# An improved Quantum Max Cut approximation
via matching

Eunou Lee Korea Institute for Advanced Study, Seoul, South Korea
<EMAIL_ADDRESS>
###### Abstract

Finding a high (or low) energy state of a given quantum Hamiltonian is a
potential area to gain a provable and practical quantum advantage. A line of
recent studies focuses on Quantum Max Cut, where one is asked to find a high
energy state of a given antiferromagnetic Heisenberg Hamiltonian. In this
work, we present a classical approximation algorithm for Quantum Max Cut that
achieves an approximation ratio of 0.584 given a generic input, and a ratio of
0.595 given a triangle-free input, outperforming the previous best algorithms
of Lee [10] (0.562, generic input) and King [8] (0.582, triangle-free input).
The algorithm is based on finding the maximum weighted matching of an input
graph and outputs a product of at most 2-qubit states, which is simpler than
the fully entangled output states of the previous best algorithms.

## Introduction

A quantum optimization problem asks one about the maximum (or minimum) of a
function that is defined over the $n$-qubit Hilbert space. In a restricted
case where the function is a sum of $k$-qubit Hamiltonians, it is well known
that the problems are in general QMA-hard, i.e. hard to solve to an inverse
polynomial precision even with a quantum computer [4]. One way to cope with
the computational hardness is to try to find approximately good solutions.

Quantum Max Cut (QMC) serves as a good benchmark problem to test our knowledge
of quantum Hamiltonian approximation. It has a simple definition, has a good
physical motivation, namely the antiferromagnetic Heisenberg model, and it
extends a classical problem of Max Cut. The task of QMC, given a positive
weighted graph $G=(V,E,W),$ is to find a maximum energy state for the
Hamiltonian

$\displaystyle H=\sum_{(i,j)\in
E}w_{ij}(I-X_{i}X_{j}-Y_{i}Y_{j}-Z_{i}Z_{j})/4,$

where $X_{i}$ is the Pauli matrix $X$ on qubit $i$ and identity on the rest.
Similarly, $Y_{i},Z_{i}$ are defined.

Most of the existing approximation algorithms for QMC follow the framework of
the seminal Goemans-Williamson algorithm [6]. The problem is first relaxed to
a semidefinite program (SDP), then the SDP is solved in polynomial time, and
finally, the SDP solution is rounded to a feasible solution of the original
problem. Since QMC maximizes over the $n$-qubit Hilbert space, the rounded
solution should be an $n$-qubit quantum state instead of an $n$-bit string as
in Max Cut. Assuming that we follow the Goemans-Williamson framework to
approximate QMC, we still have to decide three design choices: 1) what SDP to
relax the original problem to, 2) what subset of quantum states (ansatz) to
round the SDP solution to, and 3) how to round the SDP solution to an ansatz
state. None of the questions is settled as the current state of matter.

Quantum Lasserre SDP hierarchy [9, 11] is a sequence of SDPs that upper bounds
the maximum energy of a given quantum Hamiltonian. In the level-$k$ Lasserre,
up to $k$-qubit entanglement is considered. Hence the SDPs become tighter and
tighter as the level goes up in the hierarchy, eventually converging to the
given quantum Hamiltonian problem when $k=n$. The following is a way to view
the SDP construction. Fix a quantum state $|\phi\rangle$. For an $n$-qubit
Pauli matrix $P,$ define $v(P):=P|\phi\rangle.$ Then the energy of
$|\phi\rangle$ for an arbitrary Hamiltonian can be expressed as a sum of inner
products of these vectors. For example,
$\langle\phi|I-X_{i}X_{j}-Y_{i}Y_{j}-Z_{i}Z_{j}|\phi\rangle=\langle
v(I),v(I)\rangle-\langle v(X_{i}X_{j}),v(I)\rangle-\langle
v(Y_{i}Y_{j}),v(I)\rangle-\langle v(Z_{i}Z_{j}),v(I)\rangle$. Additionally, it
holds that $\|v(P)\|=1$, and $\langle v(P_{1}),v(Q_{1})\rangle=\langle
v(P_{2}),v(Q_{2})\rangle$ for all $n$-qubit Pauli matrices
$P_{1},P_{2},Q_{1},Q_{2}$ such that $P_{1}Q_{1}=P_{2}Q_{2}.$ Now forget about
$|\phi\rangle$ and maximize the energy expression in terms of $v(P)$’s, while
satisfying the inner product relations.

In the following definition, $\mathcal{P}_{k}(n)$ is the set of $n$-qubit
Pauli matrices with non-trivial terms on up to $k$ qubits.

######  Definition 1 (Level-$k$ Quantum Lasserre SDP).

$\displaystyle\text{Maximize }\sum_{(i,j)\in
E}w_{ij}v(I)\cdot(v(I)-v(X_{i}X_{j})-v(Y_{i}Y_{j})-v(Z_{i}Z_{j}))/4$ (S)
subject to $\displaystyle v(P)\in\mathbb{R}^{|\mathcal{P}_{k}(n)|},$
$\displaystyle\forall P\in\mathcal{P}_{k}(n),$ $\displaystyle v(P_{1})\cdot
v(Q_{1})=v(P_{2})\cdot v(Q_{2}),$ $\displaystyle\forall
P_{1},P_{2},Q_{1},Q_{2}\in\mathcal{P}_{k}(n)\text{ s.t.
}P_{1}Q_{1}=P_{2}Q_{2},$ $\displaystyle v(P)\cdot v(P)=1,$
$\displaystyle\forall P\in\mathcal{P}_{k}(n),$ $\displaystyle v(P)\cdot
v(Q)=0,$ $\displaystyle\forall P,Q\in\mathcal{P}_{k}(n)\text{ s.t. }PQ+QP=0.$

Every existing QMC approximation algorithm that follows the SDP rounding
framework uses a quantum Lasserre SDP. The QMC approximation algorithm of
Gharibian-Parekh [5] solves the level-1 Lasserre, and Parekh-Thompson [12],
Parekh-Thompson [13], Lee [10], and King [8] solve the level-2 Lasserre SDP.
There are more sophisticated SDPs that are aware of the $SU(2)$ symmetry in
the QMC Hamiltonian that have been developed recently [14, 15].

Once a relaxed SDP is solved, its solution is rounded to a quantum state.
Current algorithms round SDP solutions to a proper subset (ansatz) of the
$n$-qubit Hilbert space. Gharibian-Parekh [5], and Parekh-Thompson [13] round
to product states, Parekh-Thompson [12] rounds to a product of 1- and 2-qubit
states. Lee [10] and King [8] round to $n$-qubit entangled states.

In this paper, we introduce a classical algorithm that solves the level-2
Lasserre SDP, and rounds to a mixture of a product of 1-qubit states and a
product of 1- and 2-qubit states. The approximation ratio is 0.584 in general,
and 0.595 if the input graph is triangle-free, which are higher than the
previous best algorithms for general inputs (Lee [10] with a ratio of 0.562),
and triangle-free inputs (King [8] with a ratio of 0.582). Our main idea is
simple. We first observe that in the previous best algorithms, the following
Monogamy of Entanglement relation is used crucially and is in fact a main
hindrance to a higher ratio.

######  Lemma 1 (Monogamy of Entanglement, [12]).

Let $G=(V=[n],E)$ be a graph and let $(v(P))_{P\in\mathcal{P}_{2}(n)}$ be a
feasible solution to the level-2 Lasserre SDP. Define, for $i,j\in[n]$,

$\displaystyle
h_{ij}:=\frac{1}{4}v(I)\cdot(v(I)-v(X_{i}X_{j})-v(Y_{i}Y_{j})-v(Z_{i}Z_{j}))-\frac{1}{2}.$

Then for any vertex $i\in[n]$ and any $S\subseteq[n]$,

$\displaystyle\sum_{j\in S}h_{ij}\leq\frac{1}{2}.$

Especially,

$\displaystyle\sum_{j\in N(i)}\text{max}(h_{ij},0)\leq\frac{1}{2},$ (1)

where $N(i)=\\{j|(i,j)\in E\\}.$

The last statement is obtained by taking the set of edges incident to $i$ with
positive $h_{ij}$ values as $S$. Notice that $(\text{max}(2h_{e},0))_{e\in E}$
forms a feasible fractional matching. We solve the Maximum Weight Matching and
assign the best 2-qubit state, namely $(|01\rangle-|10\rangle)/\sqrt{2}$, to
matched edges. To the remaining qubits, we assign uniformly random 1-qubit
states. The resulting $n$-qubit state has low energy on edges that are not
matched. To address this issue, we mix it with a product state algorithm of
Gharibian-Parekh [5]. Below is the final algorithm.

######  Algorithm 1.

Given a weighted graph $G=(V,E,W)$ as an input,

  1. 1.

Solve SDP(S) for $k=2$ to get solution vectors
$(v(P))_{P\in\mathcal{P}_{2}(n)}$.

  2. 2.

Choose $p_{0}=\begin{cases}0.586&\text{ for a general }G,\\\ 0.674&\text{ for
a triangle-free }G.\end{cases}$

  3. 3.

With probability $p_{0}$, find a high energy product state as follows:

    1. (a)

Sample a random matrix $R$ with dimension $3\times 3|\mathcal{P}_{2}(n)|$ with
each element independently drawn from $\mathcal{N}(0,1)$.

    2. (b)

Perform Gharibian-Parekh rounding on each
$v_{i}:=(v(X_{i})\|v(Y_{i})\|v(Z_{i}))/\sqrt{3}$ to get
$u_{i}:=Rv_{i}/\|Rv_{i}\|$.

    3. (c)

Output $\rho_{1}:=\prod_{i\in
V}\frac{1}{2}(I+u_{i,1}X_{i}+u_{i,2}Y_{i}+u_{i,3}Z_{i})$.

  4. 4.

With probability $1-p_{0}$, find a high energy matching state as follows:

    1. (a)

Find the maximum weight matching $M:E\rightarrow\\{0,1\\}$ of $G$, for example
via Edmonds Algorithm [3].

    2. (b)

Output
$\rho_{2}:=\prod_{(i,j):M_{(i,j)}=1}(I-X_{i}X_{j}-Y_{i}Y_{j}-Z_{i}Z_{j})/4\prod_{i\in
U}I/2$, where $U$ is the set of unmatched qubits.

Using matching to find a good QMC state is not a new idea. Anshu, Gosset, and
Morenz [1] proved that there exists a product of 1- and 2-qubit states with
energy at least 0.55 times the maximum QMC energy via matching. Furthermore,
the Parekh-Thompson algorithm [12] finds the Maximum Weight Matching of a
subgraph of an input graph as a subroutine and outputs a product of 1- and
2-qubit states very similar to the output state of our algorithm, and gets a
ratio of 0.533. We believe that the improvement is due to the fact that we
find a matching on a whole input graph, whereas Parekh-Thompson finds matching
on a proper subgraph of the input graph, and thus not utilizing the matching
ansatz to its full potential.

We view our algorithm as a “minimal” version of the previous best algorithms
by Lee [10] and King [8]. They both start with a good product state and evolve
it to an entangled state while respecting the Monogamy of Entanglement. A
difference is that in our case Monogamy of Entanglement is satisfied in the
global sense. If our algorithm is the optimal way to exploit Monogamy of
Entanglement, a tighter SDP law is necessary to deliver a QMC approximation
algorithm with a meaningfully improved approximation ratio.

Even though Monogamy of Entanglement is tight when $h_{ij}=1/|N(i)|$ for all
$j\in N(i)$, it is easy to see that the inequality is far from tight at other
points. Suppose $h_{ij}=1/2$, with $(i,j)$ being maximally entangled. Then on
a neighbouring edge $(i,k)$, the energy is 1/4 and $h_{ik}=-1/4$. In this
case, the deviation from the upper bound grows linearly as the number of
connected edges grows.

For example, a tighter bound on 2 edges can be obtained from level-2 Lasserre
solutions as stated in the Lemma below.

Figure 1: The solid region represents a feasible area characterized by
Entanglement Convexgamy whereas the red line represents the boundary of the
feasible region by Monogamy of Entanglement.

######  Lemma 2 (Entanglement Convexgamy on 2 edges).

Consider a graph $G=(V=\\{1,2,3\\},E=\\{(1,2),(2,3)\\})$. Let $x,y$ be defined
by a feasible level-2 Lasserre SDP solution as follows:

$\displaystyle x=v(I)\cdot(v(I)-v(X_{1}X_{2})-v(Y_{1}Y_{2})-v(Z_{1}Z_{2}))/4$
$\displaystyle y=v(I)\cdot(v(I)-v(X_{2}X_{3})-v(Y_{2}Y_{3})-v(Z_{2}Z_{3})/4.$

Then $(x,y)$ is confined in the region defined by the x-axis, y-axis, and the
ellipse touching the x- and y-axis and $x+y=3/2$ as depicted in Figure 1. More
specifically, the ellipse is defined as $3(x+y-1)^{2}+(x-y)^{2}=3/4$.

###### Proof.

Let $z=v(I)\cdot(v(I)-v(X_{3}X_{1})-v(Y_{3}Y_{1})-v(Z_{3}Z_{1})/4.$ Then
$x,y,z\geq 0.$ By Theorem 4.2 of [14], $x^{2}+y^{2}+z^{2}\leq 2(xy+yz+zy),$
and $0\leq x+y+z\leq 3/2.$ The first inequality is equivalent to

$\displaystyle\frac{\sqrt{x^{2}+y^{2}+z^{2}}}{x+y+z}\leq\frac{1}{\sqrt{2}}.$

It means that if $x+y+z=c\geq 0,$ then $\sqrt{x^{2}+y^{2}+z^{2}}\leq
c/\sqrt{2}$. So $(x,y,z)$ lies the intersection of the plane $x+y+z=c$ and the
sphere of radius $c/\sqrt{2}$ centered at the origin, which is the incircle of
the triangle defined by $x+y+z\geq c$ in the region $x,y,z\geq 0$. When the
circle is projected to the $xy$-plane to give a feasible subset of $(x,y)$, we
get the ellipse inscribed in the triangle defined by $(0,0),(c,0),(0,c).$
Because $0\leq c\leq 3/2$, we prove that a feasible point is in the region
defined by the $x$-axis, $y$-axis, and the ellipse touching the $x$\- and
$y$-axis and $x+y=3/2$.

The equation of the ellipse follows by solving the inscription condition. ∎

Another issue that we are concerned with is the definition of quantum
optimization problems. A common way to define a quantum optimization problem
in the literature is to define an energy function on $n$ qubits and say “find
an algorithm that outputs a high-energy state”. A problem with this definition
is that it implies that a classical algorithm can only optimize a quantum
objective function over the boolean space because it cannot output a quantum
state outside the boolean space. A more relaxed definition that frequently
appears is to ask only for a “description” of a quantum state. This definition
accommodates a bigger family of algorithms that outputs a description of an
entangled state such as our algorithm and other previous algorithms for QMC
approximation. Now the issue is that the meaning of the word description is
vague: an output state of any quantum algorithm has a classical description,
namely the quantum algorithm itself written on paper. Therefore any quantum
algorithm is a classical algorithm if we accept this definition. To resolve
this perplexity, we propose the following definition.

######  Definition 2.

Given an objective function $f$ that maps an $n$-qubit state to a real number,
a pair of polynomial time algorithms $(P,V)$ maximizes $f$ to a value $k$ if
the following conditions hold:

  1. 1.

    1. (a)

If $\text{max }f\geq k$: $\exists$ polynomial-sized $|w\rangle$ such that
$V(|w\rangle)=1$ w.p. $\geq 2/3$,

    2. (b)

If $\text{max }f\leq k-1/p(n)$ for some polynomial $p$: $\forall$ polynomial-
sized $|w\rangle$, $V(|w\rangle)=1$ w.p. $\leq 1/3$

  2. 2.

$P$ outputs $|w^{\prime}\rangle$ such that $V(|w^{\prime}\rangle)=1$ w.p.
$\geq 2/3.$

Each of $P$ and $V$ can be either classical or quantum. Therefore according to
our definition, we can have a cc-, qc-, cq-, or qq-optimization algorithm for
a quantum optimization problem depending on whether $P$ and $V$ are classical
or quantum. Only cc-optimization algorithms for QMC are known to us so far.

We argue that finding an optimal qq-optimization algorithm for QMC is a viable
path to achieve a _provable and practical quantum advantage_. Suppose the
verifier algorithm $V$ is fixed to be classical. Then, from the prover $P$’s
side, the task is to find a bit string to convince $V$ that there is a high
energy state. Assuming there is a tight NP-hard upper bound for a classical
approximation for QMC (for example by the PCP theorem), we cannot find a
quantum prover that provably gives a higher ratio than all classical provers
unless we prove that NP is a proper subset of QMA. However, to the best of our
knowledge, no serious complexity theoretic consequences are resulted by a qq-
approximation having a higher ratio than all cc-approximation. Moreover, we
already know how to upper bound classical approximation ratios in some cases
via the PCP theorem and the Unique Games Conjecture, so we can hope to upper
bound the classical ratio for QMC. The current best upper bound on cc-
approximation of QMC is 0.97 up to a conjecture [7].

We list a few interesting open problems regarding QMC approximation:

  * •

Find a rigorous quantum approximation for QMC.

  * •

Find a heuristic quantum algorithm (e.g. VQE-based) for QMC that outperforms
rigorous classical algorithms.

  * •

Find an Entanglement Convexgamy relation from a valid SDP solution on $d$
edges. Is a higher level SDP necessary for this? How high in the hierarchy
should we go to for a tight bound?

  * •

Mixing two strategies helps boost the ratio in our case. Are there other
instances where the worst-case performance of an optimization algorithm
improves simply by mixing known strategies?

## Analysis of the algorithm

In the rest of the paper, we introduce necessary concepts regarding matching
and bound the approximation ratio of our algorithm.

######  Definition 3.

Given a weighted graph $G=(V,E,W),$ consider the following program:

$\displaystyle\text{maximize }\sum_{e\in E}w_{e}m_{e},\text{ subject to
}\sum_{e:i\in e}m_{e}\leq 1,\forall i\in V.$

Maximum Weight Fractional Matching (MWFM) $M^{f}:E\rightarrow[0,1]$ is a real-
valued optimal solution to the program, and Maximum Weight Matching (MWM)
$M:E\rightarrow\\{0,1\\}$ is a binary-valued optimal solution to the program.

From the definition, it is clear that $\sum_{e\in
E}w_{e}M^{f}_{e}\geq\sum_{e\in E}w_{e}M_{e}$.

######  Definition 4.

Given a graph $G=(V,E,W)$, define the fractional matching polytope as

$\displaystyle FM:=\\{x\in\mathbb{R}^{|E|}\ |\ 0\leq x_{e}\ \forall e\in
E,\text{ and }\sum_{e:i\in e}x_{e}\leq 1\ \forall i\in V\\}.$

######  Theorem 1.

For every vertex $u$ of the poltyope $FM$ and every $e\in E$, we have
$u_{e}\in\\{0,1/2,1\\}$. If we define $E_{1/2}(v):=\\{e|e\in E\text{ s.t.
}u_{e}=1/2\\}$, $E_{1/2}(v)$ forms a disjoint union of odd-length cycles.

The following is a well-known fact from matching theory [16]. Even though the
statement is implicit in the literature, we state and prove explicitly for
clarity.

######  Lemma 3.

Given a weighted graph $G=(V,E,W),$ and its MWFM solution $M^{f}$, and MWM
solution $M,$ the following holds:

$\displaystyle\sum_{e\in E}w_{e}M_{e}\geq\frac{2}{3}\sum_{e\in
E}w_{e}M^{f}_{e}.$

If $G$ is triangle-free,

$\displaystyle\sum_{e\in E}w_{e}M_{e}\geq\frac{4}{5}\sum_{e\in
E}w_{e}M^{f}_{e}.$

###### Proof.

There exists a vertex $u$ in the polytope $FM$ where the MWFM is realized. By
Theorem 1, we know that $u_{e}\in\\{0,1/2,1\\},\forall e\in E.$

Construct an integer matching $M^{\prime}$ as follows. If $u_{e}=1,$ then set
$M^{\prime}_{e}=1$. The remaining non-zero edges have a matching value $1/2$
on them and form disjoint odd-cycles. Let $C$ be one of such disjoint odd-
cycles that has length $k$. There are $k$ different integer matchings on the
cycle where $(k-1)/2$ edges are matched. Each edge is matched in $(k-1)/2$ of
those matchings. Therefore, on each edge $e$ of the cycle, the average
objective value from the $k$ different integer matchings is $w_{e}(k-1)/(2k)$.
Summed over edges on the cycle, this gives $(k-1)/k\cdot\sum_{e\in C}w_{e}/2$
on average. At the vertex $u,$ the objective value on the cycle is $\sum_{e\in
C}w_{e}/2$. Therefore choosing the best integer matching on $C$ as part of the
integer matching $M^{\prime}$ gives at least $(k-1)/k$ times the best
fractional matching on the cycle. Since the cycles are disjoint, they do not
interfere with other matched edges of $M^{\prime}$.

Since $k=3,5,\dots$, disjoint unions of triangles give the worst bound. Among
triangle-free graphs, disjoint unions of 5-cycles give the worst bound. ∎

######  Theorem 2 (main).

Algorithm 1 gives a 0.584-approximation for general input graphs, and gives a
0.595-approximation for triangle-free input graphs.

###### Proof.

Define

$\displaystyle H_{ij}=(I-X_{i}X_{j}-Y_{i}Y_{j}-Z_{i}Z_{j})/4,$ $\displaystyle
H=\sum_{(i,j)\in E}w_{ij}H_{ij}.$

We bound the expected energy of each case of Step 3 and 4 of the algorithm.
The subroutine of Step 3 is directly from the main algorithm of [5], where in
turn the analysis is borrowed from [2]. More precisely, the energy of
$\rho_{1}$ with respect to $H_{ij}$ is

$\displaystyle\text{Tr}\rho_{1}H_{ij}$
$\displaystyle=\frac{1}{16}\text{Tr}[(I-X\otimes X-Y\otimes Y-Z\otimes Z)$
$\displaystyle((I+u_{i,1}X+u_{i,3}Y+u_{i,3}Z)\otimes(I+u_{j,1}X+u_{j,3}Y+u_{j,3}Z))]$
$\displaystyle=\frac{1}{4}(1-u_{i}\cdot u_{j}),$

and its expected value is

$\displaystyle\text{Tr}\rho_{1}H_{ij}$ $\displaystyle=\frac{1-f_{3}(v_{i}\cdot
v_{j})}{4}$ (2)

where

$\displaystyle
f_{3}(x)=\frac{2}{3}\left(\frac{\Gamma(2)}{\Gamma(1.5)}\right)^{2}x_{2}F_{1}\left(\begin{subarray}{c}1/2,1/2\\\
\\\ 2\end{subarray};x^{2}\right).$

The estimation is from Lemma 2.1 of [2].

Now we turn to the analysis of the energy of $\rho_{2}$ from Step 4. Note that
if an edge $e$ is matched, then $\text{Tr}\rho_{2}H_{e}=1$, and if $e$ is not
matched, then $\text{Tr}\rho_{2}H_{e}=1/4.$ More succinctly,
$\text{Tr}\rho_{2}H_{e}=1/4+3M_{e}/4$. Consider the fact that
$(\max(2h_{e},0))_{e\in E}$ forms a fractional matching by Monogamy of
Entanglement (Equation (1)). Denote $x^{+}:=\text{max}(x,0)$ for
$x\in\mathbb{R}.$ Then by Lemma 3,

$\displaystyle\sum_{e\in E}w_{e}M_{e}\geq\begin{cases}\frac{4}{3}\sum_{e\in
E}w_{e}h_{e}^{+}&\text{ for general }G\\\ \\\ \frac{8}{5}\sum_{e\in
E}w_{e}h_{e}^{+}&\text{ for triangle-free }G\end{cases},$ (3)

and therefore

$\displaystyle\sum_{e\in E}w_{e}\text{Tr}\rho_{2}H_{e}=$
$\displaystyle\sum_{e\in E}w_{e}(\frac{1}{4}+\frac{3}{4}M_{e})$
$\displaystyle\geq\begin{cases}\sum_{e\in
E}w_{e}(\frac{1}{4}+h_{e}^{+})&\text{ for general }G\\\ \\\ \sum_{e\in
E}w_{e}(\frac{1}{4}+\frac{6}{5}h_{e}^{+})&\text{ for triangle-free
}G\end{cases},$ (4)

By definition,
$v(I)\cdot(v(I)-v(X_{i}X_{j})-v(Y_{i}Y_{j})-v(Z_{i}Z_{j}))/4=(1-3v_{i}\cdot
v_{j})/4=1/2+h_{ij}.$ So we have

$\displaystyle h_{ij}=-\frac{1+3v_{i}\cdot v_{j}}{4}.$ (5)

Let $\sigma$ be the density matrix of the output state of the algorithm. By
combining the energy estimation of the two cases (2) and (4), we get in the
general case,

$\displaystyle\sum_{(i,j)\in E}w_{ij}\text{Tr}\sigma H_{ij}\geq\sum_{(i,j)\in
E}w_{ij}\left[p\frac{1-f_{3}(v_{i}\cdot
v_{j})}{4}+(1-p)\left(\frac{1}{4}+\left(-\frac{1+3v_{i}\cdot
v_{j}}{4}\right)^{+}\right)\right].$

Since $-1\leq v_{i}\cdot v_{j}\leq 1/3$, find

$\displaystyle\alpha$
$\displaystyle:=\max_{p\in[0,1]}\min_{x\in[-1,1/3]}\left[p\frac{1-f_{3}(x)}{4}+(1-p)\left(\frac{1}{4}+\left(-\frac{1+3x}{4}\right)^{+}\right)\right]\left.\middle/~{}{\frac{1-3x}{4}}\right.$
$\displaystyle=0.584,$

and use the argmax $p_{0}=0.586$ as the parameter in the algorithm. This
proves the theorem for the general case since

$\displaystyle\sum_{(i,j)\in E}w_{ij}\text{Tr}\sigma H_{ij}$
$\displaystyle\geq\sum_{(i,j)\in E}w_{ij}\left[p_{0}\frac{1-f_{3}(v_{i}\cdot
v_{j})}{4}+(1-p_{0})\left(\frac{1}{4}+\left(-\frac{1+3v_{i}\cdot
v_{j}}{4}\right)^{+}\right)\right]$ $\displaystyle\geq 0.584\sum_{(i,j)\in
E}w_{ij}\frac{1-3v_{i}\cdot v_{j}}{4}\geq 0.584\lambda_{max}(H).$

The triangle-free case is proved similarly. ∎

### Acknowledgements

The author was supported by Individual Grant No.CG093801 at Korea Institute
for Advanced Study. The author appreciates helpful comments on the write-up by
Andrus Giraldo.

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# Collective coordinates for the hybrid model

C. F. S. Pereira<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Departamento de Física e Química, Universidade Federal
do Espírito Santo, Av. Fernando Ferrari, 514, Goiabeiras, 29060-900, Vitória,
ES, Brazil.  Etevaldo dos Santos Costa Filho  Departamento de Matemática da
Universidade de Aveiro and Centre for Research and Development in Mathematics
and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal  T.
Tassis  Centro de Ciências Naturais e Humanas, Universidade Federal do ABC,
Av. dos Estados, 09210-580, Santo André, São Paulo, Brazil

###### Abstract

In the present work, we carry out the study of scattering solitons for the
anti-kink/kink and kink/anti-kink configurations. Furthermore, we can observe
the same effects as those described by D. Bazeia et al. Bazeia et al. (2019).
We apply the collective coordinate approximation method to describe both
scattering configurations and verify that just as happens in the polynomial
models $\phi^{4}$ and $\phi^{6}$ Weigel (2014); Takyi and Weigel (2016);
Belova and Kudryavtsev (1988); Sugiyama (1979); Peyrard and Campbell (1983),
the method has its limitations regarding the initial scattering speeds. In
such a way that, for certain initial speeds, the solution of collective
coordinates agrees with the fullsimulation, and for other speeds, there is a
discrepancy in the solutions obtained by these two methods. We also noticed
that, considering the hybrid model, the null-vector problem persists for both
configurations, and when trying to fix it, a singularity is created in moduli-
space as well as in $\phi^{4}$ Manton et al. (2021a).

Solitons scattering, linear fluctuations, internal modes, and collective
coordinates.

##  I Introduction

In the last decades, the interest in the study of topological solitons has
grown considerably. This interest is due to the wide applicability of such
solutions in several domains of physics, which can be found, for instance, in
high energy physics Manton and Sutcliffe (2004); Rajaraman (1982), condensed
matter Schneider (1978), cosmology, gravitation Vilenkin and Shellard (1994);
Vachaspati (2006); Weinberg (2012) and even in the study of optical fiber
Mollenauer and Gordon (2006) and in biological systems Wittkowski et al.
(2014); Peyrard (2004). Thus, the so-called Derrick theorem Derrick (1964)
guarantees that the space dimension restricts the non-existence of static and
topological solutions, being possible in only $(1+1)$ dimensions. However,
there are several ways to get around this theorem and obtain topological
solutions for higher dimensions. As is the case for vortex Nielsen and Olesen
(1973) and Q-ball Bowcock et al. (2009); Lee and Pang (1992); Coleman (1985)
solutions in which gauge coupling is introduced and in skyrmion models Skyrme
(1961, 1962) in which terms derived from higher orders are entered.

As its fundamental principle, the collective coordinate method transforms a
field theory problem into one of classical mechanics. For that, all its
spatial degrees of freedom must be integrated throughout the space so that the
new Lagrangian functional will only depend upon the dynamical parameters, that
is to say, the dynamics variables of the system and its derivatives. In this
perspective, the technique has been extensively explored in the study of
scattering of solitons in $(1+1)$ dimensions, with greater emphasis on the
$\phi^{4}$ model. In recent years, several authors have been dedicated to
trying to describe the occurrence of energy transfer from translational to
vibrational modes and how this redistribution occurs so that the solitons go
directic to the asymptotic Peyrard and Campbell (1983); Belova and Kudryavtsev
(1988); Goodman and Haberman (2005); Moshir (1981); Anninos et al. (1991);
Sugiyama (1979). Concomitantly, it rekindled interest in the study of this
model since some authors identified Takyi and Weigel (2016); Weigel (2014);
Kevrekidis and Goodman (2019); Pereira et al. (2021) a crucial error in some
of the functions necessary for the lagrangian functional to be written
explicitly. The procedure was also applied to the study of the classical
2-kinks scattering in the sine-Gordon (Sutcliffe, 1993), and modified sine-
Gordon Baron and Zakrzewski (2016) models, as well as, for the nonlinear
Schrödinger model Baron et al. (2014) in which the collective coordinate was
introduced as a temporal phase.

As for the model $\phi^{6}$ in $(1+1)$ dimensions, it was already known
through numerical experiments by Dorey et al. (2011) that the scattering
between anti-kink/kink had states of resonance and along with the effect the
existence of collective vibrational modes. Thus, in Gani et al. (2014) the
authors sought to build the collective coordinates numerically for all
scattering scenarios. They realized that the anti-kink/kink case in which it
should be possible to have some information about the resonance windows.
However, for the ansatz used, it was not possible to be verified.

In the present work, we will apply the technique of collective coordinates to
study a hybrid model that was presented by D. Bazeia et al. in Bazeia et al.
(2019). This particular model has an intersection of the polynomial model
$\phi^{4}$ with $\phi^{6}$, and for specific velocities regimes, new
interesting structures can be explored. The idea of doing the study using the
method of collective coordinates aims to seek a qualitative and quantitative
treatment to understand the effects that appear in the formation of the
resonance windows. In short, we want to investigate how the energy balance
occurs between its degrees of freedom (translational and vibrational modes).

The paper is organized as follows: we start the work in Section 2 by
presenting the background and the model’s basis. Next, in Section 3, we apply
the technique of collective coordinates for scattering anti-kink/kink. In
Section 4, we construct the collective coordinates for scattering kink/anti-
kink; in section 5, we investigate the null-vector problem and finally, in
Section 6, the conclusions are presented.

##  II The model

As said before, the hybrid model, as proposed in Bazeia et al. (2019), is
given as a certain combination of the models $\phi^{4}$ and $\phi^{6}$. The
action for such model is given by

$\displaystyle
S_{Hybrid}=\int{d^{2}x}\left(\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right),$
(1) $\displaystyle
V(\phi)=\frac{\lambda\phi^{2}}{4}\left(\frac{m}{\sqrt{\lambda}}-|{\phi}|\right)^{2}.$
(2)

We deal with a real Lorentz scalar field, $\phi$, in $1+1$ dimensional
Minkowski space-time. Here, the parameters $m$ and $\lambda$ are chosen to be
real. Notice that the above potential possesses three vacuum states given by
$\phi=0$ and $\phi=\pm\dfrac{m}{\sqrt{\lambda}}=\pm\eta$, and each of these
vacua possesses multiplicity two. Moreover, these three vacua connect two
distinct topological sectors. See figure below

Figure 1: We plot here the potential $V(\phi)$.

.

The equation of motion of the field is given by

$\partial_{\mu}\partial^{\mu}\phi+\dfrac{\lambda\phi}{2}\left(\dfrac{m^{2}}{\lambda}-\dfrac{3m|\phi|}{\sqrt{\lambda}}+2\phi^{2}\right)=0,$
(3)

Without loss of generality, we will follow the convention on the literature
and set from now on $m^{2}=\lambda=2$ in order to properly do the numerical
calculations. The kink/anti-kink static configurations for each of these two
sections, connecting the vacua, are giving by the equations:

$\displaystyle\phi_{K}^{\left(0,1\right)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[1+\tanh\left(\frac{x}{2}\right)\right]\qquad\qquad\phi_{\bar{K}}^{\left(1,0\right)}=\frac{1}{2}\left[1-\tanh\left(\frac{x}{2}\right)\right]$
(4) $\displaystyle\phi_{K}^{\left(-1,0\right)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[-1+\tanh\left(\frac{x}{2}\right)\right]\qquad\qquad\phi_{\bar{K}}^{\left(0,-1\right)}=-\frac{1}{2}\left[1+\tanh\left(\frac{x}{2}\right)\right].$
(5)

Notice that such configuration is slightly similiar to the $\phi^{6}$ model
see 2. Considering linear perturbations around these static vacuum solutions,
$\phi\left(x,t\right)=\phi_{K,\bar{K}}\left(x\right)+\chi\left(x\right)\cos\left(\omega{t}\right)$,
we find a Schrödinger-like equation for the spatial perturbation component see
1 possessing the function $V_{Sch}$ as potential,

$\displaystyle
V_{Sch}=\left(-\frac{1}{2}+\frac{3}{2}\tanh^{2}\left(\frac{x}{2}\right)\right).$
(6)

Notice the existence of a map between the eigenfunctions of the present model
with those obtained through the Pöschl-Teller potential for the $\phi^{4}$
polynomial model. In this way, their eigenfrequencies are related as given by
the expression $\omega_{Hibrid}^{2}=\frac{1}{4}\omega_{\lambda\phi^{4}}^{2}$.
Therefore, the eigenfunctions associated with the translational and
vibrational modes are giving, respectively, by the equations (7),(8).

$\displaystyle\omega_{0}^{2}=0\qquad\qquad\longrightarrow\qquad\qquad\chi_{0}=\sqrt{\frac{3}{8}}\mathrm{Sech}^{2}\left(\frac{x}{2}\right),$
(7)

$\displaystyle\omega_{1}^{2}=\frac{3}{4}\qquad\qquad\longrightarrow\qquad\qquad\chi_{1}=\sqrt{\frac{3}{4}}\tanh\left(\frac{x}{2}\right)\mathrm{Sech}\left(\frac{x}{2}\right).$
(8)

This effective potential, $V_{Sch}$, which arises from the perturbations, is
shown in figure 2 together with the translational and vibrational modes.

.

Figure 2: (a) Field configuration for the kink and anti-kink b) we have the
graphic referring to the effective potential arising from linear pertubations
and the curves for the translational and vibrational mode.

##  III Collective coordinates for the anti-kink/kink scattering

###  III.1 Numerical Inputs

For all numeric simulations, we used the python library "python
scipy.integrate". Some numerical values are settled as follows. For the
spatial and temporal grid, it was sufficient to use $dx=0.05$ and
$dt=\frac{dx}{2}$. For both scattering, anti-kink/kink and kink/anti-kink, we
fix the initial positions as $x=\pm{x_{0}}$, where $x_{0}=15$. As regard the
model constants, we set $m^{2}=\lambda=2$.

###  III.2 Anti-kink/kink scattering

It is equivalent to study the phenomenology of the scattering process of kinks
in the first topological sector or the second sector. In this way, we will
only consider as reference the scattering process of kinks that occurs in the
first topological sector.

Therefore, for the scattering anti-kink/kink, we consider the following field
configuration

$\displaystyle\phi=\phi_{\bar{K}}^{\left(1,0\right)}\left(x+a(t)\right)+\phi_{K}^{\left(0,1\right)}\left(x-a(t)\right)+\xi\left(t\right)\left[\chi_{\bar{K}}^{\left(1,0\right)}\left(x+a(t)\right)-\chi_{K}^{\left(0,1\right)}\left(x-a(t)\right)\right].$
(9)

Here, the translation parameter $a(t)$ represents the relative distance
between the center of mass of the kink and the anti-kink. By it turns, the
parameter $\xi(t)$ is the amplitude of the vibrational mode. The vibrational
mode of the anti-kink/kink solution is, respectively,
$\chi_{\bar{K}}^{\left(1,0\right)}$ and $\chi_{K}^{\left(0,1\right)}$.

As it is well-known, the collective coordinates method is based on integrating
the Lagrangian density in all spatial degrees of freedom of the field
configuration (9); and hence, to get a functional which only depends on the
dynamic variables of the system and its derivatives. This is the procedure for
describing moduli-space dynamics.

In what follows, since we are studying perturbations of the vacuum, we will
neglect second-order terms of the vibrational mode amplitude
$(O\left(\xi^{2}\right))$.

$\displaystyle
L_{1}\left(a,\dot{a},\xi,\dot{\xi}\right)=\int_{-\infty}^{\infty}{dx}\left[\frac{1}{2}\left(\frac{\partial{\phi}}{\partial{t}}\right)^{2}-\frac{1}{2}\left(\frac{\partial{\phi}}{\partial{x}}\right)^{2}-\frac{1}{2}\phi^{2}\left(1-|\phi|\right)^{2}\right].$
(10)

Considering the scattering, the effective Lagrangian is given by

$\displaystyle L_{1}\left(a,\dot{a},\xi,\dot{\xi}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\bar{M}+I_{1}\left(a\right)+\xi{J_{1}\left(a\right)}+\xi^{2}{K_{1}\left(a\right)}\right]\dot{a}^{2}+\frac{1}{2}\left[{2+Q_{1}\left(a\right)}\right]\dot{\xi}^{2}+$
$\displaystyle+$
$\displaystyle\left(C_{1}\left(a\right)+\xi{N_{1}\left(a\right)}\right)\dot{a}\dot{\xi}-\left(V_{1}\left(a\right)-\xi{F_{1}\left(a\right)}+\xi^{2}{W_{1}\left(a\right)}\right),$

Here $\bar{M}=2M_{0}$, where $M_{0}$ is the rest mass of the kink.

The analytical expression of all functions above is given here,

$\displaystyle\bar{M}=\frac{1}{3}\qquad\qquad\qquad
W_{1}\left(a\to\infty\right)=\frac{3}{4}$ (12) $\displaystyle
I_{1}\left(a\right)$ $\displaystyle=$
$\displaystyle\mathrm{Csch}^{2}\left(a\right)\left(-1+a\mathrm{Coth}\left(a\right)\right)$
(13) $\displaystyle V_{1}\left(a\right)$ $\displaystyle=$
$\displaystyle-\frac{2}{3}+a+\frac{3}{\tanh\left(a\right)}-\frac{2(1+3a)}{\tanh^{2}\left(a\right)}+\frac{2a}{\tanh^{3}\left(a\right)}$
(14) $\displaystyle Q_{1}\left(a\right)$ $\displaystyle=$ $\displaystyle
6a\mathrm{Csch}\left(a\right)-12\mathrm{Coth}\left(a\right)\mathrm{Csch}\left(a\right)+12a\mathrm{Csch}^{3}\left(a\right)$
(15) $\displaystyle J_{1}\left(a\right)$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}\pi}{64}\left(19-4\mathrm{Cosh}\left(a\right)+\mathrm{Cosh}\left(2a\right)\right)\mathrm{Sech}^{4}\left(\frac{a}{2}\right)$
(16) $\displaystyle F_{1}\left(a\right)$ $\displaystyle=$
$\displaystyle\frac{3\pi\sqrt{3}}{4}\tanh^{2}\left(\frac{a}{2}\right)\mathrm{Sech}^{2}\left(\frac{a}{2}\right)\left(\mathrm{Cosh}\left(a\right)-\mathrm{Sinh}\left(a\right)\right)$
(17) $\displaystyle K_{1}\left(a\right)$ $\displaystyle=$
$\displaystyle\frac{7}{10}+\frac{3}{2}\mathrm{Csch}\left(a\right)\left(a-4\mathrm{Coth}\left(a\right)\right)+6\mathrm{Csch}^{3}\left(a\right)\left(5a-6\mathrm{Coth}\left(a\right)\right)+36a\mathrm{Csch}^{5}\left(a\right)$
(18) $\displaystyle N_{1}\left(a\right)$ $\displaystyle=$ $\displaystyle
9\mathrm{Csch}\left(a\right)+18\mathrm{Csch}^{3}\left(a\right)-3a\mathrm{Csch}\left(a\right)\mathrm{Coth}\left(a\right)-18a\mathrm{Csch}^{3}\left(a\right)\mathrm{Coth}\left(a\right)$
(19) $\displaystyle C_{1}\left(a\right)$ $\displaystyle=$
$\displaystyle-2\pi\sqrt{3}\mathrm{Csch}^{3}\left(a\right)\mathrm{Sinh}^{4}\left(\frac{a}{2}\right).$
(20)

Considering the equations of motion associated to the Lagrangian (III.2), we
need to impose that $C_{1}\left(a\right)=N_{1}\left(a\right)=0$ in order for
the system of differential equations can admit non-trivial solutions. Such
convention was made in several works in the literature, as well as considering
the asymptotic limit for the function
$W_{1}\left(a\to\infty\right)=\frac{3}{4}$ which is the value of
characteristic frequency of individual vibrational modes Pereira et al.
(2021); Kevrekidis and Goodman (2019); Takyi and Weigel (2016); Weigel (2014);
Belova and Kudryavtsev (1988); Peyrard and Campbell (1983); Sugiyama (1979).

Another aspect that has been well explored in recent works is the effort to
avoid the null-vector problem, that occurs in the $\phi^{4}$ polynomial model
for instance, and as we can see in the function
${{Q}}_{1}\left(a\left(t\right)\to{0}\right)=0$ it persists in the present
model modelManton et al. (2021a, b). It is worth mentioning that all of the
functions that come with the Lagrangean (10), except for the potential
function $V_{1}\left(a\right)$, are well behaved, both in their asymptotic
limits $a\to\pm\infty$ and in the origin $a\left(t\right)\to{0}$ (3). By its
turn, the potential $V_{1}\left(a\right)$, which is not symmetric with respect
to the origin and just as in the case of the model $\phi^{4}$, diverges when
$a\left(t\right)\to{-\infty}$.

.

Figure 3: (a) Behavior of the functions $V_{1}\left(a\right)$,
$G_{1}\left(a\right)$ and $Q_{1}\left(a\right)$ where it is straightforward to
see that $Q_{1}\left(a\to{0}\right)=-2$ and the divergence of the potential in
the limit of $a\to{-\infty}$, (b) the functions $C_{1}\left(a\right)$ and
$N_{1}\left(a\right)$, (c) the functions $J_{1}\left(a\right)$,
$F_{1}\left(a\right)$ and $K_{1}\left(a\right)$.

For the fullsimulation calculation, we will use as a definition, that is, the
average value of the position of the center of mass is written in terms of the
energy density.

$\langle x\rangle=\frac{\int dx\;x\mathcal{E}}{\int dx\;\mathcal{E}},$ (21)

where $\mathcal{E}$ is the energy density referring to the configuration (9)

$\mathcal{E}=\frac{1}{2}\left(\frac{\partial\phi}{\partial
t}\right)^{2}+\frac{1}{2}\left(\frac{\partial\phi}{\partial
x}\right)^{2}+\frac{\phi^{2}}{2}\left(1-|\phi|\right)^{2}.$ (22)

###  III.3 Analysis of the collective coordinates and fullsimulation for the
anti-kink/kink scattering

.

Figure 4: (a) Time evolution for the anti-kink/kink scattering with an initial
speed of $v_{ini}=0.201$, (b) energy transfer between the internal modes,
being $a\left(t\right)$ translation and $\xi\left(t\right)$ vibration and (c)
comparison between fullsimalation and collective coordinates for same initial
velocity $v_{ini}=0.201$.

For the configuration of anti-kink/kink, we perform the dynamics for
fullsimulation and verify that for scatterings with velocities above the
critical velocity, $v_{ini}>v_{crit}=0.2599$, we only have inelastic
collisions. That is, the anti-kink/kink perform only the first contact and
leave for asymptotics. On the other hand, for scatterings with an initial
velocity below $v_{ini}<v_{crit}=0.2599$, we have the formation of so-called
bion states, which remain emitting radiation for a long period of time until
they annihilate. Thus, we can also verify the characterization of resonance
states in which it is possible to observe the formation of a fixed number of
resonance windows that can vary from two to four 4.

Regarding the dynamics of collective coordinates, as we well know from
previous works Peyrard and Campbell (1983); Belova and Kudryavtsev (1988);
Goodman and Haberman (2005); Moshir (1981); Anninos et al. (1991); Sugiyama
(1979) this technique is an approximative method, and therefore, only for some
scattering velocities do the results agree reasonably with the fullsimulation
. In this work, it is no different, notice in 4 that we did the scattering for
an initial velocity $v_{ini}=0.201$ and the result of the dynamics in a
qualitative way agrees with the fullsimulation. Also, in figure 4, it is
observed that the number of jumps represented in the collective coordinates
should not be seen as resonance windows. Finally, we have the energy transfer
from the internal translation mode to the internal vibration mode 4.

##  IV kink/anti-kink scattering

In the present section, we will study the scattering of the kink/anti-kink
configuration. We will take as a starting point the field configuration below

$\displaystyle\phi=\phi_{K}^{\left(0,1\right)}\left(x+a(t)\right)+\phi_{\bar{K}}^{\left(1,0\right)}\left(x-a(t)\right)-1+\xi\left(t\right)\left[\chi_{K}^{\left(0,1\right)}\left(x+a(t)\right)-\chi_{\bar{K}}^{\left(1,0\right)}\left(x-a(t)\right)\right],$
(23)

with the vacuum solutions $\phi_{K}^{\left(0.1\right)}$,
$\phi_{\bar{K}}^{\left(1.0\right)}$ and their respective normal vibration
modes: $\chi_{K}^{\left(0.1\right)}$ and $\chi_{\bar{K}}^{\left(1.0\right)}$.
As in the previous case, for simplicity, we will only consider terms of up to
second order in the amplitude of the vibrational modem
$O\left(\xi^{2}\right)$.

The effective Lagrangian for the scattering is given by

$\displaystyle L_{2}\left(a,\dot{a},\xi,\dot{\xi}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\bar{M}+I_{2}\left(a\right)+\xi{J_{2}\left(a\right)}+\xi^{2}{K_{2}\left(a\right)}\right]\dot{a}^{2}+\frac{1}{2}\left[{2+Q_{2}\left(a\right)}\right]\dot{\xi}^{2}+$
$\displaystyle+$
$\displaystyle\left(C_{2}\left(a\right)+\xi{N_{2}\left(a\right)}\right)\dot{a}\dot{\xi}-\left(V_{2}\left(a\right)-\xi{F_{2}\left(a\right)}+\xi^{2}{W_{2}\left(a\right)}\right).$

The rest mass for kink and anti-kink configurations are the same. The auto-
frequencies are also identical in asymptotics and therefore
$W_{1}\left(a\to\infty\right)=W_{2}\left(a\to\infty\right)=\omega_{Hybrid}^{2}=\frac{3}{4}$
.

In this way, some functions for, both anti-kink/kink and kink/anti-kink
scattering, are equal $I_{1}\left(a\right)=I_{2}\left(a\right)$,
$V_{1}\left(a\right)=V_{2}\left(a\right)$,
$Q_{1}\left(a\right)=Q_{2}\left(a\right)$,
$K_{1}\left(a\right)=K_{2}\left(a\right)$ and
$N_{1}\left(a\right)=N_{2}\left(a\right)$. By its turn, the other functions
are anti-symmetric in relation to the scattering configuration change:
$J_{1}\left(a\right)=-J_{2}\left(a\right)$,
$F_{1}\left(a\right)=-F_{2}\left(a\right)$ e
$C_{1}\left(a\right)=-C_{2}\left(a\right)$.

An important observation is that for both scattering cases, the null-vector
problem is maintained, and, in some way, the existence of this term in the
Lagrangian of the collective coordinates is relevant for the dynamics to be
compatible with fullsimulation.

This analysis goes against how occurs the combination between the internal
vibrational modes constructed in equations (9),(23). Notice that by
construction the ansatz
$\left[\chi_{\bar{K}}^{\left(1.0\right)}\left(x+a(t)\right)-\chi_{K}^{\left(0,1\right)}\left(x-a(t)\right)\right]$,
both in the limit of $a\to{0}$ and in the limit of
$a\left(t\right)\to{\infty}$, the terms cancel each other out. Therefore, at
first, this choice is inconsistent if we consider that, at the limit
$a\left(t\right)\to{0}$, the energy exchange between the internal modes must
be a maximum, on the other hand, it is this ansatz that shows consistent with
fullsimulation and predicts the resonance windows.

On the other hand, if we consider a sum of the vibrational modes in the
ansatz,
$\left[\chi_{\bar{K}}^{\left(1,0\right)}\left(x+a(t)\right)+\chi_{K}^{\left(0,1\right)}\left(x-a(t)\right)\right]$,
we will see that in the limit $a\left(t\right)\to\infty$, the contribution of
these modes is null, as expected since the objects are free. On the other
hand, at the limit of $a\left(t\right)\to{0}$ the contribution of these
vibrational modes is maximum, as we expect. However, when these objects are
scattered, there is no resonance window, thus being incompatible with
fullsimulation.

###  IV.1 Description of the collective coordinates and fullsimulation for the
kink/anti-kink scattering

.

Figure 5: (a) Time evolution for the kink/anti-kink scattering with an initial
speed of $v_{ini}=0.124$, (b) energy transfer between the internal modes,
being $a\left(t\right)$ translation and $\xi\left(t\right)$ vibration and (c)
comparison between fullsimalation and collective coordinates for same initial
velocity $v_{ini}=0.124$.

The scattering was performed for the kink/anti-kink configuration in which we
verified the same critical velocity $v_{crit}=0.152$ obtained in D. Bazeia et
al Bazeia et al. (2019). For scatterings with an initial velocity greater than
the critical velocity $v_{ini}>v_{crit}=0.152$, we have the inelastic
collisions. On the other hand, the formation of bion states is verified for
many velocities below the critical velocity.

There are no indications of the formation of resonance windows in the present
scattering configuration, but states with only one jump. However, before the
kinks go the asymptotics, a certain number of oscillations are found. In
figure 5, we see the formation of such state for the scattering with an
initial velocity of $v_{ini}=0.124$. That is, we have the formation of three
oscillations before the kinks go the asymptotics. We also performed the
collective coordinates method for this initial velocity, and it is in
agreement with the fullsimulation 5. Furthermore, we can see in the figure 5
the energy redistribution between the internal modes.

##  V Analysis of the null-vector problem

Recently, the authors in the works Manton et al. (2021a, b) proposed the
introduction of an attenuator function $f\left(a\right)$ in the ansatz of the
collective coordinates for the potential $\phi^{4}$ Weigel (2014); Takyi and
Weigel (2016); Pereira et al. (2021) in order to solve the null-vector
problem. An interesting aspect is that, it appears the idea that when the
null-vector problem is corrected, a singularity is necessarily created in the
moduli-space. This fact was also verified for the model of the present study,
in which we performed the introduction of the function
$f\left(a\right)=\tanh\left(a\right)$ both in the ansatz of the scattering of
anti-kink /kink (25) as for the scattering kink/anti-kink (26)

$\displaystyle\phi=\phi_{\bar{K}}^{\left(1,0\right)}\left(x+a(t)\right)+\phi_{K}^{\left(0,1\right)}\left(x-a(t)\right)+\frac{\xi\left(t\right)}{f\left(a\right)}\left[\chi_{\bar{K}}^{\left(1,0\right)}\left(x+a(t)\right)-\chi_{K}^{\left(0,1\right)}\left(x-a(t)\right)\right],$
(25)

$\displaystyle\phi=\phi_{K}^{\left(0,1\right)}\left(x+a(t)\right)+\phi_{\bar{K}}^{\left(1,0\right)}\left(x-a(t)\right)-1+\frac{\xi\left(t\right)}{f\left(a\right)}\left[\chi_{K}^{\left(0,1\right)}\left(x+a(t)\right)-\chi_{\bar{K}}^{\left(1,0\right)}\left(x-a(t)\right)\right].$
(26)

The initial configuration of the field for the anti-kink/kink scattering,
given in equation (25), is represented in the figure (6). In the figure on the
right 6, we have the temporal evolution of the field in which we can see the
formation of two resonance windows. While in the left figure 6, we have the
comparison between the fullsimulation and the collective coordinates. Notice
also that, after the occurrence of the first collision, the dynamics
associated with the collective coordinate of translation is frozen in time,
just as it happens for the case $\phi^{4}$ polynomial. Such effect occurs due
to the manifestation of the singularity in the moduli-space predicted in N.
Manton et al. Manton et al. (2021a).

Figure 6: (a) anti-kink/kink scattering with an initial velocity of
$v_{ini}=0.2$ (b) Time evolution of the solution at the origin, with the
formation of two resonance windows.

Figure 7: (a) kink/ant-kink scattering with an initial velocity of
$v_{ini}=0.124$ (b) Time evolution of the solution at the origin, with the
formation of three oscillations before the separation.

Analogously, we perform the kink/anti-kink scattering for the field
configuration given by equation (26) and we can observe the temporal evolution
for the initial velocity of $v_{ini}=0.124$ and the formation of three
oscillations before the kinks come out to the asymptotic 7. For the figuration
7 once again, we have the solution freezing as an effect of the singularity.

##  VI Conclusion

We carried out the study of collective coordinates referring to the hybrid
model for both the anti-kink/kink scattering and the kink/anti-kink
scattering. We saw that it has a connection with the potentials $\phi^{4}$ and
$\phi^{6}$ and that just as happens in the case of the potential $\phi^{6}$,
it has two topological sectors. We took advantage of the mapping with the
$\phi^{4}$ polynomial model to make use of the integration technique used in
the work Pereira et al. (2021) and build the Lagrangian functional only in
terms of the dynamic parameters and their derivatives.

Since both the Lagrangian functionals for the anti-kink/kink scattering and
the kink/anti-kink scattering have remained with the null-vector problem, we
investigate the resolution of this problem by following the redefinition of
the ansatz presented by N. Manton et al. Manton et al. (2021b), and we
conclude that the moduli space becomes singular, just like the polynomial case
Manton et al. (2021a).

As a future perspective, it would be interesting to investigate whether, for
any well-behaved attenuator function in asymptotic limits that can be
introduced in the correction of the null-vector problem, necessarily produce a
singularity in the moduli-spaceManton et al. (2021a). Because as the initial
interest was to correct the null-vector problem and thus verify how the
relocation of energy between the internal modes occurs, the creation of a
singularity does not contribute to elucidating the problem since the solution
of the collective coordinates is frozen in time.

###### Acknowledgements.

This work made use of the Virgo Cluster at Cosmo-ufes/UFES, which is funded by
FAPES (Fundação de Amparo à Pesquisa e Inovação do Espírito Santo) and
administered by Renan Alves de Oliveira. C.F.S. Pereira thanks the financial
support provided by the Coordination for the Improvement of Higher Education
Personnel (CAPES). E.S.C. Filho thanks the supported by the Center for
Research and Development in Mathematics and Applications (CIDMA) through the
Portuguese Foundation for Science and Technology (FCT - Fundação para a
Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020. T.
Tassis thanks the financial support provided by the Federal University of ABC
(UFABC).

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