# Compact vortices

- 534 Downloads
- 6 Citations

## Abstract

We study a family of Maxwell–Higgs models, described by the inclusion of a function of the scalar field that represent generalized magnetic permeability. We search for vortex configurations which obey first-order differential equations that solve the equations of motion. We first deal with the asymptotic behavior of the field configurations, and then implement a numerical study of the solutions, the energy density and the magnetic field. We work with the generalized permeability having distinct profiles, giving rise to new models, and we investigate how the vortices behave, compared with the solutions of the corresponding standard models. In particular, we show how to build compact vortices, that is, vortex solutions with the energy density and magnetic field vanishing outside a compact region of the plane.

## Keywords

Vortex Energy Density Scalar Field Higgs Model Compact Interval## 1 Introduction

This work deals with vortices in generalized Maxwell–Higgs model in the three-dimensional spacetime. As is well known, vortices are planar structures of topological nature [1], and their importance in high energy physics can be found, for instance, in Refs. [2, 3]. In particular, they may appear in a phase transition during the cosmic evolution of our Universe [2]. They are also of current interest to other areas of physics; in condensed matter, they may appear in superconductors, and may also be present as magnetic domains in magnetic materials [4].

The generalized Maxwell–Higgs model in which we are interested appeared in the beginning of the 1990s, with focus on the presence of vortex solutions [5, 6]. The model includes a function \(G(|\phi |)\) of the Higgs field multiplying the Maxwell term, and for a very specific choice of this function, the generalized system supports solutions that map the vortices of the Chern–Simons–Higgs system [7, 8, 9]. The difference here is that the vortices are electrically neutral, although the magnetic flux exists and is quantized. The function \(G(|\phi |)\) can be seen as a kind of generalized *magnetic permeability*, and the limit \(G \rightarrow 1\) leads back to the standard Maxwell–Higgs model.

In this work we study the generalized model under specific circumstances, considering several new possibilities. One starts in Sect. 2, reviewing the standard Maxwell–Higgs system and introducing the generalized model, with focus on the first-order formalism which we use to describe explicit solutions of the Bogomol’nyi–Prasad–Sommerfield (BPS) type [10, 11]; see also Ref. [12]. We then investigate two new models in Sect. 3, and in Sect. 4 we investigate models that allow for the presence of compact vortices, that is, for vortex-like solutions which engender energy density and magnetic field that vanish outside a compact interval of the radial coordinate.

The motivation to study compact vortices comes from the recent advances in the study and manipulation of materials at the nanometric scale. For instance, in Ref. [13] it was experimentally observed that domain walls may modify conformation in constrained geometries, so one can also ask if the miniaturization of magnetic materials can modify the conformational structure of vortices and skyrmions [14, 15], as in the case recently investigated in [16]. In this sense, it seems of current interest to study the possibility of shrinking topological objects such as vortices to compact regions. The study of compact vortices is also part of the recent work on compact structures, such as kinks and lumps [17, 18, 19], and Q-balls [20]. These investigations are based on distinct mechanisms, and the results show that there is no obvious way to make vortices shrink to a compact region of the plane. Here, however, we follow the route proposed in [17] and show how to construct compact vortices in the generalized model of the Maxwell–Higgs type.

## 2 The model

*e*is the electric charge, and \(V(|\phi |)\) is the potential for the scalar field. We are working in the (2, 1) dimensional spacetime with Minkowski metric \(\eta _{\mu \nu }\), with diagonal elements \((1,-1,-1)\). We are also using natural units such that \(\hbar =c=1\). In the standard case, the Higgs potential has the form

*v*is another real and positive parameter that sets the scale of spontaneous symmetry breaking.

Before one moves on and introduces the new model, it is of interest to know some specific features of the standard model, in particular the dimension of the several quantities that appear in the model. Since one is working with (2, 1) spacetime dimensions, one notes that the field \(A_\mu \) has dimension of energy to the power 1 / 2 or, in short, dim\((A_\mu )=1/2\). Thus, the other quantities obey: dim\((\phi )=1/2\), dim\((e)=\mathrm{dim}(v)=1/2\), and dim\((\lambda )=1\). The model engenders spontaneous symmetry breaking and it is also known to support vortex solutions, as first studied in [1] and later in [10, 12], with focus on the presence of solutions that solve first-order differential equations.

### 2.1 Basic considerations

*r*and \(\theta \) are the radial and angular coordinates, respectively, with \(r \in [0,\infty )\) and \(\theta \in [0,2\pi )\). Also,

*n*is a nonvanishing integer, the vorticity or winding number; \(n=\pm 1,\pm 2,\ldots \). It counts how many times the scalar field winds around itself as \(\theta \) varies in the interval \([0,2\pi )\).

*r*, and \(G_g=\mathrm{d}G/\mathrm{d}g\). Also, for the field configurations given by Eq. (7), the magnetic field becomes

*B*allows that we introduce the magnetic flux, which has the form

*G*and the potential

*V*are related via the constraint (15).

### 2.2 Standard vortices

## 3 Generalized vortices

Let us now investigate some new models and their respective vortex solutions. We first suggest the function \(G(|\phi |)\) and then write the corresponding potential, in order to study the vortex solutions, energy density and magnetic field.

### 3.1 A new model

*B*are given byand

*r*, we attempt a power series solution and obtain, for positive

*n*,

*A*is to be determined numerically to match the behavior of the solutions for larger values of

*r*. To solve Eq. (26) numerically, we choose an initial value of

*A*and then integrate, searching to get the appropriate behavior for very large values of

*r*. We then repeat the procedure with a new value of

*A*until we find the correct value for

*A*, which meets the above conditions.

The results for *a*(*r*) and *g*(*r*) are shown in Fig. 2, where we also display the solutions of the Maxwell–Higgs model. We note that the solutions of the new model are larger than they appear in the standard model. Moreover, in Fig. 3 one displays the energy density \(\varepsilon \) and the magnetic field *B* of both the new and the standard Maxwell–Higgs models, for comparison. One now notes the same behavior, the energy density and the magnetic field of the new model seem to spread over a larger region in the plane.

### 3.2 Another model

*B*in the form

*B*for the model (30) and for the Chern–Simons model. As in the previous model, one notes here that the solutions, energy density and magnetic field of the new model also spread over a larger region in the plane, if compared with the Chern–Simons case.

## 4 Compact vortices

We now turn attention to the possibility of constructing compact vortices. One first notes from the results of the previous Sect. 3 that it is possible to modify the function \(G(\phi )\) in order to change the potential of the model. Thus, we get inspiration from the recent work on compact kinks [17] to describe a route to build compact vortices. We recall that in [17] one developed the possibility of changing the scalar field self-interactions, in a way capable of shrinking the solution to a compact interval of the real line. We use the same idea here, and below we illustrate this possibility introducing two distinct models.

Before going on the subject, however, one might wish to recall recent efforts to describe compact vortices. In Ref. [27] the author deals with the same issue, but there one considers a non-canonical kinetic term, leading to a different scenario. A similar investigation, with models also containing non-canonical kinetic terms has been carried out in [28]. However, one notes that both the energy density and the magnetic field do not respond as significantly as the solutions do.

These results motivate us to revisit the subject, with focus on the construction of generalized models that support genuine compact vortices, with the energy density and magnetic field vanishing outside a compact interval of the radial coordinate. We implement this possibility below, investigating two distinct models that support compact vortices.

### 4.1 A model for compact vortices

*l*is a positive real parameter, such that \(l\ge 1\). With this choice, the constraint (15) leads to the potential

*l*introduces a nice behavior, as we show in Fig. 7.

*l*, we must solve Eq. (38) numerically, since it is very complicated to find analytical solutions for the problem. Nevertheless, for a general

*n*and very large

*l*, it is possible to show that the model supports the compact solutions

*l*. We have checked that the energy density and the magnetic field tend to become compact, and for

*l*very large one gets

*compactlike*vortices. Here, the vortex becomes a compact solution, since both the energy density and magnetic field shrink to the compact interval. We illustrate this fact in Fig. 9, displaying the energy density and the magnetic field for \(n=1\), for several values of

*l*. It is interesting to see that in the compact limit the magnetic field is constant inside the compact interval, so it seems to map the magnetic field of an infinitely long solenoid. As we commented before, the discontinuity in the magnetic field in the generalized model does not modify the energy density, due to the presence of the generalized magnetic permeability.

### 4.2 Another model for compact vortices

*l*is a positive real parameter, such that \(l\ge 1\). Then the constraint given by (15) implies that the potential has the form

*l*.

*l*we focus to solve Eq. (45) numerically. However, for

*n*positive and for very large

*l*, it is possible to show that the model supports the compact solutions

*l*.

*l*. As in the previous model, the discontinuity of the magnetic field in the compact limit induces no problem here too, since it is also controlled by the presence of the generalized magnetic permeability.

## 5 Comments and conclusions

In this work we studied the presence of vortices in a generalized Maxwell–Higgs model. The main idea was to generalize the Maxwell–Higgs model in a way such that we could find first-order differential equations and explore the BPS solutions. To do this, we have changed the Maxwell term, adding to it the factor \(G(|\phi |)\), which seems to model a generalized magnetic permeability. As is well known, this modification leads to effective planar field theories that present vortex solutions which somehow describe the vortices of the models with standard Maxwell and Chern–Simons dynamics.

The results indicated the presence of the compact behavior, with the vortices shrinking to a compact interval, with the energy density and magnetic field vanishing outside the compact interval. The compact behavior appears very clearly in Figs. 8 and 9 for the model (37), and in Figs. 11 and 12, for the model (44). The two models are different from each other: the first one, described by the potential (37) is similar to the standard Maxwell–Higgs model, and the other, with potential (44), resembles the model with Chern–Simons dynamics.

We identified a new behavior, a compact behavior for the vortices that appear in the models studied in Sect. 4. This seems to be of current interest, and we hope that the above results will stimulate further research in the area, especially on the main characteristics of the solutions, and in the construction of new models. Interesting issues concern extending the current results to other topological structures, in particular to monopoles and skyrmions. The case of skyrmions is of practical interest, and the study of compact skyrmions can be used to describe new spin textures in high energy physics [29] and in magnetic materials [14, 15, 30]. Research in this direction is now under development, and we hope to report on them in the near future.

## Notes

### Acknowledgements

This work is partially supported by CNPq, Brazil. DB acknowledges support from projects CNPq:455931/2014-3 and CNPq:306614/2014-6, LL acknowledges support from projects CNPq:307111/2013-0 and CNPq:447643/2014-2, MAM thanks support from project CNPq:140735/2015-1, and RM thanks support from projects CNPq:508177/2010-3 and CNPq:455619/2014-0.

## References

- 1.H.B. Nielsen, P. Olesen, Nucl. Phys. B
**61**, 45 (1973)ADSCrossRefGoogle Scholar - 2.A. Vilenkin, E.P.S. Shellard, Cosmic Strings and Other Topological Defects. (Cambridge University Press, Cambridge, 1994)Google Scholar
- 3.N. Manton, P. Sutcliffe, Topological Solitons. (Cambridge University Press, Cambridge, 2004)Google Scholar
- 4.Hubert, A., Schäfer, R.: Magnetic Domains. The Analysis of Magnetic Microstructures. (Springer, Berlin, 1998)Google Scholar
- 5.J. Lee, S. Nam, Phys. Lett. B
**261**, 437 (1991)ADSCrossRefGoogle Scholar - 6.D. Bazeia, Phys. Rev. D
**46**, 1879 (1992)ADSCrossRefGoogle Scholar - 7.J. Hong, Y. Kim, P.Y. Pac, Phys. Rev. Lett.
**64**, 2230 (1990)ADSMathSciNetCrossRefGoogle Scholar - 8.R. Jackiw, E.J. Weinberg, Phys. Rev. Lett.
**64**, 2234 (1990)ADSMathSciNetCrossRefGoogle Scholar - 9.R. Jackiw, K. Lee, E.J. Weinberg, Phys. Rev. D
**42**, 3488 (1990)ADSMathSciNetCrossRefGoogle Scholar - 10.E.B. Bogomol’nyi, Sov. J. Nucl. Phys.
**24**, 449 (1976)MathSciNetGoogle Scholar - 11.M. Prasad, C. Sommerfield, Phys. Rev. Lett.
**35**, 760 (1975)ADSCrossRefGoogle Scholar - 12.H. Vega, F. Schaposnik, Phys. Rev. D
**14**, 1100 (1976)ADSCrossRefGoogle Scholar - 13.P.-O. Jubert, R. Allenspach, A. Bischof, Phys. Rev. B
**69**, 220410(R) (2004)ADSCrossRefGoogle Scholar - 14.A. Fert, V. Cros, J. Sampaio, Nature Nanotech.
**8**, 152 (2013)ADSCrossRefGoogle Scholar - 15.N. Romming et al., Science
**341**, 636 (2013)ADSCrossRefGoogle Scholar - 16.D. Bazeia, J.G.G.S. Ramos, E.I.B. Rodrigues, JMMM
**423**, 411 (2017)ADSCrossRefGoogle Scholar - 17.D. Bazeia, L. Losano, M.A. Marques, R. Menezes, Phys. Lett. B
**736**, 515 (2014)ADSMathSciNetCrossRefGoogle Scholar - 18.D. Bazeia, L. Losano, M.A. Marques, R. Menezes, EPL
**107**, 61001 (2014)ADSCrossRefGoogle Scholar - 19.D. Bazeia, M.A. Marques, R. Menezes, EPL
**111**, 61002 (2015)ADSCrossRefGoogle Scholar - 20.D. Bazeia, L. Losano, M.A. Marques, R. Menezes, R. da Rocha, Phys. Lett. B
**758**, 146 (2016)ADSCrossRefGoogle Scholar - 21.M.A. Anacleto, A. Ilha, J.R.S. Nascimento, R.F. Ribeiro, C. Wotzasek, Phys. Lett. B
**504**, 268 (2001)ADSMathSciNetCrossRefGoogle Scholar - 22.D. Bazeia, E. da Hora, C. dos Santos, R. Menezes, Phys. Rev. D
**81**, 125014 (2010)ADSCrossRefGoogle Scholar - 23.D. Bazeia, E. da Hora, D. Rubiera-Garcia, Phys. Rev. D
**84**, 125005 (2011)ADSCrossRefGoogle Scholar - 24.D. Bazeia, R. Casana, M.M. Ferreira Jr., E. da Hora, L. Losano, Phys. Lett. B
**727**, 548 (2013)ADSCrossRefGoogle Scholar - 25.D. Bazeia, R. Casana, M.M. Ferreira Jr., E. da Hora, EPL
**109**, 21001 (2015)ADSCrossRefGoogle Scholar - 26.H.S. Ramadhan, Phys. Lett. B
**758**, 140 (2016)ADSCrossRefGoogle Scholar - 27.E. Babichev, Phys. Rev. D
**77**, 065021 (2008)ADSCrossRefGoogle Scholar - 28.D. Bazeia, E. da Hora, R. Menezes, H.P. de Oliveira, C. dos Santos, Phys. Rev. D
**81**, 125016 (2010)ADSCrossRefGoogle Scholar - 29.C. Adam, P. Klimas, J. Sanchez-Guillen, A. Wereszczynski, Phys. Rev. D
**80**, 105013 (2009)ADSCrossRefGoogle Scholar - 30.M. Ezawa, Phys. Rev. B
**83**, 100408(R) (2011)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}