windows is approximately the same for all values of $A$. We can find an
approximate expression for the location of the windows using a simple model.
In [Campbell, Schonfeld e Wingate 1983], the authors assumed the wobbling
amplitude is small and, therefore, the relation between the wobbling amplitude
before and after the collision can be truncated at the first order. They also
assumed that collisions possess time-reversal invariance. Under these
assumptions, they were able to derive the following relation between the
amplitude before and after the collision

$S^{\prime}=-\frac{\rho}{\rho^{*}}(1+i\beta)S+i\beta S^{*}+\rho,$ (12)

where $\beta$ is a real constant and $\rho$ a complex one. Moreover, they
found that $\beta\simeq 0$ for most systems and obtained the relation

$S^{\prime}=-\frac{\rho}{\rho^{*}}S+\rho.$ (13)

Now, we can substitute eq. (10) in the relation above. It yields

$S^{\prime}=-A\exp\left[i\left(\sqrt{1-v_{i}^{2}}\frac{\omega_{D}x_{0}}{v_{i}}+\theta_{0}+2\theta_{\rho}\right)\right]+|\rho|\exp(i\theta_{\rho}),$
(14)

where $\theta_{\rho}$ is the phase of $\rho$. If the radiation is small, the
sum of the vibrational and translational energy is approximately constant.
Therefore, the maximum value of the translational energy occurs when the norm
of the vibrational amplitude has the smallest value. The norm of $S^{\prime}$
will be a minimum when the phase of the first and second terms are aligned
because, in this case, the final norm is the difference between the norms of
the two separate terms. The alignment occurs when

$\sqrt{1-v_{i}^{2}}\frac{\omega_{D}x_{0}}{v_{i}}+\theta_{0}+2\theta_{\rho}=2\pi
n+\theta_{\rho},$ (15)

for any integer $n$. We can solve this equation for $v_{i}$ to obtain

$v_{n}=\frac{\omega_{D}x_{0}}{\sqrt{(2\pi
n-\delta)^{2}+\omega_{D}^{2}x_{0}^{2}}},$ (16)

where $\delta\equiv\theta_{\rho}+\theta_{0}$.

The center of two bounce windows as a function of $n$ is shown in Fig. 10 also
for $R=1.0$. To compare the numerical values of the position with our
approximate expression above, we fit the points by a curve of the type

$v_{n}=\frac{a}{\sqrt{(2\pi n-b)^{2}+a^{2}}},$ (17)

for two constants $a$ and $b$. We find $a=9.22$, which should be compared with
the theoretical value $\omega_{D}x_{0}\simeq 11.80$. The agreement is
reasonable but not remarkable. This deviation is expected because we made many
approximations to derive eq. (16). Still, this analysis gives an approximate
picture of how the wobbling phase affects the collision.

Figure 11: Maximum value of the energy density in spacetime (a) as a function
of $v_{i}$ for $R=1.0$ and $A=0$ (b) as a function $R$ for $v_{i}=0.2$ and
$A=0$. (c) and (d) Same as before with $A=0.5$. The grey areas correspond to
resonance windows. Source: Results obtained by the author in [Campos e
Mohammadi 2021].

It is possible to study the evolution of the energy density in spacetime
during a kink-antikink collision. The energy density is defined as $e=k+u+p$
where $k=\frac{1}{2}(\partial_{t}\phi)^{2}$ is the kinetic term,
$u=\frac{1}{2}(\partial_{x}\phi)^{2}$ is the gradient term and $p=V(\phi)$ is
the potential term. Interestingly, in some recent works, it was found that the
fractal structure of kink-antikink collisions can be found by measuring the
maximum value of the energy density for the whole spacetime evolution and
plotting it as a function of the parameters [Yan et al. 2020, Gani, Marjaneh e
Saadatmand 2019]. The surprising result was that the maximum energy density is
continuous in the regions where there is separation and is chaotic in most
regions where there is annihilation.

We investigated if the same behavior could be found in kink-antikink
collisions of the double sine-Gordon model. The result is shown in Fig. 11. In
the figure, we plot the maximum values of the energy densities $e$, $k$, $u$
and $v$ as a function of $v_{i}$ for $R=1.0$ and $A=0.0$ and as a function $R$
for $v_{i}=0.2$ and $A=0.0$. Then, we repeat the procedure for $A=0.5$. In all
cases, we mark the regions where there are resonance windows, including the
one-bounce resonance windows for the case with wobbling. We find that the
dependence of the maximum value of the energy density with the parameters is
smooth in those regions. On the other hand, the behavior can be chaotic
outside these regions. This result shows that there are other ways to identify
resonant behavior besides the computation of the final velocity.

Figure 12: Wobbling frequency of the kinks after the collision in the frame
where the kink is at rest. The dashed line corresponds to the shape mode
frequency $\omega_{D}$ obtained from the stability equation. We fix $R=1.0$
and set $A=0.0$ in (a) and $A=0.1$ in (b). The colors indicate the number of
bounces during the collision according to the scheme used in previous figures.
Source: Results obtained by the author in [Campos e Mohammadi 2021].

To complete our analysis, we compute the wobbling frequency of the kinks after
the collision. We expect the kink to vibrate at the frequency $\omega_{D}$
obtained from the stability equation. However, the kinks are not static after
the collision, they are actually moving. Therefore, the vibration should be
Lorentz contracted according to eq. (9). This is precisely what we found in
our simulations. The result is shown in Fig. 12. We measure the vibration
frequency by measuring the field’s value at the position where the wobbling
amplitude is expected to be a maximum from the shape mode’s profile
$\psi_{D}$. Then, we subtract the theoretical value of the kink’s profile
$\phi_{k}$ at this position from the field’s value. We obtain the frequency of
this time series as the one which gives the largest amplitude of the Fourier
transform. After correcting for the Lorentz contraction, we obtain the
excellent agreement shown in the figure.

In the next section, we will summarize our results and conclusions and also
discuss ideas for future works.

### 4 Conclusion

We considered the double sine-Gordon model. This model is equivalent to the
sine-Gordon model for $R=0$ and in the limit $R\to\infty$. It possesses a kink
that can be described as a superposition of two sine-Gordon kinks. The kinks
always have a vibrational mode. The frequency of this mode $\omega_{D}$
decreases from the threshold value of unity to zero as $R$ increases. For
$\omega_{D}>0.5$, the vibration amplitude of a kink initially decays due to
the coupling of the first harmonic of the wobbling frequency with continuum
modes. This occurs when $R$ is not too large. As the wobbling frequency
decreases, the decay is much slower and occurs via higher harmonics.

The usual kink-antikink collision for the double sine-Gordon model exhibits
resonance windows for intermediate values of the parameter $R$ with the usual
fractal structure. However, the windows are absent near the integrable limits,
presumably due to the decoupling of the modes. Moreover, we were able to
confirm the non-monotonic behavior of the critical velocity found in [Gani et
al. 2018].

Considering collisions between wobbling kinks, we found that the collisions
exhibit one-bounce windows similarly to the $\phi^{4}$ model. These windows
appear in two ways. In the first one, they appear when the crossing curve
splits. This occurs at a point where the wobbling is out of phase at the
moment of the collision. In the second one, they appear at a point outside the
crossing region. This occurs when the wobbling is in phase at the moment of
the collision. The gradual appearance of the windows can be clearly seen in
our figures. Mapping how the region of one-bounce resonance windows moves as
the parameter $R$ is varied, we observe a spine structure. This structure is
even more intricate for large values of the wobbling amplitude.

A significant result was that, when wobbling was included, we were able to
find resonance windows much closer to the integrable limits than before, where
there was no wobbling. This suggests that the system possesses a hidden
resonant behavior when the wobbling is turned off.

We observed that the center of one-bounce windows did not depend on the
wobbling amplitude. Moreover, we were able to find an approximate expression
for the position of these windows. This was accomplished by expanding the
wobbling amplitude after the collision up to linear order on the wobbling
amplitude before the collision.

We showed that one could distinguish separation from annihilation by measuring
how the maximum value of the energy densities varies with the system’s
parameters. Finally, we found consistently that, after the collision, the kink
also vibrates at the frequency obtained from the stability analysis.

Interesting continuations to this work include the reproduction of the
wobbling kink effects using the effective collective coordinates system. Also,
it is important to understand why the resonance windows disappear near the
integrable regimes. Perhaps this could be achieved by a perturbative analysis.

### 5 Appendix: Numerical method

We simulate the system in a box with size $-100.0<x<100.0$. We apply the
method of lines to integrate the equations of motion. The space is discretized
with $N=2048$ points using periodic boundary conditions. Spatial derivatives
are computed using a Fourier Spectral Method [Trefethen 2000]. The resulting
set of ordinary differential equations is integrated using a fifth-order
Runge-Kutta method with adaptive step size and error control [Dormand e Prince
1980]. It is implemented by the odeint library in C++ [Ahnert e Mulansky
2011]. The final time of the simulation is $t=1400.0$, except for frequency
measurements, which need a more time-consuming simulation. The plots of the
final velocity as a function of the initial one are computed with separation
between points $\Delta v_{i}=10^{-5}$ and are truncated for small velocities.
To avoid that radiation cross the boundaries and interact again with the
system, we include damping for $x<-80$ and $x>80$ proportional to a bump
function with maximum value $5$ and exactly zero outside this region [Christov
et al. 2021]. The shape mode’s profile and frequency are computed using the
NDEigensystem method in Mathematica.

## 5 FERMION-KINK SYSTEMS

### 1 Definition and motivation

The second half of this thesis consists of the study of fermion-kink systems,
beginning in this section. First, let us define what a fermion-defect system
is. It is a system that describes the interaction of a topological defect and
a fermionic field. The two sets of fields are coupled via some interaction
term. In two spacetime dimensions we have fermion-kink systems. The canonical
example is the $\phi^{4}$ model coupled to a Dirac field via a Yukawa term

$\mathcal{L}=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{1}{2}(\phi^{2}-1)^{2}+i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-g\phi\bar{\psi}\psi.$
(1)

This is one of the most straightforward systems containing all these
ingredients and will be studied in section 3 and section 6. We will see that,
in general, the kink will either bind to the fermion or scatter it.

So, why do we need to study these systems? The simplest explanation is that
fermions are ubiquitous in nature and, therefore, it is likely that there
exist topological defects that interact with them. The experimental
realization of fermion-kink systems occurs in polymers, such as polyacetylene.
A revision of these results can be found in [Niemi e Semenoff 1986]. People
have also theoretically investigated interactions of fermions with vortices
[Jackiw e Rossi 1981], skyrmions [Hiller e Jordan 1986, Perapechka, Sawado e
Shnir 2018, Perapechka e Shnir 2019], solitons in a nonlinear sigma model
[Kahana, Ripka e Soni 1984], monopoles [Yamagishi 1983, Yamagishi 1983,
Rubakov 1982], and planar defects [Bazeia e Mohammadi 2018]. Thus, we imagine
that experimental realization of those systems could be found in the future.
Another theoretical application of fermion-kink systems appears in cosmology.
In some higher dimensional cosmological models, domain walls, which are the
extension of kinks to higher dimensions, present a plausible mechanism to
localize fermions in the extra dimensions [Rubakov e Shaposhnikov 1983,
Randjbar-Daemi e Shaposhnikov 2000, Koley e Kar 2005, Melfo, Pantoja e Tempo
2006].

### 2 Literature revision

One of the seminal works in the field is the reference [Jackiw e Rebbi 1976].
There, the authors observed that when fermions bind to topological defects,
they possess a zero-energy solution with charge conjugation symmetry. They
showed that this implies the defects have a fermion number of $\pm 1/2$ when
the system is quantized. This property is fascinating and is observed in
polymers due to the fermion-kink interaction [Niemi e Semenoff 1986]. Later,
it was shown that topological defects interacting with fermions could have any
fractional fermionic charge [Goldstone e Wilczek 1981, Alonso-Izquierdo et al.
2019]. Another attractive property of fermion-defect interaction is that it
can lead to domain wall superconductivity [Vachaspati 2006] and string
superconductivity [Vilenkin e Shellard 2000].

Fermion-kink systems have been investigated for several types of potentials,
and interaction terms [Shahkarami, Mohammadi e Gousheh 2011, Gousheh,
Mohammadi e Shahkarami 2013, Gousheh, Mohammadi e Shahkarami 2014, Bazeia e
Mohammadi 2017, Bazeia, Mohammadi e Moreira 2019]. An important property of
the interaction is that it is responsible for a nontrivial Casimir energy. In
those works, the energy and eigenfunctions of the fermions are found
numerically after the simplification that there is no back-reaction of the
fermion to the kink. This is a good approximation when the coupling between
the kink and fermion is weak. Recently, the interaction between kinks with
inner structure and fermions has been investigated in [Bazeia, Mohammadi e
Moreira 2021]. Interestingly, this study has applications in nanometric
electronic devices.

The system given by eq. (1) is very important because it has analytical
solutions when the back-reaction is ignored. These can be found, for instance,
in [Chu e Vachaspati 2008, Charmchi e Gousheh 2014, Charmchi e Gousheh 2014]
for both massless and massive Dirac fields. This model will be studied in
section 6 considering a wobbling kink, which is a kink with the shape mode
excited, instead of a static one. This main section is based in the following
original study the we developed [Campos e Mohammadi 2021].

Instead of studying a single kink, it is possible to compute the energy of
fermionic bound states considering a frozen kink-antikink configuration as the
background. This was done numerically in [Chu e Vachaspati 2008]. The authors
found that the energy spectrum approaches two degenerate copies of an isolated
kink configuration spectrum as the distance between the kinks increases. This
result is expected for quantum mechanical systems. Similar results were found
for the sine-Gordon potential in [Brihaye e Delsate 2008].

If one considers strong coupling between the fermion and the kink, it is
necessary to include the back-reaction of the fermion to the kink. In this
case, the problem becomes complicated and can only be solved numerically, even
for a single kink. Self-consistent solutions have been found in [Shahkarami e
Gousheh 2011, Amado e Mohammadi 2017, Klimashonok, Perapechka e Shnir 2019,
Perapechka e Shnir 2020] for kinks and in [Perapechka, Sawado e Shnir 2018,
Perapechka e Shnir 2019] for baby-skyrmions. The back-reaction can have many
interesting effects, such as creating kink-antikink pairs.

Finally, one can consider again the model (1) without back-reaction, but with
a dynamical kink-antikink configuration. The dynamical configuration can be a
kink-antikink collision as done in [Gibbons, Maeda e Takamizu 2007, Saffin e
Tranberg 2007]. The authors found that a fermion localized at the kink or the
antikink can be transferred to the other defect due to the collision. We will
consider this type of system for a slightly different model in section 7,
which is based in the following original study that we developed [Campos e
Mohammadi 2020].

### 3 Some theoretical aspects

Our goal is to study fermion-kink systems. We will start considering a
Lagrangian slightly more general than eq. (1) in two spacetime dimensions.
This is

$\mathcal{L}=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-V(\phi)+i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-g\phi\bar{\psi}\psi.$
(2)

To solve the system, we start by writing the equations of motion111The
classical limit of a Fermi field is a Grassmann number, which is
anticommuting. However, it is justifiable to consider them as c-numbers in the
equations of motion, as shown in [Rajaraman 1982].

$\displaystyle\partial_{t}^{2}\phi-\partial_{x}^{2}\phi+V^{\prime}(\phi)+g\bar{\psi}\psi=0,$
(3) $\displaystyle i\gamma^{\mu}\partial_{\mu}\psi-g\phi\psi=0.$ (4)

We will use the approximation that there is no back-reaction of the kink to
the fermion. This is done by setting the term $g\bar{\psi}\psi$ to zero in eq.
(3). This approximation is adequate when the coupling $g$ is small compared to
kink’s mass. The limit of a massive kink can be obtained by recovering the
units in the potential and setting them to large values. Considering the
approximation, the scalar field equation decouples and can be easily solved.
Furthermore, we will consider that $V(\phi)$ has more than a single vacuum. In
this case, the scalar field equation possesses a kink solution $\phi_{k}(x)$.

The next step is to solve the Dirac equation considering the kink solution as
the scalar field configuration. First, we choose the representation of the
gamma matrices as $\gamma^{0}=\sigma_{1}$ and $\gamma^{1}=i\sigma_{3}$. Then,
we use the ansatz
$\psi=e^{-iE_{n}t}\psi_{n}=e^{-iE_{n}t}(\psi_{n,+},\psi_{n,-})^{T}$. This
leads to

$\displaystyle E_{n}\psi_{n,+}+\partial_{x}\psi_{n,-}-g\phi_{k}\psi_{n,-}=0,$
(5) $\displaystyle
E_{n}\psi_{n,-}-\partial_{x}\psi_{n,+}-g\phi_{k}\psi_{n,+}=0.$ (6)

We can solve for each component, obtaining the following Schrödinger-like
equations which are uncoupled but have the same eigenvalues

$-\partial_{x}^{2}\psi_{n,\pm}+V_{\pm}\psi_{n,\pm}=E_{n}^{2}\psi_{n,\pm}.$ (7)

The obtained effective potentials are
$V_{\pm}=g(g\phi_{k}^{2}\mp\partial_{x}\phi_{k})$. As the two Hamiltonians of
the effective equations above are supersymmetric partners [Cooper, Khare e
Sukhatme 1995], the two equations are guaranteed to have the same set of
eigenvalues, except for the zero mode, which has vanishing energy. Moreover,
the corresponding eigenstates are guaranteed to solve the coupled first-order
equations as well, after some normalization.

There are a few important properties of eq. (7) and the original ones (5) and
(6). First, setting the fermion energy to zero ($E_{0}=0$), the original
equations are easily integrated, and one obtains the zero mode $\psi_{0}$.
However, due to the topological nature of the scalar field, one of the
components is set to zero because it is not normalizable. Accordingly, the
nonzero component of the zero mode is also a solution of the corresponding
component of eq. (7). Depending on the depth of the potential $V_{\pm}(x)$,
this set of equations can have more bound states. They appear in charge
conjugated pairs with energies $\pm E_{n}$. Moreover, the threshold values of
the effective potentials $V_{\pm}(x)$, which are the limits where both
negative and positive values of $x$ are large, are finite. This implies that
the equations also allow scattering states.

To make these statements more concrete, we consider the $\phi^{4}$ model as an
example. In this case $V(\phi)=\frac{1}{2}(\phi^{2}-1)^{2}$ and
$\phi_{k}=\tanh(x)$. This leads to effective potentials with Pöschl-Teller
form

$V_{\pm}=g^{2}-\frac{g(g\pm 1)}{\cosh^{2}(x)}.$ (8)

The corresponding effective Hamiltonian with these potentials have analytical
solutions which can be found, for instance, in references [Chu e Vachaspati
2008, Charmchi e Gousheh 2014]. The expressions for all the bound and
scattering states are somewhat technical, but they will be listed in section 4
for completeness. They can be written in terms of the Hypergeometric function.

Let us analyze a few properties of this model. The threshold values, in this
case, are equal to $g^{2}$ on both sides. This means that the scattering
states start at the threshold energies $E_{\text{th}}^{2}=g^{2}$, or
$E_{\text{th}}=\pm g$. It is instructive to calculate the zero mode for this
model. Setting $E_{0}=0$ in eqs. (5) and (6), we find

$\displaystyle\partial_{x}\psi_{0,-}-g\tanh(x)\psi_{0,-}=0,$ (9)
$\displaystyle-\partial_{x}\psi_{0,+}-g\tanh(x)\psi_{0,+}=0.$ (10)

These have solutions $\psi_{0,\pm}=C_{\pm}\cosh^{\mp g}(x)$. Clearly,
$\psi_{0,-}$ is not normalizable and, therefore, we set $C_{-}=0$. For $g>1$,
there are other bound states. Considering the ones with non-negative energy,
The total number of them is equal to the largest integer $n$ such that $g>n$.

Figure 1: (upper left) Effective potentials $V_{\pm}(x)$ with $g=2.0$. (upper
right) Fermion spectrum as a function of the Yukawa coupling $g$. (lower left)
Fermion zero mode and (lower right) fermion first excited state wavefunctions
considering $g=2.0$. We are considering a $\phi^{4}$ kink as the background
scalar field. Source: The author (2022).

The effective potentials, energy spectrum, and a few bound state wavefunctions
are shown in Fig. 1. Looking at the spectrum, one can see how new bound states
emerge from the threshold states as $g$ increases. Fixing $g=2.0$, we see that
the effective potential $V_{-}$ is shallower than $V_{+}$. Hence, $V_{+}$ has
one more energy eigenvalue than $V_{-}$, which corresponds to the zero mode.
For these two potentials, the wavefunction of the zero mode is shown in the
figure. We also plotted the wavefunction of the first excited state with
positive energy. One intriguing property that can be observed in the figure is
that the wavefunctions $\psi_{n,+}$ and $\psi_{n,-}$ have definite and
opposite parity. Moreover, the parity $\psi_{n,\pm}$ is inverted when $n$
increases by one because the effective potentials are even.

Before finishing this main section, we would like to discuss a condition under
which the background approximation is exact. Writing
$\psi=(u_{1}+iu_{2},v_{1}+iv_{2})^{T}$, eq. (4) becomes

$\displaystyle\dot{u}_{1}$ $\displaystyle=-\partial_{x}v_{2}+g\phi v_{2},$
(11) $\displaystyle\dot{u}_{2}$ $\displaystyle=\partial_{x}v_{1}-g\phi v_{1},$
(12) $\displaystyle\dot{v}_{1}$ $\displaystyle=\partial_{x}u_{2}+g\phi u_{2},$
(13) $\displaystyle\dot{v}_{2}$ $\displaystyle=-\partial_{x}u_{1}-g\phi
u_{1}.$ (14)

Clearly, $u_{1}$ and $v_{2}$ are decoupled from $u_{2}$ and $v_{1}$ and vice-
versa. Moreover, if we choose, for instance, an initial condition such that
$u_{2}(t=0)=v_{1}(t=0)=0$, we will have $u_{2}(t)=v_{1}(t)=0$ for all times.
As the backreaction is proportional to

$g\bar{\psi}\psi=2g(u_{1}v_{1}+u_{2}v_{2}),$ (15)

we will have that the backreaction vanishes for all times. Interestingly, the
condition $u_{2}(t=0)=v_{1}(t=0)=0$ is not rare. For instance, it is respected
when the fermion is initialized at zero mode.

In the next main section, we will study the effect of the kink’s vibration on
the fermion in the same model. In section 7, we will study in a different
model what happens to a fermion localized on a kink after a collision.

### 4 Appendix: Fermion bound states of the $\phi^{4}$ model

The fermion bound states in the presence of a $\phi^{4}$ kink are listed in
references [Chu e Vachaspati 2008, Charmchi e Gousheh 2014]. For every integer
$n<g$, the bound state solutions with positive energy are given by the
following expressions

$\displaystyle\psi_{n,+}$
$\displaystyle=N_{+}^{n}\text{sech}^{g-n}(x)F\left(-n,2g-n+1,g-n+1;\frac{1}{2}(1-\tanh(x))\right),$
(16) $\displaystyle\psi_{n,-}$
$\displaystyle=N_{-}^{n}\text{sech}^{g-n}(x)F\left(-n+1,2g-n,g-n+1;\frac{1}{2}(1-\tanh(x))\right),$
(17)

with energy $E_{n}=\sqrt{2gn-n^{2}}$. The function $F$ corresponds to the
hypergeometric function. For $g<1$, there is only the zero mode with $n=0$ and
$N_{-}^{0}=0$. A new positive energy bound state appear every time $g$ crosses
an integer value. For these states, the normalization constants obey the
relation $N_{-}^{n}/N_{+}^{n}=n/E_{n}$.

For every positive momentum $k$, the scattering states are given by the
expressions below

$\displaystyle\psi_{k,L}^{\pm}$
$\displaystyle=N_{\pm}^{k,L}\text{cosh}^{ik}(x)F\left(\frac{1}{2}-ik-\zeta_{\pm},\frac{1}{2}-ik+\zeta_{\pm},1-ik;\frac{1}{1+e^{2x}}\right),$
(18) $\displaystyle\psi_{k,R}^{\pm}$
$\displaystyle=N_{\pm}^{k,R}\text{cosh}^{ik}(x)F\left(\frac{1}{2}-ik-\zeta_{\pm},\frac{1}{2}-ik+\zeta_{\pm},1-ik;\frac{1}{1+e^{-2x}}\right),$
(19)

with energy $E_{k}=\sqrt{k^{2}+g^{2}}$. Moreover, we define $\zeta_{\pm}\equiv
g\pm\frac{1}{2}$. The normalization constants obey the relation
$N_{-}^{k,L}/N_{+}^{k,L}=-N_{-}^{k,R}/N_{+}^{k,R}=(ik+g)/E_{k}$.

## 6 FERMIONS ON WOBBLING KINKS

### 1 Overview

In this main section, we will discuss a model with a fermion coupled to a
wobbling kink background. This work resulted in the following publication
[Campos e Mohammadi 2021]. We will consider the simple realizations of a
fermion-kink system discussed in the previous main section. It consists of a
fermion in the background of a $\phi^{4}$ kink interacting via a Yukawa
coupling in two spacetime dimensions. As said before, this system has an
analytical solution for a static kink background. This can be found, for
instance, in [Chu e Vachaspati 2008, Charmchi e Gousheh 2014, Charmchi e
Gousheh 2014].

This work aims to analyze the consequences of including vibration in the kink
background, considering the abovementioned model. We studied the fermion’s
behavior coupled to a wobbling kink, which is a kink with the vibrational mode
excited. This idea was inspired by the resonant energy exchange mechanism,
which states that kinks may lose translational energy and start wobbling
during an interaction. More recently, in [Izquierdo, Queiroga-Nunes e Nieto
2021], the authors considered collisions between wobbling kinks and showed
that, in general, after a kink-antikink collision, they emerge with the
vibrational mode excited. Therefore, it is essential to consider the effect of
the vibration on any fermions that may be interacting with the kink. Moreover,
if one considers the wobbling kink as a background, it is possible to find
analytical solutions for the system’s evolution using perturbation theory. We
will show that wobbling has exciting effects.

We will consider the wobbling effect on the fermion due to both normal and
quasinormal modes. As shown in [Dorey e Romańczukiewicz 2018, Campos e
Mohammadi 2020], it is possible to modify the kink potential in order to turn
the shape mode into a quasinormal mode. This has interesting consequences for
the resonant behavior of the system. Hence, we will also investigate how this
modification can affect the fermion.

The next section will discuss the fermion-kink model for a general potential.
Moreover, we will discuss how to perform perturbation theory for this system.
Later, we will specialize in the $\phi^{4}$ model and a toy model containing
either a normal or quasinormal mode.

### 2 Model

We will consider the Lagrangian in two spacetime dimensions containing a
scalar field and a fermion field interacting via a Yukawa coupling

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)+i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-g\phi\bar{\psi}\psi.$
(1)

The potential $V(\phi)$ is chosen such that it has degenerate vacua with
vanishing energy. Therefore, the scalar field possesses a kink solution. The
Euler-Lagrange equation for the fermion field

$i\gamma^{\mu}\partial_{\mu}\psi-g\phi_{k}\psi=0,$ (2)

where we considered that the scalar field is fixed as a background and given
by the kink profile. We choose the following representation for the gamma
matrices, $\gamma^{0}=\sigma_{1}$ and $\gamma^{1}=i\sigma_{3}$. Multiplying
the Dirac equation above by $\gamma^{0}$, we obtain

$i\partial_{t}\psi=H_{0}\psi,$ (3)

where the Hamiltonian $H_{0}$ is defined as

$H_{0}=-i\sigma_{2}\partial_{x}+g\phi_{k}\sigma_{1}.$ (4)

Let us diagonalize the Hamiltonian by solving the following time-independent
equation

$H_{0}\psi_{n}=E_{n}\psi_{n}.$ (5)

Writing the eigenfunctions in term of its components as
$\psi_{n}=(\psi_{n,+},\psi_{n,-})^{T}$, we obtain eqs. (5) and (6) again. This
is expected because we are studying the same system considered in section 3.
Thus, this set of equations has analytical solutions, which are listed in
section 4.

Now, we consider wobbling kinks as a background. This can be achieved by
including a perturbation in the scalar field as follows
$\phi=\phi_{k}+A\eta_{S}\cos(\omega_{S}t)$. In this expression, $A$ is the
amplitude, and $\eta_{S}$ and $\omega_{S}$ are the shape mode’s profile and
frequency, respectively. Thus, the Dirac equation becomes

$i\partial_{t}\psi=(H_{0}+AH_{1}(t))\psi,$ (6)

where the perturbation Hamiltonian is given by
$H_{1}(t)=g\eta_{S}\cos(\omega_{S}t)\sigma_{1}$. The solution of the perturbed
equation can be written as a linear combination of the eigenfunctions of the
non-perturbed Hamiltonian with time-dependent coefficients

$\psi=\sum_{n}c_{n}(t)e^{-iE_{n}t}\psi_{n}.$ (7)

Moreover, we can expand the coefficients in powers of the perturbation
amplitude

$c_{n}(t)=c_{n}^{(0)}(t)+Ac_{n}^{(1)}(t)+\cdots.$ (8)

We choose the initial condition such that the system starts at state $i$. This
can be written as $c_{n}(0)=\delta_{ni}$. Solving in zero order, we find
$c_{n}^{(0)}(t)=\delta_{ni}$. Then, the first-order coefficients are given by
time-dependent perturbation theory as [Sakurai e Napolitano 2011]

$c_{n}^{(1)}(t)=-i\int_{0}^{t}e^{i\omega_{ni}t}\langle
n|H_{1}(t^{\prime})|i\rangle dt^{\prime},$ (9)

where we defined $\omega_{ni}=E_{n}-E_{i}$ and the matrix elements between two
states is defined as

$\langle n|\mathcal{O}|i\rangle=\int\psi_{n}^{T}\mathcal{O}\psi_{i}dx.$ (10)

The transition probability is quantified by coefficients $c_{n}(t)$, obtained
by projecting our system in the state $|n\rangle$. They are also known as
Bogoliubov coefficients. For $n\neq i$, the first-order transition probability
from the state $i$ to a state $n$ is given by

$P_{i\to n}(t)=A^{2}|c_{n}^{(1)}(t)|^{2}.$ (11)

This expression is only valid as long as $A^{2}|c_{n}^{(1)}(t)|^{2}\ll 1$. The
time integration in eq. (9) leads to

$c_{n}^{(1)}(t)=\frac{g}{2}\langle
n|\eta_{S}\sigma_{1}|i\rangle\left(\frac{1-e^{i(\omega_{ni}+\omega_{S})t}}{\omega_{ni}+\omega_{S}}+\frac{1-e^{i(\omega_{ni}-\omega_{S})t}}{\omega_{ni}-\omega_{S}}\right).$
(12)

We can define the transition rate from state $i$ to $n$ as the ratio between
the probability and the total time for long times. It is easy to show that, in
our case, it is given by Fermi’s golden rule

$R_{i\to n}=A^{2}g^{2}\frac{\pi}{2}|\langle
n|\eta_{S}\sigma_{1}|i\rangle|^{2}[\delta(\omega_{ni}+\omega_{S})+\delta(\omega_{ni}-\omega_{S})],$
(13)

for $n\neq i$. This expression must be integrated over a set of final states
to make sense of the delta functions.

In the next section, we apply this formalism to the $\phi^{4}$ model and
compare the analytical results to numerical analysis.

### 3 $\phi^{4}$ model

In this section, we specialize to the the case where the scalar field is
described by $\phi^{4}$ model. For this model, we set
$V(\phi)=\frac{1}{2}(\phi^{2}-1)^{2}$, $\phi_{k}=\tanh(x)$,
$\eta_{S}=\tanh(x)\text{sech}(x)$ and $\omega_{S}=\sqrt{3}$. The effective
fermionic potential in this case is given by

$V_{\pm}=g^{2}-\frac{g(g\pm 1)}{\cosh^{2}x},$ (14)

which has analytical solutions as shown, for instance, in [Chu e Vachaspati
2008, Charmchi e Gousheh 2014]. For completeness, the solutions as listed in
section 4. One important property of these solutions is that for $g<1$, there
is only one discrete state, the zero mode, and, as $g$ increases, a positive
and a negative discrete solutions appear every time $g$ crosses an integer
value. Moreover, there is a continuum of scattering solutions with fermion
energy obeying $|E|>g$.

Figure 1: Matrix elements of the operator $\eta_{S}\sigma_{1}$ (a) between
discrete states and (b) between discrete states and the continuum. We consider
the $\phi^{4}$ model and in (b) $g=1.6$.

(a)

(b)

Source: Results obtained by the author in [Campos e Mohammadi 2021].

The first step in the analysis is to compute the matrix elements of the
operator $\eta_{S}\sigma_{1}$. They are shown in Fig. 1. We see the matrix
elements between the discrete states in the left panel. Transitions are only
allowed when the $n$ and $i$ are both odd or even, due to the parity of the
respective eigenfunctions and the function $\eta_{S}$. For instance, the
matrix element between the zero mode and the first excited state is given by

$\langle
1|\eta_{S}\sigma_{1}|0\rangle=\int(\psi_{1,-}\eta_{S}\psi_{0,+}+\psi_{1,+}\eta_{S}\psi_{0,-})dx.$
(15)

The first term vanishes because $\psi_{1,-}$ is even, $\psi_{0,+}$ is even,
and $\eta_{S}$ is odd. The second term also vanishes because $\psi_{0,-}=0$.
The first allowed transition between positive energy states appears for $g>2$
as shown in the figure. Moreover, we can compute the transition between
discrete states and the continuum using the analytical fermionic solutions of
the $\phi^{4}$ model. In general, they are allowed, as shown in the right
panel.

Figure 2: Evolution of the transition probabilities in time for the $\phi^{4}$
model. We compute transitions from the initial states to the continuum and to
other discrete states. In (a) the transition rate to the continuum is the
slope of the dotted line. We fix the wobbling amplitude $A=0.1$.

(a) $g=0.8$

(b) $g=2.1$

Source: Results obtained by the author in [Campos e Mohammadi 2021].

We will first consider the fermion’s evolution in the presence of a wobbling
kink starting from the zero mode. Most importantly, when we specialize to this
case, the back-reaction to the kink vanishes, and the background approximation
is exact. However, we are still making approximations. We ignore the back-
reaction to obtain the form of the perturbation due to the kink’s vibration
that we are considering.

We turn on the perturbation with amplitude $A=0.1$. This value is interesting
because it is a typical vibration amplitude of a kink after colliding with an
antikink. When the perturbation starts, the fermionic density, which is
initially localized at the kink, starts to oscillate and may radiate. The
radiation is significant when the transition to the continuum is allowed by
Fermi’s golden rule, as explained below. In Fig. 2, we plot the evolution of
the transition probabilities in this case. The probabilities are computed in
two ways. First, we use the perturbation theory equations derived above.
Second, we compute the evolution of the fermionic field via numerical
integration of the equations of motion. The details of the numerical
integration are given in section 6.

For $g=0.8$, we only have the zero mode and the continuum states, and we
compute the transition rate to the continuum. This is the transition to the
set of all continuum states. The results obtained from the two methods have a
good agreement. The transition rate can be computed by integrating Fermi’s
golden rule. Due to the delta functions, the integration is only nonzero if
$g<\omega_{S}$. In this range, it yields

$R_{i\to\text{cont.}}=2A^{2}g^{2}\pi\left|\left\langle
k=\sqrt{\omega_{S}^{2}-g^{2}}\left|\eta_{S}\sigma_{1}\right|i\right\rangle\right|^{2}\frac{\omega_{S}}{\sqrt{\omega_{S}^{2}-g^{2}}}.$
(16)

In the left panel of the figure, the transition rate is shown as the slope of
the dotted line. Accordingly, the evolution of the transition probability has
the same slope on average. In the right panel, we consider $g=2.1$. the
transition probability to the second excited state has complex oscillatory
behavior, as described by eq. (12). Moreover, the transition to the continuum
is not allowed by Fermi’s golden rule in this case, and the transition
probability is bounded.

The results in this section are very relevant because they predict that
fermions will escape the kink at some rate if the kink is wobbling and if the
Yukawa coupling $g$ is not too large. Based on the fermion escape rate in
those systems, this result could be used to measure, for instance, the
wobbling amplitude or the Yukawa coupling $g$. In the next section, we will
repeat the analysis for a toy model that possesses a normal mode that can be
turned into a quasinormal one.

### 4 Toy model

The model we will consider in this section was described in detail in section
2. This model describes a kink with vibrational modes that can be turned into
quasinormal modes when the model is extended. The potential is given by eq.
(5) for the normal mode case and by eq. (6) for the quasinormal mode case. The
corresponding kinks are given by eq. (13) and eq. (14), respectively. The toy
model has one free parameter for the normal mode case that we fix at
$\gamma=3.0$. For this value of the parameter, the kink has a single shape
mode that can be found analytically as shown in section 8. For the quasinormal
case, the toy model has three free parameters that we fix at $\gamma=3.0$,
$\Delta\gamma=2.0$, and $\epsilon=0.05$. For these values of the parameters,
the shape mode of the previous case becomes a quasinormal mode.

Figure 3: Energy of fermion bound states on a kink background as a function of
the Yukawa coupling $g$. The two plots corresponds to the toy model containing
(a) a normal mode and (b) a quasinormal mode.

(a) $\epsilon=0.0$

(b) $\epsilon=0.05$

Source: Results obtained by the author in [Campos e Mohammadi 2021].

The fermionic spectra in the presence of the kinks are shown in Fig. 3 for
both normal and quasinormal mode cases. This can be easily computed
numerically after substituting both kink solutions in the effective fermionic
potential given $V_{\pm}$. This is in contrast with the $\phi^{4}$ model,
where the fermionic bound states are found analytically. Similar to the
$\phi^{4}$ model, both positive and negative energy states appear as the
Yukawa coupling $g$ is increased. In the quasinormal mode case, the kink’s
tail decays more slowly to the vacuum, and the effective potential is wider
than in the normal mode case. Therefore, the bound states appear at smaller
values of $g$.

Figure 4: Evolution of the fermion density in spacetime for the toy model with
several values of the parameters and $A=0.1$.

(a) $\epsilon=0.0$, $g=0.8$

(b) $\epsilon=0.0$, $g=2.1$

(c) $\epsilon=0.05$, $g=0.8$

(d) $\epsilon=0.05$, $g=2.1$

Source: Results obtained by the author in [Campos e Mohammadi 2021].

In Fig. 4, we show the evolution of the fermionic density obtained from
numerical integration of the equations of motion with wobbling amplitude
$A=0.1$. Similar to the $\phi^{4}$ case, we choose to initialize the fermion
at the zero mode. Hence, it starts confined at the kink, located at the
origin. Then, due to the perturbation, the system oscillates, and it is
possible to see some radiation for smaller values of $g$. In the right panels,
$g$ is larger, and the fermion density starts more localized at the origin
than the fermion density shown in the left panels. This occurs because the
unperturbed Hamiltonian becomes more confining as $g$ increases. However, the
perturbation effect as $g$ varies is more complicated, as shown in the
$\phi^{4}$ case. In fact, it can favor the escape of fermions initially, as
$g$ increases from zero. If we continue increasing $g$, it does not favor the
escape anymore, after the energy gap to the continuum becomes too large.

#### 1 Normal mode

Figure 5: Matrix elements of the operator $\eta_{S}\sigma_{1}$ (a) between
discrete states and (b) between discrete states and the continuum. We consider
the toy model with $\gamma=3.0$ and $\epsilon=0.0$. In (b), we fix $g=1.6$.

(a)

(b)

Source: Results obtained by the author in [Campos e Mohammadi 2021].

Now, we focus on the case where the kink has only normal modes. This means
that $\epsilon=0.0$ and, as mentioned before, $\gamma=3.0$. We start by
computing the matrix elements of the operator $\eta_{S}\sigma_{1}$, which in
this case must be done using the numerical values of the fermion bound states.
The result is shown in Fig. 5. The results are completely analogous to the
$\phi^{4}$ case. We also find the parity selection rule and the same
qualitative behavior of the matrix elements from discrete states to the
continuum. From this analysis, we expect that these elements should have the
same dependence with the parameters for other scalar potentials with $Z_{2}$
symmetry and containing kinks.

Figure 6: Evolution of the transition probabilities in time. We compute
transitions from the initial states to the continuum and to other discrete
states. In (a) the transition rate to the continuum is the slope of the dotted
line. We consider the toy model with $\gamma=3.0$, $\epsilon=0.0$ and $A=0.1$.

(a) $g=0.8$

(b) $g=2.1$

Source: Results obtained by the author in [Campos e Mohammadi 2021].

Next, we compute the evolution of the transition probability in time for a
fermion in the presence of a wobbling kink with amplitude $A=0.1$. This
amplitude is measured with respect to the following shape mode profile

$\eta_{S}(x)=\begin{cases}-e^{k_{S}x},&x<-L,\\\
e^{-k_{S}L}\sin(p_{S}x)/\sin(p_{S}L),&-L<x<L,\\\ e^{-k_{S}x},&x>L.\end{cases}$
(17)

This is the first odd solution listed in section 8 and the normalization was
fixed arbitrarily. In the expression above, we defined
$k_{S}\equiv\sqrt{\gamma-\omega_{S}^{2}}$ and
$p_{S}\equiv\sqrt{\omega_{S}^{2}+1}$ with $\omega_{S}$ being the solution of
the transcendental equation $k_{S}=-p_{S}\cot(p_{S}L)$. Again, we initialize
the systems with the fermion at zero mode. The result for the transition to
all continuum states and other discrete states is shown in Fig. 6. The figure
also compares the evolution computed with the perturbative analysis to
numerical simulations. Moreover, the value of the transition rate to the
continuum obtained with Fermi’s golden rule is shown as the slope of the
dotted line. Similarly to the $\phi^{4}$ case, we observe a transition to the
continuum for $g=0.8$, which agrees with Fermi’s golden rule when averaged
over large times. Considering the case with $g=2.1$, the transition to the
continuum does not exhibit a steady increase in the transition probability and
has a complicated oscillatory behavior because the energy gap between the
continuum and the zero mode is too large. Moreover, the transition probability
to the second excited state also oscillates in a complicated manner as
expected from eq. (12). Again, this result is very similar to what was
obtained in the $\phi^{4}$.

#### 2 Quasinormal mode

Now, we modify the toy model, turning the normal mode into a quasinormal mode.
The main goal of this analysis is to see how the previous results are altered
when this change is made. Again we start by computing the fermion bound states
numerically and using the result to compute the matrix elements of the
operator $\eta_{S}\sigma_{1}$. We also find the parity selection rules and the
same qualitative behavior of the matrix elements from discrete states to the
continuum. Hence, we choose not to show the results graphically.

Now, we would like to study how the fermions evolve when subject to a
quasinormal mode as a perturbation. Hence, we need to define precisely the
perturbation Hamiltonian. One possibility is to use the analytical solution of
the quasinormal mode for $\eta_{S}$ and $\omega_{S}$. This solution was
computed in section 2. It diverges for large $x$, but it describes the
behavior of the quasinormal mode for large times if carefully truncated.
However, it cannot be used to describe the transient behavior quasinormal mode
as it leaks through the potential barrier. In situations where a normal mode
turns into a quasinormal mode, the evolution of the perturbation starting with
the normal mode configuration is a good approximation of the transient
behavior that we are looking for. We will adopt this approach because we find
it more sensible. In other words, we will compute the evolution of scalar
field perturbation with the normal mode profile of the corresponding model
with the same value of $\gamma$ but with $\epsilon=0.0$ as an initial
condition and use it as the perturbation Hamiltonian. The initial profile is
given by eq. (17). Its evolution is described by the time-dependent equation
of perturbations around the kink solution given below

$\frac{\partial^{2}\eta(x,t)}{\partial
t^{2}}-\frac{\partial^{2}\eta(x,t)}{\partial x^{2}}+U(x)\eta(x,t)=0,$ (18)

where the linearized potential $U(x)$ is given by eq. (4). Then, the evolution
of the coefficients $c_{n}^{(1)}(t)$ can be computed using eq. (9) with
$H_{1}(t)=g\eta(x,t)\sigma_{1}$.

We can find an approximation for the evolution of the quasinormal mode. It
will approximately oscillate with a frequency $\Omega$ but also decay with the
rate $\Gamma$, which were found analytically in section 2. In this case, the
perturbed scalar field is approximated by $\phi=\phi_{k}+A\eta_{S}\cos(\Omega
t)e^{-\Gamma t}$, where $\eta_{S}$ is given by eq. (17). Then it is easy to
show that the coefficient $c_{n}^{(1)}(t)$ is given by

$c_{n}^{(1)}(t)=\frac{g}{2}\langle
n|\eta_{S}\sigma_{1}|i\rangle\left(\frac{1-e^{i(\omega_{ni}+\Omega+i\Gamma)t}}{\omega_{ni}+\Omega+i\Gamma}+\frac{1-e^{i(\omega_{ni}-\Omega+i\Gamma)t}}{\omega_{ni}-\Omega+i\Gamma}\right).$
(19)

This description is incomplete because the quasinormal mode will also generate
outgoing radiation as it decays, but it allows a closed expression for the
transition coefficients.

Figure 7: Evolution of the transition probabilities in time. We compute
transitions from the initial states to the continuum and to other discrete
states. The numerical evolution is compared to two results obtained from
perturbation theory. We consider the toy model with $\gamma=3.0$,
$\Delta\gamma=2.0$, $\epsilon=0.05$ and $A=0.1$.

(a) $g=0.8$

(b) $g=2.1$

Source: Results obtained by the author in [Campos e Mohammadi 2021].

After defining the perturbation, we can compute the evolution of the
transition probability with time. First, we compute the evolution numerically
using the full transient evolution of the quasinormal mode. Second, we compute
the evolution according to eq. (9) also using the full transient evolution of
the quasinormal mode. In the figure, this method is called perturbation theory
1. Finally, we compute the evolution according to eq. (19). In the figure,
this method is called perturbation theory 2. The results are shown in Fig. 7
considering the perturbation amplitude $A=0.1$. The first two methods have a
good agreement. On the other hand, the third one gives the correct behavior
but is less accurate. In the left panel, we show the evolution for $g=0.8$.
This result is similar to the cases with normal modes, but the transition
probability increases at slower rates as time evolves. This occurs because the
perturbation gets weaker as it leaks through the potential barrier of the
kink. In the right panel, we show the results for $g=2.1$. The transition
probabilities oscillate similarly to the previous cases, but the oscillations
also get weaker with time due to decreased perturbation strength.

Figure 8: Asymptotic value of the transition probability as a function of $g$.
We consider the toy model with parameters $\gamma=3.0$, $\Delta\gamma=2.0$,
$\epsilon=0.05$ and $A=0.1$. Source: Results obtained by the author in [Campos
e Mohammadi 2021].

As the perturbation gets weaker with time, the transition probability will
reach an asymptotic state if we wait long enough. We can use this value to
summarize the system’s behavior as a function of the Yukawa coupling $g$. The
result is shown in Fig. 8. We compare the results from numerical simulations
with the value of eq. (19) in the limit of very large $t$. The two results
have very similar behaviors. Observe that the asymptotic value of the
transition probability to the continuum initially increases with $g$ and then
suddenly decays to a very small value for $g\simeq\Omega$. This behavior is
reminiscent of Fermi’s golden rule, which states that the transition is
forbidden if the energy gap to the continuum is too large. Near $g=1.54$, the
second excited state appears. The asymptotic value of the transition
probability to this state quickly reaches a peak and then decays as the energy
gap between the two states gets larger.

The following section will summarize the results described above, discuss our
conclusions, and list some ideas for future works.

### 5 Conclusion

In this work, we investigated the effect of an excited vibrational mode on a
fermion that is bound to a kink. We started by considering the $\phi^{4}$
model and a toy model. Both have fermionic bound states in the presence of a
non-vibrating kink, but the former can be solved analytically, while the
latter can only be solved numerically. Then we include the vibration as a
perturbation and compute the transition between these states perturbatively.
This analysis leads to an expression for the transition probability and the
transition rate, which obeys Fermi’s golden rule.

The fermionic spectrum for the toy model is similar to the $\phi^{4}$ case.
Interestingly, the bound states in the quasinormal case appear earlier than in
the normal mode one because the effective potential is wider in this case. The
full evolution fermionic field can be computed numerically. We observe that
the fermion density is mainly located on the origin, where the kink is. The
density oscillates and can also radiate when the transition to the continuum
is allowed.

The perturbative expressions for the transition probabilities depend on matrix
elements that can be easily computed. We specialize to the case where the
fermion starts at zero mode. The perturbative analysis is compared to
numerical simulations, and the two have good agreement. We find that for small
values of the Yukawa coupling $g$, the transition to the continuum increases
with an average increasing rate given by Fermi’s golden rule, while for large
values of $g$, the probabilities are bounded. Thus, the only effect of the
perturbation is a complex oscillation. This result is significant because it
predicts that a fermion bound to a kink will have a fixed escape rate if the
kink is wobbling and the Yukawa coupling constant is not too large. This could
possibly be measured in some experimental setting of a fermion-kink system.

Finally, when considering the quasinormal mode case, we can describe the
transient behavior starting with the corresponding normal mode with
$\epsilon=0.0$ and computing the evolution of this perturbation. The
perturbation evolves according to a time-dependent equation subject to the
linearized potential $U(x)$ of the quasinormal mode. We find that the
probability reaches an asymptotic value due to the leakage of the
perturbation. When plotted as a function of the Yukawa coupling constant $g$,
this value initially increases and then sharply decreases when the gap between
the zero mode and the continuum becomes too large. This behavior is the
analogue of Fermi’s golden rule for quasinormal modes. One interesting
continuation of this work is considering excited states as the initial
condition. One could also try to formulate the problem considering the fermion
back-reaction at every step and see how the results are altered. However, this
problem is much more challenging.

### 6 Appendix: Numerical method

We simulate the system in a box with range $-100.0<x<100.0$ with spacing
$h=0.1$. Again, we apply the method of lines to integrate the equations of
motion. The partial derivatives with respect to $x$ are given by a five-point
stencil approximation

$\frac{\partial\psi}{\partial
x}=\frac{-\psi(x+2h)+8\psi(x+h)-8\psi(x-h)+\psi(x-2h)}{12h},$ (20)

where the dependence on $t$ is suppressed for brevity. Moreover, at the
boundaries, we set $\psi(x=\pm L,t)=0$. These conditions reflect the
radiation, but simulations are short enough to avoid that the boundary
conditions have any effect on the bulk evolution. At $x=-L+h$ we modify the
partial derivative in order to consider only points inside the box

$\frac{\partial\psi}{\partial
x}=\frac{\psi(-L+4h)-6\psi(-L+3h)+18\psi(-L+2h)-10\psi(-L+h)-3\psi(-L)}{12h}.$
(21)

We use a similar expression at $x=L-h$. This results in a set of first-order
ordinary differential equations for both the field $\phi$ and its time
derivative $\phi_{t}$. The equations are integrated using a fifth-order Runge-
Kutta algorithm with adaptive step size and error control [Dormand e Prince
1980]. The algorithm implemented in the solve_ivp method of the SciPy library
in Python. For the $\phi^{4}$ model and the toy model with $\epsilon=0.0$, we
ignore the slow decay of the shape mode111The amplitude of the shape mode is
not constant because it slowly emits radiation [Manton e Merabet 1997]. and we
write the scalar field as $\phi=\phi_{k}+A\eta_{S}\cos(\omega_{S}t)$. When
considering quasinormal modes, we must numerically integrate the linearized
equation for the perturbation, eq. (18). This is done using the method of
lines in a similar manner. The partial derivatives with respect to $x$ are
computed using a five-point stencil approximation, and the resulting set of
ordinary differential equations are integreted using the same Runge-Kutta
method. At the boundaries, we also set $\eta(x=\pm L,t)=0$.

For the toy model, the fermion bound states are found numerically using the
NDEigensystem method in Mathematica and the solve_bvp method of the SciPy
library in Python. The continuum states are found as described in [Gousheh,
Mohammadi e Shahkarami 2013]. For instance, the scattering state for a wave
coming from the left with wavenumber $k$ and energy $E_{k}=\sqrt{k^{2}+g^{2}}$
can be found factoring out the oscillatory behavior of the transmitted wave
contribution as follows

$\psi_{k}=e^{ikx}\chi_{k}(x).$ (22)

Then, we find the asymptotic value $\chi_{k}(\infty)$ by solving eqs. (5) and
(6) in the limit $x\to\infty$. Using $\phi_{k}(\infty)=1.0$ and
$\chi_{k}^{\prime}(\infty)=0$, one can show that

$\chi_{k}(\infty)=\begin{pmatrix}1\\\
\frac{g+ik}{\sqrt{g^{2}+k^{2}}}\end{pmatrix}.$ (23)

After that, it is a simple matter to integrate eqs. (5) and (6) backwards and
obtain $\chi_{k}(x)$. Numerically, we start the integration from $L=20.0$, as
we cannot reach infinity. Finally, the scattering functions are properly
normalized.

## 7 FERMION TRANSFER IN THE $\phi^{4}$ MODEL WITH AN IMPURITY

### 1 Overview

The work in this main section resulted in the following publication [Campos e
Mohammadi 2020]. We will discuss a model describing a fermion coupled to a
scalar field and compute the effect of a kink-antikink collision on the
fermion.

One of the earliest works to consider fermions in this scenario was [Gibbons,
Maeda e Takamizu 2007]. The authors considered a five-dimensional spacetime
where a brane, described by a $\phi^{4}$ kink, localizes the fermion in the
fifth dimension. Under a few assumptions, they show that the motion of the
fermion in the fifth dimension decouples from the other spatial dimensions and
can be effectively described by motion in $(1+1)$ dimensional spacetime. They
considered a fermion bound to the kink in the zero mode as an initial
condition. After colliding with an antikink, most fermions either stay bound
to the kink or are transferred to the antikink, and only a few are lost as
radiation. However, this is only true in the reflecting regime of kink-
antikink collisions because, when the kink and the antikink annihilate, they
form a bion. As the bion oscillates, it makes the fermions radiate at each
oscillation, and all fermions will end up radiating to infinity.

In [Saffin e Tranberg 2007], the authors made a very similar study, but with a
broader range of parameters and higher numerical precision, and found similar
results. They also included collisions of the kink with boundaries, which can
be analyzed with the same formalism. The authors did not realize in these
works that the background approximation is exact when the fermion starts at
the zero mode of the kink or antikink.

We decided to investigate fermion transfer in a different setting to derive
something new. However, finding relevant models is not an easy task because
when the kink is not centered around the origin, the effective fermionic
potential for a kink background may not allow bound states. This occurs, for
instance, for the sine-Gordon and the $\phi^{6}$ models.

In light of some recent works where the authors considered a family of models
with a half-BPS preserving impurity [Adam, Romanczukiewicz e Wereszczynski
2019, Adam et al. 2019, Adam, Queiruga e Wereszczynski 2019, Adam et al. 2019,
Manton, Oleś e Wereszczyński 2019, Adam et al. 2020], we decided to study the
fermion’s behavior in background collisions in one of these models. In
general, for topological models, Bogomol’nyi–Prasad–Sommerfield (BPS)
solutions are static and saturate the energy in a given topological sector.
Saturation means that it is the solution with the lowest energy value among
all solutions obeying the boundary conditions that define the topological
sector. Typical scalar field systems in $(1+1)$ dimensions, such as the
$\phi^{4}$ and sine-Gordon, possess BPS solutions consisting of kinks and
antikinks. When a half-BPS preserving impurity is included, one-half of the
BPS solutions disappear, and the others acquire a nontrivial structure as
opposed to the family of BPS solutions without an impurity, which is related
by a trivial translation. This allows the nontrivial evolution in the
configuration space of BPS solutions to be viewed as a multi-defect BPS
interaction. In general, multi-defect BPS interaction only appear in models in
higher spacetime dimensions, such as the abelian Higgs model and Yang-Mills-
Higgs theory, and the one-dimensional models serve as a guide to these more
complicated systems.

In the following section, we will describe the model that we investigated. It
consists of a fermion interacting with a background scalar field model with a
half-BPS preserving impurity. This scalar field model possesses BPS and non-
BPS defect solutions. In particular, we will consider, as a background,
collisions between these defects. However, the background approximation is
exact in the cases we will consider because the fermion will start at the zero
mode.

### 2 Model

#### 1 Lagrangian and field equations

We considered the following Lagrangian containing three types of terms

$\mathcal{L}=\mathcal{L}_{\text{scalar}}+\mathcal{L}_{\text{fermion}}+\mathcal{L}_{\text{int}}.$
(1)

The scalar field term consists of the $\phi^{4}$ model with a half-BPS
preserving impurity

$\mathcal{L}_{\text{scalar}}=\frac{1}{2}\phi_{t}^{2}-\frac{1}{2}\phi_{x}^{2}-U(\phi)-2\sigma\sqrt{U(\phi)}-\sqrt{2}\sigma\phi_{x}-\sigma^{2},$
(2)

where the potential is given by

$U(\phi)=\frac{1}{2}(1-\phi^{2})^{2},$ (3)

and the $\sigma$ term describes the impurity, which will be defined below. The
fermion Lagrangian has the standard massless form

$\mathcal{L}_{\text{fermion}}=i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi.$ (4)

The two fields interact via a Yukawa term

$\mathcal{L}_{\text{int}}=-g\phi\bar{\psi}\psi.$ (5)

The scalar field Lagrangian was first considered in [Adam, Romanczukiewicz e
Wereszczynski 2019] without the fermion field. Here, we will briefly summarize
some of its properties. The sigma term was added such that the system still
possesses one BPS equation, instead of two. If we write the energy of the
scalar field for a static configuration, we find

$\displaystyle E$
$\displaystyle=\int\left(\frac{1}{2}\phi_{x}^{2}+U(\phi)+2\sigma\sqrt{U(\phi)}+\sqrt{2}\sigma\phi_{x}+\sigma^{2}\right)dx,$
(6)
$\displaystyle=\int\left(\frac{1}{\sqrt{2}}\phi_{x}+\sigma+\sqrt{U}\right)^{2}dx-\sqrt{2}\int\phi_{x}\sqrt{U}dx\geq-\sqrt{2}\int_{\phi(-\infty)}^{\phi(\infty)}\sqrt{U}d\phi.$
(7)

The inequality above is clearly saturated by the BPS condition

$\phi_{x}+\sqrt{2}\sigma+(1-\phi^{2})=0.$ (8)

This solution corresponds to the antikink solution of the $\phi^{4}$ theory
without impurity. On the other hand, the kink solution can only be found
solving the full second-order field equations. To fix the impurity term, we
require that the kink solution $\phi_{K_{0}}(x)=\tanh(x)$ is still a solution
of the field equations. In this case, one finds

$\sigma=\frac{\lambda}{\cosh^{2}(x)},$ (9)

where $\lambda$ is a constant that can be anywhere in the range
$\lambda>-\sqrt{2}$. However, if we translate the kink solution, it is not a
solution anymore.

The field equations can be found via the Euler-Lagrange equations. They give

$\displaystyle\phi_{tt}-\phi_{xx}+2(\phi^{2}-1)\phi$
$\displaystyle=\frac{2\sqrt{2}\lambda}{\cosh^{2}(x)}(\phi-\tanh(x)),$ (10)
$\displaystyle i\gamma^{\mu}\partial_{\mu}\psi-g\phi\psi$ $\displaystyle=0.$
(11)

In eq. (10) we ignored the term proportional to $g\bar{\psi}\psi$ describing
fermion back-reaction on the scalar field. Moreover, we can see from the same
equation that the impurity term appearing on the right-hand side vanishes for
$\phi(x)=\tanh(x)$, as desired. In this case, we are left with the equation of
motion for the $\phi^{4}$ model, which is also solved by $\phi(x)=\tanh(x)$.

For the Dirac matrices, we consider the following representation
$\gamma^{0}=-\sigma^{2}$ and $\gamma^{1}=i\sigma^{3}$. In this case, the
fermion field splits into two decoupled Majorana fields

$\psi=\psi_{1}^{M}+i\psi_{2}^{M}.$ (12)

We will consider the field equations only for $\psi_{1}^{M}$ because they are
identical to the ones for $\psi_{2}^{M}$. Defining the components as
$\psi_{1}^{M}=(\psi_{1},\psi_{2})^{T}$, we find

$\displaystyle\partial_{t}\psi_{1}=-\partial_{x}\psi_{2}+g\phi\psi_{2},$ (13)
$\displaystyle\partial_{t}\psi_{2}=-\partial_{x}\psi_{1}-g\phi\psi_{1}.$ (14)

The representation of the gamma matrices and the notation of the Fermi field
are not the same as before, but the analysis in this main section is very
similar to what we did in section 3. Moreover, if we choose a field
configuration where $\psi_{2}^{M}=0$ as an initial condition, the back-
reaction term $g\bar{\psi}\psi$ vanishes. For this set of configurations, the
background approximation is justified.

#### 2 Scalar field solutions

Now let us review the solutions to the scalar field equations that were found
in [Adam, Romanczukiewicz e Wereszczynski 2019]. The first one we will discuss
is the kink-on-impurity $\phi_{K_{0}}=\tanh(x)$. The impurity was chosen such
that this is still a solution to the field equations, as can be seen in eq.
(10). The subscript $0$ denotes that it is bound to the impurity at the
origin.

Figure 1: Profile of BPS solutions. (Solid) Isolated lump, (Dashed) antikink
and lump, (Dotted) antikink-on-impurity. We fix $\lambda=-1.0$. Source:
Results obtained by the author in [Campos e Mohammadi 2020].

Then, we turn to the BPS solutions $\phi_{LA}(x;x_{0})$. The system does not
possess translation symmetry anymore due to impurity. However, as half of the
BPS property is preserved, there is still a generalized translational symmetry
between BPS solutions. Defining the generalized coordinate, or modulus, of the
BPS solution as the point $x_{0}$ where $\phi_{LA}(x_{0};x_{0})=0$, one can
find a family of BPS solutions. The profile is obtained integrating the BPS
equation numerically from this initial condition and, as we move in moduli
space, the solutions vary non-trivially. Due to the presence of the impurity
near the origin, the $\phi^{4}$ antikink is deformed and, in general, consists
of an antikink $A$ and a lump $L$ as shown in the dashed curve in Fig. 1. When
the modulus $x_{0}=0$, we have the antikink-on-impurity solution $A_{0}$,
which is shown in the dotted curve in Fig. 1. In the limits where
$x_{0}\to\pm\infty$, we find the lump solutions $L^{\pm}$, which are the
deformed solution corresponding to the trivial vacuum solutions $\phi=\pm 1$
in the absence of impurities. The negative solution $L^{-}$ is shown in the
solid curve in Fig. 1. The aforementioned solutions vary with the parameter
$\lambda$ and only exist for $\lambda>-\sqrt{2}$. We fixed $\lambda=-1$ for
reasons that will appear when we discuss the fermion bound states.

Besides the solutions discussed above, we can also find approximate solutions
using the additive ansatz. One prescription to construct such solutions is to
add a solution of the $\phi^{4}$ model far from the origin and a solution of
the full model with the impurity. For instance, we have the approximate
solution

$\phi(x)=\phi_{K_{0}}(x)+\phi_{A}(x;x_{0})\pm 1,$ (15)

where $\phi_{A}(x;x_{0})=-\tanh(x-x_{0})$ is the antikink solution of the
$\phi^{4}$ model. This is a good solution for $|x_{0}|\gg 1$ and the sign on
the last term depends on the sign of $x_{0}$. Of course, these solutions are
not static because there is a force between the kink-on-impurity and the
antikink. We could also consider boosted antikink solutions
$\phi_{A}(x,t;x_{0},v)=-\tanh(\gamma(x-vt-x_{0}))$ instead of the
$\phi_{A}(x;x_{0})$ in the ansatz.

Using the additive ansatz, it is possible to write an approximation to the
solution $\phi_{LA}(x;x_{0})$ for $|x_{0}|\gg 1$

$\phi(x)=\phi_{L^{\pm}}(x)+\phi_{A}(x;x_{0})\mp 1.$ (16)

Again, the sign in the equation depends on the sign of $x_{0}$. Similarly, in
the expression above, we could include a boost in the $\phi^{4}$ antikink. In
this case, and if the $\phi^{4}$ antikink velocity is small, the solution
approximates the system’s dynamics near the BPS regime. In the next section,
we will consider the evolution of the scalar field, starting with the
aforementioned approximate solutions.

#### 3 Fermion bound states

Now, let us investigate the behavior of the fermion in the presence of some
solutions of the scalar field. As mentioned before, we ignore the fermion’s
back-reaction to the kink. We start by considering the ansatz
$\psi_{1}=\eta_{+}\cos(\omega t-\theta)$ and $\psi_{2}=\eta_{-}\sin(\omega
t-\theta)$. Substituting in eqs. (13) and (14), we find two decoupled
equations for $\eta_{\pm}$

$-\partial_{x}^{2}\eta_{\pm}+g(g\phi^{2}\mp\partial_{x}\phi)\eta_{\pm}=\omega^{2}\eta_{\pm}.$
(17)

These are Scrödinger-like equations with potentials
$V_{\pm}=g(g\phi^{2}\pm\partial_{x}\phi)$. If the scalar field consists of the
$\phi^{4}$ kink, these equations have well-known analytical solutions, which
can be found, for instance, in [Chu e Vachaspati 2008, Charmchi e Gousheh
2014]. They consist of a bound state for every integer in the range $0<n<g$
and the zero mode. Moreover, if one considers boosted $\phi^{4}$ kinks, it is
necessary to boost the fermion field as well. This is achieved in the usual
way, which we will revise following [Tong 2006].

A vector is defined as being transformed under a Lorentz transformation by the
matrix

$\Lambda=\exp\left(\frac{1}{2}\Omega_{\rho\sigma}\mathcal{M}^{\rho\sigma}\right),$
(18)

where $\Omega_{\rho\sigma}$ is an antisymmetric constant matrix and
$\mathcal{M}^{\rho\sigma}$ are the Lorentz group generators in the vector
representation. They are given by

$(\mathcal{M}^{\rho\sigma})^{\mu\nu}=\eta^{\rho\mu}\eta^{\sigma\nu}-\eta^{\sigma\mu}\eta^{\rho\nu}.$
(19)

For a boost in the positive $z$ direction, we set
$\Omega_{01}=-\Omega_{10}=\chi$. After lowering one index in
$\mathcal{M}^{\rho\sigma}$, we get

${\Lambda^{\mu}}_{\nu}=\exp\begin{pmatrix}0&\chi\\\
\chi&0\end{pmatrix}=\begin{pmatrix}\cosh\chi&\sinh\chi\\\
\sinh\chi&\cosh\chi\end{pmatrix}.$ (20)

The parameter $\chi$ is known as rapidity, defined as $\chi=\tanh^{-1}v$.
Using this definition, the matrix assumes the usual form.

Now, we consider the spinor representation of the Lorentz group. In this case,
a Lorentz transformation is performed by the operator

$S[\Lambda]=\exp\left(\frac{1}{2}\Omega_{\rho\sigma}S^{\rho\sigma}\right),$
(21)

where $S^{\rho\sigma}$ are the generetors of the Lorentz group in the spinor
representation. They are given by

$S^{\rho\sigma}=\frac{1}{4}[\gamma^{\rho},\gamma^{\sigma}].$ (22)

In particular,

$S^{01}=-S^{10}=\frac{1}{4}[\gamma^{0},\gamma^{1}]=\begin{pmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{pmatrix}.$ (23)

Using the same values for $\Omega^{\rho\sigma}$, the spinor transformation
matrix becomes

$S[\Lambda]=\exp\begin{pmatrix}0&\chi/2\\\
\chi/2&0\end{pmatrix}=\begin{pmatrix}\cosh(\chi/2)&\sinh(\chi/2)\\\
\sinh(\chi/2)&\cosh(\chi/2)\end{pmatrix}.$ (24)

Finally, the full transformation will be given by

$\psi(x)\to S[\Lambda]\psi(\Lambda^{-1}x).$ (25)

We define the transformed coordinates $x^{\prime}=\gamma(x-vt)$ and
$t^{\prime}=\gamma(t-vx)$ and the full Lorentz transformation for the spinor
system is

$\displaystyle\psi_{1}^{\prime}=\cosh(\chi/2)\psi_{1}(x^{\prime},t^{\prime})+\sinh(\chi/2)\psi_{2}(x^{\prime},t^{\prime}),$
(26)
$\displaystyle\psi_{2}^{\prime}=\sinh(\chi/2)\psi_{1}(x^{\prime},t^{\prime})+\cosh(\chi/2)\psi_{2}(x^{\prime},t^{\prime}).$
(27)

Moving on to other configurations of the scalar field, we calculated the
fermion spectrum in the presence of the lump, which can only be done
numerically. The result is shown in Fig. 2. In the figure, we show all
parameter $\lambda$ values, but we fix $g=2.0$, which is the supersymmetric
value. This value is interesting because, in this case, the equation for
perturbations around the kink solutions is identical to the equation for
fermionic bound states. The spectrum of perturbations around the kink solution
was computed in [Adam, Romanczukiewicz e Wereszczynski 2019], and indeed it
coincides with the fermion spectrum that we found. For positive values of
$\lambda$, it is easy to show graphically that the potential $V_{\pm}$ has no
minimum and, therefore, has no bound states. For $\lambda<0$, a bound state
appears. The difference between the two cases is that the peak at the center
of the lump $L^{\pm}$ is inverted with respect to the value $\phi=\pm 1$ when
$\lambda$ changes sign. As $\lambda$ approaches the value $-\sqrt{2}$, the
lump consists of infinitely separated $\phi^{4}$ kink and antikink. In this
case, the spectrum consists of a zero mode and two-degenerate discrete modes
with $\omega=\sqrt{3}$. In the following analysis, we fixed $\lambda=-1$
because this is an intermediate value in the range where the lump possesses a
bound state.

Figure 2: Spectrum of fermion coupled to (a) the lump and (b) to the BPS
antikink solution with $\lambda=-1$. We set $g=2.0$, which corresponds to the
supersymmetric case. Source: Results obtained by the author in [Campos e
Mohammadi 2020].

The next configuration of the scalar field that we analyzed was the BPS
antikink $\phi_{LA}(x)$. In this case, the spectrum as a function of the
modulus $x_{0}$ is shown in Fig. 2(b) for $\lambda=-1$. For large $|x_{0}|$
the spectrum consists of the ones of a separated $\phi^{4}$ antikink and the
lump. This configuration has the zero mode and the shape mode with
$\omega=\sqrt{3}$ from the antikink and three discrete modes from the lump.
For smaller values of $|x_{0}|$, the spectrum is slightly deformed, and the
highest-energy solution disappears in the continuum.

In the next section, we will simulate the behavior of fermions during
collisions of kinks and lumps, and we will show how to use these solutions to
measure the asymptotic states of the system.

### 3 Results

#### 1 Scalar field collisions

Figure 3: Spacetime evolution of the scalar field in (a) near BPS regime and
(b) non-BPS regime. (c) and (d) Same as before for the fermion density.
Parameters are $g=2.0$, $v=0.3$ and $\lambda=-1.0$.

Source: (a) and (b) Reproduction of the results in [Adam, Romanczukiewicz e
Wereszczynski 2019]. (c) and (d) Results obtained by the author in [Campos e
Mohammadi 2020].

We will consider two distinct types of collisions in the scalar field, one
near the BPS regimes and one that is not. The first one is the evolution of
eq. (15) and the second one of eq. (16). In both cases, we will consider that
the $\phi^{4}$ antikink is boosted instead. These solutions will be used as
the initial condition, and the equations of motion will be integrated
numerically according to the numerical method described in section 5. Both
cases were analyzed in [Adam, Romanczukiewicz e Wereszczynski 2019], and here
we reproduce the results for completeness.

Considering eq. (15), it consists of $\phi^{4}$ antikink and a lump. After
boosting the antikink with $v=0.3$, we obtain the result shown in Fig. 3(a).
The antikink moves in the direction of the lump and crosses, leaving behind a
lump in the opposite vacuum. This can be written schematically as

$A+L^{-}\to L^{+}+A.$ (28)

As the collision is close to the BPS regime, we expect the antikink to move
smoothly through the lump. However, we considered $v=0.3$, which is quite far
from the small velocity regime. Therefore, we can see in the figure that the
lump is excited in the process and oscillates.

Considering eq. (15), it consists of an antikink and the kink-on-impurity.
After boosting the antikink with $v=0.3$, we obtain the result shown in Fig.
3(b). The antikink moves in the direction of the kink-on-impurity and is
reflected after a single bounce. The kink-on-impurity is tightly bound at the
impurity and remains at the origin. This can be written schematically as

$A+K_{0}\to A+K_{0}.$ (29)

This collision is far from the BPS regime, and it generates much more
radiation than the previous case, as is clearly seen in the figure. Moreover,
the kink-on-impurity and the antikink are excited in the process. Similar to
the pure $\phi^{4}$ model, for smaller velocity values, the antikink, and the
kink-on-impurity annihilate and, for other velocity values, the system can
even exhibit resonance windows. However, we will be focusing on the reflection
regime because, otherwise, the system does not have a well-defined final
state. In the following subsections, we will consider the evolution of the
fermion field in the presence of these scalar field collisions.

#### 2 Bogoliubov coefficients

Now, turning to the fermion field we are interested in the transition
probability between the allowed fermion states. After initializing the scalar
field, we set the initial state of the fermion field at one of the fermion
states bounded to one of the defects present on the scalar field at $t=0$.
This construction is possible because the defects are well separeted in the
initial condition. We call this state $j$ and its wavefunction
$\psi_{\text{in}}^{j}(t=0)$. Then, we integrate the evolution of the fermion
field according to the dirac equation subject to the time-dependent scalar
field. This solution is denoted $\psi_{\text{in}}^{j}(t)$. At a later time,
the scalar field will reach an asymptotic state composed by a different set of
defects, which possesses another set of fermion bound states. We call this new
basis set $\psi_{\text{out}}^{k}$, where $k$ can be any allowed fermion bound
state of the final configuration. We can expanded our evolved solution in
terms of the new basis set as follows

$\psi_{\text{in}}^{j}(t)=\sum_{k}B_{j\to
k}(t)\psi_{\text{out}}^{k}(t)+\text{continuum}.$ (30)

The expansion coefficients $B_{j\to k}$ are called Bogoliubov coefficients and
are precisely the transition amplitude between the initial state $j$ and the
final state $k$. They can be found by projecting the evolved state into the
new basis set

$B_{j\to
k}=(\psi_{\text{out}}^{k}(t),\psi_{\text{in}}^{j}(t))\equiv\int(\psi_{\text{out}}^{k}(t))^{T}\psi_{\text{in}}^{j}(t)dx.$
(31)

A detailed derivation of the quantum interpretation of the Bogoliubov
coefficients can be done by expanding the fermion field in terms of particle
operators. This can be found, for instance, in [Saffin e Tranberg 2007].

#### 3 Adiabatic evolution

Figure 4: Snapshots of the evolution of the (a) scalar field and (b) fermion
field for several instants during a near BPS evolution of the system.
Parameters are $v=0.02$ and $g=2.0$. Dotted curves are the static BPS
solutions and the fermion densities of the fermion zero mode of the static BPS
solutions at the same positions. The two curves are indistinguishable in this
scale. Source: Results obtained by the author in [Campos e Mohammadi 2020].

Now, we will study the system’s evolution in the near BPS regime. The initial
condition, in this case, is given by eq. (16) considering a boost in the
antikink with a small velocity $v$. Of course, we are considering that the
antikink and the lump start well separated. For small velocities, the
configuration of the scalar field is very close to the true BPS state, and a
BPS configuration can well approximate the evolution at any time. In this
case, the system is said to be evolving adiabatically. This was shown in
[Adam, Romanczukiewicz e Wereszczynski 2019]. To confirm these results we plot
the scalar field evolution for $v=0.02$ in Fig. 4(a). The solid lines are the
solution of the equations of motion at the instants where the zero of the
scalar field is at $x_{0}\simeq-5.0$, $-3.0$, $0.0$, $3.0$ and $5.0$. The
dotted lines are the BPS solution with a modulus at the same values of
$x_{0}$. As expected, both curves are indistinguishable in the scale of the
figure. This confirms that the evolution is adiabatic.

This analysis raises the question of whether the evolution of the fermion
field also evolves adiabatically near the BPS regime. To answer this question,
we start the fermion field at the fermion zero mode bound to the antikink of
the initial scalar field configuration. Then, we let it evolve according to
the Dirac equation with the time-dependent scalar field as a background. The
numerical method employed to integrate the Dirac equation is described in
section 5. We define the fermion density as

$n=\psi_{1}^{2}+\psi_{2}^{2}.$ (32)

Its configuration at the same instants of time considered above is shown in
the solid lines of Fig. 4(b). We compare these solutions to the fermion zero
mode bound of the BPS antikink solutions with a modulus at the same values of
$x_{0}$. The fermion density of these solutions is shown in dotted lines in
the figure. In this case, the solid and dotted lines are indistinguishable in
the scale of the figure. This confirms that the fermion field also evolves
adiabatically near the BPS regime.

#### 4 Relativistic evolution

Figure 5: Bogoliubov coefficients as a function of time for a collision
between (a) antikink and lump, and (b) antikink and kink-on-impurity.
Parameters are $v=0.3$, $g=2.0$ and $\lambda=-1.0$. Source: Results obtained
by the author in [Campos e Mohammadi 2020].

For the two types of collision considered for the scalar field, we will study
the evolution of the fermion field. These are the near BPS collision, which
occurs between an antikink and a lump, and the non-BPS one, which occurs
between an antikink and the kink-on-impurity. To simplify the analysis, we
will fix the initial condition of the fermion field at the fermion zero mode,
bound to the antikink, which we will denote as $A$. Other possible allowed
states111The notation for the fermionic bound states is very similar for the
notation of defects present in the scalar field. However, it is usually clear
from the context which of the two we are referring to. are the first excited
state bound to the antikink $AE$, the zero mode bound to the kink-on-impurity
$K$ as well as the first excited state $KE$ and the lowest fermion state bound
to the lump $L$. The Bogoliubov coefficients between our initial state and the
others are defined as follows

$\alpha\equiv B_{A\to A}(t),\quad\beta\equiv B_{A\to K},\quad\gamma\equiv
B_{A\to AE}(t),\quad\delta\equiv B_{A\to KE}(t),\quad\xi\equiv B_{A\to L}(t).$
(33)

Higher excited states also exist for large enough $g$, but we will not measure
the transition to these states.

The evolution of the fermion density for this initial condition is shown in
Fig. 3(c) and (d) for the near BPS and non-BPS regimes, respectively. In both
cases, the fermion density is initially localized at the antikink, which is
approaching the origin. After colliding with the lump, in the near BPS case,
and with the kink-on-impurity, in the non-BPS case, part of density stays
bound at the antikink, and part is transferred to the other defect. Moreover,
part of the fermion density is lost in the form of radiation. This effect is
more pronounced in the non-BPS collision. Interestingly, the radiation of the
fermion field is not very significant, even in the non-BPS case, and most
fermions stay bound to one of the two defects.

We can compute the evolution of the Bogoliubov coefficients with time by
calculating the inner product at each time instant according to eq. (31). The
result is shown in Fig. 5. Before the collision, we have $\alpha=1.0$ due to
our choice of initial conditions. During the collision, we cannot separate the
different defects, and the coefficients’ value is not reliable. In particular,
the coefficient $\xi$ refers to the transition to the lowest fermion state
bound to $L^{-}$ before the near-BPS collision, and at some point, we have to
change the definition to compute the transition to the lowest fermion state
bound to $L^{+}$. This is necessary because the lump adjusts to the boundaries
as the antikink passes through the origin. Finally, when the defects are
farther apart and can be distinguished again, the Bogoliubov coefficients
reach an asymptotic value that tells the transition amplitude between states
due to the collision.

An important technical detail is that the Bogoliubov coefficients oscillate
with time except for the zero mode. This can be understood by looking at our
ansatz and noticing that the fermion field oscillates with phase $\omega
t+\theta$. However, in the simulations, the value of $\theta$ is not known.
Thus, we project the fermion field onto an excited state with a fixed phase at
an arbitrary value $\theta_{0}$. Due to the field’s oscillation, the phases
will coincide once every period and, when they coincide, the coefficient will
reach a peak, which is precisely the value of the Bogoliubov coefficient.

An intriguing effect that should be noticed in Fig. 5(b) is that $\beta^{2}$
does not reach a well-defined asymptotic value because there is a small
oscillation in its value. This oscillation is accompanied by an oscillation in
the amplitude of $\delta^{2}$. Moreover, the two are negatively correlated.
This can be understood by noting that the states we are using when computing
the projection in eq. (31) are fermion bound states to static or boosted kink
and antikink as well as bound states to the lump. However, the final
configuration of the process in eq. (29) is not any of those because the kink-
on-impurity is excited and oscillates around the origin. This oscillation can
be interpreted as a perturbation around the kink fixed at the origin.
Therefore, if we project our final state on the fermion states bound to the
kink at the origin, there will be a transition between the states due to the
perturbation.

Figure 6: Final value of the Bogoliubov coefficient as a function of $g$ for
several values of $v$. We consider the near BPS collision between antikink and
lump and fix $\lambda=-1.0$. Source: Results obtained by the author in [Campos
e Mohammadi 2020].

Having said that, we will use the asymptotic value of the Bogoliubov
coefficients as a measurement of the amount of fermions that are transferred
from one defect to the other due to the collision. First, considering the near
BPS collision between the $\phi^{4}$ antikink and the lump, we find the result
shown in Fig. 6. These values are shown for relativistic velocities in the
range $v\geq 0.1$. For $v=0.1$, the value of $\alpha^{2}$ is close to $1.0$,
which means that the fermion stays at the fermion zero mode bound to the
antikink. This is expected because we are approaching the adiabatic limit
discussed previously. Moreover, for this value of $v$, a small fraction of
fermions are transferred to the lump. This fraction gets more significant as
$g$ increases. In general, as the velocity or the coupling constant increases,
less fermions stay at the zero mode bound to the antikink. In Fig. 6, we see
the $\alpha^{2}$ decreases as $v$ and $g$ increases. Accordingly, $\gamma^{2}$
has the opposite behavior, while $\xi^{2}$ has a similar tendency but has a
more complicated behavior. The amount of fermions transferred to higher
excited states or lost as radiation is quantified by the difference of the sum
$\alpha^{2}+\gamma^{2}+\xi^{2}$ from unity. In Fig. 6(d), we see that there is
less tendency for the fermions to stay at the lowest excited states as $v$
increases, presumably because the collisions are more energetic. However, we
do not see the same tendency as $g$ increases.

Figure 7: Final value of the Bogoliubov coefficient as a function of $g$ for
several values of $v$. We consider the non-BPS collision between antikink and
kink-on-impurity and fix $\lambda=-1.0$. Source: Results obtained by the
author in [Campos e Mohammadi 2020].

Then, we consider the non-BPS collision between an antikink and the kink-on-
impurity. The values of $v$ we used are also relativistic, and we set $v\geq
0.3$ to guarantee that the antikink and the kink-on-impurity will not
annihilate. The behavior of fermions when the kink and antikink annihilate is
to escape as radiation in a chaotic manner slowly. We will not discuss this
case further. The result for larger velocities is shown in Fig. 7. We show in
the plot the final values of $\alpha^{2}$, $\beta^{2}+\delta^{2}$,
$\gamma^{2}$, and the sum of the three quantities. We do not plot the terms
$\beta^{2}$ and $\delta^{2}$ separately because they oscillate, as discussed
previously, while the sum has a well-defined value. It is clear that in the
non-BPS case, the system’s behavior is more complicated and very sensitive to
the parameters. The behavior of $\alpha^{2}$ and $\beta^{2}+\delta^{2}$ are
approximately sinusoidal and negatively correlated. This approximate behavior
was justified in [Gibbons, Maeda e Takamizu 2007] by solving the Dirac
equation with an ansatz for the fermion field symmetric with respect to $x$,
with time-dependent amplitude and phase, and interacting with a constant
scalar field at its maximum value during the collision.

In Fig. 7(c), we show the values of $\gamma^{2}$. They have a complicated
dependence on $g$. The fraction of fermions that are transferred to this state
is small in general but can be significant in a few cases. Again, we quantify
the amount of fermions that is not transferred to higher excited states and
not lost as radiation by the sum $\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}$.
It is plotted in Fig. 7(d). We can also see that the sum tends to decrease for
larger values of $v$, where the collision is more energetic.

We will discuss and summarize our results and conclusions in the following
sections and mention some ideas for future projects.

### 4 Conclusion

In this work, we considered a scalar field with a half-BPS preserving defect
coupled to a fermion field via a Yukawa coupling. The scalar field possesses
BPS solutions as well as non-BPS solutions. The BPS ones can be interpreted as
being composed of two defects: a $\phi^{4}$ antikink and a lump. When the
fermion couples to the scalar field, it gains states bound to the defect
configurations of the system considered.

Among the many possible collisions of the scalar field, we considered a near-
BPS one, which occurs between the antikink and the lump, and a non-BPS one,
which occurs between the antikink and the kink-on-impurity. Then, we coupled
the fermion to the scalar field, starting with the fermion bound to the
antikink at the zero mode. For the near-BPS collision, the scalar field
configuration and the fermion evolve adiabatically in moduli space for non-
relativistic velocities.

We found that after the collision, most fermions stay bound to one of the two
defects at the final state. Only a tiny fraction is lost as radiation,
especially for the near-BPS regime. Then, we quantified these transitions via
the Bogoliubov coefficients, which reach an asymptotic value for late times.
This asymptotic value measures the transition probabilities between defects
due to the whole collision process. They have complicated behavior, but it is
possible to see a few tendencies. For the near-BPS case, the collision is
almost adiabatic for small $v$ and $g$. As these quantities increase, the
probability of being transferred to the other defect or to move to an excited
state at the same defect also increases. Moreover, we computed the difference
of the sum of the probability to stay at the lowest excited states from unity.
We found that the amount of fermions transferred to higher excited states or
lost as radiation increases as the collision becomes more energetic. For the
non-BPS, the behavior is much less regular, but the amount of fermions that
are transferred to higher excited states or lost as radiation also tends to
increase as the collision becomes more energetic.

This work is most relevant because, if we view the BPS antikink and a multi-
defect configuration composed of a $\phi^{4}$ antikink and a lump, we can use
our analysis to guide the study of higher dimensional systems, where multi-
defect BPS configurations can exist as well. A natural continuation of this
work is to consider situations where there is a back-reaction of the fermion
to the scalar field and investigate how the inclusion of this term affects
kink-antikink collisions. Moreover, it would be interesting to analyze whether
it is possible to define Bogoliubov coefficients in this case as well.

### 5 Appendix: Numerical method

To integrate the equations of motions (10) and (11), we employ the method of
lines. It consists of discretizing space with spacings $h=0.01$. The scalar
field at the gridpoints $x_{i}$ is denoted by $\phi_{i}(t)$, where
$i=0,1,\ldots,N$. We make similar definitions for the fermion field. The
spatial derivatives are discretized according to a second-order finite
difference method. For the fermion field, we need a first derivative, which
reads

$\frac{\partial\psi_{i}}{\partial x}=\frac{\psi_{i+1}-\psi_{i-1}}{2h}.$ (34)

For the scalar field, we need a second derivative, which reads

$\frac{\partial^{2}\phi_{i}}{\partial
x^{2}}=\frac{\phi_{i+1}-2\phi_{i}+\phi_{i-1}}{h^{2}}.$ (35)

In the expressions above, we omitted the time dependence for simplicity. This
results in a set of first-order ordinary differential equaitons for a vector
containing the values at the grid points of the components of the fermion
field, the scalar field $\phi$ and its derivative $\phi_{t}$. Note, that we
need to define an auxiliary field equal to $\phi_{t}$ to transform the
equation of motion for $\phi_{i}(t)$, which is second-order, into two first-
order equations. The equations are integrated using a fourth-order Runge-Kutta
method with step size $\tau=0.01$. We fix the boundaries at $x=\pm 100.0$,
which gives $N=20000$. The boundary conditions are
$\psi_{0}(t)=\psi_{N}(t)=0.0$, $\phi_{0}(t)=1.0$ and, depending on the case
considered, we can have either $\phi_{N}(t)=\pm 1.0$. The final time of the
integration is in the range $100.0<t_{f}<400.0$. These values are short
enough, and therefore, the boundary does not interfere with the bulk evolution
of the system. The fermion field is normalized to one, and the time
integration is approximately unitary and conserves the normalization during
the whole evolution of the system up to a relative error of order $10^{-5}$.

## 8 FINAL REMARKS

We started this thesis by defining solitons, solitary waves and topological
defects, and giving a brief historical introduction to these concepts. Then,
we discussed important theoretical and applied works in the literature. In
section 2, we discussed some general properties of kinks and made a careful
revision of kink-antikink collisions and the resonant energy exchange
mechanism.

In section 3, we described a model that we constructed such that the
vibrational mode of the kink becomes a quasinormal mode. Our toy model made it
possible to compute the frequency of the quasinormal mode and the decay rate
analytically because the stability potential was constant in different
domains. In most models, it is impossible to perform this task, and numerical
evaluation of the quasinormal mode can be tricky. Moreover, we were able to
confirm previous results in the literature, such as the gradual disappearance
of the resonant structure. The toy model potential was constructed using a
continuous and piecewise-defined function with a continuous derivative. This
is an exciting possibility that has been little explored. This prescription
could be used to construct new topological models as done, for instance, in
[Basak, Roy e Kar 2021]. One interesting idea for future work is to study the
collective coordinates effective model of this toy model because the kink and
vibrational modes are simple. The collective coordinates method could also be
applied for newly constructed toy models.

In section 4, we considered a scalar field with a family of polynomial
potentials of higher order. The potentials in this family are analogous to the
$\phi^{4}$ one with two minima. The crucial difference is that, for some
values of the parameters, the first contribution when we expand the potential
around the vacua is not quadratic. It can be quartic, sextic, and so on. In
this case, we obtain a kink with long-range tails on both sides of the kink,
which decay as a power-law. The long-range character of the tails makes it
impossible to use the usual ansatz to initialize the system. Therefore, it is
necessary to introduce a minimization procedure to find a suitable initial
condition. The procedure consists of finding a configuration that obeys the
equations of motion of a traveling wave as close as possible. This is done
starting with an initial guess and using a least-square minimization
algorithm. Similarly, the velocity field must obey the zero-mode equation of a
traveling wave as close as possible.

After correctly initializing the system, we found the kink-antikink
annihilated directly into radiation for velocities below an ultra-relativistic
critical value. One interesting continuation of this work is to apply the
minimization procedure to other families of models containing kinks with long-
range tails. In particular, one could construct a model that starts from the
$\phi^{4}$ model and gradually turns into a model with long-range tails. This
way, it should be possible to understand how the critical velocity becomes
ultra-relativistic, and the bion formation is suppressed. Finally, it should
be interesting to study the interaction of kinks with long-range tails in the
presence of boundaries obeying Neumann boundary conditions or any other types
of boundary conditions. Interactions of kinks with boundaries have very rich
physics and have not been fully explored yet.

In section 5, we investigated collisions of the double sine-Gordon model. We
simulated kink-antikink collisions in this model for a wide range of model
parameters and indicated the regions where annihilation and separation occur
after one or more bounces. Surprisingly, the multi-bounce region disappears as
we approach the integrable limits of the model, despite the presence of a
vibrational mode. Then, we simulated collisions where the kink and the
antikink have their vibrational mode symmetrically excited beforehand. This
excitation led to the appearance of one-bounce resonance windows and a more
complex fractal pattern of higher-resonance windows. After characterizing the
behavior of each region in parameter space, we found that a spine structure
appears in the region of separation after one bounce. Interestingly, the
excitation increased the range in parameter space where higher-bounce
resonance windows occur, showing that the resonance structure was only hidden
near the integrable limits. Finally, we were able to find an approximate model
to the position of one-bounce resonance windows and to characterize the
separation regions using the maximum value of the energy densities.

One fascinating continuation to the work in section 5 is to study other
modifications of the sine-Gordon model. Also, one could try to understand the
behavior near the integrable limits analytically using the perturbation
theory. Finally, it should also be possible to compare the simulations results
to an effective model using collective coordinates. However, the effective
Lagrangian can only be found numerically in this case.

In section 6, we introduced the concept of kinks coupled with fermions. Then,
we discussed important results in the literature of fermion-kink systems and
listed a few general analytical properties relevant to those systems.

In section 7, we analyzed the effect of the shape mode excitation on a fermion
that is bound to a kink. This system is very important because the kink is
expected to be excited after an interaction. In this case, the equations of
motion can be solved analytically using perturbation theory. However, it is
necessary to neglect the fermion back-reaction on the scalar field in order to
do so. We considered both the $\phi^{4}$ model and the toy model used in
section 3, which possesses quasinormal modes. The fermionic bound states can
be found analytically in the former and can only be found numerically in the
latter. According to our results, the fermion should escape the kink at a
constant rate whenever the transition to the continuum is allowed by Fermi’s
golden rule. This is an interesting prediction that could be verified or used
to constraint the wobbling amplitude of a kink or the value of the Yukawa
coupling constant. The transition rate diminishes with time when the shape
mode is turned into a quasinormal mode, and the transition probabilities reach
an asymptotic value. Interestingly, the behavior of the asymptotic value can
be understood as being reminiscent of Fermi’s golden rule.

An exciting continuation of this work is to study fermions interacting with
kinks with an inner structure, such as the double sine-Gordon. This could be
done with and without wobbling. Moreover, one could consistently include the
fermion back-reaction and see how the results are modified. Of course, when
fermion back-reaction is considered, it opens many possibilities for future
works as very few works in the literature include this effect.

Finally, in section 8, we considered defect collisions of a scalar field model
with a half-BPS preserving impurity. The scalar field was used as a background
for a fermionic field which evolves as the defects collide. We considered two
types of collisions. The first one was a collision between the antikink and
the lump, which occurs near the BPS sector of the system. The second one was
between the antikink and the kink-on-impurity, which occurs far from the BPS
sector of the system. We started the fermion at the zero mode of the antikink
and measured the fermion transfer between the defects participating in the
collision process. Interestingly, the background approximation is exact for
this initial condition. One of the main results of this work was to conclude
that near the BPS regime, the evolution becomes adiabatic, and there is no
fermion transfer. For relativistic collisions, the behavior is more
complicated, but, in general, the fermion is more likely to be transferred to
the other defect, move to a higher excited state at the same defect, or be
lost as radiation.

An interesting continuation of this work is to consider cases where the
fermion back-reaction is non-vanishing and compute the evolution consistently.
For strong coupling constants, the effect of this term can be considerable.
Thus, the fermion back-reaction to the scalar field should be considered
whenever possible. However, the problem becomes considerably more complicated
and more numerically involved. For this reason, the study of fermions without
back-reaction can be used to give some intuition on the expected behavior of
the system. An interesting question in light of the work [Adam et al. 2019] is
whether a fermion excitation could also lead to the appearance of spectral
walls when the back-reaction is included. We plan to study this and other
questions in future works.

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