# BPS chiral vortices in a Maxwell-Higgs electrodynamics

J. Andrade<EMAIL_ADDRESS><EMAIL_ADDRESS>Rodolfo Casana
<EMAIL_ADDRESS><EMAIL_ADDRESS>Departamento de Física,
Universidade Federal do Maranhão, 65080-805, São Luís, Maranhão, Brazil.  E.
da Hora<EMAIL_ADDRESS><EMAIL_ADDRESS>Coordenadoria
Interdisciplinar de Ciência e Tecnologia, Universidade Federal do Maranhão,
65080-805, São Luís, Maranhão, Brazil.

###### Abstract

We investigate the existence of BPS structures in a Maxwell-Higgs
electrodynamics immersed within a chiral medium, whose electromagnetic
properties are described by both a the Chern-Simons term and a neutral scalar
field. The implementation of the Bogomol’nyi-Prasad-Sommerfield’s technique
provides the BPS potential and the self-dual equations whose solutions
saturate the Bogomol’nyi bound. In such a context, we look for vortices in two
chiral media: the first one engenders localized vortices with an exponential
decay similar to that of the Abrikosov-Nielsen-Olesen solutions, whereas the
second medium generates delocalized profiles whose tail follows a power-law
decay. Once we have solved the BPS systems, we comment on the effects induced
by the presence of the chiral medium on the Maxwell-Higgs vortices.

##  I Introduction

In the context of classical field theories, configurations with nontrivial
topology are solutions of highly nonlinear second-order field equations driven
by the presence of a symmetry-breaking potential which characterizes a phase
transition [1]. However, under exceptional circumstances, they can also be
studied as the solutions of a set of coupled first-order differential
equations obtained from the minimization of the effective energy functional
through the Bogomol’nyi-Prasad-Sommerfield (BPS) formalism [2]. Other methods
which also lead to the first-order equations include the study of the
conservation of the energy-momentum tensor [3] and the so-called On-Shell
Method [4]. In this scenario, topologically nontrivial configurations in a
planar space (so-called vortices) were obtained not only in the Maxwell-Higgs
(MH) model [5], but also in the Chern-Simons-Higgs (CSH) [6] and in the
Maxwell-Chern-Simons-Higgs (MCSH) [7] ones.

Another interesting issue is the existence of first-order vortices in the
context of enlarged gauged models. In this case, it is worthwhile to highlight
that the usual Maxwell-Higgs model itself was extended to accommodate an
$SO(3)$ group via the inclusion of an extra scalar sector [12]. In addition, a
recent work [13] has verified that such enlarged theory indeed supports first-
order vortices with internal structures, which may find relevant applications
in connection with the study of metamaterials [14]. Beyond this point, such
vortices were also shown to occur in an extended Maxwell-$CP(2)$ model [15].

The present manuscript aims to investigate the existence of BPS configurations
within a chiral electrodynamics defined by a generalized Maxwell-Chern-Simons
model, the chiral one prototype. For our purpose, we have organized our
results as follows: In the Sec. II, we define a chiral electrodynamics coupled
to the Higgs field and a real scalar field in the context of an extended
Maxwell-Chern-Simons-Higgs theory. We then focus our attention on the
construction of the BPS structure inherent to the resulting chiral
electrodynamics, from which we find the self-dual equations and the respective
Bogomol’nyi bound for the total energy. Next, in the Sec. III, the system
formed by the BPS equations and the Gauss law are solved through a finite-
difference technique to obtain chiral vortices for two different choices of a
dielectric function. We depict the relevant field profiles and identify the
effects of the chiral medium. In the Section IV, we end our manuscript by
summarizing our results and enunciating our perspectives regarding future
investigations.

##  II The Maxwell-Higgs electrodynamics in a chiral medium

It is well-known that in $(1+3)$-dimensions, the following Lagrangian density
describes the electromagnetic field in the vacuum [16]

$\mathcal{L}_{\text{v}}=-\frac{1}{4}F_{\hat{\mu}\hat{\nu}}F^{\hat{\mu}\hat{\nu}}\text{,}$
(1)

while, in an electromagnetic medium, the constitutive relations between the
electric and magnetic fields are expressed by the tensor
$\mathcal{C}_{\hat{\mu}\hat{\nu}}$, i.e.,

$\mathcal{C}_{\hat{\mu}\hat{\nu}}\equiv\mathcal{C}_{\hat{\mu}\hat{\nu}}\left(A_{\hat{\mu}},F_{\hat{\mu}\hat{\nu}}\right)\text{,}$
(2)

where $A_{\hat{\mu}}$ is the electromagnetic field and
$F_{\hat{\mu}\hat{\nu}}=\partial_{\hat{\mu}}A_{\hat{\nu}}-\partial_{\hat{\nu}}A_{\hat{\mu}}$
represents its strength tensor. Then, the Lagrangian density which describes
the electromagnetic field in the medium in the presence of a source becomes

$\mathcal{L}_{\text{m}}=-\frac{1}{4}\mathcal{C}_{\hat{\mu}\hat{\nu}}F^{\hat{\mu}\hat{\nu}}+A_{\hat{\mu}}J^{\hat{\mu}}\text{.}$
(3)

A constitutive tensor $\mathcal{C}_{\hat{\mu}\hat{\nu}}$ engendering a chiral
medium [17] is given by

$\mathcal{C}_{\mu\nu}=F_{\hat{\mu}\hat{\nu}}+\frac{1}{4}\epsilon_{\hat{\mu}\hat{\nu}{\hat{\rho}}{\hat{\sigma}}}b^{\hat{\rho}}A^{\hat{\sigma}}\text{,}$
(4)

from which one gets that a resulting Lagrangian density describing a chiral
electrodynamics reads

$\mathcal{L}_{\text{m}}=-\frac{1}{4}F_{\hat{\mu}\hat{\nu}}F^{\hat{\mu}\hat{\nu}}-\frac{1}{4}\epsilon_{\hat{\mu}\hat{\nu}{\hat{\rho}}{\hat{\sigma}}}b^{\hat{\rho}}A^{\hat{\sigma}}F^{\hat{\mu}\hat{\nu}}+A_{\hat{\mu}}J^{\hat{\mu}}\text{,}$
(5)

where $\epsilon^{\hat{\mu}\hat{\nu}{\hat{\rho}}{\hat{\sigma}}}$ is the Levi-
Civita tensor (with $\epsilon^{0123}=+1$) and
$b^{\hat{\mu}}=\left(b_{0},\mathbf{b}\right)$ represents a vector whose
component $b_{0}$ is related to the chiral magnetic effect [18, 19], while the
vector $\mathbf{b}$ is associated to the anomalous Hall effect [20, 21, 22,
23, 24] or also can be related to the generation of an anomalous charge
density [25]. In the context of Lorentz-violating field theories, the vector
$b^{\hat{\mu}}$ becomes the Carroll-Field-Jackiw background
$\left(k_{AF}\right)^{\hat{\mu}}$, from which the corresponding model is named
the Maxwell-Carroll-Field-Jackiw electrodynamics (MCFJ) [26]. Furthermore, the
axion electrodynamics is also described by the Lagrangian density (3) when the
field $b_{\hat{\mu}}=\partial_{\hat{\mu}}\theta$, where $\theta$ is a
pseudoscalar function identified as the axion field [27, 28]. Recently, the
MCFJ electrodynamics was devoted to analyzing the effects of the chiral medium
on Cerenkov’s radiation [29] and the cold plasma modes [30, 31].

We now consider the electromagnetic field in a chiral medium interacting with
the Higgs field $\phi$ and a real scalar field $\chi$ through the Lagrangian
density

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{G(\chi)}{4}F_{\hat{\mu}\hat{\nu}}F^{\hat{\mu}\hat{\nu}}-\frac{1}{4}\epsilon_{\hat{\mu}\hat{\nu}{\hat{\rho}}{\hat{\sigma}}}b^{\hat{\rho}}A^{\hat{\sigma}}F^{\hat{\mu}\hat{\nu}}$
(6)
$\displaystyle+\left|D_{\hat{\mu}}\phi\right|^{2}+\frac{1}{2}\partial_{\hat{\mu}}\chi\partial^{\hat{\mu}}\chi-V\left(\left|\phi\right|,\chi\right)\text{,}$

where $G(\chi)$ plays the role of a dielectric function, whereas
$D_{\hat{\mu}}\phi$ is the minimal covariant derivative which couples the
electromagnetic and Higgs fields, i.e.,

$D_{\hat{\mu}}\phi=\partial_{\hat{\mu}}\phi-ieA_{\hat{\mu}}\phi\text{,}$ (7)

with $e$ being the electromagnetic coupling constant.

Here, we point out that when $G(\chi)\equiv 1$, the field $\chi$ decouples of
the model, therefore originating the chiral Maxwell-Higgs electrodynamics or,
in a Lorentz-violating context, the Maxwell-Carroll-Field-Jackiw-Higgs theory
[32]. The Reference [33] has shown the existence of a BPS structure coming
from a $(1+2)$-dimensional version of the Maxwell-Carroll-Field-Jackiw-Higgs
model described by the Lagrangian density [34]

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{\kappa}{4}\,\epsilon^{\nu\rho\sigma}A_{\nu}F_{\rho\sigma}+\left|D_{\mu}\phi\right|^{2}$
$\displaystyle+\frac{1}{2}\partial_{\mu}\Psi\partial^{\mu}\Psi-e^{2}\Psi^{2}\left|\phi\right|^{2}-U\left(\left|\phi\right|,\Psi\right)$
$\displaystyle-\frac{1}{2}\epsilon^{\mu\rho\sigma}\left(k_{AF}\right)_{\mu}A_{\rho}\partial_{\sigma}\Psi-\frac{1}{2}\epsilon^{\mu\rho\sigma}\left(k_{AF}\right)_{\mu}\Psi~{}\partial_{\rho}A_{\sigma},$

where $\kappa=-\left(k_{AF}\right)_{\hat{3}}$ plays the role of a Chern-Simons
coupling constant, $\epsilon^{\mu\nu\rho}$ is the $(1+2)$-dimensional Levi-
Civita tensor ($\epsilon^{012}=+1$), $\Psi=A_{\hat{3}}$ stands for a real
scalar field, and $U\left(\left|\phi\right|,\Psi\right)$ represents the
potential. In such a context, $\left(k_{AF}\right)_{\mu}$ is the
$(1+2)$-dimensional CFJ vector background which couples the real and gauge
fields. We highlight that here $\Psi$ plays the role of the neutral field [7]
introduced into the Maxwell-Chern-Simons-Higgs model to attain a BPS
structure.

We aim to show the existence of a BPS structure (able to generate chiral
vortices) starting from a $(1+2)$-dimensional version of the model (6)
attained via the same dimensional reduction procedure used in the Ref. [33]
and by considering $\left(k_{AF}\right)_{\mu}=0$ [35]. This way, we get the
model described by the Lagrangian density

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{G(\chi)}{4}F_{\mu\nu}F^{\mu\nu}-\frac{\kappa}{4}\epsilon^{\mu\nu\kappa}A_{\mu}F_{\nu\kappa}+\left|D_{\mu}\phi\right|^{2}$
(9)
$\displaystyle+\frac{G(\chi)}{2}\partial_{\mu}\Psi\partial^{\mu}\Psi-e^{2}\Psi^{2}\left|\phi\right|^{2}$
$\displaystyle+\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi-U\left(\left|\phi\right|,\Psi,\chi\right)\text{,}$

which will be the starting point for our study. We immediately recognize it as
representing an enlarged version of the self-dual MCSH scenario, where
$U\left(\left|\phi\right|,\Psi,\chi\right)$ defines a potential to be
determined by the BPS formalism itself. As we will below, the function
$G(\chi)$, beyond specifying the electromagnetic properties of the medium, is
also responsible for modifying the profiles of the electric and magnetic
fields inherent to the vortex configurations. Also, we observe that when
$G(\chi)=1$, the field $\chi$ decouples from the gauge sector and, as a
consequence, the Lagrangian density (9) reduces to that of the MCSH model in
an electromagnetic vacuum. Moreover, we point out that the presence of the
field $\chi$ has already been considered in the context of topological
vortices with internal structures in both the Maxwell-Higgs and the
Maxwell-$CP(2)$ models.

The Euler-Lagrange equation obtained from the Lagrangian density (9) for the
gauge field reads

$\partial_{\alpha}\left(GF^{\alpha\mu}\right)-\frac{\kappa}{2}\epsilon^{\mu\alpha\beta}F_{\alpha\beta}=eJ^{\mu}\text{,}$
(10)

where $J^{\mu}$ is the conserved current density, i.e.,

$J^{\mu}=i\left(D^{\mu}\phi\right)^{\ast}\phi-i\phi^{\ast}\left(D^{\mu}\phi\right)\text{.}$
(11)

Similarly, the field equation for the Higgs sector becomes

$D_{\alpha}D^{\alpha}\phi+e^{2}\Psi^{2}\phi+\frac{\partial
U}{\partial\phi^{\ast}}=0\text{,}$ (12)

whereas the ones for the fields $\Psi$ and $\chi$ are, respectively,

$\partial_{\alpha}\left(G\partial^{\alpha}\Psi\right)+2e^{2}\Psi\left|\phi\right|^{2}+\frac{\partial
U}{\partial\Psi}=0\text{,}$ (13)
$\square\chi+\left(\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\partial_{\mu}\Psi\partial^{\mu}\Psi\right)\frac{\partial
G}{\partial\chi}+\frac{\partial U}{\partial\chi}=0\text{,}$ (14)

where $\square$ stands for the d’Alembertian operator.

Along this manuscript, we are interested in stationary configurations. In this
situation, the Gauss and Ampère laws can be written, respectively, as

$\displaystyle\displaystyle\partial_{j}\left(G\partial_{j}A_{0}\right)+\kappa
B=2e^{2}A_{0}\left|\phi\right|^{2}\text{,}$ (15)
$\displaystyle\displaystyle\partial_{k}\left(GB\right)+\kappa\partial_{k}A_{0}=-e\epsilon_{kj}J_{j}\text{,}$
(16)

where we have defined the magnetic field as
$B=F_{12}=\epsilon_{ij}\partial_{i}A_{j}$. The presence of the Chern-Simons
term in the Gauss law allows us to conclude that the resulting solutions
possess a nonvanishing electric charge and magnetic flux.

Similarly, the stationary equation for the Higgs field reads

$D_{i}D_{i}\phi-e^{2}\Psi^{2}\phi-\frac{\partial
U}{\partial\phi^{\ast}}=0\text{,}$ (17)

while the ones for $\Psi$ and $\chi$ assume the forms

$\displaystyle\displaystyle\partial_{i}\left(G\partial_{i}\Psi\right)-2e^{2}\Psi\left|\phi\right|^{2}-\frac{\partial
U}{\partial\Psi}=0\text{,}$ (18)
$\displaystyle\displaystyle\partial_{i}\partial_{i}\chi-\frac{1}{2}\left[B^{2}-\left(\partial_{i}A_{0}\right)^{2}+\left(\partial_{i}\Psi\right)^{2}\right]\frac{\partial
G}{\partial\chi}-\frac{\partial U}{\partial\chi}=0\text{,}$ (19)

respectively.

###  II.1 The BPS formalism

We focus our attention on those solutions satisfying a particular set of
differential equations, i.e., the so-called BPS or self-dual ones. These
equations arise through the implementation of the BPS prescription in the
total energy of the model (9), the starting-point being the expression for the
stationary energy density, i.e.,

$\displaystyle\varepsilon$ $\displaystyle=$
$\displaystyle\frac{1}{2}GB^{2}+\left|D_{i}\phi\right|^{2}+\frac{1}{2}G\left(\partial_{i}A_{0}\right)^{2}+\frac{1}{2}G\left(\partial_{i}\Psi\right)^{2}$
(20)
$\displaystyle+e^{2}\left(A_{0}\right)^{2}\left|\phi\right|^{2}+e^{2}\Psi^{2}\left|\phi\right|^{2}+\frac{1}{2}\left(\partial_{i}\chi\right)^{2}+U\text{,}$

from which one gets the following boundary conditions:

$\displaystyle\displaystyle\lim_{\left|\mathbf{x}\right|\rightarrow\infty}\sqrt{G}B=0\text{
\ \ and \ \
}\lim_{\left|\mathbf{x}\right|\rightarrow\infty}D_{i}\phi=0\text{,}$ (21)
$\displaystyle\displaystyle\lim_{\left|\mathbf{x}\right|\rightarrow\infty}\sqrt{G}\partial_{i}A_{0}=0\text{
\ \ and \ \
}\lim_{\left|\mathbf{x}\right|\rightarrow\infty}\sqrt{G}\partial_{i}\Psi=0\text{,}$
(22)
$\displaystyle\displaystyle\lim_{\left|\mathbf{x}\right|\rightarrow\infty}A_{0}=0\text{
\ \ and \ \ }\lim_{\left|\mathbf{x}\right|\rightarrow\infty}\Psi=0\text{,}$
(23)
$\displaystyle\displaystyle\lim_{\left|\mathbf{x}\right|\rightarrow\infty}\chi=0\text{
\ \ and \ \ }\lim_{\left|\mathbf{x}\right|\rightarrow\infty}U=0\text{,}$ (24)

which must be satisfied in order to ensure a finite energy.

The corresponding total energy reads as

$E=\int d^{2}\mathbf{x}~{}\varepsilon\text{,}$ (25)

which, after some algebraic manipulations, can be written in the form (here,
we have introduced the auxiliary functions
$\Sigma\equiv\Sigma\left(\phi,\Psi\right)$ and $F_{i}\equiv
F_{i}\left(\chi\right)$, for the sake of convenience)

$\displaystyle E$ $\displaystyle=$ $\displaystyle\int
d^{2}\mathbf{x}\left[\frac{\left(GB\mp\Sigma\right)^{2}}{2G}+\left|D_{\pm}\phi\right|^{2}\mp\frac{1}{2}\epsilon_{ij}\partial_{i}J_{j}\right.$
(26) $\displaystyle\hskip
28.45274pt+\frac{G}{2}\left(\partial_{i}A_{0}\mp\partial_{i}\Psi\right)^{2}+e^{2}\left(A_{0}\mp\Psi\right)^{2}\left|\phi\right|^{2}$
$\displaystyle\hskip
28.45274pt+\frac{1}{2}\left(\partial_{i}\chi\mp\epsilon_{ij}\frac{\partial
F_{j}}{\partial\chi}\right)^{2}\pm\epsilon_{ij}\partial_{i}F_{j}$
$\displaystyle\hskip 28.45274pt\pm
B\left(\Sigma+e\left|\phi\right|^{2}+\kappa\Psi\right)\pm\partial_{j}\left(\Psi
G\partial_{j}A_{0}\right)$ $\displaystyle\hskip
28.45274pt\left.+U-\frac{\Sigma^{2}}{2G}-\frac{1}{2}\left(\frac{\partial
F_{j}}{\partial\chi}\right)^{2}\right]\text{,}$

where we have used not only the well-known identity

$\left|D_{i}\phi\right|^{2}=\left|D_{\pm}\phi\right|^{2}\pm
eB\left|\phi\right|^{2}\mp\frac{1}{2}\epsilon_{ij}\partial_{i}J_{j}\text{,}$
(27)

but also the Gauss’ law (15).

Now, to continue with the implementation of the BPS formalism, it is useful to
observe that, in view of the conditions (21) and (22), those integrations
involving total derivatives do not contribute to the calculation of the total
energy. In addition, the factor which multiplies the magnetic field allows us
to fix the function $\Sigma$ by setting it as equal to $ev^{2}$, then we get

$\Sigma=e\left(v^{2}-\left|\phi\right|^{2}\right)-\kappa\Psi\text{.}$ (28)

Furthermore, by setting the last row of the Eq. (26) as equal to $0$, one gets
the BPS potential

$U=\frac{\left[e\left(v^{2}-\left|\phi\right|^{2}\right)-\kappa\Psi\right]^{2}}{2G}+\frac{1}{2}\left(\frac{\partial
F_{j}}{\partial\chi}\right)^{2}\text{,}$ (29)

where the first term in the right-hand side (with $G=1$) corresponds to the
potential which allows the standard Maxwell-Chern-Simons-Higgs model to be
self-dual [7]. In our enlarged case, the last term stands for the potential
related to the field $\chi$, with $F_{i}(\chi)$ being called a superpotential.

In view of the considerations above, the total energy (26) can be written as

$E=E_{BPS}+E_{\text{quad}}\geq E_{BPS}\text{,}$ (30)

where $E_{BPS}$ is the Bogomol’ny bound given by

$E_{BPS}=\pm ev^{2}\int d^{2}\mathbf{x~{}}B\pm\int
d^{2}\mathbf{x~{}}\epsilon_{ij}\partial_{i}F_{j}\geq 0\text{,}$ (31)

and $E_{\text{quad}}$ represents the contribution of all the quadratic terms,
i.e.,

$\displaystyle E_{\text{quad}}$ $\displaystyle=$ $\displaystyle\int
d^{2}\mathbf{x}\left\\{\frac{\left[GB\mp
e\left(v^{2}-\left|\phi\right|^{2}\right)\pm\kappa\Psi\right]^{2}}{2G}\right.$
(32) $\displaystyle\hskip
14.22636pt+\left|D_{\pm}\phi\right|^{2}+\frac{1}{2}\left(\partial_{i}\chi\mp\epsilon_{ij}\frac{\partial
F_{j}}{\partial\chi}\right)^{2}$ $\displaystyle\hskip
14.22636pt\left.+\frac{G}{2}\left(\partial_{i}A_{0}\mp\partial_{i}\Psi\right)^{2}+e^{2}\left(A_{0}\mp\Psi\right)^{2}\left|\phi\right|^{2}\right\\}\text{.}\quad\quad$

We observe that, when $E_{\text{quad}}=0$, the total energy attains its lower-
bound, i.e., $E=E_{BPS}$. In such a situation, the fields satisfy the
following set of differential equations:

$\displaystyle\displaystyle D_{\pm}\phi=0\text{,}$ (33)
$\displaystyle\displaystyle GB=\pm
e\left(v^{2}-\left|\phi\right|^{2}\right)\mp\kappa\Psi\text{,}$ (34)
$\displaystyle\displaystyle\partial_{i}\chi=\pm\epsilon_{ij}\frac{\partial
F_{j}}{\partial\chi}\text{,}$ (35)
$\displaystyle\displaystyle\partial_{i}\Psi=\pm\partial_{i}A_{0}\text{ \ \ and
\ \ }\Psi=\pm A_{0}\text{,}$ (36)

which are the self-dual BPS equations whose solutions provide solitonic
configurations with an energy which saturates a lower-bound, therefore
guaranteeing the stability of the corresponding structures. It is worthwhile
to comment that the set of Eqs. (33)-(36) is equivalent to that of Euler-
Lagrange equations (16)-(19) associated with Lagrangian density (9) whether we
consider the BPS potential (29).

The last two BPS equations in Eq. (36) are identically satisfied by $\Psi=\pm
A_{0}$, from which the remaining equations assume the form

$\displaystyle\displaystyle D_{\pm}\phi=0\text{,}$ (37)
$\displaystyle\displaystyle GB=\pm
e\left(v^{2}-\left|\phi\right|^{2}\right)-\kappa A_{0}\text{,}$ (38)
$\displaystyle\displaystyle\partial_{i}\chi=\pm\epsilon_{ij}\frac{\partial
F_{j}}{\partial\chi}\text{,}$ (39)

which must be studied together with

$\partial_{j}\left(G\partial_{j}A_{0}\right)+\kappa
B=2e^{2}A_{0}\left|\phi\right|^{2}\text{,}$ (40)

which is the Gauss law of the model.

###  II.2 The vortex ansatz

We look for vortex configurations via standard map for the Higgs field,

$\phi\left(r,\theta\right)=vg(r)e^{in\theta}\text{,}$ (41)

where $r$ and $\theta$ are the polar coordinates and the integer $n=\pm 1,$
$\pm 2,$ $\pm 3,$ $...$ represents the winding number of the final solutions.

In addition, for the gauge field, we set

$A_{i}=\varepsilon_{ij}\frac{x_{j}}{er^{2}}\left[a(r)-n\right]\text{,}$ (42)

where $\varepsilon_{ij}$ is the Levi-Civita tensor and $x_{j}$ expresses the
cartesian coordinates. It is interesting to note that the combination between
the Eqs. (40) and (42) leads to the conclusion that the electric potential
must depend on the radial coordinate only, i.e., $A_{0}=A_{0}(r)$.

Now, based on the Eqs. (41) and (42), it is reasonable to suppose that the
scalar field $\chi$ and the superpotential $F_{i}$ can be parametrized as

$\chi=\chi(r)\text{ \ \ and \ \
}F_{i}=-\varepsilon_{ij}\frac{x_{j}}{r^{2}}W\left(\chi\right)\text{,}$ (43)

while the field profiles are expected to be regular functions satisfying the
following boundary conditions at $r=0$:

$g(0)=0\text{, \ \ }a(0)=n\text{ \ \ and \ \ }A_{0}(0)=cte\text{,}$ (44)
$\chi(0)=\chi_{0}\text{ \ \ and \ \ }W(\chi_{0})=W_{0}\text{,}$ (45)

and (here, prime denotes the derivative with respect to $r$)

$g(\infty)=1\text{, \ \ }a(\infty)=0\text{ \ \ and \ \
}A_{0}^{\prime}(\infty)=0\text{,}$ (46) $\chi(\infty)=\chi_{\infty}\text{ \ \
and \ \ }W(\chi_{\infty})=W_{\infty}\text{,}$ (47)

which hold in the asymptotic limit $r\rightarrow\infty$ .

It proves useful to write the rotationally symmetric expressions for both the
magnetic and electric fields, i.e.,

$B(r)=-\frac{1}{er}\frac{da}{dr}\text{ \ \ and \ \
}E(r)=-\frac{dA_{0}}{dr}\text{,}$ (48)

which were obtained directly from the Eq. (42).

By using the boundary conditions defined previously, we perform the integrals
which appear in the Eq. (31) in order to calculate the value of the total BPS
energy explicitly. As a result, we find that the Bogomol’nyi bound for the
chiral Maxwell-Higgs electrodynamics reads

$E_{BPS}=\pm\left(2\pi v^{2}n+2\pi\Delta W\right)\geq 0\text{,}$ (49)

with $\Delta W=W_{\infty}-W_{0}$. In the expression above, the upper (lower)
sign holds for positive (negative) values of both $n$ and $\Delta W$. In what
follows, we analyze the case with $n>0$ and $\Delta W>0$ only, for the sake of
simplicity.

The magnetic flux $\Phi$ can be calculated as

$\Phi=\int B\,d^{2}\mathbf{x}=\frac{2\pi}{e}n\text{,}$ (50)

while the total electric charge $Q$ of the chiral vortices is obtained as the
integral of the Gauss law, therefore reading

$Q=\kappa\Phi\text{.}$ (51)

It is well known that the presence of the Chern-Simons term generates a
nonvanishing quantized angular momentum, i.e.,

$J=-\int\varepsilon_{ij}x_{i}T_{0j}\,d^{2}\mathbf{x}=\frac{\pi\kappa}{e^{2}}n^{2}=\frac{1}{4\pi}Q\Phi\text{,}$
(52)

which has precisely the same value as the self-dual vortices of the pure
Chern-Simons-Higgs model [6]. This result does not depend on the function
$G(\chi)$ whenever it satisfies the finite energy conditions given in the Eqs.
(21) and (22). The quantized angular momentum tells us that the chiral
vortices also have an anyonic behavior due to the presence of the Chern-Simons
term [36].

The rotationally symmetric version of the BPS Eqs. (37) and (38) and of the
Gauss law (40) can be written, respectively, as

$\frac{dg}{dr}=\frac{ag}{r}\text{,}$ (53)
$GB=ev^{2}\left(1-g^{2}\right)-\kappa A_{0}\text{,}$ (54)
$\frac{1}{r}\frac{d}{dr}\left(rG\frac{dA_{0}}{dr}\right)+\kappa
B=2e^{2}v^{2}g^{2}A_{0}\text{,}$ (55)

while the Eq. (39) can be verified to assume the form

$\frac{d\chi}{dr}=\frac{1}{r}\frac{dW}{d\chi}\text{,}$ (56)

which, as we discuss separately below, plays a central role in the
construction of the BPS chiral vortices.

Now, whether we use the BPS expressions, the Eq. (20) for the BPS energy
density can be written as

$\varepsilon_{BPS}=\varepsilon_{G}+\varepsilon_{\chi}\text{,}$ (57)

where $\varepsilon_{G}$ represents the contribution due to the vortex alone,
i.e.,

$\varepsilon_{G}=GB^{2}+G\left(\frac{dA_{0}}{dr}\right)^{2}+2v^{2}e^{2}g^{2}\left(A_{0}\right)^{2}+2v^{2}\frac{a^{2}g^{2}}{r^{2}}\text{,}$
(58)

while $\varepsilon_{\chi}$ corresponds to that exclusively related to the
source field $\chi$,

$\varepsilon_{\chi}=\left(\frac{d\chi}{dr}\right)^{2}=\frac{1}{r^{2}}\left(\frac{dW}{d\chi}\right)^{2}\text{.}$
(59)

It is interesting to perceive that Eq. (56) for the field $\chi$ does not
contain the other sectors of the model, therefore depending solely on the form
of the superpotential $W(\chi)$. This fact allows us to solve it separately by
choosing the superpotential ${W}(\chi)$ adequately. Once $\chi$ is known, we
must selected the functional form of $G(\chi)$ and then proceed to the
resolution of the set formed by the Eqs. (53), (54) and (55). The next Section
is devoted to this aim.

##  III Some chiral vortices in the Maxwell-Higgs scenario

We now investigate the construction of specific BPS scenarios and their
respective numerical solutions. In this sense, we first choose the
superpotential as

$W(\chi)=\chi-\frac{1}{3}\chi^{3}\text{,}$ (60)

which was used recently as an attempt to understand planar skyrmion-like
solitons [37] and the behavior of massless Dirac fermions in a skyrmion-like
background [38].

In view of this choice, the first-order equation (56) for the source field can
be written in the form

$\frac{d\chi}{dr}=\pm\frac{1}{r}\left(1-\chi^{2}\right)\text{,}$ (61)

whose exact solution reads

$\chi(r)=\pm\frac{r^{2}-r_{0}^{2}}{r^{2}+r_{0}^{2}}\text{,}$ (62)

where $r_{0}$ represents an arbitrary positive constant such that
$\chi(r_{0})=0$. We note that this solution attains the boundary values
$\chi_{0}=\mp 1$ and $\chi_{\infty}=\pm 1$. Our analysis only considers the
upper sign, as previously commented after the Eq. (49).

In the sequence, we choose a particular expression for the dielectric function
$G(\chi)$ which characterizes the chiral medium. In the following sections, we
separate our study into two different scenarios.

###  III.1 Abrikosov-Nielsen-Olesen-like chiral vortices

We consider the dielectric function $G(\chi)$ as

$G(\chi)=\frac{1}{\chi^{2}}=\frac{\left(r^{2}+r_{0}^{2}\right)^{2}}{\left(r^{2}-r_{0}^{2}\right)^{2}}\text{,}$
(63)

where we have already used the Eq. (62) for $\chi\left(r\right)$. We note that
$r_{0}=0$ leads to $G\equiv 1$. As a consequence, the set of equations (53),
(54) and (55) correspond to those of the standard MCSH electrodynamics.
Furthermore, for $r_{0}>0$, the dielectric function behaves as $G(r=0)=1$ and
$G(r\rightarrow\infty)\rightarrow 1$, which means that the medium does not
change sensibly the behavior of the field profiles around the boundary
conditions, i.e., they are expected to exhibit a behavior which is similar to
that of the BPS solutions of the MCSH model. Nevertheless, the numerical
analysis reveals that, along the radial coordinate, the dielectric function
can alter the shape of the field profiles in a significant way.

We now use (63) to rewrite the BPS equations (53) and (54) as

$\frac{dg}{dr}=\frac{ag}{r}\text{,}$ (64)
$B=-\frac{1}{er}\frac{da}{dr}=\frac{\left(r^{2}-r_{0}^{2}\right)^{2}}{\left(r^{2}+r_{0}^{2}\right)^{2}}\left[ev^{2}\left(1-g^{2}\right)-\kappa
A_{0}\right]\\!\text{,}$ (65)

while the Gauss law (55) assumes the form

$\frac{1}{r}\frac{d}{dr}\left[\frac{\left(r^{2}+r_{0}^{2}\right)^{2}r}{\left(r^{2}-r_{0}^{2}\right)^{2}}\frac{dA_{0}}{dr}\right]+\kappa
B=2e^{2}v^{2}g^{2}A_{0}\text{.}$ (66)

Here, we pay attention to the fact that, at $r=r_{0}$, the function $G(\chi)$
in (63) diverges. Consequently, to attain a finite energy density (58), both
the magnetic and electric fields must converge to zero at $r=r_{0}$ faster
than $G(\chi)$ diverges to counterbalance such a singularity. The numerical
simulations show that it is exactly what happens, this way manifesting how the
presence of the chiral medium can affect the field configurations.

Figure 1: Numerical solutions for the Higgs profile $g(r)$ obtained by solving
the system of equations (64), (65) and (66) concerning the dielectric function
(63) for $r_{0}=1$ (dash-dotted blue line) and $r_{0}=2$ (dashed red line).
The solid black line illustrates the solution for $G(\chi)\equiv 1$, which we
have depicted here for the sake of comparison.

Figure 2: Numerical solutions for the gauge profile function $a(r)$ (top) and
the magnetic field $B(r)$ (bottom). Conventions as in the Fig. 1.

We now verify how the dielectric medium changes the way the fields of the
model behave near the boundaries. By solving the Eqs. (64), (65) and (66)
around the boundary values (44) and (46) such a goal is attained. By
considering $N>0$, we write below the behaviors near the origin of the field
profiles; this way, the function $g(r)$ reads

$\displaystyle g(r)$ $\displaystyle\approx$ $\displaystyle
g_{N}r^{N}-\frac{eB_{0}g_{N}r^{N+2}}{4}+\frac{e^{2}v^{2}g_{N}^{3}r^{3N+2}}{4\left(N+1\right)^{2}}$
(67)
$\displaystyle+\left(\frac{eB_{0}}{8}-\frac{\kappa^{2}}{16}+\frac{1}{r_{0}^{2}}\right)\frac{eB_{0}g_{N}r^{N+4}}{4}\text{,}$

whereas, the profile $a(r)$ behaves as

$\displaystyle a(r)$ $\displaystyle\approx$ $\displaystyle
N-\frac{eB_{0}}{2}r^{2}-eB_{0}\left(\frac{\kappa^{2}}{16}-\frac{1}{r_{0}^{2}}\right)r^{4}$
(68)
$\displaystyle+\frac{e^{2}v^{2}g_{N}^{2}r^{2N+2}}{2\left(N+1\right)}\text{,}$

and respective magnetic field becomes

$B(r)\approx
B_{0}+4B_{0}\left(\frac{\kappa^{2}}{16}-\frac{1}{r_{0}^{2}}\right)r^{2}-ev^{2}g_{N}^{2}r^{2N}\text{.}$
(69)

Besides, the scalar potential reads

$\displaystyle A_{0}(r)$ $\displaystyle\approx$ $\displaystyle
A_{0}(0)-\frac{\kappa B_{0}}{4}r^{2}+\frac{3\kappa
B_{0}}{4}\left(-\frac{\kappa^{2}}{48}+\frac{1}{r_{0}^{2}}\right)r^{4}$ (70)
$\displaystyle+\frac{ev^{2}[2eA_{0}(0)+\kappa]g_{N}^{2}r^{2N+2}}{4\left(N+1\right)^{2}}\text{,}$

which leads to the electric field

$\displaystyle E(r)$ $\displaystyle\approx$ $\displaystyle\frac{\kappa
B_{0}}{2}r+\left(\frac{\kappa^{3}B_{0}}{16}-\frac{3\kappa
B_{0}}{r_{0}^{2}}\right)r^{3}$ (71)
$\displaystyle-\frac{ev^{2}[2eA_{0}(0)+\kappa]g_{N}^{2}r^{2N+1}}{2\left(N+1\right)}\text{.}$

Figure 3: Numerical solutions for the scalar potential $A_{0}(r)$ (top) and
the electric field $E(r)$ (bottom). Conventions as in the Fig. 1.

In the expressions above, the values of $g_{N}>0$, $B_{0}=B(r=0)$ and
$A_{0}(0)=A_{0}(r=0)$ depend on the coupling constants and parameters of the
model. In particular, the quantity $B_{0}$ is defined as

$B_{0}=v^{2}e-\kappa A_{0}(0)$ (72)

and stands for the value of the magnetic field at the origin, while $A_{0}(0)$
is the amplitude of the scalar potential at $r=0$. For instance, for fixed
$e=v=\kappa=1$ and $N=1$, the dependence on $r_{0}$ of both $B(0)$ and
$A_{0}(0)$ are depicted in the Fig. 5 below.

In addition, for the energy density $\varepsilon_{G}$, we obtain

$\displaystyle\varepsilon_{G}(r)$ $\displaystyle=$ $\displaystyle
B_{0}^{2}+2N^{2}v^{2}g_{N}^{2}r^{2N-2}+2e^{2}v^{2}(A_{0}(0))^{2}g_{N}^{2}r^{2N}$
(73)
$\displaystyle+\left(\frac{3\kappa^{2}B_{0}^{2}}{4}-\frac{4B_{0}^{2}}{r_{0}^{2}}\right)r^{2}$
$\displaystyle-\left(N^{2}+2N+2\right)ev^{2}B_{0}g_{N}^{2}r^{2N}\text{,}$

from which the dependence of $\varepsilon_{G}(0)$ on $r_{0}$, again for
$e=v=\kappa=1$ and $N=1$, is shown in the Fig. 5.

In general, when very close to the origin, the behaviors of the profiles
resemble those of BPS solutions inherent to the MCSH electrodynamics, as
expected.

On the other hand, in the asymptotic limit $r\rightarrow\infty$, the fields
behave as

$\displaystyle g(r)$ $\displaystyle\approx$ $\displaystyle
1-C_{\infty}\frac{e^{-Mr}}{\sqrt{r}}\text{,}$ (74) $\displaystyle a(r)$
$\displaystyle\approx$ $\displaystyle MC_{\infty}\sqrt{r}e^{-Mr}\text{,}$ (75)
$\displaystyle A_{0}(r)$ $\displaystyle\approx$
$\displaystyle\frac{MC_{\infty}}{e}\frac{e^{-Mr}}{\sqrt{r}}\text{,}$ (76)

therefore preserving the typical behavior which characterizes the Abrikosov-
Nielsen-Olesen’s vortices. Further, the parameter $M$ represents the BPS mass
of the bosonic fields and is given by

$M=\frac{1}{2}\sqrt{8e^{2}v^{2}+\kappa^{2}}-\frac{|\kappa|}{2}\text{,}$ (77)

i.e., the exact definition inherent to the BPS vortices obtained in the
Maxwell-Chern-Simons-Higgs model. Then, as previously pointed out, the
dielectric function (63) does not change sensibly how the fields approach the
vacuum.

Figure 4: Numerical solutions for the energy density $\varepsilon_{G}(r)$ of
the BPS configurations. Conventions as in the Fig. 1.

#### Numerical analysis – ANO-like chiral vortices:

Hereafter, we fix the constants of the model as $e=v=1$, $\kappa=1$, $n=1$,
and select $r_{0}=1$ and $r_{0}=2$ to solve, through the finite-difference
technique and according to the boundary conditions (44) and (46), the set
defined by the Eqs. (64), (65) and (66), from which we compare the new
solutions with the ones attained when $G\equiv 1$. The figures 1, 2, 3 and 4
below show the numerical solutions for the profile functions $g(r)$, $a(r)$
and the magnetic field $B(r)$, the scalar potential $A_{0}(r)$ and the
electric $E(r)$, and the energy distribution $\varepsilon_{G}$. The solid
black line represents the solution for $G\equiv 1$. Further, in the figures 5
and 6, we have depicted the dependence of $A_{0}(r=0)$, $B(r=0)$ and
$\varepsilon_{G}(r=0)$ on the dielectric parameter $r_{0}$. We have also
studied how the maximums (and their respective localizations along the radial
axis) of both the electric and magnetic fields depend on the dielectric
parameter.

The Higgs profile function $g(r)$ grows monotonically according to the
boundary values (44) and (46), see the Fig. 1. The numerical analysis reveals
that the size of the core inherent to these profiles becomes larger whether
the dielectric parameter increases within the interval $0\leq
r_{0}<r^{\ast(1)}$. In particular, when $r_{0}=r^{\ast(1)}$, the core attains
its maximum width. On the other hand, for $r_{0}>r^{\ast(1)}$, the width of
the core diminishes until it achieves the value obtained when $G\equiv 1$.
Here, for instance, whether we define the width of the profile as $\approx
63.2\%$ of its height, then the maximum width of the profile is attained when
$r^{\ast(1)}\approx 1.95$.

Figure 5: Numerical results for the quantities $A_{0}(0)$, $B(0)$ and
$\varepsilon_{G}(0)$ as functions of the dielectric parameter $r_{0}$. In this
Figure, $E_{1,\text{max}}$ and $r_{1,E_{\text{max}}}$ represent the amplitude
of the first maximum of the electric field and its position along the radial
coordinate, respectively, also as functions of $r_{0}$. We have obtained these
results for the dielectric function $G(\chi)$ as given in Eq. (63).

Figure 2 displays the solutions for both the gauge profile $a(r)$ and the
magnetic field $B(r)$. The gauge profiles present a similar dependence with
$r_{0}$ compared to the found for the Higgs sector. The novelty here is the
development of a plateau around the point $r=r_{0}$ that becomes larger
whenever the dielectric parameter increases. At the same time, the value of
$a(r_{0})$ diminishes continuously. On the other hand, the magnetic field
profiles do not vanish at the origin, whose amplitudes at $r=0$ vary with
$r_{0}$ as shown by the blue line in Fig. 5. Note that, for sufficiently large
values of $r_{0}$, the value of $B(0)$ converges to that obtained when
$G\equiv 1$. Furthermore, the magnetic field vanishing at $r=r_{0}$ develops a
second maximum whose amplitude $B_{2,\text{max}}$ and its location
$r_{2,B_{m_{ax}}}>r_{0}$, as functions of $r_{0}$, are both illustrated,
respectively, by the black and red lines in the Fig. 6. Moreover, for $r_{0}$
sufficiently large, we observe that the second maximum decreases exponentially
with $r_{0}$ (i.e., $\sim\exp(-0.687r_{0})$), while, in contrast, the
respective location grows linearly with $r_{0}$.

The scalar potential $A_{0}(r)$ at the origin is nonzero whose amplitude
denoted by $A_{0}(0)$ depends on $r_{0}$ such as depicted by the red line in
the Fig. 5. Along the radial coordinate, $A_{0}(r)$ behaves as the gauge
profile $a(r)$ in the sense that it also develops a plateau centered at
$r=r_{0}$; see the Fig. 3. Similarly, as it happens with the magnetic field,
the resulting electric field also vanishes at $r=r_{0}$; consequently, it
exhibits two different maximums: the first one $E_{1,\text{max}}$ located at
$r_{1,E_{\text{max}}}$, which is within the interval $0\leq r<r_{0}$, and the
second maximum $E_{2,\text{max}}$ positioned at $r_{2,E_{\text{max}}}$ on the
region $r>r_{0}$. In contrast, the electric field without the dielectric
function possesses a unique maximum located at $r_{1}^{\ast}=1.076$, whose
amplitude is $\thickapprox 0.186$, see the black line in the bottom picture of
the Fig. 3. In Figure 5, both $E_{1,\text{max}}$ and $r_{1,E_{\text{max}}}$
are depicted as functions of $r_{0}$, whence we observe that for $r_{0}$
sufficiently large both converge to the values attained when $G(\chi)\equiv
1$. On the other hand, the green and the blue lines appearing in Fig. 6
illustrate, respectively, how the amplitude $E_{2,\text{max}}$ and its
localization $r_{2,E_{\text{max}}}$ vary with the dielectric parameter
$r_{0}$. We notice that, for sufficiently large values of $r_{0}$, the second
maximum decreases exponentially with $r_{0}$ (i.e., $\sim\exp(-0.687r_{0}$)),
while its position increases linearly.

Figure 6: Numerical results to the quantities $B_{2,\text{max}}$ and
$E_{2,\text{max}}$ (i.e., the amplitudes of the second maximums of the
magnetic and the electric fields, respectively), together with their
respective positions along the radial coordinate (namely,
$r_{2,B_{\text{max}}}$ and $r_{2,E_{\text{max}}}$, respectively). We have
obtained these results for the dielectric function $G(\chi)$ as given in Eq.
(63).

The numerical results for the energy density $\varepsilon_{G}(r)$ are shown in
Fig 4. Such as it happens when $G(\chi)\equiv 1$, the value at $r=r_{0}$
denoted as $\varepsilon_{G}(0)$ is nonnull, but now it depends on the
dielectric parameter $r_{0}$ such as depicted as the black line in the Fig. 5.
Besides, we also observe that there is a value $r^{\ast}_{\text{en}}$ such
that for $0<r_{0}<r_{\text{en}}^{\ast}$ a local minimum arises around
$r=r_{0}$. Simultaneously, a second maximum appears on the right-hand side of
$r_{0}$. However, both extremes disappear for $r_{0}>r_{\text{en}}^{\ast}$,
such that, for sufficiently large values of $r_{0}$, the profiles attain the
same format as in the case with $G(\chi)\equiv 1$.

Figure 7: Numerical solutions for the Higgs profile $g(r)$ obtained by solving
the system of equations (80), (79) and (81) regarding the dielectric function
(78)for $r_{0}=1$ (long-dashed gray line) and $r_{0}=2$ (dotted orange line).
Again, the solid black line represents the solution for $G(\chi)\equiv 1$,
which we have depicted here for the sake of comparison.

###  III.2 Delocalized chiral vortices

In what follows, we choose the dielectric function as

$G(\chi)=\frac{1}{1-\chi^{2}}=\frac{\left(r^{2}+r_{0}^{2}\right)^{2}}{4r^{2}r_{0}^{2}}\text{,}$
(78)

where we have already considered the solution (62), which is finite along the
radial axis but diverges at $r=0$ and when $r\rightarrow\infty$. Therefore, to
avoid the energy density (58) being singular, both the magnetic field and
electric field must vanish at the boundaries more rapidly than the dielectric
function diverges in such a way that the total energy converges to the finite
value given by (49).

By using (78), the self-dual equations (53) and (54) assume the form

$\frac{dg}{dr}=\frac{ag}{r}\text{,}$ (79)
$B=-\frac{1}{er}\frac{da}{dr}=\frac{4r^{2}r_{0}^{2}}{\left(r^{2}+r_{0}^{2}\right)^{2}}\left[ev^{2}\left(1-g^{2}\right)-\kappa
A_{0}\right]\\!\text{,}$ (80)

while the Gauss law (55) can be rewritten as

$\frac{1}{r}\frac{d}{dr}\left[\frac{\left(r^{2}+r_{0}^{2}\right)^{2}}{4r_{0}^{2}r}\frac{dA_{0}}{dr}\right]+\kappa
B=2e^{2}v^{2}g^{2}A_{0}\text{.}$ (81)

We now investigate how the dielectric medium affects the behavior of the
fields close to the boundary values given by the Eqs. (44) and (46). With this
purpose in mind, we solve the equations (79), (80) and (81) around the
boundaries. In this case, for $n=N>0$, we get that, near the origin, the Higgs
profile behaves as

$\displaystyle g(r)$ $\displaystyle\approx$ $\displaystyle
g_{N}r^{N}-\frac{eB_{0}g_{N}r^{N+4}}{4r_{0}^{2}}+\frac{2eB_{0}g_{N}r^{N+6}}{9r_{0}^{4}}$
(82)
$\displaystyle+\frac{e^{2}v^{2}g_{N}^{3}r^{3N+4}}{\left(N+2\right)^{2}r_{0}^{2}}\text{,}$

where $g_{N}>0$ and $B_{0}$ is defined as in the Eq. (72). We observe that the
next-to-leading-order contribution is proportional to $r^{N+4}$, whereas the
one in (67) is to $r^{N+2}$.

Figure 8: Numerical solution for the gauge profile $a(r)$ (top) and the
magnetic field $B(r)$ (bottom). Conventions as in the Fig. 7.

In the meanwhile, the gauge profile $a(r)$ reads

$a(r)\approx
N-\frac{eB_{0}r^{4}}{r_{0}^{2}}+\frac{3eB_{0}r^{6}}{4r_{0}^{4}}+\frac{2e^{2}v^{2}g_{N}^{2}r^{2N+4}}{\left(N+2\right)r_{0}^{2}}\text{,}$
(83)

from which we see that it converges faster to the value $N$ than the one in
(68). The corresponding magnetic field behaving as

$\displaystyle B(r)$ $\displaystyle\approx$
$\displaystyle\frac{4B_{0}}{r_{0}^{2}}r^{2}-\frac{8B_{0}}{r_{0}^{4}}r^{4}+\frac{12B_{0}}{r_{0}^{6}}r^{6}$
(84)
$\displaystyle-\frac{4ev^{2}}{r_{0}^{2}}g_{N}^{2}r^{2N+2}+\frac{8ev^{2}}{r_{0}^{4}}g_{N}^{2}r^{2N+4}\text{,}$

vanishes at $r=0$, contrasting the one in (69) and acquiring a maximum outside
from the origin; see Fig 8.

Figure 9: Numerical solutions for the scalar potential $A_{0}(r)$ (top) and
the electric field $E(r)$ (bottom). Conventions as in the Fig. 7.

The scalar potential near the origin reads

$\displaystyle A_{0}(r)$ $\displaystyle\approx$ $\displaystyle
A_{0}(0)-\frac{2\kappa B_{0}r^{6}}{3r_{0}^{4}}+\frac{5\kappa
B_{0}r^{8}}{3r_{0}^{6}}$ (85)
$\displaystyle+\frac{2e^{2}v^{2}A_{0}(0)g_{N}^{2}r^{2N+4}}{\left(N+1\right)\left(N+2\right)r_{0}^{2}}+\frac{4ev^{2}{\kappa}g_{N}^{2}r^{2N+6}}{\left(N+2\right)\left(N+3\right)r_{0}^{4}}$
$\displaystyle-\frac{4e^{2}v^{2}A_{0}(0)g_{N}^{2}r^{2N+6}}{\left(N+1\right)\left(N+3\right)r_{0}^{4}}\text{,}$

(where $A_{0}(0)$ represents the scalar potential value at $r=0$) whose next-
to-leading-order contribution is proportional to $r^{6}$, whereas the one in
(70) is to $r^{2}$. Furthermore, for fixed $e=v=\kappa=1$ and $N=1$, the
amplitude $A_{0}(r=0)$ vs. $r_{0}$ behaves differently from the one shown in
the Fig. 5, see the Fig. 11. Furthermore, the electric field is given by

$\displaystyle E(r)$ $\displaystyle\approx$ $\displaystyle\frac{4\kappa
B_{0}r^{5}}{r_{0}^{4}}-\frac{40\kappa
B_{0}r^{7}}{3r_{0}^{6}}-\frac{4e^{2}v^{2}A_{0}(0)g_{N}^{2}r^{2N+3}}{\left(N+1\right)r_{0}^{2}}$
(86)
$\displaystyle+\frac{8e^{2}v^{2}A_{0}(0)g_{N}^{2}r^{2N+5}}{\left(N+1\right)r_{0}^{4}}-\frac{8ev^{2}{\kappa}g_{N}^{2}r^{2N+5}}{\left(N+2\right)r_{0}^{4}}\text{,}$

which converges faster to zero at the origin than that in Eq. (71).

Finally, near the origin, the energy density $\varepsilon_{G}(r)$ can be
expressed as

$\displaystyle\varepsilon_{G}(r)$ $\displaystyle\approx$ $\displaystyle
2N^{2}v^{2}g_{N}^{2}r^{2N-2}+2e^{2}v^{2}A_{0}(0)g_{N}^{2}r^{2N}$ (87)
$\displaystyle+\frac{4B_{0}^{2}}{r_{0}^{2}}r^{2}-\frac{8B_{0}^{2}}{r_{0}^{4}}r^{4}$
$\displaystyle-\left(N^{2}+4N+8\right)\frac{ev^{2}B_{0}}{r_{0}^{2}}g_{N}^{2}r^{2N+2}\text{,}$

which contains more contributions associated with the dielectric parameter
$r_{0}$ than the solution in the Eq. (73). The amplitude of
$\varepsilon_{G}(r)$ at origin, $\varepsilon_{G}(0)$, in terms of $r_{0}$ is
depicted in the Fig. 11.

Thus, we observe that close to the origin, the behavior of the field profiles
shows the strong influence of the dielectric function (78) through the
parameter $r_{0}$, resulting in very different behavior when compared with
that of the BPS solutions of electrodynamics MCSH, as the numerical analysis
below demonstrates.

Figure 10: Numerical solutions for the energy density $\varepsilon_{G}(r)$ of
the BPS configurations. Conventions as in the Fig. 7.

In the limit $r\rightarrow\infty$, the profile functions read

$\displaystyle g(r)$ $\displaystyle\approx$ $\displaystyle
1-C_{\infty}r^{-\Lambda}\text{,}$ (88) $\displaystyle a(r)$
$\displaystyle\approx$ $\displaystyle\Lambda C_{\infty}r^{-\Lambda}\text{,}$
(89) $\displaystyle A_{0}(r)$ $\displaystyle\approx$
$\displaystyle-\frac{\Lambda
C_{\infty}}{2e\kappa\left(r_{0}\right)^{2}}r^{-\Lambda}\text{,}$ (90)

where $C_{\infty}$ is a positive integration constant. Here, we have
introduced the parameter $\Lambda$, i.e.,

$\Lambda=1+\sqrt{1+8\left(evr_{0}\right)^{2}}\text{,}$ (91)

which reveals how the power-law decay depends on $r_{0}$. We here highlight
that for $r\rightarrow\infty$, the presence of the dielectric medium ruled by
$G(\chi)$ as defined in (78) has radically changed the behavior of the field
profiles, whose tails now decay by following a power-law, in contrast to the
canonical behavior (i.e., the exponential decay one).

The literature shows that the London limit [39] provides the behavior of the
fields of a vortex via the full Ginzburg-Landau model, therefore predicting
that the magnetic field varies monotonically and that, at large distances, is
exponentially localized. However, very recently, it has been discovered that
some vortex solutions have a delocalized magnetic field whose profiles decay
slowly. These delocalized vortices whose tails follow a power-law decay have
arisen during the study of the magnetic field delocalization in two-component
superconductors [40], and also in the context of $k$-generalized Maxwell-Higgs
electrodynamics [41]. Furthermore, such behavior has also been reported in
diamagnetic vortices generated within a Chern-Simons theory [42].

Figure 11: Numerical results for the quantities $A_{0}(0)$,
$\varepsilon_{G}(0)$, and $E_{\text{max}}$ (the last one represents the
maximum value of the electric field). We have obtained these results for the
dielectric function $G(\chi)$ as given in the Eq. (78).

#### Numerical analysis – Delocalized chiral vortices:

We again fix the constants of the model as $e=v=1$, $\kappa=1$, $n=1$, via
which we solve the set defined by the Eqs. (79), (80) and (81) according to
the boundary conditions (44) and (46) through the finite-difference technique,
for $r_{0}=1$ and $r_{0}=2$. We then compare the new solutions with the ones
attained when $G\equiv 1$ (which appear as the solid black lines in the plots
below). The figures 7, 8, 9 and 10 depict, respectively, the numerical results
for the Higgs profile $g(r)$, the vector gauge profile $a(r)$ and the magnetic
field $B(r)$, the scalar potential $A_{0}(r)$ and the electric $E(r)$ field,
and the energy distribution $\varepsilon_{G}$. Furthermore, in the figures 11
and 12, we analyze the dependence of the amplitudes at $r=0$ of the scalar
potential and energy density on the dielectric parameter $r_{0}$. We also look
at how the maxima of the electric and magnetic fields (and their respective
localizations along the radial coordinate) depend on the dielectric parameter.

The Higgs profiles $g(r)$ grow monotonically according to the boundary values
(44) and (46), see the Fig. 7. The numerical analysis indicates that the core
size inherent to these profiles is larger than those related to $G(\chi)\equiv
1$ for all values of $r_{0}$. Whether the value of the dielectric parameter
increases within the interval $0\leq r_{0}\leq r^{\ast(2)}$, the core
diminishes continuously until its minimum width attained for
$r_{0}=r^{\ast(2)}$. In addition, when $r_{0}>r^{\ast(2)}$, the width of the
core grows slowly. Here, for instance, whether one considers the width of the
profile as $\approx 63.2\%$ of its height, the minimum width is attained for
$r^{\ast(2)}\approx 1.1$.

Figure 8 displays the solutions to both the gauge profile $a(r)$ and the
magnetic field $B(r)$. In this case, the behavior of $a(r)$ on the parameter
$r_{0}$ is analogous to previously described for the Higgs field. On the other
hand, the magnetic field vanishing at the origin develops a global maximum,
$B_{\text{max}}$, whose amplitude varies with $r_{0}$ as shown by the red line
in the Fig. 12. It is worthwhile to note that for sufficiently large values of
$r_{0}$, the amplitude $B_{\text{max}}$ decays as $r_{0}^{-1}$ and its
localization, $r_{B_{\text{max}}}$, increases as $r_{0}^{1/2}$ along the
radial coordinate, see the blue line in the Fig. 12.

Figure 12: Numerical results for the quantity $B_{\text{max}}$ (the maximum
amplitude of the magnetic field) and $r_{B_{\text{max}}}$ (the position of
$B_{\text{max}}$). Here, $r_{E_{\text{max}}}$ represents the point at which
the electric field achieves its maximum. We have obtained these results for
the dielectric function $G(\chi)$ as given in the Eq. (78).

The results, which refer to the scalar potential $A_{0}(r)$ and the electric
field, appear in Fig. 9. The amplitude $A_{0}(r=0)$ again depends on the
dielectric parameter $r_{0}$, see the red line in the Fig. 11. However, it now
behaves in a different way than that related to the previous case; see Fig 5.
The maximum amplitude of the electric field, $E_{\text{max}}$, versus the
dielectric parameter $r_{0}$ is depicted in the Fig. 11, while its radial
localization, $r_{E_{\text{max}}}$, which increases as $r_{0}^{1/2}$ for
sufficiently large $r_{0}$, is shown as the black line in the Fig. 12.

The profiles of the energy density $\varepsilon_{G}(r)$ are plotted in Fig 12.
The profile does not form a lump (as it happens when $G(\chi)\equiv 1$) but
now develops a ring-like shape, such as it happens for the winding number
$n\geq 2$ in the context of Maxwell-Higgs electrodynamics. Furthermore, the
dependence of $\varepsilon_{G}(r=0)$ on $r_{0}$ is depicted as the black line
in the Fig. 11, from which we see that it differs entirely from the one
obtained in the first case, as shown by the Fig. 5.

##  IV Summary and perspectives

We have investigated the formation of BPS solitons in a chiral medium ruled by
an extended Maxwell-Chern-Simons electrodynamics coupled to the Higgs field
and an extra real scalar field $\chi$ which controls the dielectric medium. In
such a context, we have focused our attention on the BPS structure of the
chiral model in order to attain the self-dual BPS equations whose solutions
saturate the Bogomol’nyi bound. In the sequence, we have observed that the
self-dual equation for the field $\chi$ does not involve the others sectors of
the theory and depends solely on the particular choice for the superpotential
$F_{i}(\chi)$. Nevertheless, the influence of the field $\chi$ on the overall
scenario occurs through the dielectric function $G(\chi)$, which appears in
the equations for the gauge sector.

To describe the BPS vortices in the chiral medium, we have used the usual
radially symmetric ansatz for the Higgs and the gauge fields. Inspired by such
maps, we have also parameterized the superpotentials $F_{i}(\chi)$ in terms of
a function $W(r)$. We have solved the self-dual equation for $\chi(r)$
analytically and introduced an specific expression for the dielectric function
$G(\chi)$ which becomes responsible for the changes on the profiles of the
chiral vortices. To better understand the influence of the kink engendered by
the field $\chi$ through $G(\chi)$ on the vortex profiles, we have split our
investigation into two cases based on the functional forms chosen for the
dielectric function.

In all cases, the dielectric function satisfies the conditions
$\sqrt{G}B\rightarrow 0$ and $\sqrt{G}\partial_{i}A_{0}\rightarrow 0$ in the
limit $|\mathbf{x}|\rightarrow\infty$, which ensures the convergence of the
total energy of the self-dual vortex to the Bogomol’nyi bound. The numerical
analysis has revealed that the dielectric causes different effects on the
corresponding vortex profiles in comparison to those obtained without it
(i.e., via $G=1$). Our first choice, $G(\chi)=\chi^{-2}$, affects the behavior
of the fields along the radial coordinate, except near the boundary values. In
particular, the magnetic and the electric fields vanish at $r=r_{0}$, from
which the resulting configurations engender new ringlike profiles.
Nevertheless, the tail of the chiral configurations decays like that of the
Abrikosov-Nielsen-Olesen’s vortex. On the other hand, the second choice
$G(\chi)=(1-\chi^{2})^{-1}$ changes the behavior of the fields near the
boundaries. Consequently, the magnetic sector and the energy density develop
ringlike formats. Nevertheless, the electric field preserves the same ring
profile as in the canonical MCSH model. On the other hand, this time, the
chiral vortex tail follows a power-law decay, which contrasts the exponential
law inherent to the Abrikosov-Nielsen-Olesen’s vortex.

An interesting perspective is the application of the present idea to the study
of the existence of chiral self-dual solitons in the context of $CP(2)$ or
Skyrme models in the presence of a magnetic impurity, see the references [43]
and [44], for instance. We are investigating these issues and will report the
results in future contribution.

###### Acknowledgements.

This study was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, the Conselho
Nacional de Desenvolvimento Científico e Tecnológico - Brasil (CNPq) and the
Fundação de Amparo à Pesquisa e ao Desenvolvimento Científico e Tecnológico do
Maranhão - Brasil (FAPEMA). J. A. thanks the full support from CAPES. R. C.
acknowledges the support from the grants CNPq/306724/2019-7,
FAPEMA/universal-00812/19, and FAPEMA/APP-12299/22. E. H. thanks the support
from the grants CNPq/307545/2016-4 and FAPEMA/COOPI/07838/17.

## References

  * [1] N. Manton and P. Sutcliffe, Topological Solitons (Cambridge University Press, Cambridge, England, 2004).
  * [2] E. Bogomol’nyi, Sov. J. Nucl. Phys. 24, 449 (1976). M. Prasad and C. Sommerfield, Phys. Rev. Lett. 35, 760 (1975).
  * [3] H. J. de Vega and F. A. Schaposnik, Phys. Rev. D 14, 1100 (1976).
  * [4] A. N. Atmaja and H. S. Ramadhan, Phys. Rev. D 90, 105009 (2014). A. N. Atmaja, H. S. Ramadhan and E. da Hora, J. High Energy Phys. 1602, 117 (2016).
  * [5] H. B. Nielsen and P. Olesen, Nucl. Phys. B 61, 45 (1973).
  * [6] R. Jackiw and E. J. Weinberg, Phys. Rev. Lett. 64, 2234 (1990). R. Jackiw, K. Lee and E. J. Weinberg, Phys. Rev. D 42, 3488 (1990).
  * [7] C. Lee, K. Lee and H. Min, Phys. Lett. B 252, 79 (1990).
  * [8] A. Yu. Loginov, Phys. Rev. D 93, 065009 (2016).
  * [9] R. Casana, M. L. Dias and E. da Hora, Phys. Lett. B 768, 254 (2017).
  * [10] V. Almeida, R. Casana and E. da Hora, Phys. Rev. D 97, 016013 (2018). R. Casana, M. L. Dias and E. da Hora, Phys. Rev. D 98, 056011 (2018).
  * [11] R. Casana, N. H. Gonzalez-Gutierrez and E. da Hora, Europhys. Lett. 127, 61001 (2019).
  * [12] E. Witten, Nucl. Phys. B 249, 557 (1985). M. Shifman, Phys. Rev. D 87, 025025 (2013). A. Peterson, M. Shifman and G. Tallarita, Annals Phys. 353, 48 (2014); Annals Phys. 363, 515 (2015).
  * [13] D. Bazeia, M. A. Marques and R. Menezes, Phys. Lett. B 780, 485 (2018).
  * [14] R. A. Shelby, D. R. Smith, S. Schultz, Science 292, 77 (2001). S. A. Ramakrishna, Rep. Prog. Phys. 68, 449 (2005). C. Caloz, Mater. Today 12, 12 (2009).
  * [15] J. Andrade, R. Casana, E. da Hora and C. dos Santos, Phys. Rev. D 99, 056014 (2019).
  * [16] In what follows, we use $\eta^{\hat{\mu}\hat{\nu}}=(+---)$ as the metric signature of the $(1+3)$-dimensional flat spacetime, together with the natural units system, for the sake of simplicity.
  * [17] Zebin Qiu, Gaoqing Cao and Xu-Guang Huang, Phys. Rev. D 95, 036002 (2017).
  * [18] D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A 803, 227 (2008).
  * [19] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D 78, 074033 (2008).
  * [20] F. D. M. Haldane, Phys. Rev. Lett. 93, 206602 (2004).
  * [21] D. Xiao, M. C. Chang and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
  * [22] A. A. Zyuzin and A. A. Burkov, Phys. Rev. B 86, 115133 (2012).
  * [23] C.-X. Liu, P. Ye and X.-L. Qi, Phys. Rev. B 87, 235306 (2013).
  * [24] X. G. Huang, Sci. Rep. 6, 20601 (2016).
  * [25] F. Wilczek, Phys. Rev. Lett. 58, 1799 (1987).
  * [26] S. M. Carroll, G. B. Field and R. Jackiw, Phys. Rev. D 41, 1231 (1990).
  * [27] M. E. Tobar, B. T. McAllister and M. Goryachev, Phys. Dark Universe 26, 100339 (2019).
  * [28] A. Sekine and K. Nomura, J. Appl. Phys. 129, 141101 (2021).
  * [29] Eduardo Barredo-Alamilla, Luis F. Urrutia, and Manoel M. Ferreira, Jr., Phys. Rev. D 107, 096024 (2023).
  * [30] Filipe S. Ribeiro, Pedro D. S. Silva, and Manoel M. Ferreira, Jr., Phys. Rev. D 107, 096018 (2023).
  * [31] Filipe S. Ribeiro, Pedro D. S. Silva, and Manoel M. Ferreira, Jr., Cold plasma modes in the chiral Maxwell-Carroll-Field-Jackiw electrodynamics, https://doi.org/10.48550/arXiv.2312.13196.
  * [32] H. Belich, M. M. Ferreira Jr. and J. A. Helayel-Neto, Eur. Phys. J. C 38, 511 (2005).
  * [33] R. Casana, M. M. Ferreira Jr., E. da Hora and A. B. F. Neves, Eur. Phys. J. C 74, 3064 (2014).
  * [34] In what follows, we use $\eta^{\mu\nu}=(+--)$ as the metric signature of the $(1+2)$-dimensional flat spacetime, together with the natural units system.
  * [35] The analysis of the BPS structure for $(k_{AF})_{\mu}\neq 0$ is in development, and the results will be reported elsewhere.
  * [36] A. Polyakov, Mod. Phys. Lett. A 3, 325 (1988). Y. H. Chen, F. Wilczek, E. Witten and B. Halperin, Int. J. Mod. Phys. B 3, 1001 (1989).
  * [37] D. Bazeia, M. M. Doria and E. I. B. Rodrigues, Phys. Lett. A 380, 1947 (2016). D. Bazeia, J. G. G. S. Ramos and E. I. B. Rodrigues, J. Magnetism Magnetic Materials 423, 411 (2017).
  * [38] D. Bazeia and A. Mohammadi, Phys. Lett. B 779, 420 (2018).
  * [39] E. Babaev, Phys. Rev. Lett. 89, 067001 (2002).
  * [40] E. Babev, J. Jäykkä and M. Speight, Phys. Rev. Lett. 103, 237002 (2009).
  * [41] R. Casana, A. Cavalcante and E. da Hora, J. High Energ. Phys. 2016, 51 (2016).
  * [42] M. M. Anber, Y. Burnier, E. Sabancilar and M. Shaposhnikov, Phys. Rev. D 92, 085049 (2015).
  * [43] D. Tong and K. Wong, J. High Energy Phys. 1401, 090 (2014).
  * [44] A. Cockburn, S. Krusch and A. A. Muhamed, J. Math. Phys. 58, 063509 (2017). J. Ashcroft and S. Krusch, Phys. Rev. D 101, 025004 (2020).