11institutetext: Departamento de Ciência da Computação, Universidade de
Brasília, Brazil 11email: laugustogarcia<EMAIL_ADDRESS>22institutetext: Instituto de Computação, Universidade Federal Fluminense,
Niterói, Brazil 22email<EMAIL_ADDRESS>
# A barrier for further approximating Sorting By Transpositions

Luiz Augusto G. da Silva 11 Luis Antonio B. Kowada 22 Maria Emília M. T.
Walter 11

###### Abstract

The Transposition Distance Problem (TDP) is a classical problem in genome
rearrangements which seeks to determine the minimum number of transpositions
needed to transform a linear chromosome into another represented by the
permutations $\pi$ and $\sigma$, respectively. This paper focuses on the
equivalent problem of Sorting By Transpositions (SBT), where $\sigma$ is the
identity permutation $\iota$. Specifically, we investigate palisades, a family
of permutations that are “hard” to sort, as they require numerous
transpositions above the celebrated lower bound devised by Bafna and Pevzner.
By determining the transposition distance of palisades, we were able to
provide the exact transposition diameter for $3$-permutations (TD3), a special
subset of the Symmetric Group $S_{n}$, essential for the study of approximate
solutions for SBT using the simplification technique. The exact value for TD3
has remained unknown since Elias and Hartman showed an upper bound for it.
Another consequence of determining the transposition distance of palisades is
that, using as lower bound the one by Bafna and Pevzner, it is impossible to
guarantee approximation ratios lower than $1.375$ when approximating SBT. This
finding has significant implications for the study of SBT, as this problem has
been subject of intense research efforts for the past 25 years.

##  1 Introduction

Mutational events involving large portions of DNA in a chromosome are studied
in the area of genome rearrangements. These are rare events and their study is
crucial for understanding the evolutionary history of organisms [10, 13, 14,
16]. One of these events is _transposition_ , which informally corresponds to
the exchange of positions of two adjacent blocks of genes in a linear
chromosome. Transposition can also be seen as cutting out a gene block and
then pasting it elsewhere (in the same chromosome). Then the question arises:
what is the minimum number $t$ of transpositions required to transform one
linear chromosome, represented by a permutation $\pi$, into another one
represented by $\sigma$, both consisting of the same set of genes? This
classical problem is referred to as Transposition Distance Problem (TDP) and,
if we consider the target chromosome $\sigma$ as being the identity $\iota$,
then TDP turns into the equivalent problem of Sorting By Transpositions (SBT).
In this case, the number $t$ is called the _transposition distance_ of $\pi$
and is denoted by $d_{t}(\pi)$.

SBT is a $\mathcal{NP}$-hard problem [3] and the first approximate algorithm
for solving it was devised by Bafna and Pevzner [2], with an approximation
ratio of $1.5$. Later, Elias and Hartman [6] devised an $1.375$-approximation
algorithm running in quadratic time, using a technique called _simplification_
, which transforms an arbitrary permutation provided as input into a new one,
seemingly easier to handle with. In 2022, Silva et al. [15] showed that a side
effect of simplification could make the algorithm of Elias and Hartman [6]
need one extra transposition above the approximation ratio of $1.375$ and so
they devised a new $1.375$-approximation algorithm, not using simplification,
but with a high time complexity of $O(n^{6})$. Recently, Alexandrino et al.
[1] published a result making it possible to approximate SBT with a $1.375$
ratio, without extra transpositions, in $O(n^{5})$ time.

In this work, we consider the palisades, a family of permutations which are
“hard” to sort, in the sense that, to be sorted, numerous transpositions above
the well-known lower bound established by Bafna and Pevzner [2] are required.
By determining the transposition distance of palisades, we are able to
determine the exact value for the transposition diameter for $3$-permutations
(denoted by $TD3$), a special subset of the Symmetric Group $S_{n}$, essential
for the study of approximate solutions for SBT using the simplification
technique [9, 8]. This problem has been open since Elias and Hartman [6] have
shown an upper bound for it. Another surprising consequence of the result on
the transposition distance of palisades, as we will show, is that, using as
lower bound, the one by Bafna and Pevzner [2], it is impossible to devise
approximate solutions for SBT with approximation ratios lower than $1.375$.

We organise the text as follows. First, we briefly review the basic concepts
of permutation groups needed to understand the algebraic formalism used in
this work. Next, we will review SBT using the algebraic approach introduced by
Silva et al. [15]. Subsequently, we will present our findings regarding
palisades and, lastly, we include a brief discussion on the results and
conclude. A very brief overview on the cycle graph [2] is given in Appendix
0.A.

##  2 Permutation groups

A permutation group is a group of bijections from a set to itself (i.e.,
permutations), whose group operation is the usual function composition. The
Symmetric Group $S_{n}$ is the permutation group consisting of all
permutations on a finite set $E$ containing $n$ symbols. Let $\alpha$ and
$\beta$ be two permutations in $S_{n}$. The product $\alpha\beta$ is obtained
by composing the two permutation as functions. That is, the result of
$\alpha\beta$ is the permutation in which each element $x$ of $E$ is mapped to
$\alpha(\beta(x))$.

A _cycle_ $\gamma$ is a sequence $(c_{1}\;c_{2}\dots c_{\kappa})$ of symbols
of $E$ such that $\alpha(c_{1})=c_{2}$, $\alpha(c_{2})=c_{3}$, …,
$\alpha(c_{\kappa-1})=c_{\kappa}$, $\alpha(c_{\kappa})=c_{1}$. In other words,
the symbols in $\gamma$ are cyclically permuted by $\alpha$. Due to its cyclic
nature, $\gamma$ can be written starting by any of the symbols moved by it. If
two cycles have no common symbols, then they commute, meaning that the order
in which we compose them does not matter. In this case, these two cycles are
said to be _disjoint_. A _disjoint cycle decomposition_ of $\alpha$ is a
(unique) representation of $\alpha$ as a product of disjoint cycles.
Additionally, the number $\kappa$ is the _length_ of $\gamma$ or,
alternatively, $\gamma$ is a $\kappa$-cycle. If $x\in E$ is an element such
that $\alpha(x)=x$, then $x$ is said to be a _fixed element_. Fixed elements
are sometimes omitted when representing a permutation as a product of cycles,
but when necessary, a fixed element $x$ is represented by the $1$-cycle $(x)$.
The identity permutation $\iota$ is the permutation which fixes all $E$
elements.

###### Example 1

The permutation $(1\;2\;3)(5\;7)\in S_{7}$ is the permutation which sends $1$
to $2$, $2$ to $3$, $3$ to $1$, $5$ to $7$ and $7$ to $5$, and leaves all the
other symbols fixed.

It is well known that any permutation on $S_{n}$ can be written as a (not
unique) product of $2$-cycles. Furthermore, if a permutation can be written as
a product of an even (odd) number of $2$-cycles, then it is impossible to
write the same permutation using a product of an odd (even) number of
$2$-cycles. Proofs for these results can be found in algebra textbooks [7, 5].

###### Example 2

The permutation $(1\;3\;2)(4\;5\;7)$ can be written as a product of four
$2$-cycle as $(1\;3)(3\;2)(4\;5)(5\;7)$. But this same permutation can be
written as a product of $2$-cycles in infinitely many other ways.

Lastly, we highlight that many results in permutation groups consider the
parity of the permutations, defined as follows. Suppose we can write $\alpha$
as a product of $p$ $2$-cycles. In this case, $\alpha$ is an _even (odd)_
permutation if $p$ is even (odd). An important result concerning the parity of
permutations is that, assuming that $\alpha$ and $\beta$ have the same parity,
then the product $\alpha\beta$ is even [5].

##  3 Preliminaries

The order in which a set of $n$ genes appear along a linear chromosome, in the
context of genome rearrangements, is usually represented by a permutation. For
instance, given a set of $n$ genes labelled $1$ through $n$, a permutation of
these genes can be represented as a sequence of integers
$\pi=[\pi_{1}\;\pi_{2}\dots\pi_{n}]$, where each $\pi_{i}$, $i\in[1,n]$,
indicates the position of gene $i$ in the chromosome. In our approach, the
chromosome $\pi$ is represented by the $(n+1)$-cycle
$\bar{\pi}=(0\;\pi_{1}\;\pi_{2}\dots\pi_{n})$. In its turn, the identity
permutation $\iota$ is represented by $\bar{\iota}=(0\;1\;2\dots n)$. A
transposition exchanging the positions of two adjacent blocks of genes is
modelled by the multiplication of a $3$-cycle
$\tau=(\pi_{i}\;\pi_{j}\;\pi_{k})$ with
$\bar{\pi}=(\pi_{0}\;\pi_{1}\dots\pi_{i-1}\;\pi_{i}\;\pi_{i+1}\dots\pi_{j-1}\;\pi_{j}\;\pi_{j+1}\dots\pi_{k-1}\;\pi_{k}\dots\pi_{n})$,
resulting the $(n+1)$-cycle
$(\pi_{0}\;\pi_{1}\dots\pi_{i-1}\;\pi_{j}\;\pi_{j+1}\dots\pi_{k-1}\;\pi_{i}\;\pi_{i+1}\dots\pi_{j-1}\;\pi_{k}\dots\pi_{n})$.
When performing this multiplication, we can say that we are “applying” $\tau$
on $\bar{\pi}$ and, for this reason, $\tau$ is _applicable_ (to $\bar{\pi}$).
However, we remark that there is a condition for a $3$-cycle
$\tau=(\pi_{i}\;\pi_{j}\;\pi_{k})$ to be applicable, which is that the symbols
$\pi_{i}$, $\pi_{j}$, and $\pi_{k}$ must be in the same cyclic order in both
$\tau$ and $\bar{\pi}$ (as in the definition above). Otherwise, the product
$\tau\bar{\pi}$ will not be an $(n+1)$-cycle and consequently, will not model
a chromosome.

###### Example 3

Let $\bar{\pi}=(0\;4\;5\;2\;3\;7\;6\;1\;8)$. Applying the $3$-cycle
$\tau=(2\;6\;8)$ to $\bar{\pi}$ simulates a transposition as
$\tau\bar{\pi}=(0\;4\;5\;6\;1\;2\;3\;7\;8)$. However, the $3$-cycle
$\tau^{\prime}=(2\;4\;7)$ cannot be applied to $\bar{\pi}$. This is because
$\tau^{\prime}\bar{\pi}$ does not correspond to a transposition in our
approach, as $\tau^{\prime}\bar{\pi}=(0\;7\;8)(1\;4\;5\;6)(2\;3)$, which is
not a $(n+1)$-cycle.

The problem of Sorting By Transpositions is therefore the problem of, given a
$(n+1)$-cycle $\bar{\pi}$, finding the minimum $t$, denoted by
$d_{t}(\bar{\pi})$, such that

$\tau_{t}\dots\tau_{2}\tau_{1}\bar{\pi}=\bar{\iota},$ (1)

where $\tau_{1}$ is a $3$-cycle applicable to $\bar{\pi}$ and each $\tau_{i}$,
$i\in[2,t]$, is a $3$-cycle applicable to
$\tau_{i-1}\dots\tau_{2}\tau_{1}\bar{\pi}$.

###  3.1 Lower bound for transposition distance

Mira and Meidanis [12] devised a measure for an even permutation $\alpha\in
S_{n}$ called $3$-norm, denoted as $\lvert\lvert\alpha\rvert\rvert 3$. The
$3$-norm of $\alpha$ is defined as the minimum $\ell$ such that
$\beta_{\ell}\dots\beta_{2}\beta_{1}=\alpha$, where each $\beta_{i}$,
$i\in[1,\ell]$, is a $3$-cycle. Let ${c^{\circ}}_{odd}(\alpha)$ denote the
number of odd-length cycles (including $1$-cycles) in $\alpha$. Based on this,
Mira and Meidanis [12] proved the following lemma:

######  Lemma 1 (Mira and Meidanis [12])

$\lvert\lvert\alpha\rvert\rvert_{3}=\frac{n-{c^{\circ}}_{odd}(\alpha)}{2}$.

Now let us exam Equation 1. It can be rewritten as:

$\tau_{t}\dots\tau_{2}\tau_{1}=\bar{\iota}\bar{\pi}^{-1}.$ (2)

This means that the product of the $3$-cycles that transform $\bar{\pi}$ into
$\bar{\iota}$ is equal to the product $\bar{\iota}\bar{\pi}^{-1}$.
Furthermore, since $\bar{\pi}$ and $\bar{\iota}$ are both $(n+1)$-cycles, the
product $\bar{\iota}{\bar{\pi}}^{-1}$ is always even. As a result, we can
state a lower bound for SBT as follows:

######  Lemma 2 (Mira and Meidanis [12])

$\displaystyle d_{t}(\bar{\pi})$
$\displaystyle\geq\lvert\lvert\bar{\iota}\bar{\pi}^{-1}\rvert\rvert 3$
$\displaystyle\geq\frac{n+1-{c^{\circ}}_{odd}(\bar{\iota}\bar{\pi}^{-1})}{2}.$

Observe that the maximum number of cycles in $\bar{\iota}{\bar{\pi}}^{-1}$ is
achieved if and only if $\bar{\iota}{\bar{\pi}}^{-1}$ is the identity
permutation $\iota$. In this case, $\iota$ has $n+1$ cycles, all of which are
odd-length (notably, they are all of length $1$).

Lastly, Silva et al.[15] have shown that there is a one-to-one mapping from
$\bar{\iota}{\bar{\pi}}^{-1}$ to the cycle graph [2] of $\pi$. Therefore, the
lower bound stated above is equivalent to the well-known one devised by Bafna
and Pevzner [2], which is formalised in Appendix 0.A (Theorem 0.A.1).

###  3.2 Cycles of $\bar{\iota}{\bar{\pi}}^{-1}$

While the concepts in this section apply to any $(n+1)$-cycle $\bar{\pi}$, as
previously defined by Silva et al. [15], in the specific context of this paper
$\bar{\pi}$ will typically be a $3$-permutation. A $(n+1)$-cycle $\bar{\pi}$,
$(n+1)\equiv 3\pmod{0}$, is a $3$-permutation when
$\bar{\iota}{\bar{\pi}}^{-1}$ consists only of $1$\- or $3$-cycles.

Let $\gamma=(a\dots b\dots c\dots)$ be one of the disjoint cycles of
$\bar{\iota}{\bar{\pi}}^{-1}$. If $\bar{\pi}^{-1}=(a\dots c\dots b$ $\dots)$,
meaning that the symbols $a$, $b$, and $c$ appear in $\gamma$ in a cyclic
order different from the one in $\bar{\pi}^{-1}$, then we define $\gamma$ as
an _oriented cycle_. On the other hand, if there is no oriented triplet in
$\gamma$, then $\gamma$ is called an _unoriented cycle_.

Let $\delta=(a\;b\dots)$ and $\epsilon=(d\;e\dots)$ be two cycles of
$\bar{\iota}{\bar{\pi}}^{-1}$. If $\bar{\pi}^{-1}=(a\dots e\dots$ $b\dots
d\dots)$, meaning that the symbols of pairs $(a,b)$ and $(d,e)$ appear in an
alternating order in $\bar{\pi}^{-1}$, we say that $\delta$ and $\epsilon$ are
_intersecting_ cycles. A special case is when $\delta=(a\;b\;c\dots)$ and
$\epsilon=(d\;e\;f\dots)$ are such that $\bar{\pi}^{-1}=(a\dots e\dots b\dots
f\dots c\dots d\dots)$, implying that the symbols of triplets $(a,b,c)$ and
$(d,e,f)$ appear in an alternating order in $\bar{\pi}^{-1}$. Under this
circumstance, $\delta$ and $\epsilon$ are considered to be _interleaving_
cycles.

###### Example 4

Let $\bar{\pi}=(0\;7\;6\;1\;9\;5\;8\;4\;3\;2)$ and
$\bar{\iota}{\bar{\pi}}^{-1}=(0\;3\;5)(1\;7)(2\;4\;9)(6\;8)$. The cycles
$(1\;7)$ and $(6\;8)$ are examples of intersecting cycles, while $(0\;3\;5)$
and $(2\;4\;9)$ are interleaving cycles.

From Equation 2, we have that
$\bar{\iota}{\bar{\pi}}^{-1}{\tau_{1}}^{-1}\dots\tau_{t}^{-1}=\iota$, meaning
that the application of transpositions $\tau_{1}$, $\dots$, $\tau_{t}$ sorting
$\bar{\pi}$ (i.e., transforming $\bar{\pi}$ into $\bar{\iota}$) can be seen as
the incremental multiplication of $\bar{\iota}{\bar{\pi}}^{-1}$ by
${\tau_{1}}^{-1}$, $\dots$, ${\tau_{t}}^{-1}$. Let us denote the difference
${c^{\circ}}_{odd}(\bar{\iota}{\bar{\pi}}^{-1}\tau^{-1})-{c^{\circ}}_{odd}(\bar{\iota}{\bar{\pi}}^{-1})$
as $\Delta{c^{\circ}}_{odd}(\bar{\iota}{\bar{\pi}}^{-1},\tau)$.

######  Proposition 1 (Meidanis, Dias, and Mira [11, 12])

If $\tau$ is an applicable $3$-cycle, then
$\Delta{c^{\circ}}_{odd}(\bar{\iota}{\bar{\pi}}^{-1},\tau)\in\\{-2,0,2\\}$.

An applicable $3$-cycle $\tau$ such that
$\Delta{c^{\circ}}_{odd}(\bar{\iota}{\bar{\pi}}^{-1},\tau)=\mu$ is referred to
as a $\mu$-move. Proposition 1 identifies three types of moves, namely
$(-2)$-, $0$\- and $2$-move. An $(x,y)$-sequence, where $x\geq y$, denotes a
sequence of $x$ applicable $3$-cycles $\tau_{1}$, $\dots$, $\tau_{x}$ with at
least $y$ of them being $2$-moves. If $x\leq a$ and
$\frac{x}{y}\leq\frac{a}{b}$, then the $(x,y)$-sequence is called a
$\frac{a}{b}$-sequence.

A _configuration_ $\Gamma$ is a disjoint product of cycles of
$\bar{\iota}{\bar{\pi}}^{-1}$. We say $\Gamma$ is _connected_ if, for any two
cycles $\gamma_{1}$ and $\gamma_{m}$ of $\Gamma$, there are cycles
$\gamma_{2},\dots,\gamma_{m-1}$ in $\Gamma$ such that for each $i\in[1,m-1]$,
$\gamma_{i}$ intersects or interleaves with $\gamma_{i+1}$. $\Gamma$ is
considered a _component_ if it consists of only one oriented cycle not
intersecting or interleaving with any other cycle of
$\bar{\iota}{\bar{\pi}}^{-1}$; or it consists of a maximal connected
configuration of $\bar{\iota}{\bar{\pi}}^{-1}$. Lastly, we define an
_unoriented interleaving pair_ as a product $\gamma\delta$ of two $3$-cycles
such that $\gamma\delta$ is a component of $\bar{\iota}{\bar{\pi}}^{-1}$ and
both $\gamma$ and $\delta$ are unoriented and interleave with each other.

###### Example 5

Consider $\bar{\pi}=(0\;3\;2\;1\;4\;6\;5)$ with
$\bar{\iota}{\bar{\pi}}^{-1}=(0\;6\;5)(1\;3)(2\;4)$. $(0\;6\;5)$ and
$(1\;3)(2\;4)$ are the two components of $\bar{\iota}{\bar{\pi}}^{-1}$.

###### Example 6

Let $\bar{\pi}=(0\;5\;4\;3\;8\;7\;6\;2\;1\;9\;14\;13\;12\;11\;10)$ with
$\bar{\iota}{\bar{\pi}}^{-1}=(0\;11\;13)(1\;3\;5)(2\;7\;9)(4\;6\;8)(10\;12\;14)$.
The products $(0\;11\;13)(10\;12\;14)$ and $(1\;3\;5)(2\;7\;9)(4\;6\;8)$ are
both components of $\bar{\iota}{\bar{\pi}}^{-1}$. In particular,
$(0\;11\;13)(10\;12$ $\;14)$ is an unoriented interleaving pair.

Finally, let $\bar{\pi}$ be a $(n+1)$-cycle. We say $u\geq 2$ components are
_cleared_ when we are able to apply a $\frac{3}{2}$-sequence
$\tau_{1},\dots,\tau_{s}$ on $\bar{\pi}$ so that
$\lvert\lvert\bar{\iota}{\bar{\pi}}^{-1}\tau_{1}^{-1}\dots\tau_{s}^{-1}\rvert\rvert$
$=\lvert\lvert\bar{\iota}{\bar{\pi}}^{-1}\rvert\rvert-2u$.

##  4 Palisades and their transposition distance

This section introduces the concept of palisades, a family of permutations
that are “hard” to sort, as they require a significant number of
transpositions above the lower bound by Bafna and Pevzner (Theorem 0.A.1) to
be sorted. In this regard, this section aims to define their transposition
distance.

A permutation $\bar{\pi}$ is said to be a _palisade_ if all components of
$\bar{\iota}{\bar{\pi}}^{-1}$ are unoriented interleaving pairs. If
$\bar{\iota}{\bar{\pi}}^{-1}$ consists of $\phi$ unoriented interleaving
pairs, then we refer to $\bar{\pi}$ as a palisade of size $\phi$, or a _$\phi$
-palisade_.

###### Example 7

The permutations
$(0\;5\;4\;3\;2\;1\;6\;11\;10\;9\;8\;7\;12\;17\;16\;15\;14\;13)$ and
$(0\;11\;16\;15\;14\;13\;12\;17\;10\;9\;8\;1\;6\;5\;4\;3\;2\;7)$ are examples
of palisades. Their cycle graphs [2] are depicted in Appendix 0.A (figures 1
and 2, respectively).

The next lemma can be demonstrated by observing that, for clearing two and
three unoriented interleaving pairs, respectively, in the analysis carried out
to prove the algorithms of Elias and Hartman [6] and of Silva et al. [15], no
$(4,3)$\- and no $(8,6)$-sequence were found.

######  Lemma 3 (Elias and Hartman [6], Silva et al. [15])

There is no $(4,3)$\- and $(8,6)$-sequence of transpositions able to clear two
and three unoriented interleaving pairs on a palisade, respectively.

We can use the lower bound based on the counting of hurdles by Christie [4] to
establish a tighter transposition distance lower bound for palisades, setting
the initial conditions required for determining the transposition distance of
palisades. A _hurdle_ is a component of $\bar{\iota}{\bar{\pi}}^{-1}$
consisting only of odd-length cycles that cannot be cleared using only
$2$-moves. It is easy to see that the number of hurdles in a $\phi$-palisade
is $\phi$. Let $\bar{\pi}$ be a $\phi$-palisade, then

######  Lemma 4 (Christie [4])

$d_{t}(\bar{\pi})\geq 2\phi+\lceil\frac{\phi}{2}\rceil$,

where $2\phi$ is $3$-norm of $\bar{\iota}{\bar{\pi}}^{-1}$ and
$\lceil\frac{\phi}{2}\rceil$ is the minimum number of $0$-moves required to
sort $\bar{\pi}$ (i.e. one $0$-move for each pair of unoriented interleaving
pairs).

###### Theorem 4.1

If $\bar{\pi}$ is $\phi$-palisade, then
$d_{t}(\bar{\pi})=\Bigl{\lceil}\frac{11\phi}{4}\Bigr{\rceil}=\Bigl{\lceil}\frac{11(n+1)}{24}\Bigr{\rceil}$.

###### Proof

Let us consider the lower bound established by Lemma 4. To sort $\bar{\pi}$
using as few as $2\phi+\lceil\frac{\phi}{2}\rceil$ moves, it should be
possible to clear two unoriented interleaving pairs using only one $0$-move
(i.e., using a $(5,4)$-sequence, which includes a $(4,3)$-sequence). However,
this is not possible due to Lemma 3. Therefore, the lower bound can be raised
to $2\phi+2\lfloor\frac{\phi}{3}\rfloor+\phi\bmod 3$, meaning that at least
two $0$-moves are required to clear three unoriented interleaving pairs — the
remaining ones require one $0$-move each. Nonetheless, this is not possible
either, since Lemma 3 also states that there is no $(8,6)$-sequence to apply
on $\bar{\pi}$ clearing three unoriented interleaving pairs. As a result, the
lower bound of $\phi$ can be raised again, using similar reasoning, to
$2\phi+3\lfloor\frac{\phi}{4}\rfloor+\phi\bmod 4$. Luckily, in this case,
$(11,8)$-sequences [6, 15] can be optimally applied clearing four pairs at
once using three $0$-moves. Based on this, we can conclude that
$d_{t}(\bar{\pi})=2\phi+3\lfloor\frac{\phi}{4}\rfloor+\phi\bmod 4$. Now, it
remains to show the equality:

$2\phi+3\Bigl{\lfloor}\frac{\phi}{4}\Bigr{\rfloor}+\phi\bmod
4=\Bigl{\lceil}\frac{11\phi}{4}\Bigr{\rceil}.$ (3)

For this, we will examine the congruence of $\phi$ modulo $4$, as follows.

  1. 1.

$\phi\equiv 0\pmod{4}$. In this case, $\phi=4m$ for some integer $m$. Thus,

$\displaystyle 2\phi+3\Bigl{\lfloor}\frac{\phi}{4}\Bigr{\rfloor}+\phi\bmod 4$
$\displaystyle=2(4m)+3\Bigl{\lfloor}\frac{4m}{4}\Bigr{\rfloor}$
$\displaystyle=11m=\Bigl{\lceil}\frac{44m}{4}\Bigr{\rceil}=\Bigl{\lceil}\frac{11\phi}{4}\Bigr{\rceil}.$

  2. 2.

$\phi\equiv 1\pmod{4}$. In this case, $\phi=4m+1$ for some integer $m$. Thus,

$\displaystyle 2\phi+3\Bigl{\lfloor}\frac{\phi}{4}\Bigr{\rfloor}+\phi\bmod 4$
$\displaystyle=2(4m+1)+3\Bigl{\lfloor}\frac{4m+1}{4}\Bigr{\rfloor}+1$
$\displaystyle=11m+3$
$\displaystyle=\Bigl{\lceil}\frac{44m+11}{4}\Bigr{\rceil}$
$\displaystyle=\Bigl{\lceil}\frac{11(4m+1)}{4}\Bigr{\rceil}$
$\displaystyle=\Bigl{\lceil}\frac{11\phi}{4}\Bigr{\rceil}.$

  3. 3.

$\phi\equiv 2\pmod{4}$. In this case, $\phi=4m+2$ for some integer $m$. Thus,

$\displaystyle 2\phi+3\Bigl{\lfloor}\frac{\phi}{4}\Bigr{\rfloor}+\phi\bmod 4$
$\displaystyle=2(4m+2)+3\Bigl{\lfloor}\frac{4m+2}{4}\Bigr{\rfloor}+2$
$\displaystyle=11m+6$
$\displaystyle=\Bigl{\lceil}\frac{44m+22}{4}\Bigr{\rceil}$
$\displaystyle=\Bigl{\lceil}\frac{11(4m+2)}{4}\Bigr{\rceil}$
$\displaystyle=\Bigl{\lceil}\frac{11\phi}{4}\Bigr{\rceil}$

  4. 4.

$\phi\equiv 3\pmod{4}$. Finally, we have $\phi=4m+3$ for some integer $m$.
Thus,

$\displaystyle 2\phi+3\Bigl{\lfloor}\frac{\phi}{4}\Bigr{\rfloor}+\phi\bmod 4$
$\displaystyle=2(4m+3)+3\Bigl{\lfloor}\frac{4m+3}{4}\Bigr{\rfloor}+3$
$\displaystyle=11m+9$
$\displaystyle=\Bigl{\lceil}\frac{44m+33}{4}\Bigr{\rceil}$
$\displaystyle=\Bigl{\lceil}\frac{11(4m+3)}{4}\Bigr{\rceil}$
$\displaystyle=\Bigl{\lceil}\frac{11\phi}{4}\Bigr{\rceil}.$

Therefore, Equation 3 holds for all possible values of $\phi\bmod 4$, and
hence for all $\phi$. To complete the proof, remark that a $\phi$-palisade is
a permutation on $(n+1)\equiv 0\pmod{6}$ symbols. Thus,
$\lceil\frac{11\phi}{4}\rceil=\lceil\frac{11(n+1)}{24}\rceil$.

##  5 Transposition diameter for $3$-permutations

In this section, Theorem 4.1 is used to provide the exact value for $TD3(n)$.

Silva et al. [15] introduced a new upper bound for the transposition distance,
valid for any $\bar{\pi}$, which can be stated as follows.

######  Theorem 5.1 (Silva et. al [15])

$d_{t}(\bar{\pi})\leq
11\Bigl{\lfloor}\frac{\lvert\lvert\bar{\iota}{\bar{\pi}}^{-1}\rvert\rvert_{3}}{8}\Bigr{\rfloor}+\Bigl{\lfloor}\frac{3(\lvert\lvert\bar{\iota}{\bar{\pi}}^{-1}\rvert\rvert_{3}\bmod
8)}{2}\Bigr{\rfloor}$.

Using this upper bound, we can devise a new upper bound for the transposition
diameter for $3$-permutations (it should be noted that a $3$-permutation is a
permutation on $(n+1)\equiv 0\pmod{3}$ symbols).

###### Corollary 1

$TD3(n)\leq
11\Bigl{\lfloor}\frac{n+1}{24}\Bigr{\rfloor}+\Bigl{\lfloor}\frac{(n+1)\bmod
24}{2}\Bigr{\rfloor}$.

Let $\bar{\pi}$ be a $3$-permutation. We say $\bar{\pi}$ is _diametral_ if
$d_{t}(\bar{\pi})$ is equal to $TD3(n)$. One way to show that $\bar{\pi}$ is
diametral is by showing that $d_{t}(\bar{\pi})$ is equal to the upper bound of
the transposition diameter stated above. This observation is useful for
proving the next theorem, which provides the exact value for $TD3(n)$.

###### Theorem 5.2

$\displaystyle TD3(n)=$
$\displaystyle\Bigl{\lceil}\frac{11(n+1)}{24}\Bigr{\rceil},(n+1)\equiv
0\pmod{6}$ (4) $\displaystyle TD3(n)=$
$\displaystyle\Bigl{\lceil}\frac{11(n-2)}{24}\Bigr{\rceil}+1,(n+1)\equiv
3\pmod{6}$ (5)

###### Proof

We divide the proof into two parts.

  1. 1.

If $(n+1)\equiv 0\pmod{6}$, then the palisades on $n+1$ symbols are diametral.
The proof is by showing that their transposition distances are equal to the
upper bound established by Corollary 1. That is, we have to show that

$\Bigl{\lceil}\frac{11(n+1)}{24}\Bigr{\rceil}=11\Bigl{\lfloor}\frac{n+1}{24}\Bigr{\rfloor}+\Bigl{\lfloor}\frac{(n+1)\bmod
24}{2}\Bigr{\rfloor}.$ (6)

We begin by noting that the fact $(n+1)\equiv 0\pmod{6}$ implies $n+1=6m$, for
some $m$ integer. By replacing $n+1$ with $6m$ in the left-hand side of
Equation 6, we obtain:

$\lceil\frac{11(n+1)}{24}\rceil=\lceil\frac{11m}{4}\rceil.$

The same substitution made on the right-hand side yields:

$\displaystyle
11\Bigl{\lfloor}\frac{n+1}{24}\Bigr{\rfloor}+\Bigl{\lfloor}\frac{(n+1)\bmod
24}{2}\Bigr{\rfloor}$
$\displaystyle=11\Bigl{\lfloor}\frac{6m}{24}\Bigr{\rfloor}+\Bigl{\lfloor}\frac{6m-24\Bigl{\lfloor}\frac{6m}{24}\Bigr{\rfloor}}{2}\Bigr{\rfloor}$
$\displaystyle=11\Bigl{\lfloor}\frac{m}{4}\Bigr{\rfloor}+\Bigl{\lfloor}3m-12\Bigl{\lfloor}\frac{m}{4}\Bigr{\rfloor}\Bigr{\rfloor}$
$\displaystyle=11\Bigl{\lfloor}\frac{m}{4}\Bigr{\rfloor}+3m-12\Bigl{\lfloor}\frac{m}{4}\Bigr{\rfloor}$
$\displaystyle=3m-\Bigl{\lfloor}\frac{m}{4}\Bigr{\rfloor}.$

Now it remains to show that the equality below holds:

$3m-\Bigl{\lfloor}\frac{m}{4}\Bigr{\rfloor}=\Bigl{\lceil}\frac{11m}{4}\Bigr{\rceil}.$
(7)

We will divide the analysis into two cases, next.

    1. (a)

$m\equiv 0\pmod{4}$. In this case, $m=4l$, for some integer $l$. Replacing $m$
with $4l$ in Equation 7 gives:

$\displaystyle
12l-\Bigl{\lfloor}\frac{4l}{4}\Bigr{\rfloor}=\Bigl{\lceil}\frac{11(4l)}{4}\Bigr{\rceil}=11l.$

    2. (b)

$m\not\equiv 0\pmod{4}$. In this case, $m=4l+r$ for some integer $l$ and
$r\in[1,3]$. By rewriting the left-hand side of Equation 7, we have:

$3m-\Bigl{\lfloor}\frac{m}{4}\Bigr{\rfloor}=3(4l+r)-l=11l+3r.$

On the right-hand side,

$\Bigl{\lceil}\frac{11m}{4}\Bigr{\rceil}=\Bigl{\lceil}11l+\frac{11r}{4}\Bigr{\rceil}=11l+\Bigl{\lceil}\frac{11r}{4}\Bigr{\rceil}.$

Finally, note that $\lceil\frac{11r}{4}\rceil=3r$ for all $r\in[1,3]$.

  2. 2.

Now we consider the case $(n+1)\equiv 3\pmod{6}$. In this scenario, we have
identified another family of $3$-permutations that consists of
$\frac{(n-2)}{6}$ unoriented interleaving pairs and an oriented $3$-cycle not
intersecting with any other cycle of $\bar{\iota}{\bar{\pi}}^{-1}$. The
transposition distances of these permutations are
$\lceil\frac{11(n-2)}{24}\rceil+1$, and we can prove that they are diametral
(Figure 3 shows the cycle graph [2] of such a permutation). For this, we have
to demonstrate the following equality:

$\Bigl{\lceil}\frac{11(n-2)}{24}\Bigr{\rceil}+1=11\Bigl{\lfloor}\frac{n+1}{24}\Bigr{\rfloor}+\Bigl{\lfloor}\frac{(n+1)\bmod
24}{2}\Bigr{\rfloor}.$ (8)

Note that if $(n+1)\equiv 3\pmod{6}$, then $n+1=6m+3$ for some integer $m\geq
1$. By substituting this expression for $n+1$ into the left-hand side of
Equation 8, we obtain:

$\displaystyle\Bigl{\lceil}\frac{11(n-2)}{24}\Bigr{\rceil}+1$
$\displaystyle=\Bigl{\lceil}\frac{11m}{4}\Bigr{\rceil}+1.$

The same substitution on right-hand side gives:

$\displaystyle
11\Bigl{\lfloor}\frac{n+1}{24}\Bigr{\rfloor}+\Bigl{\lfloor}\frac{(n+1)\bmod
24}{2}\Bigr{\rfloor}$
$\displaystyle=11\Bigl{\lfloor}\frac{6m+3}{24}\Bigr{\rfloor}+\Bigl{\lfloor}\frac{6m+3-24\Bigl{\lfloor}\frac{6m+3}{24}\Bigr{\rfloor}}{2}\Bigr{\rfloor}$
$\displaystyle=11\Bigl{\lfloor}\frac{6m+3}{24}\Bigr{\rfloor}+\Bigl{\lfloor}\frac{6m+3}{2}\Bigr{\rfloor}-12\Bigl{\lfloor}\frac{6m+3}{24}\Bigr{\rfloor}$
$\displaystyle=\Bigl{\lfloor}\frac{6m+3}{2}\Bigr{\rfloor}-\Bigl{\lfloor}\frac{6m+3}{24}\Bigr{\rfloor}$
$\displaystyle=\Bigl{\lfloor}3m+\frac{3}{2}\Bigr{\rfloor}-\Bigl{\lfloor}\frac{m}{4}+\frac{3}{24}\Bigr{\rfloor}$
$\displaystyle=3m-\Bigl{\lfloor}\frac{m}{4}+\frac{3}{24}\Bigr{\rfloor}+1$

Our focus now is to show that:

$3m-\Bigl{\lfloor}\frac{m}{4}+\frac{3}{24}\Bigr{\rfloor}+1=\Bigl{\lceil}\frac{11m}{4}\Bigr{\rceil}+1.$
(9)

For this, let us analyse the congruence of $m$ modulo $4$.

    1. (a)

$m\equiv 0\pmod{4}$. In this case, $m=4l$, for some integer $l$. By rewriting
the terms, we have:

$\displaystyle 3m-\Bigl{\lfloor}\frac{m}{4}+\frac{3}{24}\Bigr{\rfloor}+1$
$\displaystyle=12l+1-\Bigl{\lfloor}l+\frac{3}{24}\Bigr{\rfloor}$
$\displaystyle=11l+1$
$\displaystyle=\Bigl{\lceil}\frac{11m}{4}\Bigr{\rceil}+1.$

Therefore, Equation 9 holds.

    2. (b)

$m\not\equiv 0\pmod{4}$. In this case, $m=4l+r$ for some integer $l$ and
$r\in[1,3]$. Rewriting the left-hand side of Equation 9 gives:

$\displaystyle\Bigl{\lceil}\frac{11m}{4}\Bigr{\rceil}+1$
$\displaystyle=\Bigl{\lceil}\frac{11(4l+r)}{4}\Bigr{\rceil}+1$
$\displaystyle=\Bigl{\lceil}11l+\frac{11r}{4}\Bigr{\rceil}+1$
$\displaystyle=11l+\Bigl{\lceil}\frac{11r}{4}\Bigr{\rceil}+1.$

On the right-hand side, we obtain:

$\displaystyle 3m-\Bigl{\lfloor}\frac{m}{4}+\frac{3}{24}\Bigr{\rfloor}+1$
$\displaystyle=3(4l+r)-\Bigl{\lfloor}\frac{4l+r}{4}+\frac{3}{24}\Bigr{\rfloor}+1$
$\displaystyle=12l+3r-\Bigl{\lfloor}l+\frac{r}{4}+\frac{3}{24}\Bigr{\rfloor}+1$
$\displaystyle=12l+3r-(l+\Bigl{\lfloor}\frac{r}{4}+\frac{3}{24}\Bigr{\rfloor})+1$
$\displaystyle=11l+3r+1.$

Note that the term $\lfloor\frac{r}{4}+\frac{3}{24}\rfloor$ in the expression
above was cancelled since it always evaluates to $0$. The observation that
$3r=\lceil\frac{11r}{4}\rceil$ holds for all $r\in[1,3]$ concludes the proof.

##  6 A barrier for approximating SBT

We observe that, when sorting a permutation $\bar{\pi}$ by applying sequences
of transpositions obtained by a polynomial-time routine, palisades of any
sizes $\phi$,
$0<2\phi\leq\lvert\lvert\bar{\iota}{\bar{\pi}}^{-1}\rvert\rvert_{3}$, may
appear. The lowest achievable approximation ratio when sorting a palisade —
using Bafna and Pevzner’s [2] as lower bound, is

$\bigg{(}\Bigl{\lceil}\frac{11(n+1)}{24}\Bigr{\rceil}\mathbin{/}\frac{n+1}{3}\bigg{)}\geq\bigg{(}\frac{33}{24}=1.375\bigg{)},$

where $\frac{n+1}{3}$ is the $3$-norm of a palisade on $n+1$ symbols. This
remark has a striking implication that the palisades are an obstacle to
approximating SBT with approximation ratios lower than $1.375$.

###### Corollary 2

It is impossible to guarantee an approximation ratio lower than $1.375$ when
sorting by transpositions using as lower bound, the one devised by Bafna and
Pevzner [2].

To illustrate Corollary 2, we provide the following example: an arbitrary
permutation in $S_{n}$ on which we apply a sequence of $2$-moves resulting in
a palisade. In this example, it is worth noting that, based only on the most
apparent properties of the cycles of $\bar{\iota}{\bar{\pi}}^{-1}$ (such as
cycle parity), we cannot set a lower bound for $d_{t}(\bar{\pi})$ that is
tighter than the one proposed by Bafna and Pevzner [2].

###### Example 8

Consider
$\bar{\pi}=(0\;23\;22\;21\;1\;6\;5\;11\;20\;10\;9\;8\;13\;4\;3\;7\;12\;18\;2\;17\;16$
$\;15\;14\;19)$ (cycle graph [2] depicted in Figure 4). We can apply on
$\bar{\pi}$ a sequence of three $2$-moves, namely: $\tau_{1}=(1\;20\;10)$,
$\tau_{2}=(2\;17\;7)$ and $\tau_{3}=(4\;18\;11)$, giving
$\tau_{3}\tau_{2}\tau_{1}\bar{\pi}=(0\;23\;22\;21\;20\;1\;6\;5\;4\;3\;2\;7\;12\;11\;10\;9\;8\;13\;18\;17\;16\;15\;14\;19)$,
which is a $4$-palisade.

We highlight that attempting to determine in advance whether a sequence of
transpositions will result in a palisade would be impractical. In this regard,
we recall that, even for a $3$-permutation, deciding whether or not a single
$0$-move is required for a sorting, is $\mathcal{NP}$-hard [3].

##  7 Conclusions

This study focused on palisades, a family of permutations that are
particularly “hard” to sort. We demonstrated that for $(n+1)\equiv 0\pmod{6}$,
palisades qualify as diametral permutations for the $3$-permutations subset of
$S_{n}$. Moreover, we have shown that there exists another family of diametral
$3$-permutations for $(n+1)\equiv 3\pmod{6}$ (a $3$-permutation is a
permutation on $(n+1)\equiv 0\pmod{3}$). Elias and Hartman [6] had previously
established an upper bound for the transposition diameter for
$3$-permutations. In this work, we have provided the exact value.

Interestingly, our findings also indicate that palisades pose an obstacle to
achieving approximation ratios lower than $1.375$ for SBT. This partly answers
a question raised by Bulteau, Fertin and Rusu [3] about the feasibility of
devising a polynomial-time scheme for approximating SBT. Such a scheme is not
possible using as lower bound, the well-known one by Bafna and Pevzner [2].
Therefore, for the study of approximate solutions for SBT to advance, it is
crucial to find new lower bounds for the transposition distance.

##  Appendix 0.A Cycle graph

The cycle graph is a commonly used graphical representation for permutations
in the context of SBT [2]. To construct the cycle graph of
$\pi=[\pi_{1}\;\pi_{2}\dots\pi_{n}]$, two extra symbols $\pi_{0}=0$ and
$\pi_{n+1}=n+1$ are added to it. Thus, _cycle graph_ of $\pi$, denoted by
$G(\pi)$, is a directed graph consisting of the set of vertices
$\\{+0,\;-1,\;+1,\;-2,\;+2,\;\dots,$ $\;-n,\;+n,\;-(n+1)\\}$ and a set of
coloured edges (which can be black or gray). The black edges connect
$-\pi_{i}$ to $+\pi_{i-1}$ for all $i\in[1,n+1]$. The gray edges connect
vertex $+i$ to vertex $-(i+1)$ for all $i\in[0,n]$. The black edges represent
the current state of genes in the chromosome of $\pi$, while the gray edges
indicate the desired order in the second permutation $\iota=[1\;2\dots n]$.
Figures 1, 2 and 3 provides examples of cycle graphs.

Each vertex in $G(\pi)$ has in-degree and out-degree equal to $1$, with one
black edge entering and one gray edge leaving. This induces a unique cycle
decomposition in $G(\pi)$. A _$\kappa$ -cycle_ is a cycle $C$ in $G(\pi)$ with
$\kappa$ black edges. If $\kappa$ is _even (odd)_ , we also say that $C$ is an
_even_ (_odd_) cycle.

A transposition $\tau(i,j,k)$, with $1\leq i<j<k\leq n+1$, exchanges the block
of symbols corresponding to interval $[\pi_{i},\pi_{j-1}]$ with the block of
symbols corresponding to interval $[\pi_{j},\pi_{k-1}]$. Thus, applying
$\tau(i,j,k)$ on $\pi$, denoted by $\tau(i,j,k)\cdot\pi$ or just
$\tau\cdot\pi$, yields the new permutation
$[\pi_{1}\dots\pi_{i-1}\;\pi_{j}\dots\pi_{k-1}$
$\;\pi_{i}\dots\pi_{j-1}\;\pi_{k}\dots\pi_{n}]$. Let $c_{odd}(\pi)$ be the
number of odd cycles in $G(\pi)$ and let $\Delta
c_{odd}(\pi,\tau)=c_{odd}(\tau\cdot\pi)-c_{odd}(\pi)$ denote the change in the
number of odd cycles in $G(\pi)$ after the application of a transposition
$\tau$. The following lower bound, formulated by Bafna and Pevzner [2], is the
foundation of the most prevalent approximation algorithms for SBT.

######  Theorem 0.A.1 (Bafna and Pevzner [2])

$d(\pi)\geq\frac{n+1-c_{odd}(\pi)}{2}$.

Figure 1: Cycle graph of the $3$-palisade
$[5\;4\;3\;2\;1\;6\;11\;10\;9\;8\;7\;12\;17\;16\;15\;14\;$ $13]$. Figure 2:
Cycle graph of the $3$-palisade
$[11\;16\;15\;14\;13\;12\;17\;10\;9\;8\;1\;6\;5\;4\;3\;2$ $\;7]$. Figure 3:
Cycle graph of the diametral $3$-permutation
$[8\;13\;12\;11\;10\;9\;14\;7\;6\;5\;$ $1\;3\;2\;4]$. Figure 4: Cycle graph of
$[23\;22\;21\;1\;6\;5\;11\;20\;10\;9\;8\;13\;4\;3\;7\;12\;18\;2\;17\;16\;15\;$
$14\;19]$.

## References

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Topological excitations in a spin-orbit coupled spin-1 Bose-Einstein
condensate under sinusoidally varying magnetic fields

Arpana Saboo1$\star$, Soumyadeep Halder1, Subrata Das1,2 and Sonjoy
Majumder1$\dagger$

1 Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur,
West Bengal 721302, India

2 Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA

$\star$<EMAIL_ADDRESS>, $\dagger$<EMAIL_ADDRESS>
## Abstract

We present a theoretical study of a spin-orbit coupled spin-1 Bose-Einstein
condensate under the influence of sinusoidally varying magnetic fields. In the
ground state of the ferromagnetic spin-1 condensate, we investigate
topological excitations in the system arising due to the combined effect of
Rashba spin-orbit coupling and an in-plane sinusoidally varying magnetic
field. In this work, we offer a comparative study for various choices of
magnetic fields in the $x\rm{-}y$ plane. For a fixed field strength, the spin-
orbit coupled system sustains a rich variety of exotic vortex structures
ranging from vortex-antivortex lattices to vortex clusters as we increase the
coupling strength.

##  1 Introduction

The experimental realization of spin-orbit coupling (SOC) in ultracold atomic
physics [1, 2, 3, 4, 5, 6, 7, 8, 9] stands as a transformative milestone,
opening new avenues for exploration at the quantum frontier. This
groundbreaking achievement propels the field into uncharted territories where
the delicate interplay between particle spins and their orbital motion can be
precisely controlled and manipulated [10, 11, 12]. Typically, the order
parameter of the spin-orbit (SO) coupled trapped Bose-Einstein condensates
(BECs) [3] exhibit a very rich variety of exotic quantum states such as vortex
lattice [13, 14], half-quantum vortex [3, 15], vortex neclace [16], skyrmion
[17, 18, 19], domain wall [2], solitons [20, 21, 22], knots [23, 24],
supersolid stripe phase [25, 26, 27], spin and field squeezing [28, 29] etc.
Moreover in recent times, research has been extended in new directions such as
SO-induced supersolidity [30, 27], spin-tensor momentum coupling [31], spin-
orbital angular momentum coupling [32], etc. The introduction of such
couplings has led to many interesting quantum states with novel topological
textures [33, 34]. Recently, the splitting of vortex cores [35] and the first-
order phase transitions [36] have been experimentally observed. These
topological defects in ultracold condensates are vital for drawing analogies
with those of condensed matter systems.

In this light, spinor condensates [37, 38, 11, 39] emerge as exceptional
platforms within ultracold atomic physics, serving as a prolific area of
research for the investigation of solid-state-like properties [40]. These
remarkable quantum systems, exhibiting a multitude of spin degrees of freedom,
provide an intriguing bridge between the microscopic world of ultracold atoms
and the condensed matter physics of solids. Spinor condensates offer a
controlled environment to explore phenomena such as magnetic ordering [41],
topological excitations [21] and defects [42], and quantum phase transitions
[43, 44], shedding light on the intricate connection between quantum gases and
condensed matter systems. This convergence of ultracold atomic physics and
solid-state physics not only enriches our understanding of fundamental physics
but also holds the promise of unveiling new quantum phenomena with potential
applications in quantum information processing and beyond [45].

On the other hand, recent literature shows that in-plane magnetic fields [46,
47, 48, 49, 50, 51, 52, 53, 54, 55] can be significantly used in investigating
the equisite many-body physics of ultracold atoms. Relevant studies
demonstrate that in-plane gradient magnetic fields can be used to achieve SOC
and spin Hall states in BECs [56]. These magnetic fields play a vital role in
creating synthetic non-Abelian gauge fields [57, 58]. Varying magnetic fields
can be accredited for the observation of a wide assortment of topological
defects, such as magnetic monopoles [50, 59], quantum knots [23], skyrmions
[18], etc. The ground state of spinor BECs in the gradient field has been
investigated to exhibit the central Mermin-Ho vortex with symmetrical vortex
lattices [60]. The combined effect of SOC and gradient magnetic field on such
systems resulted in skyrmion chains [18, 61] in the ground state order
parameter. To the best of our knowledge, the effect of a sinusoidally varying
magnetic field on SO-coupled BECs has not been given due attention. In this
work, we study the combined effect of a sinusoidally varying magnetic field
and SOC on the ground state of a spin-1 BEC. Here, we consider an SO-coupled
spin-1 BEC subject to various combinations of sinusoidally varying magnetic
fields in th $x\rm{-}y$ plane. We offer a comparative study between the
various choices of sinusoidally varying magnetic fields, thereby highlighting
the different topological structures so obtained for each case.

This paper is organised as follows: In section 2, we introduce our model
system and discuss its mean-field ground state under different situations for
the different combinations of sinusoidally varying in-plane magnetic field. In
section 3, we discuss the results obtained by numerically solving the Gross-
Pitaevskii (GP) equations and investigate the topological structures
(vortices) generated in the system as a consequence of the interplay between
SOC, sinusoidally varying magnetic field with the spin-1 order parameter. In
section 4, we summarize our findings and provide an outreach for future scope.

##  2 Model

In the mean-field, we consider a quasi-2D Rashba SO-coupled ferromagnetic
spin-1 BEC of $N$ identical atoms in an harmonic trap in a sinusoidally
varying magnetic field. The effective Hamiltonian [62] in the Gross-Pitaevskii
form can be given as $H=H_{0}+H_{int}$,

$H_{0}=\int
d\mathbf{r}\Psi^{\dagger}\left[-\frac{\hbar^{2}\nabla^{2}}{2M}+V(\mathbf{r})+\nu_{\rm{soc}}+g_{F}\mu_{B}\mathbf{B(\mathbf{r}).f}\right]\Psi$
(1) $H_{int}=\int
d\mathbf{r}\left(\frac{1}{2}c_{0}n^{2}+\frac{1}{2}c_{2}\langle\hat{F}\rangle^{2}\right)$
(2)

where
$\Psi=[\Psi_{1}(\mathbf{r}),\Psi_{0}(\mathbf{r}),\Psi_{-1}(\mathbf{r})]^{T}$
with $\mathbf{r}=x\hat{x}+y\hat{y}$ is the spinor wave function which
satisfies the normalization criteria, $\int d\mathbf{r}\Psi^{\dagger}\Psi=N$,
and $M$ is the atomic mass.
$V(\mathbf{r})=\frac{1}{2}M\omega_{\perp}^{2}(x^{2}+y^{2})$ is the 2D harmonic
trap with $\omega_{\perp}$ being the radial trapping frequency and
$l=\sqrt{\hbar/M\omega_{\perp}}$ being the harmonic oscillator length. The
quasi-2D geometry is achieved by imposing a tight binding along the axial
direction with trap aspect ratio, $\lambda=\omega_{z}/\omega_{\perp}\gg 1$.
Here, $n=n_{1}+n_{0}+n_{-1}=\sum_{m}\Psi^{\dagger}_{m}\psi_{m}$, $m=\pm 1,0$
is the atomic density, while $\langle\hat{F}\rangle=\Psi^{\dagger}\hat{F}\Psi$
is the spin density. $\hat{F}=(F_{x},F_{y},F_{z})$ represents the spin density
vector defined as $F_{\alpha}=\Psi^{\dagger}f_{\alpha}\Psi\quad(\alpha=x,y,z)$
with $f=(f_{x},f_{y},f_{z})$ is the irreducible representation of the $3\times
3$ Pauli spin matrices. $H_{int}$ denotes the standard density-density and
spin-exchange interactions are characterised by $c_{0}$ and $c_{2}$
parameters, respectively. For the ferromagnetic ground state, $c_{2}<0$ while
$c_{2}>0$ for polar ground state. For the interaction terms, the coupling
parameters are given as $c_{0}=\frac{4\pi\hbar^{2}(a_{0}+2a_{2})}{3M}$ and
$c_{2}=\frac{4\pi\hbar^{2}(a_{2}-a_{0})}{3M}$, where $a_{0,2}$ are the
$s\rm{-}$wave scattering lengths for the spin channels with total spins $0$
and $2$, respectively.

The Rashba SOC interaction [1, 3] is given as
$\nu_{SOC}=\gamma(f_{x}p_{y}-f_{y}p_{x})$ where $\gamma$ is the SOC strength
and $(p_{x},p_{y})$ represents the momentum in the quasi-2D space. Note that
this SOC is SU(2) type, therefore the states $\Psi_{1}$ and $\Psi_{-1}$ cannot
be coupled directly. In real experiments, the SU(2) SOC Hamiltonian can be
achieved by Raman coupling three laser beams of different polarizations and
frequencies. The external magnetic field is given as
$B(\mathbf{r})=(B_{x}\hat{x}-B_{y}\hat{y})$ in the 2D plane. Here,
$B_{x},B_{y}$ are sinusoidally varying magnetic fields along the $x$ and $y$
directions. We absorb the negative sign in the direction and denote
$B(\mathbf{r})=(B_{x},B_{y})$. The Lande’s g-factor $g_{F}=-\frac{1}{2}$ and
$\mu_{B}$ is the Bohr’s magneton.

The ground state and dynamics of the system can be described by the following
non-dimensionalised coupled GP equations,

$\displaystyle\rm{i}\frac{\partial{\psi_{1}}}{\partial{t}}$
$\displaystyle=\left[-\frac{1}{2}\nabla^{2}+V+c_{0}|\psi|^{2}+c_{2}\left(|\psi_{1}|^{2}+|\psi_{0}|^{2}-|\psi_{\rm{-}1}|^{2}\right)\right]\psi_{1}+c_{2}\psi_{\rm{-}1}^{*}\psi_{0}^{2}$
$\displaystyle+(B_{x}+\rm{i}B_{y})\psi_{0}-\rm{i}{\gamma}\left(\frac{\partial}{\partial
y}+\rm{i}\frac{\partial}{\partial x}\right)\psi_{0}$ (3)
$\displaystyle\rm{i}\frac{\partial{\psi_{0}}}{\partial{t}}$
$\displaystyle=\left[-\frac{1}{2}\nabla^{2}+V+c_{0}|\psi|^{2}+c_{2}\left(|\psi_{1}|^{2}+|\psi_{\rm{-}1}|^{2}\right)\right]\psi_{0}+2c_{2}\psi_{1}\psi_{0}^{*}\psi_{\rm{-}1}$
$\displaystyle+(B_{x}-\rm{i}B_{y})\psi_{1}+(B_{x}+\rm{i}B_{y})\psi_{\rm{-}1}-\rm{i}{\gamma}\left(\frac{\partial}{\partial
y}-\rm{i}\frac{\partial}{\partial
x}\right)\psi_{1}-\rm{i}{\gamma}\left(\frac{\partial}{\partial
y}+\rm{i}\frac{\partial}{\partial x}\right)\psi_{\rm{-}1}$ (4)
$\displaystyle\rm{i}\frac{\partial{\psi_{\rm{-}1}}}{\partial{t}}$
$\displaystyle=\left[-\frac{1}{2}\nabla^{2}+V+c_{0}|\psi|^{2}+c_{2}\left(-|\psi_{1}|^{2}+|\psi_{0}|^{2}+|\psi_{\rm{-}1}|^{2}\right)\right]\psi_{\rm{-}1}+c_{2}\psi_{1}^{*}\psi_{0}^{2}$
$\displaystyle+(B_{x}-\rm{i}B_{y})\psi_{0}-\rm{i}{\gamma}\left(\frac{\partial}{\partial
y}-\rm{i}\frac{\partial}{\partial x}\right)\psi_{0}$ (5)

where $\psi_{m}=N^{-1/2}l\Psi_{m}\quad(m=\pm 1,0)$ denotes the dimensionless
$m$th component wave function and $|\psi|^{2}=\sum_{m}|\psi_{m}|^{2}$ is the
total particle density confined in dimensionless external potential
$V=(x^{2}+y^{2})/2$. The contact interaction parameters in the dimensionless
form are given as $c_{0}=2N\sqrt{2\pi\lambda}(a_{0}+2a_{2})/3l$ and
$c_{2}=2N\sqrt{2\pi\lambda}(a_{2}-a_{0})/3l$, respectively. $B,{\gamma}$ are
the dimensionless magnetic field and SOC strengths, respectively. In our
numerical calculations, the lengths, time, energy(interaction and SOC) and
magnetic field are measured in the units of
$l,1/\omega_{\perp},\hbar\omega_{\perp}$ and
$(g_{F}\mu_{B}l)/\hbar\omega_{\perp}$, respectively.

The vorticity [63] of the $m-$th component can be measured as

$\Omega_{m}=\nabla\times J_{m}$ (6)

with the probability current density,

$J_{m}=\frac{\rm{i}\hbar}{2M}(\psi_{m}\nabla\psi^{*}_{m}-\psi^{*}_{m}\nabla\psi_{m})$
(7)

The topological excitation of the sytem arising due to the formation of
vortices is characterised by the spin texture [64]. For a spin-1 BEC, the spin
texture can be written as [65, 66],

$S_{\alpha}=\sum_{m,n=0,\pm
1}\psi^{*}_{m}(f_{\alpha})_{m,n}\psi_{n}/|\psi|^{2}\quad(\alpha=x,y,z)$ (8)

The topological charge density which characterises the topological structure
of the system is given as,

$q(\mathbf{r})=\frac{1}{4\pi}\mathbf{s}\cdot\left(\frac{\partial\mathbf{s}}{\partial
x}\times\frac{\partial\mathbf{s}}{\partial y}\right)$ (9)

and the total topological charge $Q$ is defined as

$Q=\int q(\mathbf{r})dxdy$ (10)

Here $\mathbf{s}=S/|S|$ and the topological charge $|Q|$ is unchanged under
any exchange of $S_{x},S_{y},S_{z}$.

##  3 Results and Discussions

We investigate the ground state structures of the spin-orbit coupled
ferromagnetic spin-1 BEC under the influence of external in-plane magnetic
field. In this section, we first discuss the effect of a polarized magnetic
field in the $x$ or $y$ direction. The ground state density profile and the
spin textures are shown for cases both with and without SOC. Thereafter we
discuss cases for sinusoidally varying in-plane magnetic fields. For each
combination of external magnetic field $B(\mathbf{r})=(B_{x},B_{y})$, we
present the ground state structures with and without SOC.

We offer a detailed numerical study to analyze the effects of system
parameters on the ground state structures. The luxury of the present system
lies in the large number of tunable parameters, which includes the spin-
independent and spin-exchange contact interactions, SOC strength, magnetic
field strength. To focus our attention on the effects of SOC and in-plane
magnetic field, we have considered $10^{4}$ number of $\rm{{}^{87}{Rb}}$ atoms
confined in a quasi-2D harmonic trap $V(\mathbf{r})$ with transverse trapping
frequency, $\omega_{\perp}=2\pi\times 20$ Hz with trap aspect ratio,
$\lambda=\omega_{z}/\omega_{\perp}=20$. The $s\rm{-}$wave scattering lengths
for channels of total spin 0 and 2 are $a_{0}=101.8a_{B}$ and
$a_{2}=100.4a_{B}$ respectively, where $a_{B}$ is the Bohr radius. With these
parameters [67], using the split-step Crank-Nicolson scheme [68, 69, 70], we
numerically solve the governing GP equations (2-2) for imaginary time
propagation to generate the ground state solutions for diverse choices of
magnetic fields for numerous SO coupling strengths. All of our simulations run
from a spatial extent of $-20l$ to $20l$ in both $x$ and $y$ directions with
$2001\times 2001$ grid points. The employed spatial discretization refers to
$\Delta x=\Delta y=0.02l$, with time step of numerical integration $\delta
t=(2\times 10^{-4})/\omega_{\perp}$.

###  3.1 Polarized gradient magnetic field

For this case, we choose a polarized magnetic field either in $x$ or $y$
direction. In this section, we draw comparisons between two choices, i.e.,
$x\rm{-}$polarised $B_{x}=B_{0}x$ or $y\rm{-}$polarised $B_{y}=B_{0}y$ for
equal magnetic field strength $B_{0}=0.25$. However, to ensure that $B$ is
divergenceless, a magnetic field along the $z-$ direction, i.e.,
$B_{z}=-B_{0}z$. As we ensure a very tight binding along the $z-$direction,
the magnetic field along this direction will not affect the Quasi-2D
condensates. The motivation to explore this case is to underline the impact of
such direction-dependent in-plane field on the SO-coupled BEC. The ground
state density distributions $n_{1}=|\psi_{1}|^{2}$, $n_{0}=|\psi_{0}|^{2}$ and
$n_{-1}=|\psi_{-1}|^{2}$ for the $x-$polarised case are shown in Fig.
1($a1-a3$) and that for the $y-$polarised case are shown in Fig. 1($c1-c3)$.
In the absence of SOC, the density $n_{0}$ is fragmented into two halves about
the $x-$axis with both components being phase coherent individually, while the
densities $n_{1}$ and $n_{-1}$ have a overall disk-shaped condensate with high
densities at the $x=0$ line. However, for the $y-$polarised case, the density
modulations are observed along the $y=0$ line. Due to the intricate interplay
between the spin-1 order parameter and the magnetic field bias, the density
profiles $n_{\pm 1}$ and $n_{0}$ show reverse signatures, i.e., the central
high density peak is observed in $n_{0}$ while the fragments are observed in
$n_{1},n_{-1}$. In the absence of SOC, there are no vortices formed in the
system as the polarised gradient magnetic field doesn’t have any saddle point.
However, the momentum space density profiles (as shown in Fig. 1($b1-b3$) and
Fig. 1($d1-d3$) for the $x-$polarised and $y-$polarised case, respectively),
emphasizes the unique structure of the condensates owing to the direction-
dependent density modulations. These density modulations can be associated
with the momentum distribution. Along with the central peak at
$k_{x},k_{y}=0$, there exists some higher $k_{x}$ value for the $x-$ polarised
magnetic field and similarly, some higher $k_{y}$ value for the $y-$ polarised
case. These $k_{i}$ values could be associated to a density wave with
wavelength $\lambda_{i}=2\pi/k_{i}\quad i=x,y$. In other words, extra peaks in
momemtum distribution at $k_{x},k_{y}\neq 0$ corresponds to density
modulations in the real space, such that the wavelength $\lambda_{i}$ of this
modulation is inversely $\propto k_{i}$.

Figure 1: The ground state densities and momentum distributions for the spin-1
BEC of $10^{4}$ $\rm{{}^{87}{Rb}}$ atoms without SOC. The first and second
columns show the density ($a1-a3$) and momentum ($b1-b3$) distributions of the
$m_{F}=1$, $m_{F}=0$ and $m_{F}=-1$ components for the $x-$ polarised magnetic
field $B=B_{0}x$. The third and fourth columns represent the density ($c1-c3$)
and momentum ($d1-d3$) distributions for the $y-$ polarised magnetic field
$B=B_{0}y$. The magnetic field strength $B_{0}=0.25$ for both cases. The
momentum space densities have been normalised by dividing the maximum value.
The unit length is the oscillator length $l$.

Now, with SOC in play, our investigation centers to unravel the competition
between the SOC and external magnetic field. Depending on the choices of the
external magnetic field applied, the ground state structures for the SO-
coupled spin-1 BEC are different. We show the ground state structures for the
SOC strength $\gamma=0.5$ for the $x-$polarised magnetic field in figure 2.
The ground state densities $n_{\pm 1},n_{0}$ exhibit modulations along the
$x=0$ line as shown in Fig. 2 ($a1-a3$). Our numerical simulations show that
the formation of vortices in the three components is pertinent to the
interplay of SOC and the external magnetic field. The SOC interacts with the
magnetic field in such a way that the effective magnetic field is modified. In
addition to this, as the SOC couples the intrinsic spin to the particles’
motion, it imparts some extra angular momentum to the system which results in
the formation of vortices in the system. Although there are no saddle points
in the magnetic field, these vortices appear at the $x=0$

Figure 2: The ground state structures for the spin-orbit coupled spin-1 BEC of
$10^{4}$ $\rm{{}^{87}{Rb}}$ atoms with SOC strength $\gamma=0.5$ in the the
$x-$ polarised magnetic field $B=B_{0}x$ with strength $B_{0}=0.25$. The first
column ($a1-a3$) shows the density distributions, second column ($b1-b3$)
shows the vorticity profiles and the third column ($c1-c3$) shows the momentum
distributions of the $m_{F}=1$, $m_{F}=0$ and $m_{F}=-1$ components. The
momentum space densities have been normalised by dividing the maximum value.
The unit length is the oscillator length $l$. Figure 3: ($a$) shows the
topological charge distribution, ($b$) shows the spin texture of the system
and ($c$) shows the local amplification of the spin texture for the SO-coupled
spin-1 BEC shown in Fig. 2. The unit length is the oscillator length $l$.

line under the influence of SOC. The vorticity profile associated to each
component is shown in Fig. 2 ($b1-b3$). The vortices are slightly deformed
along the $y-$direction as the component densities are modulated along the
$x=0$ line. The k-space density profiles (as shown in Fig. 2 (c1-c3)) depict
that apart from the central peaks, some extra momenta predominantly along the
$k_{x}-$direction is observed. These extra peaks at $k_{x}\neq 0$ corresponds
to the density modulations along the $x-$ direction. This is due to the fact
that SOC influences the distribution of particles in the k-space, similar to
the impact of SOC as seen in the distribution of electronic states as a
function of momentum in solid-state physics [71].

The combined effect of the in-plane direction dependent gradient field and the
SOC yield intricate spin textures in the spin-1 BEC as shown in Fig. 3 ($b$)
with corresponding topological charge distribution $q$ shown in Fig. 3($a$).
We can identify the presence of a skyrmion at the center of the vortex core,
from the spin texture. The local amplification of the skyrmion spin texture is
shown in Fig. 3 ($c$).

For the SO-coupled spin-1 BEC in $y-$polarised gradient magnetic field, we can
analogously understand the density modulations along the $y=0$ line for all
the three components. However, these density modulations converge into the
formation of anti-vortices in the system, and the corresponding spin textures
have anti-skyrmions along the $x-$axis. It is interesting to note that the
state which had fragments in the absence of SOC, now has more vortices or
anti-vortices as we observe for the case of $\gamma=0.5$. As we increase the
SOC strength $\gamma$, the number of vortices increases. However, for the
polarised magnetic field, they appear in the same alignment, with the vortex
spacing being reduced.

###  3.2 Cross-polarised sinusoidally varying in-plane magnetic fields

Thus far, we have discussed how impactful the polarised gradient field can be
on the ground state structure of a SO-coupled spin-1 BEC. To further exploit
this direction-dependent interplay of external magnetic field and SOC, we now
choose sinusoidally varying in-plane magnetic fields in both $x$ and $y$
directions. In general, to make the magnetic field divergenceless, we also
apply a magnetic field $B_{z}=-B_{0}zf(x,y)$ in the $z$-direction, where
$f(x,y)$ are sinusoidal functions in the $x-y$ plane. However, as we are in
the Quasi-2D geometry, for tight binding along the $z-$direction, $B_{z}$ will
not affect the spinor condensate. In this section, we center our focus only on
the in-plane sinusoidal magnetic field. Experimentally, a polarised sinusoidal
field can be realised using a similar technique discussed in [18], shown for
the magnetic field in the vertical direction. Campbell et al has also
demonstrated a similar in-plane sinusoidal field in their work [72]. We
discuss below, various combinations of $B(\mathbf{r})=(B_{x},B_{y})$ in
different cases.

####  3.2.1 Case-i@ Sine-Sine magnetic field
$B(\mathbf{r})=B_{0}(\rm{sin}x,\rm{sin}y)$

For the present case, we choose the magnetic fields as sine functions in both
directions, i.e., $B_{x}=B_{0}\mathrm{sin}x$ and $B_{y}=B_{0}\mathrm{sin}y$
with equal magnetic strength $B_{0}=0.25$ in both directions. We first present
the ground state structures for the no SOC case in Fig. 4. The density
profiles $n_{1},n_{0}$ and $n_{-1}$ of the condensates $m_{F}=1,m_{F}=0$ and
$m_{F}=-1$, respectively are shown in Fig. 4 ($a1-a3$). The sine-sine type of
magnetic field results in the density modulations in $m_{F}=\pm 1$ components
with the appearance of topological exciations in the form of a triad of
alternating vortex-anti-vortex along both $x-$ and $y-$axes. This is due to
the fact that the applied magnetic field from both directions, generate
certain saddlepoint structures such that the nature of the triads in $m_{F}=1$
and $m_{F}=-1$ components are complementary to each other. The opposite nature
of the triads is due to the fact that the magnetic field appears as conjugate
for the $m_{F}=1$ and $m_{F}=-1$ components in the GP equations (2-2).
Furthermore, combining with the spin-mixing term, the magnetic field yields
vortex brights in the $m_{F}=0$ component at

Figure 4: The ground state structures for spin-1 BEC of $10^{4}$
$\rm{{}^{87}{Rb}}$ atoms without SOC in the sine-sine magnetic field
$B_{x}=B_{0}\mathrm{sin}x$ and $B_{y}=B_{0}\mathrm{sin}y$ with strength
$B_{0}=0.25$. The first column ($a1-a3$) shows the density distributions,
second column ($b1-b3$) shows the vorticity profiles and the third column
($c1-c3$) shows the momentum distributions of the $m_{F}=1$, $m_{F}=0$ and
$m_{F}=-1$ components. The momentum space densities have been normalised by
dividing the maximum value. The unit length is the oscillator length $l$.

the loci (saddle points of $B(\mathbf{r})$) of vortices in the $m_{F}=\pm 1$
components so as to conserve the overall angular momentum of the spin-1 BEC in
the absence of SOC. Thus, the $m_{F}=0$ component has no vortices, instead a
high density peak is observed at the center of the condensate. The vorticity
profiles associated to the condensate densities are depicted in Fig. 4
($b1-b3$). These vortices or anti-vortices are coreless as there is no phase
singularity in the total density of the system. The k-space density profiles
(see Fig. 4 ($c1-c3$)) support observed density modulations in the system. For
the $m_{F}=1$ and $m_{F}=-1$ components, apart from the central peak, we
observe finite elongations in both the $k_{x}$ and $k_{y}$ directions which
correspond to the density modulations with wavelength inversely $\propto
k_{x},k_{y}$.

Since, in the absence of SOC, the overall angular momentum of the spin-1
system is conserved, the corresponding topological charge of the system is
$|Q|=0$.

In the presence of SOC, the ground state structures emerge as a result of the
interplay

Figure 5: The ground state structures for spin-orbit coupled spin-1 BEC of
$10^{4}$ $\rm{{}^{87}{Rb}}$ atoms with SOC strength $\gamma=0.5$ in the sine-
sine magnetic field $B_{x}=B_{0}\mathrm{sin}x$ and $B_{y}=B_{0}\mathrm{sin}y$
with strength $B_{0}=0.25$. The first column ($a1-a3$) shows the density
distributions, second column ($b1-b3$) shows the vorticity profiles and the
third column ($c1-c3$) shows the momentum distributions of the $m_{F}=1$,
$m_{F}=0$ and $m_{F}=-1$ components. The momentum space densities have been
normalised by dividing the maximum value. The unit length is the oscillator
length $l$.

among the contributions from the magnetic field, spin-orbit coupling, and the
spinor order parameter. For SOC strength, $\gamma=0.5$, we show the density
profiles of the $m_{F}=1,m_{F}=0$ and $m_{F}=-1$ components in Fig. 5
($a1-a3$). The system experiences an induced anisotropy as a result of
rotational symmetry breaking due to SOC. This induces distinctive ground state
structures, giving rise to the formation of unique configurations such as
vortices, skyrmions, and unconventional spin textures. The vorticity profiles
(see Fig. 5 ($b1-b3$)) depicts the existence of vortices, anti-vortices in the
spin-1 condensate. Due to the induced anisotropy, the vortices or anti-
vortices are not only deformed in shape, but also biased in opposite axes for
the $m_{F}=1$ and $m_{F}=-1$ components, respectively. The $m_{F}=0$ component
experiences density modulations in both directions as a consequence of the
spin-exchange interaction, SOC and external magnetic field.

The k-space density structures (as shown in Fig. 5 ($c1-c3$)) are also
anisotropically

Figure 6: ($a$) shows the topological charge distribution, ($b$) shows the
spin texture of the system and ($c$) shows the local amplification of the spin
texture for the SO-coupled spin-1 BEC shown in Fig. 5. The unit length is the
oscillator length $l$. Figure 7: The ground state structures for spin-1 BEC of
$10^{4}$ $\rm{{}^{87}{Rb}}$ atoms without SOC in the cosine-cosine magnetic
field $B_{x}=B_{0}\mathrm{cos}x$ and $B_{y}=B_{0}\mathrm{cos}y$ with strength
$B_{0}=0.25$. The first column ($a1-a3$) shows the density distributions,
second column ($b1-b3$) shows the vorticity profiles and the third column
($c1-c3$) shows the momentum distributions of the $m_{F}=1$, $m_{F}=0$ and
$m_{F}=-1$ components. The momentum space densities have been normalised by
dividing the maximum value. The unit length is the oscillator length $l$.

elongated in the preferred direction of high densities, thereby justifying the
density modulations, arising due to the combined outcome of SOC and the sine-
sine magnetic field. Analogous to the previously discussed situations, here
also we can associate a density modulation with wavelength $\lambda_{i}$ for
additional momentum peak at $k_{i}\neq 0,\quad i=(x,y)$.

We show for the above discussed case, the topological charge distribution in
Fig. 6 ($a$) and associated spin texture in Fig. 6 ($b$). The overall spin
texture has both skyrmions and anti-skyrmions along the two orthogonal axes.
Fig. 6 ($c-d$) are the local amplification of the spin textures showing a
skyrmion and anti-skyrmion, respectively.

####  3.2.2 Case-ii@ Cosine-Cosine magnetic field
$B(\mathbf{r})=B_{0}(\mathrm{cos}x,\mathrm{cos}y)$

Here, we choose the magnetic fields as cosine functions in both directions,
i.e., $B_{x}=B_{0}\mathrm{cos}x$ and $B_{y}=B_{0}\mathrm{cos}y$ with equal
magnetic strength $B_{0}=0.25$ in both directions. Analogously, we first
attempt to understand the ground state structures due to this magnetic field
independent of the SO-coupling. The component-wise density profiles are shown
in Fig. 7 ($a1-a3$). Much likely the sine-sine magnetic field case, we observe
for this case also, due to the interaction of magnetic fields in both
directions, a certain stable saddlepoint structure is formed, owing to which a
combination of equal vortices and anti-vortices are being formed in the
$m_{F}=1$ and $m_{F}=-1$ components such that the overall conservation of
angular momentum is obeyed in the spin-1 system. The spin-exchange interaction
along with the effect of magnetic field, yields vortex brights in the
$m_{F}=0$ component. These local high densities in the $m_{F}=0$ component
compensate for the vortices or anti-vortices in the adjacent two components.
The vorticity profiles associated to the component densities are presented in
Fig. 7 ($b1-b3$), which clearly explains the angular momentum conservation in
the system. The topological excitations arise in the system in the form of
exotic vortex-anti-vortex configuration at the saddle points of the external
magnetic field. The density modulations due to such exotic structures can be
explained by the k-space density profiles (see Fig. 7 ($c1-c3$)). For this
case also, apart from the central peak, there exists extra momenta both in
$k_{x},k_{y}$ conforming the density modulations with wavlength inversely
$\propto$ non-zero momentum peaks in both directions. The topological charge
is zero for the non SOC case to respect the angular momentum conservation.

Now, we discuss further the effect of the cosine-cosine magnetic field on the
SO-coupled spin-1 BEC. For a weak SOC with strength $\gamma=0.5$, the
rotational symmetry of the system breaks giving rise to certain topological
excitations in the system. The density profiles of the SO-coupled spin-1 BEC
are shown in Fig. 8 ($a1-a3$). Due to the symmetry-breaking, an anisotropy is
induced in the system which renders a very unique density distribution to
three components of the spin-1 BEC. Trivially, due to the spin-mixing
interaction, the $m_{F}=1$ and $m_{F}=-1$ components complement each other,
with the presence of vortices in one at the loci of high density in the other
component. Moreover, the vortices in one are compensated by the anti-vortices
in the other. In the $m_{F}=0$ component, criss-cross vortex and vortex-
brights are observed at the loci of local high densities at the $m_{F}=\pm 1$
components, as a consequence of the density modulations due to the combined
effect of SOC, external magnetic field and spin-mixing dynamics in the system.
However, these vortices in the $m_{F}=0$ component have opposite vorticity in
the stable state configuration. The vorticity profiles as shown in Fig. 8
($b1-b3$) gives a clear picture about the topological excitations in the
system. In Fig. 8 ($c1-c3$), we present the k-space densities of the three
components of the spin-1 BEC. Here also, apart from the central peak, extra
momenta along the $k_{x},k_{y}$ directions are observed. These corresponds to
the density modulations. Interestingly, for the $m_{F}=0$ component, the
k-space density is modulated obliquely as the local high densities are in that
direction.

We present the associated topological charge distribution in Fig. 9 ($a$) and
the spin texture is shown in Fig. 9 ($b$). For the diagonal vortices in the
$m_{F}=0$ component, as the cores are symmetric, the skyrmion and anti-
skyrmion structures are circular. However, for elongated

Figure 8: The ground state structures for spin-orbit coupled spin-1 BEC of
$10^{4}$ $\rm{{}^{87}{Rb}}$ atoms with SOC strength $\gamma=0.5$ in the
cosine-cosine magnetic field $B_{x}=B_{0}\mathrm{cos}x$ and
$B_{y}=B_{0}\mathrm{cos}y$ with strength $B_{0}=0.25$. The first column
($a1-a3$) shows the density distributions, second column ($b1-b3$) shows the
vorticity profiles and the third column ($c1-c3$) shows the momentum
distributions of the $m_{F}=1$, $m_{F}=0$ and $m_{F}=-1$ components. The
momentum space densities have been normalised by dividing the maximum value.
The unit length is the oscillator length $l$. Figure 9: ($a$) shows the
topological charge distribution, ($b$) shows the spin texture of the system
and ($c$) shows the local amplification of the spin texture for the SO-coupled
spin-1 BEC shown in Fig. 8. The unit length is the oscillator length $l$.
Figure 10: The ground state structures for spin-1 BEC of $10^{4}$
$\rm{{}^{87}{Rb}}$ atoms without SOC in the sine-cosine magnetic field
$B_{x}=B_{0}\mathrm{sin}x$ and $B_{y}=B_{0}\mathrm{cos}y$ with strength
$B_{0}=0.25$. The first column ($a1-a3$) shows the density distributions,
second column ($b1-b3$) shows the vorticity profiles and the third column
($c1-c3$) shows the momentum distributions of the $m_{F}=1$, $m_{F}=0$ and
$m_{F}=-1$ components. The momentum space densities have been normalised by
dividing the maximum value. The unit length is the oscillator length $l$.

vortices or anti-vortices, their corresponding topological charge
distributions are also elliptical. We show in Fig. 9 ($c$), the local
amplification of the spin texture for the region of the diagonal skyrmion and
anti-skyrmion.

It is paramount to note that in the absence of SOC, the ground state
structures for the $sine-sine$ case and the $cosine-cosine$ case can be
analogously explained by taking into account the fact that the topological
excitations are observed for certain saddle points in the magnetic fields. In
this light, it becomes interesting to study the ground state structures for
cross-fields, sine-cosine and/or cosine-sine fields.

####  3.2.3 Case-iii@ Sine-Cosine magnetic field
$B(\mathbf{r})=B_{0}(\rm{sin}x,\rm{cos}y)$

In this case, we study the ground state structures for the spin-1 BEC with the
external magnetic field chosen as $B_{x}=B_{0}\mathrm{sin}x$ and
$B_{y}=B_{0}\mathrm{cos}y$ with equal magnetic strength $B_{0}=0.25$. For

Figure 11: The ground state structures for spin-orbit coupled spin-1 BEC of
$10^{4}$ $\rm{{}^{87}{Rb}}$ atoms with SOC strength $\gamma=0.5$ in the sine-
cosine magnetic field $B_{x}=B_{0}\mathrm{sin}x$ and
$B_{y}=B_{0}\mathrm{cos}y$ with strength $B_{0}=0.25$. The first column
($a1-a3$) shows the density distributions, second column ($b1-b3$) shows the
vorticity profiles and the third column ($c1-c3$) shows the momentum
distributions of the $m_{F}=1$, $m_{F}=0$ and $m_{F}=-1$ components. The
momentum space densities have been normalised by dividing the maximum value.
The unit length is the oscillator length $l$.

this combination of magnetic fields, in the absence of SOC, the ground state
structures exhibit density modulations due to the resultant interaction of the
sine-cosine magnetic fields in the $x,y$ directions, respectively. We present
the ground state density profiles of the $m_{F}=1$, $m_{F}=0$ and $m_{F}=-1$
components in Fig. 10 ($a1-a3$). In the absence of SOC, although the overall
angular momentum is conserved, signature of topological excitations in the
form of vortices and anti-vortices are observed in the $m_{F}=1$ and
$m_{F}=-1$ components. These are arranged in pairs of alternating vorticity
along the $y-$direction and as alternating vortex-anti-vortex triads in the
$x-$direction. Due to different magnetic fields in different directions, the
local high densities in the $m_{F}=0$ component (which arrise as an outcome of
the spin-exchange interaction of $m_{F}=0$ component with the $m_{F}=\pm 1$
components), are anisotropic in nature. The vorticity profile (see Fig. 10
($b1-b3$)) confirms the complementary response of the $m_{F}=1$ and $m_{F}=-1$
components to the external magnetic field. The momentum distribution is shown
in Fig. 10 ($c1-c3$). For this case also, along with the central peak, extra
momentum both in $k_{x},k_{y}$ directions is observed. However, as their
magnitudes are not same in orthogonal directions, the density modulations have
different wavelengths along different axes.

In the presence of spin-orbit coupling, the system experiences symmetry
breaking, due to which an anisotropy is introduced in the spin-1 system under
the influence of magnetic field. For this case, the three components undergo
density modulations in both directions, however differently for different
components. The density as well as vortex distributions are highly anisotropic
in nature due to the combined effect of SOC-induced anisotropy and different
sinusoidal fileds in different directions. We present the component-wise
density profiles in Fig. 11 ($a1-a3$). For the $m_{F}=1$ component, there is a
local high density surrounded by vortices in the system, while for the
$m_{F}=-1$ component, there is a vortex surrounded by adjacent local
densities. The $m_{F}=0$ component has a unique configuration for which a
local high density is observed, which is surrounded by anti-vortices along the
$x-$direction and an adjacent vortex along the $y-$direction. The
corresponding vorticity profiles are given in Fig. 11 ($b1-b3$). For this case
also, the coreless vortices have deformed elliptical shape as a consequence of
induced anisotropy in the system. The momentum distributions as shown in Fig.
11 ($c1-c3$) show the effect of anisotropy in the density modulations in the
spin-1 system. The k-space densities are elongated in opposite directions for
the $m_{F}=1$ and $m_{F}=-1$ components, respectively, i.e, for the $m_{F}=1$
case, the central peak is slightly elongated along the $x-$axis while for the
$m_{F}=-1$ component the central density peak is slightly elongated in the
$y-$direction. This is to respect the density modulations with different
wavelengths in the anisotropic system. In addition to this, the extra momenta
in the $k_{x},k_{y}$ directions are not equivalent. For the $m_{F}=0$
component, the momentum distribution is symmetric however not the same about
the $k_{x},k_{y}$ axis.

The corresponding topological charge distribution is shown in Fig. 12 ($a$).
The presence of skyrmions and anti-skyrmions is confirmed by the spin texture
of the system shown in Fig. 12 ($b$). Because of the combined effect of
anisotropy and different magnetic fields along different directions, the spin
texture has both skyrmions and anti-skyrmions. Fig. 12 ($c$) shows the local
amplification of the spin texture showcasing the central vortex-anti-vortex
triad.

Figure 12: ($a$) shows the topological charge distribution, ($b$) shows the
spin texture of the system and ($c$) shows the local amplification of the spin
texture for the SO-coupled spin-1 BEC shown in Fig. 11. The unit length is the
oscillator length $l$.

####  3.2.4 Case-iv@ Cosine-Sine magnetic field
$B(\mathbf{r})=B_{0}(\rm{cos}x,\rm{sin}y)$

For the present case, we propose an alternate choice of cross-magnetic field
in the $x-y$ plane, i.e., $B_{x}=B_{0}\mathrm{cos}x$ and
$B_{y}=B_{0}\mathrm{sin}x$. Here, we attempt to highlight the striking
similarities and subtle differences of this present case with the previous
case (discussed in section 3.2.3). It is trivial to understand that for this
case also, the ground state structures are direction-dependent

Figure 13: The ground state structures for spin-orbit coupled spin-1 BEC of
$10^{4}$ $\rm{{}^{87}{Rb}}$ atoms with SOC strength $\gamma=0.5$ in the sine-
cosine magnetic field $B_{x}=B_{0}\mathrm{cos}x$ and
$B_{y}=B_{0}\mathrm{sin}y$ with strength $B_{0}=0.25$. The first column
($a1-a3$) shows the density distributions, second column ($b1-b3$) shows the
vorticity profiles and the third column ($c1-c3$) shows the momentum
distributions of the $m_{F}=1$, $m_{F}=0$ and $m_{F}=-1$ components. The
momentum space densities have been normalised by dividing the maximum value.
The unit length is the oscillator length $l$.

as a result of choosing different magnetic fields in different directions.
Importantly, as the magnetic fields along the $x,y-$directions are swapped,
their effect on the spin-1 ground state will therefore, be also swapped. In
the absence of SOC, the density profiles, associated vorticities and momentum
space densities of the three components are interchanged along the $x-y$ axes.

However, for the SO-coupled spin-1 BEC in the cosine-sine magnetic field, the
ground state structures are not simply the transpose of that of the sine-
cosine case. The ground state structure which is an outcome of the intricate
interplay between the external magnetic field, SOC and the spinor order
parameter is being presented in Fig. 14 for the SOC strength $\gamma=0.5$. The

Figure 14: ($a$) shows the topological charge distribution, ($b$) shows the
spin texture of the system and ($c$) shows the local amplification of the spin
texture for the SO-coupled spin-1 BEC shown in Fig. 13. The unit length is the
oscillator length $l$.

SOC results in rotational symmetry-breaking of the system rendering the system
an induced anisotropy. The density profiles (see Fig. 13 ($a1-a3$)) vouch for
the experienced anisotropy in the system. The $m_{F}=1$ component has a unique
configuration of vortices along with adjacent local high densities in the
system. The $m_{F}=0$ component has a linear chain of vortices in the
$y-$direction with an additional anti-vortex in the $x-$direction. The
$m_{F}=-1$ component has a local high density surrounded by vortex structures
on adjacent sides. The vorticity profile as shown in Fig. 13 ($b1-b3$) depicts
the nature of this vortices or anti-vortices in the system. The coreless
vortices have a slightly distorted shape due to the symmetry breaking. The
momentum distributions are presented in Fig. 13 ($c1-c3$). Due to the induced
anisotropy in the system, the k-space densities have visible elongations along
a particular direction. In addition, there exists certain extra momenta in the
$k_{x},k_{y}$ direction which is not symmetric.

The topological charge distribution is shown in Fig. 14 ($a$). The skyrmions
and anti-skyrmions are observed in different axes. The spin texture is shown
in Fig. 14 ($b$). Aside from the central triad formed with alternating vortex-
anti-vortex configuration, a few other spin textures can be observed owing to
the coreless vortices in the system. Fig. 14 ($c$) shows the local
amplification of the spin texture showing the central triad in the system.

##  4 Conclusion

In conclusion, the interplay between spin-orbit coupling and an external in-
plane magnetic field with sinusoidal variation in the $x-y$ direction in a
spin-1 BEC introduces an additional layer of complexity to an already
intricate system. The sinusoidal variation of the magnetic field in the $x-y$
plane provides a unique spatial modulation that further influences the ground
state of the condensate. The topological excitations in the form of vortex
formation occurs at the saddle points of the in-plane magnetic field. Along
with the spin-exchange interaction and spin-mixing dynamics, the magnetic
field exhibits conjugate vortices and anti-vortices in the $m_{F}=\pm 1$
components and vortex brights in the $m_{F}=0$ component. These density
modulations correspond to a wavelength $\lambda$ which is inversely
proportional to the non-zero momentum values in the k-space densities.

In this work, we have attempted to showcase numerous cases with various
combinations of sinusoidal fields in the $x-y$ direction. We have discussed
the ground state structures for all of these choices both with and without the
SOC, highlighting the subtle differences in each case. The SOC induces an
anisotropy in the system which breaks the rotational symmetry of the system.
As a consequence of which, the overall angular momentum is not conserved in
the system. Furthermore, the combination of SOC and the spatially varying
magnetic field gives rise to spatially modulated spin textures, including the
formation of unique skyrmions and anti-skyrmions. These topological defects
represent fascinating quantum states within the condensate, showcasing the
intricate interplay between the periodicity induced by the sinusoidal magnetic
field and the coupling of spin and motion.

The sinusoidal variation introduces periodicity into the system, creating an
opportunity to engineer and manipulate the condensate’s properties. This
perocidity of magnetic field can be exploited to see the effect of various
combinations of in-plane magentic fields. Beyond this, the dynamics of a SO-
coupled spin-1 BEC in a polarised sinusoidal magnetic field can be explored.
Moreover, it will be interesting to look into the skyrmion dynamics of the SO-
coupled system. The ability to tune the sinusoidal variation of the magnetic
field in a controlled environment not only provides a deeper understanding of
the quantum phases emerging from the interplay but also opens up possibilities
for designing novel quantum devices and applications. As investigations into
this intricate interplay continue, we anticipate the discovery of new
spatially modulated quantum states, offering exciting prospects for both
fundamental research and technological advancements.

## Acknowledgements

We thank Hari Sadhan Ghosh for many insightful discussions. We acknowledge the
National Supercomputing Mission (NSM) for providing computing resources of
PARAM Shakti at IIT Kharagpur, which is implemented by C-DAC and supported by
the Ministry of Electronics and Information Technology (MeitY) and Department
of Science and Technology (DST), Government of India.

##### Funding information

A.S. gratefully acknowledges the support from the Prime Minister’s Research
Fellowship (PMRF), India. S.H. acknowledges the MHRD, Govt. of India for the
research fellowship. S.D. acknowledges support from AFOSR FA9550-23-1-0034.

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