# Highly-enhanced propagation of long-range kinks in heterogeneous media

Jorge A. González  Department of Physics, Florida International University,
Miami, Florida 33199, USA.  Alberto Bellorín  Escuela de Física, Facultad de
Ciencias, Universidad Central de Venezuela, AP 47586, Caracas 1041-A,
Venezuela  Luis E. Guerrero  Departamento de Física, Universidad Simón
Bolívar, Apartado 89000, Caracas 1080-A, Venezuela  Salvador Jiménez
Departamento de Matemática Aplicada a las TIC, ETSI Telecomunicación,
Universidad Politécnica de Madrid, 28040-Madrid, Spain  Juan F. Marín
Departamento de Física, Facultad de Ciencias Naturales, Matemática y del Medio
Ambiente, Universidad Tecnológica Metropolitana, Las Palmeras 3360, Ñuñoa
780-0003, Santiago, Chile.

###### Abstract

We investigate a field-theoretical model that describes the interaction
between kinks and antikinks and between kinks and other heterogeneous fields
and impurities. We show that the long-range kink can tunnel through a barrier
created by heterogeneous fields and impurities even when the energy of the
center of mass of the kink is less than the height of the energy barrier. We
also study the conditions under which the kink can pass freely through a
disordered medium. We introduce the concept of “effective translational
symmetry”. We compare our results with those from recent papers published in
this journal, where the Bogomol’nyi-Prasad-Sommerfield property is discussed.

##  I Introduction

Self-dual impurity models have gained significant interest in the recent
literature Adam et al. (2019a, b); Adam and Wereszczynski (2018); Adam et al.
(2019c). In Ref. Adam et al. (2019a), the $\phi^{4}$ model is coupled to an
impurity in a way that preserves one-half of the so-called Bogomol’nyi-Prasad-
Sommerfield (BPS) property. Although the impurity breaks the translational
invariance, Adam et al. argue that this symmetry is somehow restored in the
BPS sector, where the energy of the soliton-impurity solution does not depend
on their mutual distance. Furthermore, they discuss the existence of certain
“generalized translational symmetry” that provides a motion on moduli space
that transforms one BPS solution into another. This avenue of research aims to
formulate a model that allows for domain walls that do not get stuck in
impurities.

The existence of the generalized translational symmetry would imply that the
binding energy between the kink and the impurity is zero, and the kink may be
translated freely through the wire, although it changes its shape during this
translation. The impurity is coupled to different terms in the Lagrangian in a
non-conventional way. Therefore, soliton-impurity solutions with zero binding
energy will exist, which results in a zero static force between these solitons
and the impurity. A natural question is whether it is possible to realize such
a symmetry generalization experimentally.

In the system discussed in Refs. Adam et al. (2019a, b), infinitely many
energetically equivalent solutions describe a kink at an arbitrary distance
from the impurity. However, and this is a crucial point, when the kink moves,
it is no longer a BPS state. Adam et al. argue that at low speed, the domain
wall dynamics may be approximated by a sequence of BPS states (i.e., a
geodesic motion on the moduli space) describing an adiabatic motion. Even if
this model cannot be realized experimentally, the mere formulation of the
problem is an important step.

Another recent development is the explosion of papers dedicated to kinks with
power-law tails (see, for example, Refs. Christov et al. (2019); Campos and
Mohammadi (2021); Kumar et al. (2021); Khare and Saxena (2019, 2022a)).
González and Estrada-Sarlabous González and Estrada-Sarlabous (1989)
investigated the behavior of kink-antikink long-range interaction forces for
the first time. After that, several papers presented applications of the
original results Guerrero et al. (1997); González and Martín-Landrove (1994);
Mello et al. (1998); González and Martín-Landrove (2002). A whole “long-range”
movement is gaining strength throughout the physics community González and
Martín-Landrove (1994); González and de A Mello (1996); González et al.
(1996); Guerrero et al. (1997); Guerrero and González (1998); González et al.
(1998); Mello et al. (1998); González and Oliveira (1999); González and
Martín-Landrove (2002); Bazeia et al. (2018); Manton (2019); Christov et al.
(2019); d’Ornellas (2020); Andrade et al. (2020); Campos and Mohammadi (2021);
Kumar et al. (2021); Khare and Saxena (2019, 2022b, 2022a); Manton (2023);
Bazeia et al. (2023a, b); Andrade et al. (2023). Many researchers are
discovering new properties and applications of long-range solitons Campos and
Mohammadi (2021); Kumar et al. (2021); Khare and Saxena (2019, 2022a).

This article will show that long-range kinks can pass freely through an
impurity that will be effectively “transparent” if some conditions are
provided. Also, we will find the conditions under which the long-range kinks
can move through an utterly disordered medium as if there are no obstacles.
Finally, we will introduce the concept of “effective translational symmetry”
based on our findings. Thus, we will be combining ideas from these two current
developments: (i) how to design a wire where impurities are effectively
transparent for topological defects and (ii) kinks with long-range
interactions. The article is organized as follows. In Section II, we briefly
review the models of long-range kinks that will be the focus of this study. In
Section III, we demonstrate that the tunneling of kinks can be highly enhanced
by increasing their long-range character. We show in Section IV that long-
range kinks can travel even through a disordered distribution of impurities.
In Section V, we discuss the implications of our results in other physical
systems, whereas in Section VI, we discuss experimental and technological
applications. In Section VII, we give a final discussion of our results and
conclude in Section VIII.

##  II Preliminaries

###  II.1 Modeling long-range kinks in interaction with other fields

In this section, we introduce the field equations that will be studied in the
present article and briefly review their properties. Consider a real scalar
field $\phi(x,\,t)$, whose dynamic is described by the Lagrangian density

$\mathcal{L}=\frac{1}{2}\left(\frac{\partial\phi}{\partial
t}\right)^{2}-\frac{1}{2}\left(\frac{\partial\phi}{\partial
x}\right)^{2}-\mathcal{U}(\phi).$ (1)

The field equation will be

$\frac{\partial^{2}\phi}{\partial t^{2}}-\frac{\partial^{2}\phi}{\partial
x^{2}}=-\frac{\hbox{d}\mathcal{U}(\phi)}{\hbox{d}\phi}.$ (2)

The potential $\mathcal{U}(\phi)$ is an analytical function of $\phi$ having
three local extrema: two minima at $\phi_{1}$ and $\phi_{3}$, and a maximum at
$\phi_{2}$ with $\phi_{1}<\phi_{2}<\phi_{3}$, as shown in Fig. 1. If the
potential $\mathcal{U}(\phi)$ fulfills the condition
$\mathcal{U}(\phi_{1})=\mathcal{U}(\phi_{3})$, Eq. (2) has solutions of kink
and antikink types which can move with any constant velocity $0\leq|v|\leq 1$.

The existence of a difference between $\mathcal{U}(\phi_{1})$ and
$\mathcal{U}(\phi_{3})$ (i.e.,
$\Delta:=\mathcal{U}(\phi_{1})-\mathcal{U}(\phi_{3})\neq 0$) is equivalent to
the action of an effective external force on the solitons Gonzalez and Hołyst
(1987); Marín (2021). If $\Delta>0$ ($\Delta<0$), there is a force acting on
the kink in the negative (positive) direction of the $x$-axis. When the
initial state is that of an antikink, the existence of $\Delta>0$ ($\Delta<0$)
implies the action of a force on the antikink in the positive (negative)
direction of the $x$-axis.

For some of the phenomena investigated in the present work, we will consider
that $\mathcal{U}(\phi)$ has three minima and two maxima (see Fig. 2). Some of
our results are obtained for a sine-Gordon-like periodic potential
$\mathcal{U}(\phi)$ (see Fig. 3). All these potentials are known to support
kink and antikink solutions connecting the equilibrium states of the
underlying potential. We will also investigate the motion of kinks in
heterogeneous media, which can be pursued considering the following
modification of Eq. (2),

$\frac{\partial^{2}\phi}{\partial t^{2}}-\frac{\partial^{2}\phi}{\partial
x^{2}}+\frac{\hbox{d}\mathcal{U}(\phi)}{\hbox{d}\phi}=F(x).$ (3)

where $F(x)$ accounts for the presence of heterogeneous fields.

Figure 1: Bistable potentials supporting kink-antikink solutions connecting
(a) two degenerate stable states $\phi_{1}$ and $\phi_{3}$, and (b) one global
minimum $\phi_{1}$ (true-vacuum state) and one metastable state $\phi_{3}$
(false-vacuum state). In both cases, stable states are separated by a barrier
at $\phi=\phi_{2}$.

The behavior of the kink in the tails (i.e., the asymptotic behavior of kinks
at infinity) governs how the kink interacts with other structures and fields
placed at some distance from the kink. The most studied systems in the
literature, such as the $\phi^{4}$ and the sine-Gordon models, have the
property that the tails of kink have an exponential behavior at infinity. Such
exponential behavior is deeply rooted in the fact that the kink is connecting
_Morse critical points_ : in the Taylor expansion of the potential
$\mathcal{U}(\phi)$ in a neighborhood of the critical point, the first term
different from zero is of the second degree. Nevertheless, when the first term
different from zero in the Taylor expansion is of higher order, kink solutions
tend in a slower way to the critical values, giving rise to the phenomenon of
long-range interactions between kinks González and Estrada-Sarlabous (1989).

Figure 2: Potentials with three stable states, $\phi_{1}$, $\phi_{3}$, and
$\phi_{5}$, separated by two barriers at $\phi_{2}$ and $\phi_{4}$. In (a),
the minima are degenerated, whereas in (b), the minimum at $\phi=\phi_{3}$ has
more potential energy than the other two.

Figure 3: Periodic sine-Gordon-like potential with multiple degenerate
equilibrium states separated by multiple barriers.

####  II.1.1 Long-range kink-antikink interactions

Let the degree of the first term different from zero in the Taylor expansion
of $\mathcal{U}(\phi)$ in the neighborhood of the minima $\phi_{1}$ and
$\phi_{3}$ be $2n$ (with $n>1$). Then, for large values of $x$, it is known
that the solution fulfills the relation González and Estrada-Sarlabous (1989)

$\phi-\phi_{j}\sim x^{k},$ (4)

where $k=1/(1-n)$, with $j=1,3$. For $\Delta\neq 0$, Eq. (1) has stationary
states in the form of bell-shaped solutions, as shown in figure 4, which are
kink-antikink equilibrium states Gonzalez and Hołyst (1987). These states
exist due to a balance between the mutual attracting force between the kink
and the antikink, which, at a certain critical distance, is balanced by the
effective external force (generated by the fact that $\Delta\neq 0$) acting on
the kink and antikink in opposite directions Gonzalez and Hołyst (1987);
González and Estrada-Sarlabous (1989). Similar arguments explained the
formation and stability of similar bell-like solutions in one González et al.
(2006) and two spatial dimensions García-Ñustes et al. (2017); Marín (2018);
Castro-Montes et al. (2020); Marín (2021). The bell solutions are unstable
equilibrium states. However, the fact that this equilibrium consideration
exists (at least mathematically) allows calculating the distance $d$ between
the inflection points of the function $\phi(x)$ describing the bell, which can
be obtained from the formula Gonzalez and Hołyst (1987); González and Estrada-
Sarlabous (1989)

$d=\sqrt{2}\int_{\phi_{2}}^{\phi_{3}^{*}}\frac{\mbox{d}\phi}{\sqrt{\mathcal{U}(\phi)-\mathcal{U}(\phi_{3})}}.$
(5)

Equation (5) implicitly contains the function $\Delta(d)$, which measures the
attracting interaction force between the kink and the antikink.

Following Ref. González and Estrada-Sarlabous (1989), we will investigate
equation (5) to determine the properties of function $\Delta(d)$. Consider
equilibrium kink-antikink configurations with $\Delta\neq 0$ where the
involved solitons have the same asymptotic behavior for $\Delta\neq 0$. Thus,
we assume that the perturbation of the potential that produced the difference
$\Delta\neq 0$ holds without alterations of the order $2n$ of the minima.
Without loss of generality we put $\phi_{1}=0$ and $\phi_{3}=1$ 111This can be
obtained by making an affine transformation
$\phi\to(\phi-\phi_{1})/(\phi_{3}-\phi_{1})$). In these new coordinates we
have $\Delta_{1}=\mathcal{U}(\phi_{1})-\mathcal{U}(\phi_{3})$, with
$\Delta\sim\Delta_{1}$. and re-write equation (5) in the following form

$d=\sqrt{2}\int_{\phi_{2}}^{\phi_{3}^{*}}\frac{\mbox{d}\phi}{\sqrt{\mathcal{U}(\phi)-\Delta}}.$
(6)

After some algebra González and Estrada-Sarlabous (1989), one obtains

$d=\mathcal{O}(-\ln\Delta),\quad\text{for }n=1,$ (7a)
$d=\mathcal{O}\left(\Delta^{\frac{1-n}{2n}}\right),\quad\text{if }n\geq 2.$
(7b)

Figure 4: Antikink-kink equilibrium state forming a bell-like solution of Eq.
(1). The function $\Delta(d)$ defined in the text, where $d$ is the distance
separating the inflection points, gives a measure of the kink-antikink
interaction force.

For potentials $\mathcal{U}(\phi)$ in which the minima are parabolic, the
interaction force decreases exponentially with the increase of the distance.
On the other hand, when the minima of $\mathcal{U}(\phi)$ behave as
$\mathcal{U}(\phi)\sim(\phi-\phi_{i})^{2n}$, $n>1$, the interaction force
decreases with distance as a power law

$\Delta\sim d^{\frac{2n}{1-n}}.$ (8)

Kink-antikink interactions have a long-range character in potentials where
$n>1$. The interaction force behaves as $\Delta\sim d^{-2}$ when $n\to\infty$.

####  II.1.2 Long-range kink-kink interaction

When we have a potential with three minima or more, as shown in figures 2 and
3, kink-kink structures are possible (see Fig. 5). In the case of Fig. 2a, the
two kinks are repealing each other, and they will just move away. However, the
situation depicted in Fig. 2b leads to the existence of a kink-kink stationary
structure, which is stable. Define

$\Delta_{13}:=\mathcal{U}(\phi_{1})-\mathcal{U}(\phi_{3}),$ (9a)
$\Delta_{35}:=\mathcal{U}(\phi_{3})-\mathcal{U}(\phi_{5}).$ (9b)

We assume that $|\Delta|=|\Delta_{13}|=|\Delta_{35}|$. Let us call kink13 the
soliton that connects the vacua $\phi_{1}$ and $\phi_{3}$
($\phi(x)\to\phi_{3}$ as $x\to\infty$ and $\phi(x)\to\phi_{1}$ as
$x\to-\infty$), and let us call kink35 the soliton that connects the vacua
$\phi_{3}$ and $\phi_{5}$ ($\phi(x)\to\phi_{5}$ as $x\to\infty$ and
$\phi(x)\to\phi_{3}$ as $x\to-\infty$). The fact that $\Delta_{13}>0$ and
$\Delta_{35}<0$ means that there exists the equivalent of an external force
acting on the kink13 to the “right” and there exists an external force acting
on the kink35 to the “left”. Recall that kink13 and kink35 repel each other
with a force that depends on the distance between their centers of mass. There
is always a distance $d$ at which the two forces acting on kink13 and the two
acting on kink35 cancel out.

Figure 5: Kink-kink equilibrium state of Eq. (1) when the underlying potential
have three minima or more. The function $\Delta(d)$ gives a measure of the
kink-antikink interaction force.

Function $\Delta(d)$ will provide the law that governs the kink-kink
interactions. To calculate $\Delta(d)$, we need a modified potential
$\mathcal{U}(\phi)$ like the one shown in Fig. 2b. The stable stationary kink-
kink structure shown in Fig. 5 due to the potential from Fig. 2b will provide
the function $\Delta(d)$. The stationary kink-kink structure is a solution to
the equation

$\frac{\mbox{d}^{2}\phi}{\mbox{d}x^{2}}-\frac{\mbox{d}\mathcal{U}}{\mbox{d}\phi}=0.$
(10)

This expression can be integrated,

$\frac{1}{2}\left(\frac{\mbox{d}\phi}{\mbox{d}x}\right)^{2}-\mathcal{U}\left(\phi(x)\right)=\Delta.$
(11)

Thus, the equation that connects $\Delta$, $d$, and the shape of the kink-kink
stationary structure is the following,

$d=\sqrt{2}\int_{\phi_{3}}^{\phi_{4}}\frac{\mbox{d}\phi}{\sqrt{\mathcal{U}(\phi)+\left|\Delta\right|}}.$
(12)

After some algebra, we again get Eq. (7) for the distance $d$. Thus, in
potentials where $n>1$, the kink-kink repulsive force decreases with distance
as the power law given in Eq. (8). For the kink-kink repulsion, the force
behaves as $\Delta\sim d^{-2}$ when $n\to\infty$ as before. The coefficient is
larger for the kink-kink interaction than for the kink-antikink interaction.
It depends on the specific potential $\mathcal{U}(\phi)$. As d’Ornellas has
shown d’Ornellas (2020), the modified $\mathcal{U}(\phi)$ method can yield the
exact coefficient. So when we use the concrete potential in the integral (12),
we can obtain the correct coefficients. We have checked all these calculations
using the collective-coordinate calculation of the acceleration and the
modified potential techniques. Since our formulas are general, our
calculations do not have contradictions after considering other particular
cases Manton (2019); Christov et al. (2019); d’Ornellas (2020).

###  II.2 Kink tunneling

Figure 6: Bistable and monostable potential wells. These potentials are
created by the perturbation $F(x)$. They are not related to the potential
$\mathcal{U}(\phi)$.

Kink tunneling was discovered in the perturbed $\phi^{4}$ theory González et
al. (1999), but it can occur in other models. The main idea in kink tunneling
is that a kink is not a point particle Gonzalez and Hołyst (1987); González et
al. (1998). Consider the general heterogeneous Klein-Gordon equation (3) and
imagine a kink moving in a potential landscape created by $F(x)$. The field
$F(x)$ can create potential wells and barriers for the motion of the kink
González et al. (2007). For instance, when $F(x)$ has three zeros, there is
the possibility that the kink will be moving in a bistable potential, as shown
in Fig. 6a. Collective coordinate techniques do not work here González and de
A. Mello (1996). In some cases of $F(x)$, the collective coordinate techniques
can predict that the kink is trapped inside the potential well on the left of
Fig. 6a. However, the actual dynamic of the kink can be entirely different
González and Hołyst (1992); González et al. (1998); González and de A. Mello
(1996); Gonzalez and Hołyst (1987).

Imagine there is a stationary solution $\phi_{k}(x)$ of Eq. (3) in such a way
that the kink center of mass is at point $x=0$, which is usually a zero of
$F(x)$ (or a zero of $F(x)$ is close to $x=0$). To know if the point $x=0$ is
a stable equilibrium for the kink (or not), we must solve the complete
stability problem using the exact solution $\phi_{k}(x)$ to the exact
nonlinear partial differential equation (3). For this purpose, we express the
spatiotemporal dynamics as

$\phi(x,\,t)=\phi_{k}(x)+f(x)e^{\lambda t}.$ (13)

The complete spectral problem is

$\hat{L}f=\Gamma f,$ (14)

where

$\hat{L}=-\frac{\mbox{d}^{2}}{\mbox{d}x^{2}}+\left.\frac{\mbox{d}^{2}\mathcal{U}(\phi)}{\mbox{d}\phi^{2}}\right|_{\phi=\phi_{k}(x)},\quad\Gamma=-\lambda^{2}.$
(15)

The translational invariance is lost in Eq. (3), but the nodeless mode
$f_{o}(x)\sim\mbox{d}\phi_{k}(x)/\mbox{d}x$ is still the translational mode!
The eigenvalue $\Gamma_{0}=-\lambda^{2}\neq 0$ corresponding to the
translational mode, i.e., the Goldstone mode, is not zero anymore, but it will
decide the stability of the equilibrium position González and Hołyst (1992);
González and de A. Mello (1996); González et al. (1998); Castro-Montes et al.
(2020); Marín (2021).

There are situations where there are three equilibrium positions for the kink,
as in Fig. 6a. The kink will be trapped near the point $x=x_{1}$, not moving
to the right. Under certain conditions, although there is an $F(x)$ in Eq. (3)
creating obstacles for the motion of the kink, the kink does not feel the
finiteness of the barrier as in Fig. 6a. The potential $V(x)$ for the kink
will be like that shown in Fig. 6b. Thus, the equilibrium position $x=0$ is
now stable.

Imagine now that we construct a potential $V(x)$ similar to that shown in Fig.
6a with the difference that at point $x=x_{3}$ there is no local minimum, and
the function $V(x)$ will continue decreasing monotonously, as shown in Fig.
6c. If the condition for the stability of point $x=x_{2}$ is satisfied, then
the following phenomenon will occur. A kink at the bottom of the left
potential well (at point $x=x_{1}$) will move to the right. It will cross the
barrier, and it will escape the potential well despite the fact that its
center of mass has less energy than the barrier (see Fig. 6d). Of course, as
for any tunneling, the height and the width of the barrier play an important
role in this phenomenon.

##  III Long-range kink tunneling

In this section, we will show that the tunneling of long-range kinks can be
highly enhanced by increasing their long-range character. The nonlocal
properties of soliton dynamics González et al. (1998); Gonzalez and Hołyst
(1987); González et al. (1999) can be observed in all their extraordinary
splendor during the propagation of long-range kinks in heterogeneous systems.
As quantum tunneling is considered a non-equilibrium process, we will see that
long-range kink tunneling is also a non-equilibrium process.

We performed numerical simulations of the following equations,

$\partial_{tt}\phi-\partial_{xx}\phi+\frac{1}{2}\phi(\phi^{2}-1)^{2n-1}=F(x),$
(16)

and

$\partial_{tt}\phi-\partial_{xx}\phi+2\sin^{2n-1}\left(\frac{\phi}{2}\right)\cos\left(\frac{\phi}{2}\right)=F(x).$
(17)

The potential for Eq. (16) is $\mathcal{U}(\phi)=(1/8n)(\phi^{2}-1)^{2n}$ and
for Eq. (17) is $\mathcal{U}(\phi)=(2/n)\sin^{2n}(\phi/2)$. Notice that,
according to the results briefly reviewed in the previous section, for $n>1$,
both equations (16) and (17) possess long-range kink solutions. When $n=1$,
Eq. (16) will recover the $\phi^{4}$ equation, and Eq. (17) will be equivalent
to the sine-Gordon model.

Figure 7: A long-range kink tunneling through a potential barrier. Results
obtained from the numerical simulation of Eq. (17) with $n=2$ and the initial
conditions given in Eq. (18) under the heterogeneous force of Eq. (19) with
$x_{0}=2$, $\alpha=0.01$, and $\beta=0.007$.

Figure 7 shows a long-range kink tunneling through a potential barrier. This
simulation was performed using Eq. (17) with $n=2$. The initial conditions
were

$\phi(x,0)=2\arctan\left[\left(\frac{\sqrt{2}}{2}\right)(x+1)\right]+\pi,$
(18a) $\partial_{t}\phi(x,0)=0.$ (18b)

The heterogeneous field was

$F(x)=-\alpha\mbox{sech}^{2}(x+x_{0})+\beta\mbox{sech}^{2}(x)-\alpha\mbox{sech}^{2}(x-x_{0}),$
(19)

with $x_{0}=2$, $\alpha=0.01$ and $\beta=0.007$. The comparison between the
dynamics of a conventional $\phi^{4}$ kink and a long-range kink can be seen
in Fig. 8. Note that the $\phi^{4}$ kink remains trapped inside the potential,
while the long-range kink can tunnel through the barrier. The potential well
is defined between the points $x=-6$ and $x=0$. The barrier is defined between
the points $x=0$ and $x=6$. Figure 8a is built with the simulation of the
$\phi^{4}$ equation (16) with $n=1$. Figure 8b corresponds to Eq. (16) with
$n=9$. We use the same force of Eq. (19) for both simulations with
$\alpha=0.08$, $\beta=0.02$, and $x_{0}=6$. The initial position of the kinks
can be any point in the interval $-6<x<0$. The long-range kink will always
spontaneously escape through the barrier. The $\phi^{4}$ kink will remain
confined inside the potential for all the initial conditions. This computer
experiment was also done with strong damping, and the long-range kink was able
to cross the barrier even under these conditions. Thus, this very robust
phenomenon can occur even when there is a strong interaction with a
macroscopic environment.

Figure 8: Comparison between the dynamics of (a) a conventional ($n=1$) and
(b) a long-range ($n=9$) $\phi^{4}$ kink. Results obtained from the numerical
simulation of Eq. (16) under the heterogeneous force of Eq. (19) with
$\alpha=0.08$, $\beta=0.02$, and $x_{0}=6$.

Figure 9 shows the tunneling of a long-range kink through a wider barrier of
width $w=10$. Here we are simulating Eq. (17) with $n=9$, under the force of
Eq. (19) with $\alpha=0.1$, $\beta=0.02$, and $x_{0}=10$.

Figure 9: Numerical simulation of the tunneling of a long-range kink in Eq.
(17) with $n=9$, under the force of Eq. (19) with $\alpha=0.1$, $\beta=0.02$,
and $x_{0}=10$.

The difference between two long-range kinks with different extents is shown in
Fig. 10. A long-range kink with $n=2$ from Eq. (17) collides with an extensive
barrier. The long-range kink rebounds after colliding with the barrier. It is
not able to penetrate such a strong obstacle. Nevertheless, the $n=9$ long-
range kink (also from Eq. (17)) is tunneling through this very long wall, and
we can see it moving on the other side of the barrier.

The spatiotemporal dynamics of an $n=9$ long-range kink can be observed in
Fig. 11. As we will discuss later, conventional fluxons in Josephson junctions
would never penetrate a barrier like this. On the other hand, long-range
fluxons created in a Josephson junction with nonlocal electrodynamics could
cross the kink of barriers.

Figure 10: Position of the center of mass of two long-range kinks with $n=2$
(blue dashed lines) and $n=9$ (red solid line). Numerical simulations obtained
from Eq. (17).

Figure 11: Spatiotemporal dynamics of a long-range kink tunneling through a
barrier. Numerical simulations obtained from Eq. (17).

Figure 12 shows the motion of a long-range kink in a potential $V(x)$ with a
very long barrier. Conventional kinks, like those from the $\phi^{4}$ and
sine-Gordon models, and all the other long-range kinks with shorter extents,
cannot penetrate this barrier.

Figure 12: Numerical simulations of the dynamics of a long-range kink,
represented here with a red dot, moving in a potential with a wide barrier.
The kink can tunnel through the barrier.

The tunneling problem can be solved using energy and net-force considerations:
the net force acting on the long-range kink decides if the kink will move
through the barrier. Indeed, tunneling is a non-equilibrium process that
depends on the net force and how the long-range kink interacts with the force.

The general topological form of the force $F(x)$ for which the kink-stability
problem must be solved is schematized in Figure 13. It can be proved that the
stability condition of the translational mode for the equilibrium position
$x_{0}^{*}$ depends on the properties of the zones $F_{01}(x)$, $F_{m1}(x)$,
$F_{02}(x)$, and $F_{2m}(x)$. The general conditions for long-range kink
tunneling can be expressed as a competition between the negative zones,
$F_{01}(x)$ and $F_{02}(x)$, and the positive zones, $F_{m1}(x)$ and
$F_{m1}(x)$.

Figure 13: General topological form of the force $F(x)$ in the kink-stability
analysis.

The conditions for kink tunneling can be obtained using several methods, such
as stability theory, energy considerations, positive work conditions, and the
formalism of equilibrium-nonequilibrium transitions. All these methods yield
similar results. The most powerful long-range kink tunneling condition is the
following,

$E_{P}(x_{0})>E_{P}(x_{z}),$ (20)

where $x_{z}$ is any point in the interval $x_{0}<x_{z}<x_{2}$ (see Fig. 14).
The function $E_{P}(X)$ is defined as

$E_{P}(X)=-\int_{-\infty}^{+\infty}\mbox{d}x\,F(x)\phi_{s}(x-X).$ (21)

Figure 14: Typical structure of the heterogeneous force for observing long-
range kink tunneling.

Our findings show that a long-range kink of arbitrarily low energy can
penetrate any finite barrier if the kink extent is sufficiently long. In Fig.
14, we can see the typical $F(x)$ structure for observing long-range kink
tunneling. We remark that even when the total energy of the center of mass of
the kink is less than the height of the energy barrier, kink tunneling can
occur. Moreover, even if the total initial energy of the center of mass of the
kink is precisely zero, long-range kink tunneling can happen. Consider a
randomly disordered $F(x)$ as that shown in Fig. 15. If for every basic block
structure containing a negative minimum, a positive maximum, and another
negative minimum (as in Fig. 14), the tunneling condition of Eq. (20) is
satisfied, then the long-range kink can pass freely through the disordered
zone.

Figure 15: Illustration of a randomly disordered force with a distribution of
maxima and minima. Tunneling of long-range kinks is possible if the condition
of Eq. (20) is satisfied for every structure with a negative minimum, a
positive maximum, and a negative minimum as in Fig. 14.

##  IV Kink propagation in disordered media

Consider the heterogeneous kink-bearing nonlinear partial differential
equation (3), with

$F(x)=\sum_{i=1}^{N}\alpha_{i}\mbox{sech}^{2}\left[\beta_{i}(x-x_{i})\right],$
(22)

where $\alpha_{i}$, $\beta_{i}$ and $x_{i}$ can be random numbers. We assume
that these parameters do not take values that lead to instabilities of the
shape modes of kinks so that the kink will conserve its integrity as an
entity. Thus, Eq. (3) can describe the dynamics of a kink in a disordered
system. If the condition of Eq. (20) is satisfied for every basic block
structure containing a negative local minimum, a positive local maximum, and
another negative minimum (as in Fig. 14), the long-range kink can pass freely
through the disordered zone as if there are no obstacles there.

Figure 16: Numerical simulation of a long-range kink moving through an array
of impurities.

Figure 16 shows a long-range kink ($n=5$) moving through an array of
impurities. This kink moves almost with constant velocity. Remarkably, the
kink does not feel any obstacle in its way.

##  V Implications in other systems

###  V.1 The quantum long-range kink

Our results suggest that quantum tunneling of a kink can be enhanced if the
system parameters are changed so that it will support long-range kinks. First,
the long-range kink tunneling phenomenon would shorten the wall and reduce the
height of the barrier. In some situations, long-range kink tunneling can
suppress the barrier completely. Second, the kink mass will be smaller. Using
results from Ref. Khare and Saxena (2022a), we can calculate the kink mass and
obtain

$M_{k}(n)=\frac{2^{(4m+1)}\left[(2m)!\right]^{2}}{4(m+1)\left[(4m+1)!\right]},$
(23)

where $m=n-1$, $n=1,\,2,\,\ldots$. The kink mass decreases as $n$ increases,
so the longer the extent of the kink is, the more quantum the kink is.

###  V.2 Kac-Baker interactions

Another example of physical systems relevant to ours is the kink dynamics
governed by the sine-Gordon equation with the Kac-Baker long-range interaction
potential Baker (1961); Kac and Helfand (1963); Sarker and Krumhansl (1981);
Ishimori (1982); Pokrovsky and Virosztek (1983); Remoissenet and Flytzanis
(1985); Ferrer (1989). For instance, the Hamiltonian of a system of particles
of mass $m$ placed on a one-dimensional lattice can be the following

$H=\frac{1}{2}m\sum_{i}\phi_{i}^{2}+\sum_{i}\mathcal{U}(\phi_{i})+\frac{1}{2}\sum_{i\neq
j}V_{ij}\left(\phi_{i}-\phi_{j}\right)^{2},$ (24)

where each particle is lying on an on-site potential $\mathcal{U}(\phi_{i})$,
where $i$ and $j$ are the lattice points, and $V_{ij}$ is the Kac-Baker
potential,

$V_{ij}=\frac{c(1-r)}{2r}r^{|i-j|},$ (25)

where $c$ is a constant and $r$ is the number of neighboring interactions
Kittel and McEuen (2018).

Our investigation leads to the following conclusion. The extent of the kink,
hereby denoted as $L_{\mbox{\footnotesize K}}$, goes to infinity as the
parameter $n$, which defines the range of the interaction, is increased. This
can be seen in the following estimate for the kink extent,

$L_{\mbox{\footnotesize K}}^{2}=\frac{C(1+r)b^{2}}{\alpha(1-r)^{2}}.$ (26)

Indeed, as $r\to 1$, $L_{\mbox{\footnotesize K}}\to\infty$. These kinks will
behave as long-range kinks. Note that we do not have to change the sine-Gordon
substrate potential $\mathcal{U}(\phi)\sim(1-\cos\phi)$ to observe long-range
features.

###  V.3 Non-local Josephson electrodynamics

Long-range kinks exist in the context of non-local Josephson electrodynamics
Abdumalikov et al. (2006). Indeed, the long-range kinks studied in this
article are also solutions to non-local integrodifferential equations
governing the dynamics of long ultranarrow Josephson junctions like the
following

$\frac{\partial^{2}\phi}{\partial t^{2}}+\alpha\,\frac{\partial\phi}{\partial
t}-\frac{\partial}{\partial
x}\int\mbox{d}x^{\prime}Q(x,\,x^{\prime})\frac{\partial\phi}{\partial
x^{\prime}}+\sin\phi=F(x),$ (27)

written here in dimensionless form. The function $Q(x,\,x^{\prime})$ is the
non-locality kernel,

$Q(x)=\left(\frac{1}{\pi\lambda_{L}}\right)\mbox{K}_{0}\left(\frac{|x|}{\lambda_{L}}\right),$
(28)

where $\lambda_{L}$ is the London penetration depth, and $\mbox{K}_{0}$ is the
modified Bessel function. The kink solution is

$\phi(x,\tilde{\lambda}_{J})=2\arctan\left(\frac{x}{\tilde{\lambda}_{J}}\right)+\pi,$
(29)

which is a long-range kink. There are experimental situations where the
dynamics of fluxons in ultranarrow long Josephson junctions are described by
equation (27) Ivanchenko and Soboleva (1990); Grønbech-Jensen et al. (1990);
Grønbech-Jensen and Samuelsen (1995, 2002); Abdumalikov et al. (2006).
Sometimes, the interaction between coupled long Josephson junctions is so
strong that the nonlocality leads to phenomena where the fluxons behave like
Coulomb particles in the presence of an external field with a potential like
$V(r)\sim 1/r$ Ivanchenko and Soboleva (1990). We believe the fluxons can have
this behavior in stacks of ultrathin-long Josephson junctions.

We hypothesize that the strong interaction between the junctions in stacks of
ultranarrow Josephson junctions will lead to Ivanchenko’s non-local
electrodynamics. In this case, the interaction energy between the Josephson
vortices behaves as follows Ivanchenko and Soboleva (1990),

$\mathcal{E}_{\mbox{\footnotesize
int}}\simeq\frac{8\pi\sigma_{1}\sigma_{2}}{\gamma(\Delta x)},$ (30)

where $\Delta x$ is the distance between the vortices. The above result means
these topological objects will behave as our long-range kinks when $n\gg 1$.
The experimental and technological significance of this observation will be
discussed in the following section.

##  VI Applications: Experiments and technology

Our results have applications in spintronics, Josephson junction circuits, new
superconductivity technology, quantum information technology, and quantum
computing. In Ref. Brooke et al. (2001), the experimental macroscopic quantum
tunneling of a domain wall is reported. They discuss their results in the
context of the quantum tunneling escape rate

$\Gamma=f_{m}\,e^{-2b\sqrt{\frac{2mE_{0}}{\hbar^{2}}}},$ (31)

where $b$ is the width of the barrier, $E_{0}$ is the height of the barrier,
and $m$ is the domain wall mass. The potential is similar to that shown in
Fig. 7. When we carefully study the original data from Ref. Brooke et al.
(2001), we observe that the domain wall behaves like our long-range kink. They
report on a disordered magnetic system for which it is possible to adjust the
parameters of the system with a knob in the laboratory. The authors can
control the domain wall mass, the extent of the kink, and many other physical
quantities. When they change some fields, they can broaden the domain walls
(at some critical values of the parameters, the domain wall can fill the
entire system, and its mass approaches zero). If suitable experimental
conditions are introduced into $LiH_{0}F_{4}$, the domain wall dynamics can
pass from classical (large $m$) to quantum mechanical (small $m$). They can
observe the trapped domain walls at the fixed pinning centers (the disorder
can be ideal for pinning domain walls). The kink tunneling mechanism can serve
to depin the domain walls despite the existence of the disorder. As parameters
that increase the extent of the kink and decrease the domain wall mass are
changed appropriately, the measured tunneling rate has been observed to
increase. The general mobility of the domain wall through the disorder will
increase as the extent of the kink is increased exactly in the same way as our
long-range kinks.

The superconductors, the superinsulators, and the transitions that lead to
these states of matter have been active areas of research for the last several
decades Rotschild et al. (2006); Grimshaw et al. (2022); Arkhipov et al.
(2022). Nevertheless, some fundamental questions remain unsolved. In
particular, the nature of the insulating phase, which shows unconventional
transport properties, is unclear. One exciting scenario suggests that this
phase incorporates superconducting fluctuating islands embedded in an
insulating matrix. These phenomena lead to very exciting physics governed by
the interplay between disorder and superconductivity.

Josephson junction arrays, i.e., networks of superconducting islands weakly
coupled by tunnel junctions, are among the physical systems used to study the
aforementioned phenomena. One of the parameters controlling the
superconductor-insulator transition in Josephson arrays is the ratio
$E_{J}/E_{C}$, where $E_{J}$ is the Josephson coupling energy, and $E_{c}$
corresponds to the energy needed to add an extra charge to a neutral island.
Another relevant parameter is the dimensionless resistance $R/R_{Q}$, where
$R_{Q}=h/(2e)^{2}$. In addition, $L$ will be the length of the system. Several
works in the literature have studied Josephson arrays, where superconducting
grains (or islands) are connected through tunnel junctions. The dynamics of
excess cooper pairs in the array are described using charge kinks created by
the polarization of the grains Hermon et al. (1996).

Let us consider a ring-shaped 1-D array of serially coupled Josephson
junctions biased by an external flux through its center. Typically, the width
of the junction is $d\simeq 2\hbox{nm}$. Using results from Ref. Hermon et al.
(1996) in combination with our findings in this work, we can make the
following statement. If the kink extent is larger than the circumference of
the array, quantum phenomena such as persistent currents are predicted. As we
estimate that the characteristic kink extent in non-local Josephson arrays can
be of the order of several millimeters, this is a macroscopic quantum
phenomenon. This is just another manifestation of long-range quantum kinks
moving smoothly through different media. A long-range Cooper-pair charge kink
with a very large extent implies an extremely long-range Josephson coupling.
This could explain the recently measured experimental data where extremely
long-range Josephson coupling was observed Sanchez-Manzano et al. (2022a). In
some situations, the charge kink becomes lighter than the plasmons Hermon et
al. (1996). Recall that the mass of the kink depends on its extent. In that
case, the kink takes the role of the fundamental quantum of the system.
However, it loses its correspondence to the classical particle configuration.
In a sense, the charge kinks become “too quantum”. Thus, any classical theory
cannot describe the long-range charge kink (with a large kink extent and very
small mass). All the processes linked to the kink will be very macroscopic
quantum phenomena. The persistent current oscillations phenomenon manifests
the Aharonov-Bohm effect, where a charged particle encircles a flux tube
Aharonov and Bohm (1959). Weak spatial heterogeneity in the array, e.g., non-
identical grains of junctions or disordered grains, leads to persistent
current oscillations as a function of the external flux. The amplitude of the
persistent current of one charge kink decreases as the heterogeneity
increases. This “damping” of the current can be minimized if the charge kink
is long-range. Under certain conditions, the long-range Cooper pair charge
kink would not feel the heterogeneity.

Several experiments Haviland et al. (1989); Delsing et al. (1994); Hermon et
al. (1996); Haviland and Delsing (1996); Sambandamurthy et al. (2004, 2005);
Dubi et al. (2007); Baturina et al. (2007); Fistul et al. (2008) have shown
that in thin superconducting films, disorder creates a droplet-like electronic
texture (superconducting islands immersed into a standard matrix). Tuning
disorder can drive the system from superconducting to insulating behavior.
Consider an array of superconducting islands, each coupled to its nearest
neighbors by Josephson weak links. Several parameters are relevant for the
quantum phase transitions: the Josephson coupling energy of the two adjacent
superconducting islands ($E_{J}$), the charging energy ($E_{C}$), the length
of the sample $L$, and the dimensionless conductance

$g=\frac{2\pi\hbar}{e^{2}R}.$ (32)

The inequalities $E_{J}\gg E_{C}$ and $g\gg 1$ lead to the insulator-
superconductor transition. The activation of Cooper pair charge kinks mediates
the charge transfer. According to some experimental and theoretical evidence
Haviland et al. (1989); Delsing et al. (1994); Hermon et al. (1996); Haviland
and Delsing (1996); Sambandamurthy et al. (2004, 2005); Dubi et al. (2007);
Baturina et al. (2007); Fistul et al. (2008) and references quoted therein and
our results in this work, the long-range Cooper-pair charge kink can spread
over several superconducting islands, and the long-range Cooper pair kinks can
be the ultimate charge carriers. Thus, they can lead to a transition to
superconductivity. The extent of the kink is proportional to the screening
length in the system. In some cases, they are equal. At the same time, much of
the picture we have about these materials holds, provided the screening length
in the system exceeds the sample size. So, to satisfy these conditions, the
extent of the kink must exceed the sample size. Hence, this must be a very
long-range Cooper-pair charge kink. The Cooper pair is transferred across the
effective junction in the form of a charge kink spread over the array Mooij et
al. (1990). Only a long-range kink can be spread over the whole array. So, the
Cooper-pair charge kink must be a long-range kink.

The conditions for the superconducting and insulator states can be formulated
in terms of the super propagation or localization of the Cooper-pair charge
kinks. If the disorder exponentially localizes the charge kinks, we observe
the superinsulator state Baturina et al. (2007). On the other hand, if the
charge kinks undergo a regime of super propagation, where the kinks do not
feel the disorder, then we can observe the superconducting state. Consider a
Josephson junction array as that presented in Ref. Hermon et al. (1996). Our
prediction is the following. For $\pi(E_{J}/E_{C})^{1/2}=2.5$, we start with a
kink with a very small extent ($L_{SE}\gtrapprox 0$). Then, we keep increasing
the extent of the kink until $L_{SE}\approx L$. This process will lead to an
insulator-superconductor transition. We estimate that this will hold even in
the presence of disorder and dissipation. The inverse process will lead to a
superconductor-insulator transition.

##  VII Discussion

Our results show that the solitary waves are long-range kinks when one of the
following conditions is fulfilled:

  * •

The potential $\mathcal{U}(\phi)$ possesses the property
$\left.\mbox{d}^{2}\mathcal{U}/\mbox{d}\phi^{2}\right|_{\phi=\phi_{1,3}}\simeq
0$.

  * •

Nonlocal electrodynamics rules in the physical system.

  * •

The interaction between the elements of the lattice is described by a Kac-
Baker potential.

  * •

The extent of the soliton is of the same order of magnitude as the length of
the system or sample.

This means these kink-like objects can move relatively easily, even in a
medium with impurities and heterogeneity. For example, if these kinks are
charged objects, this enhanced propagation can lead to an enormous increase in
conduction.

DNA kinks have been shown to exhibit long-range interactions Mello et al.
(1998); González and Martín-Landrove (1994, 2002). Thus, they can have
increased mobility. This effect can have biological relevance.

There are magnetic systems where long-range kinks have been observed, and
recently, long-range skyrmions have been studied Kosevich et al. (1990); Shnir
(2021). In Ref. Wallraff et al. (2003), the authors report the experimental
observation of macroscopic quantum tunneling of a fluxon in a long Josephson
junction. These findings have been controversial Grønbech-Jensen and Cirillo
(2004); Blackburn et al. (2016). Our results show that a normal fluxon can
tunnel through a barrier with a width of the order of
$20\mu\mbox{m}$-$100\mu\mbox{m}$. On the other hand, long-range fluxons
supported on the non-local electrodynamics of some ultranarrow Josephson
junctions can cross much wider barriers ($L_{B}\simeq 5\mbox{mm}$).

Several experimental works have reported the giant enhancement of the quantum
tunneling rate in stacks of Josephson junctions Jin et al. (2006); Koyama and
Machida (2008). Some researchers have invoked the presence of non-local
electrodynamics. However, these works have failed to use the fact that these
fluxons are long-range kinks. Our findings indicate that there is still much
more room for improving the macroscopic tunneling rate in these stacks of
Josephson junctions. Ordinary point particles and kinks cannot cross very wide
barriers. The fact that standard quantum tunneling can only be observed in
systems with very narrow barriers has been used to discard quantum effects as
an explanation of some transport measurements in charge density wave materials
Rojo-Bravo et al. (2016). The argument is due to experimental evidence, where
quantum current oscillations have been observed in samples as long as several
millimeters or even a couple of centimeters. Our results demonstrate that
long-range quantum kinks can tunnel through very wide barriers or very long
structures filled with obstacles.

Charged Cooper-pair kinks have been subject to intense experimental study. If
we can engineer long-range Cooper-pair kinks, this could help the development
of new materials for superconducting technology. In addition, the recently
reported extremely long-range Josephson coupling Sanchez-Manzano et al.
(2022b) could indicate that long-range kinks can play an important role in
modern quantum technology.

The exact condition for an effective translational symmetry would be

$\int_{-\infty}^{\infty}\mbox{d}x\,F(x)\phi_{s}(x-x_{0})=\mbox{constant}.$
(33)

However, condition (33) does not need to be satisfied exactly. That is, exact
tuning is not necessary. Condition (33) can also be replaced by

$\frac{\mbox{d}E_{p}(x_{0})}{\mbox{d}x_{0}}\simeq 0.$ (34)

When the long-range kink tunneling condition is satisfied for every structure
with a negative minimum, a positive maximum, and another negative minimum in
$F(x)$, the kink behaves as if there are no obstacles in its way. The values
of the parameters of $F(x)$ for which this phenomenon can occur form a
continuum set of points in the parameter space. This phenomenon is robust.

Effective translational symmetry means that flat potential energy is not
required. The function $F(x)$ in Fig. 16 is an example that describes a
completely disordered medium. The tunneling condition is satisfied for every
region containing two minimums and one maximum. However, the exact condition
(33) does not hold. Nevertheless, the long-range kink moves approximately with
constant velocity from point $x=25$ to point $x=100$, an interval where the
medium is completely disordered. Figure 17 shows the effective potential
energy for a long-range kink with $n\gg 1$ moving in a medium with five
localized impurities located at points $x=-6$, $x=-2$, $x=0$, $x=2$, and
$x=6$. This is a striking example of an almost exactly-satisfied condition for
“effective translational symmetry”.

Figure 17: Instance of an effective translational symmetry. Potential energy
for a long-range kink with $n\gg 1$ in a medium with five localized
impurities. Here, $x_{0}$ is the position of the kink.

Combining the BPS property with the long-range property can lead to perfect
long-range kink propagation. For instance, consider a real system where the
BPS condition is approximately satisfied. Consider also that the system
supports the existence of long-range kinks. Then, the kink may move freely
through the medium despite the impurities. Moreover, the low-velocity
adiabatic motion condition can be dropped. We have obtained similar results
for parametrically perturbed equations like the following González et al.
(2007)

$\frac{\partial^{2}\phi}{\partial t^{2}}-\frac{\partial^{2}\phi}{\partial
x^{2}}+\left[1+g(x)\right]\frac{\mbox{d}\mathcal{U}(\phi)}{\mbox{d}\phi}=0.$
(35)

##  VIII Conclusions

We have investigated very general perturbed Klein-Gordon equations that can
support kink-like solutions. We have found general conditions for the
existence of long-range kinks. External fields and heterogeneity in the
parameters of the system can create potential wells and barriers for the
motion of the kinks. Under certain conditions, the kink can tunnel through the
barrier even when the initial energy of the kink is less than the height of
the energy barrier. When the moving object is a long-range kink, it can cross
very wide barriers. Some long-range kinks can move in a completely disordered
medium as if there are no obstacles there.

The phenomena reported in this work can be observed and used to design new
technologies in several physical systems, like conducting polymers, charge
density waves, long Josephson junctions, Josephson junction arrays, DNA,
proteins, superconductors, superinsulators, topological superconductors, and
insulators, among others.

###### Acknowledgements.

This work started when J.F.M. was a lecturer and researcher at Laboratoire
Gulliver, École Supérieure de Physique et de Chimie Industrielles (ESPCI-PSL),
Paris, France. J.F.M. thanks Olivier Dauchot, Alexandre Allauzen, and
Laboratoire Gulliver for their hospitality while part of this research was
done.

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