BPS solitons with internal structure in the gauged $O(3)$ sigma model

Vortex solutions, in the field theory context, were found in the Maxwell-Higgs model by Nielsen and Olesen, relating them to the Nambu string in the strong-coupling realm Nielsen_1973 , besides in the nonrelativistic limit the Abrikosov ABRIKOSOV ; Ginzburg_1955 superconducting vortices emerge naturally. In particular, the type-II superconductors present vortex states possessing quantized magnetic flux, which was confirmed experimentally by Essmann and Träuble Essmann_1967_experiment . An interesting fact about soliton solutions is that in some special situations can be obtained via a system of first-order differential equations attained employing a technique known as the Bogomol’nyi-Prasad-Sommerfield (BPS) formalism, also determining the minimum value of the system energy (the Bogomol’nyi bound) Prasad_1975 ; Bogo_1976 . In connection with the Abrikosov study, the BPS limit is the interface between the two superconductor phases, such as it was studied by Bogomol’nyi in Bogo_1976 and by de Vega and Schaposnik de_Vega_1976 using an alternative technique. Furthermore, the existence of vortex solutions supporting both electric and magnetic fields also have proposed in scenarios involving the Chern-Simons action Zhang_1989 ; Jackiw_1990 ; Jackiw_1990_2 ; Lee_1990 ; Ghosh_1994 .

Vortex-like structures also emerge in the (1+2)-dimensional gauged sigma $O(3)$ model. The nonlinear sigma model has aroused the interest of lots of researchers due to its wide range of applications in condensed matter physics Polyakov:1975yp ; Rajaraman:1982is ; 9780852742310 . Moreover, there is a close connection between the $O(3)$ and $CP(1)$ models, such as shown in Refs. Eichenherr_1978 ; Golo_1978 ; Witten_1979 . Despite the model possesses a self-dual structure, the resulting topological solitons are scale-invariant, consequently do not represent particles in the context of Quantum Field Theory Leese_1990 .

A first intent, proposed by Schroers, breaks the scale invariance coupling minimally the sigma field to the Maxwell gauge field and introducing a potential that preserves the self-dual structure Schroers_1995 . This way, his approach generated a new class of topological solitons with nonquantized magnetic flux. Subsequently, Ghosh Ghosh_1996 studied this new type of soliton by coupling the sigma field to the $U(1)$ Chern-Simons gauge field and, as expected, the solitons engendered - topological or nontopological - are electrically charged. For both models, the homotopy group $\pi_{2}(S^{2})$ characterizes the soliton solutions. A second approach by Mukherjee explores the breaking of the scale-invariance by adopting a self-interaction potential leading to the spontaneous breaking of the Abelian local gauge symmetry. Consequently, the topological solitons possessing quantized magnetic flux are classified through the fundamental homotopy group $\pi_{1}(S^{1})$ Mukherjee_1997 ; Mukherjee_1998 . These solitons also were investigated in scenarios where the sigma field is coupled nonminimally to a gauge field Cavalcante_2000 and in Lorentz-violating sigma-models CASANA . The BPS structure of the gauged $O(3)$ sigma model has also discussed in Ref. stepien .

Moreover, other new vortex solutions are found promoting the extension of the $U(1)$ symmetry, for example, in the Maxwell-Higgs model Witten_1985 ; Peterson_2015 . There are other interesting examples in the literature as the extended groups $U(1)\times Z_{2}$ Maxwell_Higgs_SE and $CP(2)\times Z_{2}$ Andrade_2019 , being obtained through the introduction of a real scalar field, allowing the description of self-dual vortices in a dielectric medium. Such new objects can be of great utility in the study of metamaterials Shelby_2001 ; Ramakrishna_2005 ; Caloz_2009 .

Motivated by these discussions, we are searching for the occurrence of such first-order structures, but now in the context of the gauged nonlinear sigma $O(3)$ model, where its subgroup $SO(2)$ is enlarged as $SO(2)\times Z_{2}$. The corresponding extra scalar field is coupled to the gauge one by mean of a generalized dielectric function multiplying the Maxwell term. We present our results as follows: In Sec. II, we introduce the model, and next, by considering rotationally symmetric configurations, we implement the BPS formalism that provides the energy lower bound and the corresponding first-order equations. In Sec. III, we discuss three distinct scenarios by selecting the dielectric medium. After solving the BPS system of equations, we highlight the main new characteristics presented by the self-dual solitons. Lastly, in Sec. IV, we make our final comments and conclusions.

Our starting point is a $(2+1)$-dimensional extended gauged $O(3)$ sigma model defined by the following Lagrangian density:

where the sigma field $\vec{\phi}=(\phi_{1},\phi_{2},\phi_{3})$ is a triplet of real scalar fields whose norm is fixed to be $\vec{\phi}\cdot\vec{\phi}=1$. The tensor $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the electromagnetic field strength tensor of the $U(1)$ gauge field $A_{\mu}$. The sigma and gauge fields are minimally coupling via the covariant derivative defined as

with $\hat{n}$ being an unit vector in the internal space. The self-interacting potential $V(\phi_{n},\chi)$ is always a nonnegative function and stands for some appropriate interaction between the sigma field and the neutral scalar field $\chi$. Further, we have supposed this latter field coupled to gauge sector through a dielectric function $\Sigma(\chi)$, which is also a nonnegative real function. It is worthwhile to point out in absence of the scalar field $\chi$ we recover the standard gauged $O(3)$ sigma model studied in Ref. Mukherjee_1998 .

To investigate stationary solitons solutions we shall take $\hat{n}=(0,0,1)$ and assume henceforth the well-known hedgehog ansatz for $\vec{\phi}$,

Furthermore, for the gauge field components and the neutral scalar field we set

respectively. The quantity $N$ is a nonnull integer denoting the winding number (or topological degree). The real functions $f(r)$, $A_{0}(r)$, $a(r)$, and $\chi(r)$ are well-behaved satisfying appropriated boundary conditions.

We now present the stationary Euler-Lagrange equations associated with the Lagrangian density (1). The Gauss law is given by

where the prime symbol stands for the derivative with respect to the radial coordinate $r$. We observe the gauge condition $A_{0}=0$ identically satisfies the Gauss law allowing us to choose this condition for our following analysis. This way, the resulting configurations are magnetic flux carriers possessing a null electric charge.

The respective Ampère law reads

where the magnetic field, under the gauge field parametrization, becomes

Under the gauge condition $A_{0}=0$, the field equations for the sigma and scalar field profiles, $f(r)$ and $\chi(r)$, are

and

respectively. Above, we have defined $V_{f}=\partial_{f}V$, $\Sigma_{\chi}=d\Sigma/d\chi$ and $V_{\chi}=\partial_{\chi}V$.

At the origin, the remaining functions must satisfy the following boundary conditions

whereas for the asymptotic limit we require,

where $\chi_{0}$ and $\chi_{\infty}$ are finite constants. Of course, the set of boundary conditions established above is consistent with the vacuum configurations of the fields and ensures the finiteness of the energy.

The corresponding energy density is

To implement the BPS procedure Bogo_1976 , we introduce two auxiliary functions $U\equiv U(f)$ and $\mathcal{W}\equiv\mathcal{W}(\chi)$, such that Eq. (12) can be rewritten as

being $\mathcal{W}_{\chi}=\partial_{\chi}\mathcal{W}$. The two first terms in the third row of Eq. (13) can be rewritten as a total derivative by assuming the constraint $U^{\prime}=(\cos f)^{\prime}$, allowing to obtain the explicit form for $U(f)$:

where, without loss of generality, we have taken the integration constant to be zero.

Also, we choose the self-dual potential as

The function $\mathcal{W}(\chi)$ acts as a “superpotential” for the scalar field $\chi$, allowing us to find solutions satisfying first-order differential equations. It is important to mention that the presence of the neutral scalar field $\chi$ in Eq. (1) demands the insertion of a term, being an explicit function of the radial coordinate $r$, into the potential $V(f,\chi)$ whose finality is the full implementation of the BPS formalism. The effects of that dependence first were studied in Theocharis_2003 . Later, in Ref. Bazeia_theorem was considered to circumvent the Derrick-Hobart theorem Hobart_1963 ; Derrick_1964 . Furthermore, it already has been used in different contexts, e.g., Maxwell-Higgs Maxwell_Higgs_SE ; B_Research_2019 , magnetic monopoles Monopoles_SE , and gauged $CP(2)$ Andrade_2019 .

By using the two last considerations above the energy density becomes

where we have defined the BPS energy density as

This way, from (16), we write the total energy as follows

with the BPS total energy (Bogomol’nyi bound) given by

where we have used the boundary conditions (10) and (11), and also defined $\Delta\mathcal{W}=W(\chi_{\infty})-\mathcal{W}(\chi_{0})$. The upper (lower) sign describes the self-dual solitons (antisolitons) corresponding to $N$ and $\Delta\mathcal{W}$ positive (negative) quantities. Although we have not chosen until then the explicit form for the superpotential $\mathcal{W}\left(\chi\right)$ and dielectric function $\Sigma(\chi)$, we emphasize that the BPS total energy depends only on the boundary conditions of $\mathcal{W}(\chi)$, besides, of course, the winding number $N$.

Coming back to Eq. (18), the second term reads

such that the total energy (18) has a bound $E\geq E_{{}_{\text{BPS}}}$ which is saturated when $\bar{E}=0$, i.e., the solutions obeys the first-order differential equations

This set is called self-dual or BPS equations, where the (lower) upper sign stands for (anti) solitons with ($N<0$) $N>0$. Therefore, the BPS system ensures the energy lower bound and the stability of the corresponding field configurations. A comment not least is that indeed the Eqs. (21)-(23) satisfy the set of Euler-Lagrange equations associated with Lagrangian density (1) as well.

Furthermore, by using the BPS equations, the BPS energy density (17) can be rewritten in the form

where we have defined

respectively. The $\varepsilon_{{}_{\Sigma}}$ standing for the energy density associated with the new solitons while $\varepsilon_{\chi}$ is the contribution belonging to the kink $\chi$ that, according to the BPS equation (23), becomes independent of the other fields. This situation allows us to choose some convenient $\chi$-field configurations sourcing the dielectric medium driving the new soliton configurations.

At this stage, we observe both the dielectric function and the self-dual equation (23) do not include either the sigma field and the gauge field. Such a situation enables us to study interesting physical scenarios by adequately selecting the dielectric function and the superpotential. In the next section, we shall address some interesting scenarios by solving the Eq. (23) for a given superpotential $\mathcal{W}(\chi)$ and choosing some different dielectric functions $\Sigma(\chi)$.

We shall consider some internal structure scenarios by choosing a specific form of the $\chi$ field that introduces additional nonlinearities to the original sigma model, allowing us to analyze how the shape of the original solitons is modified. For this purpose, we consider the following superpotential:

where $\alpha$ is a positive parameter. The particular case $\alpha=1$ has been previously approached in different contexts as global defect structures Bazeia_theorem , skyrmion-like configurations Bazeia_2016 ; Bazeia_2017 , massless Dirac fermions Bazeia_2018 , magnetic monopoles Monopoles_SE and vortices with internal structures Maxwell_Higgs_SE ; Andrade_2019 . On the other hand, arbitrary values of $\alpha$ were used to discuss the solutions into a multilayered structure B_Research_2019 .

Then, by assuming the superpotential (26), the BPS equation (23) results

which implies in the exact kink-like solution

where $r_{0}$ is an arbitrary positive constant. Besides the solution satisfies $\chi\left(r_{0}\right)=0$, it fixes the boundary conditions for the neutral field: $\chi(0)=\chi_{0}=\mp 1$ and $\chi(\infty)=\chi_{\infty}=\pm 1$.

Under such considerations, the BPS bound for the energy (19) becomes

where the second term is the contribution from the neutral scalar field. Similarly, the magnetic flux reads

For our study, in the remaining of the manuscript, we only consider the soliton solutions, i.e., $N>0$.

We have studied the existence of new topological magnetic solitons living into a dielectric medium $\Sigma$. To describe these structures, we have considered a gauged sigma model containing an extra real scalar field (source field $\chi$) that characterizes the dielectric properties of the medium, $\Sigma\equiv\Sigma(\chi)$. Stable solitons are obtained by following the BPS technique, whose implementation is only possible by introducing a superpotential $\cal{W}(\chi)$ in the source sector, allowing us to decouple it from both the gauge and sigma sectors.

Even without knowing an explicit form of both the dielectric function $\Sigma(\chi)$ and superpotential $\mathcal{W}(\chi)$, we have found first-order equations, which are also solutions of the Euler-Lagrange ones. Because of the arbitrariness of these functions, we point out that the proposed model can fit a variety of physical environments. In our analysis, we have adopted the superpotential (26) that supports kink-like solutions for the source field that allows us to investigate solitons with internal structure in three distinct scenarios. The superpotential introduces the parameter $\alpha>0$ and, by solving the BPS equation (23), we obtain the kink solution (28) presenting a second parameter $r_{0}$, which characterizes the kink radius where $\chi(r_{0})=0$. Further, the dielectric function $\Sigma(\chi)$, theoretically, could introduce a set of additional parameters.

In the first scenario, we only study the case $\alpha=1$ because we have perceived there are not solutions valid for $\alpha\geq 2$. Here, the kink radius $r_{0}$ indicates the position of the maximum amplitude attained by both the magnetic field and energy density $\varepsilon_{{}_{\Sigma}}$ for sufficiently large winding numbers. Additionally, we found that the profiles are quite similar to the ones of the Chern-Simons vortices Jackiw_1990 , the ones in some generalized Maxwell-Higgs model Maxwell_Higgs_SE , and the solitons with an internal structure of the gauged $CP(2)$ model studied in Ref. Andrade_2019 .

In the second scenario, we have discussed the formation of solitons whose magnetic field has a distribution composed by a core with a maximum amplitude around which is formed a ring with a maximum having a lower intensity. The magnetic field is null at $r=r_{0}$, so it separates the core from the ring. Furthermore, the magnetic field profiles in the range $0\leq r\leq r_{0}$ remind the ones found by Nielsen-Olesen Nielsen_1973 , whereas for $r>r_{0}$ resemble the behavior of the Chern-Simons vortices, including typical exponential decay, here occurring yet in the presence of the dielectric medium. Also, we have realized the parameter $\alpha$ plays the role of controlling the core size and the maximum intensity of the external ring. On the other hand, the profiles of the energy density $\varepsilon_{{}_{\Sigma}}$, for sufficiently large values of $N$, also vanish at $r_{0}$ and present a behavior very similar to the one described for the magnetic field.

In the third and last scenario, we have introduced an oscillating dielectric function controlled via two extra parameters ($m\in\mathbb{Z}$ and $\lambda\in\mathbb{R}$) to describe a multilayered system. The value $\lambda=0$ generates a dielectric system producing a more notable formation of the internal structures than for $\lambda\neq 0$. In the latter case, the emergent internal structure becomes more relevant for small values of $\lambda$ than the big ones, such that, for sufficiently large values, the dielectric effects diminish rapidly and, so we recover the realm of the usual gauged sigma model. In both cases, the arising of $2m$-plateaus along the gauge field profile implies in the same number of magnetic rings surrounding the soliton-core.

Finally, we are studying the possible existence of BPS charged vortices (maybe behaving as anions) in the presence of dielectric media in the realm of the gauged $O(3)$ sigma model with Chern-Simons term Ghosh_1996 . Advances in this direction will report elsewhere.