Electrically charged localized structures

From Eq. (3b), we obtain the electric field

Also, knowing that the scalar electric fields do not depend on $\theta$, Eq. (3a) can be rewritten as

In this case, both E and $\phi$ do not depend on $\theta$, and so the above equation becomes, after using the expression for the electric field in Eq. (9),

This is a second order differential equation with nonlinearities introduced by the dielectric function $\varepsilon(\phi)$.

We now follow the lines of Ref. bogo to find a first-order formalism for the above model. One can use Eq. (4) to write the energy density as $\rho=\rho_{f}+\rho_{c}$, where

with the indices $f$ and $c$ standing for field and charge, respectively. The total energy can be written in the form $E=E_{f}+E_{c}$, as the integral of the above energy densities, respectively.

The key point here is that one can introduce an auxiliary function $W=W(\phi)$ to write $\rho_{f}$ in the form

Notice the presence of the factor $1/r$ in the last term; it is important since under integration to get the energy, the line element $dx\,dy=r\,dr\,d\theta$ makes $dW/dr$ be a surface term after integration. When the dielectric function is given by

one can show that the energy $E_{f}$ is bounded, i.e., $E_{f}\geq E_{B}$, where

For solutions obeying the first-order equations

we get to the minimum energy case, with $E_{f}=E_{B}$. The total energy is $E=E_{B}+E_{c}$, where $E_{c}=2\pi A_{0}(r=0)$ is the energy due to the electric charge. One can show that the above equation is compatible with the equation of motion (11). We further notice that equations of the above form, with the $1/r$ factor, were considered before in Refs. prl ; global , and they engender scale invariance. Moreover, the above first-order equations have two signs; they are related by the change $r\to 1/r$, so we only deal with the positive sign from now on. We further notice that equations similar to the above first order equations (16) appeared before in internal ; research , for instance, in the study of vortices with internal structure.

The first order equation allows that we write the energy density in Eq. (12a) in the form

Since $\textbf{E}=-\nabla A_{0}$, we can use Eqs. (9), (14) and (16) with positive sign to obtain

such that the energy of the charge, $E_{c}$, is written as

and it can be calculated straightforwardly, by just knowing the value of $\phi(r)$ at $r=0$.

Before illustrating the general results, let us notice that if we consider $\varepsilon(\phi)$ such that $\varepsilon(0)=1$, the model (1) becomes the standard model, with the electric field correctly given by (9) with $\varepsilon(0)=1$ in the standard case. However, when one implements the first order framework to get minimum energy configurations, the first order equations (16) are mandatory, and they impose that the constant field $\phi=\bar{\phi}$ has to obey $W_{\phi}(\bar{\phi})=0$, meaning that a constant scalar field configuration cannot be chosen at will anymore. A direct consequence of this appears in Eq. (14): any constant field $\bar{\phi}$ that obeys the first order equations (16) induces a divergence in the dielectric function $\varepsilon(\phi)$, and this imposes that the electric field in Eq. (9) has to vanish in this case to regulate the energy of the system.

Another interesting issue arises from the first order equations (16); as it was shown in Ref. prl , if one changes $r\to e^{x}$ the equations (16) become

which are first order equations that describe static configurations of a $(1,1)$ dimensional scalar field theory with potential

In this sense, if one chooses

where $\lambda$ and $a$ are real parameters, we get the scalar field model

with the potential

and the first order equations

with the standard solutions $\phi(x)=\pm a\tanh[\lambda(x-x_{0})]$. The above $(1,1)$ dimensional scalar field model is the prototype of the Higgs field; it is driven by a double-well potential that engenders spontaneous symmetry breaking and has important consequences in nonlinear science. In this sense, the scalar field which drives the symmetry breaking may be seen as an order parameter similar to the spontaneous polarization which is studied, for instance, in ferroelectric materials; see, e.g., Refs. lgd ; Chem ; ferro . We notice that the parameters $\lambda$ and $a$ which we added in (22) control the energy barrier in the double-well potential of the scalar field, so they are of direct interest in applications in ferroelectrics lgd ; Chem ; ferro , for instance. For simplicity, however, we will take $\lambda=a=1$ in this work.

Motivated by the above results, let us now illustrate the planar model with $W(\phi)$ as in Eq. (22) (with $\lambda=a=1$). Here the dielectric function in Eq. (14) is then written as

and Eq. (16) gives

It is solved by

which has energy density

In Fig. 1, we display the solution of the above first-order equation and its energy density. It engenders scale invariance, so we use the condition $\phi(1)=0$, for simplicity. From Eqs. (15) and (19), we get that the energy is $E=E_{B}+E_{c}$, where $E_{B}=8\pi/3$ and $E_{c}=4\pi/3$, which matches with the numerical integration of the energy density displayed in Fig. 1. In order to have a better view of the localized solution, in Fig. 2 we display the energy density (29) in the $(r,\theta)$ plane. We further notice that it is controlled by the dielectric function $\varepsilon(\phi)$ that couples the gauge field to the scalar field.

As we have already commented on, the scalar field model described by (22) develops spontaneous symmetry breaking and can be related to the potential based on the phenomenological Landau-Ginzburg-Devonshire or LGD theory of ferroelectrics lgd ; Chem ; ferro . The recent measurements ferro of the intrinsic double-well energy landscape in a thin layer of ferroelectric material integrated into a heterostructure with a second dielectric layer, suggest that the negative capacitance negative of the dielectric material has its origin in the energy barrier of the double-well potential. The negative capacitance can be used to provide, for instance, voltage amplification for low power nanoscale devices; this can be done, for instance, after trading a standard insulator with a ferroelectric insulator of appropriate thickness nega .

We now focus on the second model. Here, considering the presence of the same electric charge at the origin, the electric field has the same form of Eq. (9) with $\varepsilon(\phi)\to\varepsilon(\phi,\chi)$, so the equations of motion that describe the scalar fields are

The energy density for the solutions of the above equations can be written as $\rho=\rho_{f}+\rho_{c}$, where

As before, here we also introduce an auxiliary function $W=W(\phi,\chi)$ to write $\rho_{f}$ as

We choose the dielectric function in the form

to make the energy bounded, i.e., $E_{f}\geq E_{B}$, where

such that $\Delta W=W(\phi(\infty),\chi(\infty))-W(\phi(0),\chi(0))$. One can show that the bound is saturated to $E_{f}=E_{B}$ if the following first-order equations are satisfied

It is not hard to show that solutions of the above first order equations also solve the equations of motion (30). In this situation, the total energy is $E=E_{B}+E_{c}$, where $E_{c}=2\pi A_{0}(r=0)$. The upper and lower signs are related by the change $r\to 1/r$, so we only consider the positive sign from now on. By using the above equations, one can write the energy density (31a) as the sum of two contributions, in the form $\rho_{f}=\rho_{1}+\rho_{2}$, where

The first-order equations (35) with positive sign may be combined with Eqs. (9) under the change $\varepsilon(\phi)\to\varepsilon(\phi,\chi)$ and (33) to give

The energy associated to the electric charge is

We remark that the energy contributions presented in (34) and (38) do not depend on $f(\chi)$. Also, the function $W=W(\phi,\chi)$ depends on both $\phi$ and $\chi$, so the first order equations (35) are coupled and must be solved simultaneously, in general. However, an interesting case arises for $W(\phi,\chi)=g(\phi)+h(\chi)$. In this specific situation, the first order equation (35a) can be solved independently, and this simplifies the calculation importantly. This allows that we add another integration constant in the problem, leading to a more general situation. For simplicity, however, we will not consider this possibility in the present work. As in the previous model, here we also have to be careful with the choice of constant fields, $\bar{\chi}$ and $\bar{\phi}$, since in the first order framework they have to obey first order differential equations. This issue is similar to the one discussed before in the paragraph below Eq. (19), so we do not comment on it anymore.

To illustrate the new possibility, we take

where $\alpha$ is a positive real parameter. From Eqs. (35a) and (36b), we get the solution, $\chi(r)$, and the energy density $\rho_{2}$, in the form

Their profiles for $\alpha=1$ can be seen in Fig. 1, where they appear as the dotted lines. Next, we use the above solution to feed the function $f(\chi)$, which we firstly consider as $f(\chi)=1/\chi^{2}$. This is perhaps the simplest choice, which is not negative, is unity for $\bar{\chi}=\pm 1$, as required by the choice of $h(\chi)$ in Eq. (39), and engenders the appropriate profile. To find the behavior of $\phi(r)$, we choose $g(\phi)$ in the form

The dielectric function becomes

and the first order equation (55b) now changes to

It supports the analytical solution

The energy density (36a) takes the form

The energy of this model is $E=E_{B}+E_{c}$, where $E_{B}=8\pi(1+\alpha)/3$ and $E_{c}=4\pi(1+\alpha)/3$, matching with Eqs. (34) and (38). The solution (44) and the above energy density are displayed in Fig. 3. Notice that the solution presents a plateau around $r=1$ that gets wider as $\alpha$ decreases. This introduces a hole in the energy density at this point, which becomes more evident as $\alpha$ decreases. In order to highlight this feature, we plot the above energy density in the plane in Fig. 4. The structure presents a hole at the center and a ring around it, which is more visible as $\alpha$ increases. This behavior is controlled by the dielectric function $\varepsilon(\phi)$.

The model described by Eq. (41) with $f(\chi)=\chi^{-2}$ supports rings around the central hole in the energy density. We keep the same $g(\phi)$ in Eq. (41) with the $\chi$ field as in Eq. (40), but now we consider $f(\chi)=\sec^{2}(n\pi\chi)$, where $n$ is a natural number. This is another choice, which is also not negative, unity for $\bar{\chi}=\pm 1$, as required by the choice of $h(\chi)$ in Eq. (39) above, and engenders the interesting wavelike profile which will bring internal modification in the electric structure. We also remark that, since $W(\phi,\chi)$ does not change, the energy remains the same, i.e., $E=E_{B}+E_{c}$, where $E_{B}=8\pi(1+\alpha)/3$ and $E_{c}=4\pi(1+\alpha)/3$. In this case, we have to change $\chi^{2}\to\cos^{2}(n\pi\chi)$ in the factor that multiplies $(1-\phi^{2})^{2}$ in the dielectric function shown in Eq. (42). Moreover, the first order equation (55b) takes the form

It supports the solution

where $\text{Ci}(z)$ denotes the cosine integral function. The energy density in Eq. (36a) takes the form

Since we have already seen how the parameter $\alpha$ modifies the configurations, we fix $\alpha=3$ and plot the above solution and energy density for $n=1$ and $2$ in Fig. 5. We see that, as $n$ increases, the solution exhibits more and more plateaux which appear in the energy density as valleys, points in which $\rho=0$. Including the central valley, one gets $2n+1$ valley in the solutions. To illustrate this feature, we plot the above energy density in the plane for the very same values of $\alpha$ and $n$ in Fig. 6. The localized configuration develops an interesting internal multilayered ringlike structure.

Let us now focus on the presence of electrically charged localized structures in the model (1) in $(3,1)$ dimensions. Here we first notice that Eqs. (2)-(4) are also valid in this case, and more, there is no magnetic field. However, in the spatial case the electric field changes to

In the presence of spherical symmetry, the equation of motion (3a) for the scalar field becomes

Combining this with the above electric field, we obtain

We use the Eq. (4) in order to write the energy density as $\rho=\rho_{f}+\rho_{c}$, where

As before, the indices $f$ and $c$ stand for field and charge, respectively. Following similar steps of the previous Sec. II.1, one can write the dielectric function as in Eq. (14) and show that $E_{f}=4\pi|W(\phi(r\to\infty))-W(\phi(r=0))|$ is the minimum energy if the solutions obey the first order equation

We notice that first order equations similar to the above ones appeared before in mo1 ; mo2 , in the study of magnetic monopoles with internal structure in three spatial dimensions.

Regarding the function $A_{0}$, it has the very same expression displayed in Eq. (18), so $E_{c}=4\pi|W(\phi(r=0))|$. Considering the function in Eq. (22), one can show that the above equation with negative sign, which we choose for convenience, supports the solution and energy density

We plot them in Fig. 7. The planar section of the energy density passing through the center of the structure is similar to the configuration displayed in Fig. 2, so we do not display it here. One can show that the total energy is $E=E_{f}+E_{c}$, where $E_{f}=E_{c}=8\pi/3$.

We now turn attention to the second model, which is defined in Eq. (5). We can also develop a first order framework in $(3,1)$ dimensions. In this case, one gets the very same dielectric function in Eq. (33) and, also, the first order equations (35) with the change $r\to r^{2}$ in the right hand side of the equations, that is,

Considering the $\chi$ field described by the same $h(\chi)$ in Eq. (39), we obtain the following solution and associated energy density

They can be seen in Fig. 7 for $\alpha=1$. By using the above solution, one can use the $g(\phi)$ in Eq. (41) with $f(\phi)=\sec^{2}(n\pi\chi)$ to obtain

such that its corresponding contribution in the energy density is

By using the Bogomol’nyi bound, one gets the energy $E=E_{B}+E_{c}$, where $E_{B}=E_{c}=8\pi(1+\alpha)/3$. In Fig. 8, we display the solution (57) and the above energy density. We see that the most external ring is almost invisible. In general, the energy density engenders $n+1$ rings, whose intensities are controlled by $\alpha$. To highlight the ringlike profile of the energy density, we plot the planar section of the energy density passing through the center of the structure in Fig. 9. As both Figs. 7 and 8 show, the shell structure in the spatial case is different from the ringlike structure in the planar case.