortex solutions in the Abelian Higgs Model with a neutralscalar
Prabal Adhikari, Jaehong Choi
St. Olaf College, 1520 St. Olaf Avenue, Northfield, MN 55057We construct an extension of the Abelian Higgs model, which consistsof a complex scalar field by including an additional real, electromagneticallyneutral scalar field. We couple this real scalar field to the complex scalarfield via a quartic coupling and investigate U (1) vortex solutions in this“extended Abelian Higgs Model”. Since this model has two additionalhomogeneous ground states, the U (1) vortices that can form in this modelhave a richer structure than in the Abelian Higgs Model. We also findthe “phase diagram” of the model showing the parameter space in whichthe real scalar particle condenses in the vortex state while having a zerovacuum expectation value in the homogeneous ground state.
1. Introduction
Magnetic U (1) vortices have been of great interest both theoretically andexperimentally since they were first predicted by Abrikosov [1] in Ginzburg-Landau theory [2] back in the fifties. Since then they have been studiedextensively not only in the context of non-relativistic condensed mattersystems [3] but also in particle physics [4, 5, 6, 7]. It is interesting to notethat a mathematical isomorphism between Ginzburg-Landau theory andthe Abelian Higgs model was formally established [8] in the seventies. Morerecently, there has been increasing interest in the properties of QuantumChromodynamics (QCD) in the presence of large magnetic fields [9]. Thisinterest stems from the presence of large magnetic fields in a variety ofphysical systems including magnetars [10] and in quark gluon plasma [11,12, 13] at RHIC.Additionally, there has also been increasing interest in systems of mesonsin the presence of magnetic fields. It was shown that ρ -mesons may condensein the QCD vacuum at extremely large magnetic fields [14, 15, 16]. Theresulting condensed system consists of charged ρ ± mesons and a neutral ρ meson and in the presence of magnetic field they can form a ground state (1) a r X i v : . [ h e p - t h ] M a r Vortex printed on March 6, 2017 consisting of a magnetic vortex lattice, in which not only the charged ρ -mesons condense but also the neutral meson superfluid condenses. On theother hand, there are systems such as low-energy QCD described by chiralperturbation theory, which at finite isospin and in the presence of externalmagnetic fields behaves like a type-II superconductor. Chiral perturbationtheory has three mesons, namely the charged pions π ± and the neutral pion, π . However, in the magnetic vortices that form at moderate external fieldsonly the charged pions ( π ± ) condense in the vortex state and the neutralpions are absent in the vortices [17]. This is also true in the context of thelinear sigma model [18].The physics encoded within chiral perturbation theory is constrained bythe symmetries and symmetry breaking properties of QCD. In this paper, weconsider a less restrictive model: an extension of the Abelian Higgs modelconsisting of charged scalars coupled to a charge-less scalar via a quarticcoupling and investigate vortex solutions within this model. In particularwe are interested in studying the possibility of neutral scalar condensation.Of course there is the trivial scenario, where the real scalar field has anon-zero vacuum expectation value (vev) and as condenses in a vortex.But a slightly more non-trivial scenario is when the real scalar field hasa zero vacuum expectation value, yet condenses in the vortex phase. Theadvantage of using such a model is that there are parameters such as thescalar masses and strength of the interaction between the scalars that canbe controlled less restrictively.The paper is organized as follows: in section 2, we construct the “ex-tended Abelian Higgs model” and find the homogeneous ground state of thetheory. In section 3, we consider the behavior of the vortex asymptoticallyfar away from the vortex center. In section 4, we show the full numericalvortex solutions for a wide range of parameters and interactions between thecharged and the neutral scalars. Finally, we construct a “phase diagram”showing the region in parameter space, where neutral pion condenses whilehaving a zero vacuum expectation value.
2. The extended Abelian Higgs Model
We begin by constructing the Lagrangian for an extended version ofthe standard Abelian Higgs model that not only contains a set of chargedscalars (Φ, Φ † ) but also a real scalar field ( φ ) that couples to the complexscalar field via an interaction term of the form f | Φ | | φ | , where f is thestrength of the coupling between the real and complex scalar fields. Notethat the two fields cannot couple via electromagnetic interactions since thereal scalar is electromagnetically neutral. Therefore, the fields must couplevia strong interactions - it is worth pointing out that in chiral perturbation, ortex printed on March 6, 2017 the two charged pions ( π ± ) and the neutral pion ( π ) interact with eachother through both momentum-dependent and momentum-independent in-teractions [19].The effective Lagrangian for the extended Abelian Higgs Model is then: L = − F ij F ij − | ( (cid:126) ∇ + ie (cid:126)A )Φ | −
12 ( (cid:126) ∇ φ ) − V ( | Φ | , | φ | ) , (1)where V ( | Φ | , | φ | ) = a | Φ | + b | Φ | + c | φ | + d | φ | + f | Φ | | φ | . (2)In the Lagrangian, F ij ≡ ∂ i A j − ∂ i A j is the electromagnetic tensor with A i representing the spatial components of the vector potential. Φ is a complexscalar field and φ is a real scalar field, which is electromagnetically neutral. ± e is the charge associated with the complex scalar fields. Finally, note thatthe Lagrangian is invariant under local U(1) gauge transformations, i.e.Φ → e i Λ( (cid:126)x ) Φ e (cid:126)A → e (cid:126)A + (cid:126) ∇ Λ( (cid:126)x ) , (3)and under | φ | → −| φ | and | Φ | → −| Φ | . In this section, we find the ground state of the model in the absence of aphoton field, i.e. (cid:126)A = 0. The ground state must be homogeneous, thereforewe begin by minimizing the potential. In order to do so, we introduce thenotation: X ≡ | Φ | , Y ≡ | φ | . (4)and minimize the potential with respect to X and Y . We get ∂V∂X = a + 2 bX + f Y = 0 ∂V∂Y = c + 2 dY + f X = 0 . (5)The extremum of the potential then occurs when X ext = cf − ad bd − f and Y ext = af − bc bd − f , (6)where X ext and Y ext are the magnitude squared of the fields Φ and φ (re-spectively) at the extremum of the potential V in Eq. (2). The extremum is Vortex printed on March 6, 2017 a locally stable minimum if the following conditions are satisfied (accordingto the second derivative test): ∂ V∂X = 2 b > D ≡ ∂ V∂X ∂ V∂Y − (cid:18) ∂ V∂X∂Y (cid:19) = 4 bd − f > , (8)which automatically implies that d ≥ f is positive definite.Since by definition, X min ≥ Y min ≥ D ≡ bd − f >
0, it followsthat the model possesses the following vacuum expectation values in theirground states. The vevs of Φ and φ are denoted by v Φ and v φ respectively:1. normal vacuum: v Φ = v φ = 0 if cf ≤ ad and af ≤ bc (9)2. charged superfluid: v Φ = (cid:114) − a b , v φ = 0 if cf > ad and af ≤ bc (10)3. neutral superfluid: v Φ = 0 , v φ = (cid:114) − c d if cf ≤ ad and af > bc (11)4. charged and neutral superfluids v Φ = (cid:115) cf − ad bd − f , v φ = (cid:115) af − bc bd − f if cf > ad and af > bc . (12)Note that the vacuum expectation values in the normal and charged super-fluid state are consistent with the Abelian Higgs Model [5, 6] as expected.Also, local stability requires D > b > d >
3. Vortex solutions
In this section, we consider single vortex solutions within the extendedAbelian Higgs model. We assume time-independence, i.e. ∂ A i = 0 and ortex printed on March 6, 2017 that the zeroth component of the vector potential, A = 0. In other words,we will only concern ourselves with magnetic fields in this paper and ignoreelectric fields.It is important to note that when considering the thermodynamics onehas to be careful not to ignore the energy density associated with the electro-magnetic charge densities that condense in the charged superfluid state. Itcan be ignored as long as the electromagnetic energy density associated withthe charged superfluids is small compared to the strength of the remaininginteractions, see Ref. [17] for a full discussion in the context of chiral per-turbation theory. In this paper, we will only consider single vortex solutionsand leave the thermodynamic discussion to future work. Vortices are known to be the ground state in the presence of “moderate”external magnetic fields in Ginzburg Landau theory [1] (and therefore theAbelian Higgs model [8]). Vortex solutions are cylindrically symmetric andobey the following ansatz: A r ( (cid:126)x ) = A z ( (cid:126)x ) = 0 , A θ ( (cid:126)x ) = A ( r )Φ( (cid:126)x ) = | Φ( r ) | exp( iχ ) where χ = nθφ ( (cid:126)x ) = φ ( r ) , (13)where θ is the polar angle and n is an integer. Note that only the radialand longitudinal components of the vector potential are zero and the polarcomponent only depends on the radial distance from the center of the vortex.The polar dependence of the the charged scalar field is what gives rise tothe vortex quantization condition. The electromagnetic current is as follows: j i = ie (Φ † ∂ i Φ − Φ ∂ i Φ † ) + 2 e A i | Φ | . (14)(Note that it is proportional to the electromagnetic charge of the complexscalar as expected.) Using the vortex ansatz, it is straightforward to showthat the radial and longitudinal components of the current are both zero. j z = 0 , j r = ie (Φ † ∂ r Φ − Φ ∂ r Φ † ) = 0 . (15) Vortex printed on March 6, 2017 j θ = ie (cid:18) Φ † r ∂ θ Φ − Φ 1 r ∂ θ Φ † (cid:19) + 2 e A ( r ) | Φ | = − e | Φ( r ) | ∂ θ χr + 2 e A ( r ) | Φ( r ) | = 2 e | Φ( r ) | (cid:18) A ( r ) − πner (cid:19) . (16)The polar component of the electromagnetic current is non-zero but mustvanish at infinity (i.e. r → ∞ ) giving the magnetic flux quantization con-dition, which is identical to the Abelian Higgs model since the new scalarfield is charge-less and hence does not contribute to the electromagneticcurrent [4]. (cid:90) C ∞ (cid:126)A · d(cid:126)l = (cid:90) (cid:126)B · d(cid:126)a = 2 nπe , (17)where the line integral in the first term is performed at infinity. Asymptot-ically, the polar component of the vector potential has the following form:lim r →∞ A ( r ) = ner , (18)which is consistent with the flux quantization condition of Eq. (17). In order to analyze the vortex asymptotics, we begin with the equationsof motion − ∂ i F ik = j k ( ∂ i + ieA i ) | Φ | + 2 a | Φ | + 4 b | Φ | + 2 f | Φ || φ | = 0 ∂ i ∂ i | φ | + 2 c | φ | + 4 d | φ | + 2 f | Φ | | φ | = 0 , (19)which can be simplified in cylindrical coordinates using the vortex ansatz: − ∂ A∂ r − r ∂A∂r + A ( r ) r = j θ = 2 e | Φ( r ) | (cid:16) A ( r ) − ner (cid:17) (20) − ∂ | Φ | ∂ r − r ∂ | Φ | ∂r + (cid:16) nr − eA ( r ) (cid:17) | Φ | + 2 a | Φ | + 4 b | Φ | + 2 f | φ | | Φ | = 0(21) − ∂ | φ | ∂ r − r ∂ | φ | ∂r + 2 c | φ | + 4 d | φ | + 2 f | Φ | | φ | = 0 (22) ortex printed on March 6, 2017 While the equations above cannot be solved exactly, it is possible to deducethe behavior close to the center of the vortex ( r → r → A ( r ) ∼ r n with n ≥ r → | φ ( r ) | ∼ r n with n ≥ r → | Φ( r ) | ∼ r n with n ≥ , (23)and also asymptotically far away from the vortex:lim r →∞ A ( r ) = ner lim r →∞ | Φ( r ) | = v Φ lim r →∞ | φ ( r ) | = v φ . (24)Furthermore, it is possible to find how the photon and scalar fields approachtheir asymptotically values. We begin with A ( r ) written in the followingform: A ( r ) = ner + δA ( r ) . (25)Assuming | Φ( r ) | = v Φ , the equation of motion for A ( r ) can be solved to get: δA ( r ) = c A K ( √ v Φ er ) + c B I ( √ v Φ er ) , (26)where the c i s are arbitrary constants and K and I are modified Besselfunctions of the second and first kinds. We can set c B = 0 noting that theBessel function of the second kind, I , has the wrong asymptotic behav-ior . Therefore, the polar component of the vector potential, A ( r ), has thefollowing asymptotic form: A ( r ) = ner + c A K ( √ v Φ er )= ner + ˜ c A e −√ v Φ er (cid:18) √ r + O (cid:18) r / (cid:19)(cid:19) , (27) Note that the modified Bessel functions have the following asymptotic properties: K ( mx ) = e − mx (cid:18)(cid:114) π mx + O (cid:18) mx ) / (cid:19)(cid:19) I ( mx ) = e − mx (cid:32) − i (cid:114) πmx + O (cid:18) mx ) / (cid:19)(cid:33) + e mx (cid:18) πmx + O (cid:18) mx ) / (cid:19)(cid:19) Vortex printed on March 6, 2017 with ˜ c A ≡ c A / (cid:113) πev Φ with c A being an undetermined constant.Next, we proceed to find the asymptotic forms of both the scalar fields, | Φ( r ) | and | φ ( r ) | . In order to do so, we first define: | Φ( r ) | = v Φ + δ Φ( r ) | φ ( r ) | = v φ + δφ ( r ) , (28)where v Φ and v φ are the vevs in the ground state and δ Φ and δφ are thefluctuations around the vevs. Then equations of motion for δ Φ( r ) and δφ ( r )are as follows: δ Φ (cid:48)(cid:48) ( r ) + δ Φ (cid:48) ( r ) r − n δ Φ( r ) − n δφ ( r ) = v Φ e δA ( r ) δφ (cid:48)(cid:48) ( r ) + δφ (cid:48) ( r ) r − n δφ ( r ) − n δ Φ( r ) = 0 , (29)where the n i s are defined as: n ≡ a + 12 bv + 2 f v φ n = n ≡ f v φ v Φ n ≡ c + 12 dv φ + 2 f v . (30)We can then rewrite the homogeneous coupled differential equation of Eq. (29)in the following form: (cid:18) δ Φ (cid:48)(cid:48) ( r ) + δ Φ (cid:48) ( r ) /rδφ (cid:48)(cid:48) ( r ) + δφ (cid:48) ( r ) /r (cid:19) = (cid:18) n n n n (cid:19) (cid:18) δ Φ( r ) δφ ( r ) (cid:19) . (31)The above matrix equation can be solved exactly; in order to do so notethat the matrix with n i s in Eq. (31) has the following eigenvalues and eigen-vectors λ ± = 12 (cid:16) n + n ± (cid:112) ( n − n ) + 4 n n (cid:17) (cid:126)v ± = 1 (cid:113) s ± (cid:18) s ± (cid:19) s ± = n − n ± (cid:112) ( n − n ) + 4 n n n , (32)where (cid:126)v ± are the normalized eigenvectors with eigenvalues λ ± . Then wedefine the matrix P = s − √ s − s + √ s √ s − √ s , (33) ortex printed on March 6, 2017 and the matrix N = (cid:18) n n n n (cid:19) , (34)such that P − N P = D , where D = (cid:18) λ − λ + . (cid:19) , (35)and (cid:18) δ ˜Φ( r ) δ ˜ φ ( r ) (cid:19) = P − (cid:18) δ Φ( r ) δφ ( r ) (cid:19) (36)with P − = √ s − s − − s + − s + √ s − s − − s + − √ s s − − s + s − √ s s − − s + . (37)The coupled equations of Eq. (31) then decouple in the new basis: δ ˜Φ (cid:48)(cid:48) ( r ) + δ ˜Φ (cid:48) ( r ) r = λ − ˜Φ( r ) δ ˜ φ (cid:48)(cid:48) ( r ) + δ ˜ φ (cid:48) ( r ) r = λ + ˜ φ ( r ) . (38)The solutions of the equations then are δ ˜Φ( r ) = c Φ K ( (cid:112) λ − r ) + d Φ I ( (cid:112) λ − r ) δ ˜ φ ( r ) = c φ K ( (cid:112) λ + r ) + d φ I ( (cid:112) λ + r ) , (39)where K and I are different Bessel functions of the second and first kindsrespectively. As before, we set d φ = d Φ = 0 since I has the wrong asymp-totic properties . Finally then we have solutions to the homogeneous cou-pled equation: (cid:18) δ Φ( r ) δφ ( r ) (cid:19) = P (cid:18) δ ˜Φ( r ) δ ˜ φ ( r ) (cid:19) = P (cid:18) c Φ K ( (cid:112) λ − r ) c φ K ( (cid:112) λ + r ) (cid:19) . (40)The particular solutions of this equation have been calculated in Ref. [20],we simply quote the result here: (cid:18) δ Φ p ( r ) δφ p ( r ) (cid:19) ≈ (cid:18) g Φ g φ (cid:19) e − αr r (41) Note that for asymptotically large x , K ( mx ) has the same behavior as K ( mx ) atleading order. But I ( mx ) is different from I ( mx ) by a a complex conjugation, i.e. i → − i at leading order. Vortex printed on March 6, 2017 with corrections of O (cid:0) r (cid:1) . Plugging the ansatz into the coupled differentialequation of Eq. (29), we get for α , g Φ and g φ : α = 2 √ ev Φ g Φ = ˜ c A e v Φ (cid:0) n − ( α ) (cid:1) n n − n n + ( n + n ) (cid:0) α (cid:1) − (cid:0) α (cid:1) g φ = − ˜ c A e n v Φ n n − n n + ( n + n ) (cid:0) α (cid:1) − (cid:0) α (cid:1) . (42)Therefore, the full solution to the coupled equations of Eq. (29) is: δ Φ( r ) ≈ g Φ e − αr r + s − p c Φ K ( (cid:112) λ − r ) + s + p c φ K ( (cid:112) λ + r ) δφ ( r ) ≈ g φ e − αr r + p c Φ K ( (cid:112) λ − r ) + p c φ K ( (cid:112) λ + r ) , (43)where p ij are the elements of the matrix P and g Φ , g φ , c Φ and c φ areundetermined constants. However, note that the solution quoted in Eq. (43)is not valid if the vev of φ is zero. The case with v φ = 0 is discussed towardsthe end of the next section.
4. Numerical Vortex Solutions
In this section, we present the single vortex solutions (with unit flux of πe ) generated numerically. Before we proceed with a discussion of these so-lutions, we note that the mass of the scalars and the photon in the extendedAbelian Higgs model are m Φ = √ a, m φ = √ c and m (cid:126)A = ev Φ . (44)It is interesting to note that the mass scales that affect the asymptoticbehavior of the charged and neutral scalar condensates are λ ± and α =2 √ m (cid:126)A . The mass scale that affects the asymptotic behavior of the vectorpotential is α , not m (cid:126)A .In our numerical vortex solutions, we use e = 1 and sgn( a ) = − | a | is used to set the scale in the problem.The following dimensionless quantities are used in our numerical plots: r = (cid:112) | a | r, ¯Φ = Φ (cid:112) | a | , ¯ φ = φ (cid:112) | a | , B = B | a | . (45)Note that (cid:126)B = (cid:126) ∇ × (cid:126)A , B ≡ | (cid:126)B | and that b , d and f are dimensionlessparameters. ortex printed on March 6, 2017 v Φ = v φ (cid:54) = 0Here we consider numerically generated vortex solutions assuming c ≡ ca = 1 and b = d such that the vevs, v Φ and v φ are identical. We choose b = 40 and vary f between −
75 and 75. f controls the strength and nature(attractive, repulsive or neither) of the quartic interaction between the Φand φ fields. Figs. 1, 2, 3, 4 and 5 show the resulting vortex structure.For f >
0, the interaction between the charged scalar and the neutralscalar fields is repulsive and the resulting solutions are such that the scalarfields approach their vevs from opposite sides as shown in Figs. 1 and 2.For the choice of parameters in Fig. 1, v Φ = v φ = √ and λ − < α < λ + ,suggesting that the behavior asymptotically far from the vortex center iscontrolled by λ − . Furthermore, s − is negative, which is consistent with thefact that two scalar fields approach their vevs from opposite directions.In Fig. 2, v Φ = v φ = √ and α < λ − < λ + . Therefore, α determinesthe asymptotic behavior of both the scalar fields. From Eq. 42, we notethat sgn( g Φ ) = − sgn( g φ ), which is consistent with the fact that Φ and φ approach v Φ and v φ respectively from opposite sides. See Eq. 43.Fig. 3 shows the result for f = 0 in which case v Φ = v φ = √ . In thiscase, g φ = 0, g Φ (cid:54) = 0 and α < λ + = λ − , which is the asymptotic behaviorof Φ and φ are different. The asymptotic behavior of Φ is controlled by α and that of φ by λ + .For f <
0, the interaction between the scalar fields is attractive and theresulting vortex solutions approach the vevs from the same direction. InFig. 4 the vevs are v Φ = v φ = √ and in Fig. 5 v Φ = v φ = √ . In bothcases, sgn( g Φ ) = sgn( g φ ) consistent with numerical results that show thescalar fields approaching their vevs from the same side. v Φ (cid:54) = 0 , v φ = 0Finally, we consider the extended Abelian Higgs model with the param-eters chosen such that the vev of φ is zero, v φ = 0, but that of Φ is not, v Φ (cid:54) = 0, with particular interest in the possibility of real scalar ( φ ) conden-sation within single vortices ( n = 1). We choose b = d but a (cid:54) = c . Figs. 6,7 and 8 show the resulting vortices for b = 10 and c = ca = for f > f < v φ (cid:54) = 0. The resulting asymptotic behavior of thescalar fields is not given by Eq. (43). It is easy to see from Eq. (30) that n = n = 0 and the resulting matrix N of Eq. (34) is diagonal with thenon-zero elements being n = 2 a + 12 bv n = 2 c + 2 f v . (46) Vortex printed on March 6, 2017
Noting that λ + = n , λ − = n and g φ = 0, the resulting solutions for δ Φand δφ are: δ Φ( r ) ≈ g Φ e − αr r + c Φ K ( (cid:112) λ + r ) δφ ( r ) ≈ c φ K ( (cid:112) λ − r ) . (47)It is immediately obviously then that the asymptotics of the φ field in Figs. 6,7 and 8 are controlled by λ − (or n ). On the other hand, for the Φ fieldasymptotics depends on both λ + (or n ) and α . However, it is straight-forward to check that α < λ + and as such α determines the asymptoticbehavior of Φ. In Figs. 9 and 10, we show the resulting “phase diagram”assuming c = and c = respectively and b = d . The fine black, meshregion shows the region where D < v φ (cid:54) = 0and v Φ (cid:54) = 0. In the gray shaded and the solid white regions, v Φ (cid:54) = 0 but v φ = 0 with the difference being that in the gray shaded region, real scalarscondense in the vortex phase. Finally, note that the region with the realscalar condensed phase is smaller when c = compared to the case when c = .
5. Conclusion and Future Work
In this paper, we have constructed an extension of the Abelian Higgsmodel consisting of an additional real scalar field coupled to the chargedscalars via a quartic coupling that preserves the U (1) symmetry of theAbelian Higgs model. The focus of this paper was to study the possibilityof real scalar condensation in single vortices with one unit ( n = 1) of mag-netic flux (which equals πe ) when the vev associated with the real scalarfield is non-zero while that associated with the complex scalar is non-zero.We constructed a “phase diagram” showing the region in parameter spacewith real scalar condensation. In future work, we will consider the ther-modynamics of the extended Abelian Higgs model in the presence of anexternal magnetic field and consider the structure of Abrikosov lattices andthe resulting phase diagram as a function of the external magnetic field andmodel parameters. REFERENCES [1] A.A. Abrikosov, Zh. Eksp. Teor. Fiz. , 1442 (1957); Sov. Phys. JETP ,1174 (1957). ortex printed on March 6, 2017 , 1064 (1950); L.D.Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546.[3] A.J. Leggett, Rev. Mod. Phys. , 331 (1975).[4] H.B. Nielsen, and P. Olesen, Nucl. Phys. B , 45 (1973).[5] N.K. Nielsen, and P. Olesen, Nucl. Phys. B144 (2), 376 (1978).[6] J. Ambjørn and P. Olesen, Nucl. Phys. B (2), 265 (1980).[7] H.J. De Vega, and F. A. Schaposnik, Phys. Rev.
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Vortex printed on March 6, 2017
Appendix A
List of Figures r | Φ | , | ϕ | , B Fig. 1: The solid (blue) curve represents Φ, the dotted (red) curve represents φ and the dashed (orange) curve represents the magnetic field, B , in a singlevortex with n = 1 for b = 40, f = 75. ortex printed on March 6, 2017 r | Φ | , | ϕ | , B Fig. 2: The solid (blue) curve represents Φ, the dotted (red) curve represents φ and the dashed (orange) curve represents the magnetic field, B , in a singlevortex with n = 1 for b = 40, f = 60. Vortex printed on March 6, 2017 r | Φ | , | ϕ | , B Fig. 3: The solid (blue) curve represents Φ, the dotted (red) curve represents φ and the dashed (orange) curve represents the magnetic field, B , in a singlevortex with n = 1 for for b = 40, f = 0. ortex printed on March 6, 2017 r | Φ | , | ϕ | , B Fig. 4: The solid (blue) curve represents Φ, the dotted (red) curve represents φ and the dashed (orange) curve represents the magnetic field, B , in a singlevortex with n = 1 for b = 40, f = − Vortex printed on March 6, 2017 r | Φ | , | ϕ | , B Fig. 5: The solid (blue) curve represents Φ, the dotted (red) curve represents φ and the dashed (orange) curve represents the magnetic field, B , in a singlevortex with n = 1 for b = 40, f = − ortex printed on March 6, 2017 r | Φ | , | ϕ | , B Fig. 6: The solid (blue) curve represents Φ, the dotted (red) curve represents φ and the dashed (orange) curve represents the magnetic field, B , in a singlevortex with n = 1 for b = 10, c = ca = , d = b , f = 16 . Vortex printed on March 6, 2017 r | Φ | , | ϕ | , B Fig. 7: The solid (blue) curve represents Φ, the dotted (red) curve represents φ and the dashed (orange) curve represents the magnetic field, B , in a singlevortex with n = 1 for b = 10, c = ca = , d = b , f = 17 . ortex printed on March 6, 2017 r | Φ | , | ϕ | , B Fig. 8: The solid (blue) curve represents Φ, the dotted (red) curve represents φ and the dashed (orange) curve represents the magnetic field, B , in a singlevortex with n = 1 for b = 10, c = ca = , d = b , f = 18 . Vortex printed on March 6, 2017 f Fig. 9: “Phase diagram” for ¯ c ≡ ca = and d = b . In the gray, solidregion the real scalar field condenses in the vortex phase but has a zerovev. The region with the black, fine mesh has no stable minimum since D ≡ bd − f <
0. The coarse, grey mesh shows the region with af < bc ,or f < b , where both Φ and φ have non-zero vevs. In the white region,real scalar remains uncondensed in the vortex state as in the homogeneousground state. ortex printed on March 6, 2017 f Fig. 10: “Phase diagram” for c = ca = and d = b . In the gray, solid regionthe real scalar field condenses in the vortex but has a zero vev. The regionwith the black, fine mesh has no stable minimum since D ≡ bd − f < af < bc , or f < b , whereboth Φ and φφ