# Joint Network Coding and Machine Learning for Error-prone Wireless
Broadcast1

Dong Nguyen3, Canh Nguyen1, Thuan Duong-Ba2, Hung Nguyen3, Anh Nguyen3, and
Tuan Tran24 1This research will be presented in IEEE-CCWC 20172Corresponding
author 1FPT University, Hanoi, Vietnam
Email<EMAIL_ADDRESS>2Hanoi University of Science and Technology,
Hanoi, Vietnam
Email<EMAIL_ADDRESS>3Saolasoft Inc., CO 880211, USA
Email: {dnguyen, lnguyen<EMAIL_ADDRESS>4Sullivan University, KY
40205, USA
Email<EMAIL_ADDRESS>
###### Abstract

Reliable broadcasting data to multiple receivers over lossy wireless channels
is challenging due to the heterogeneity of the wireless link conditions.
Automatic Repeat-reQuest (ARQ) based retransmission schemes are bandwidth
inefficient due to data duplication at receivers. Network coding (NC) has been
shown to be a promising technique for improving network bandwidth efficiency
by combining multiple lost data packets for retransmission. However, it is
challenging to accurately determine which lost packets should be combined
together due to disrupted feedback channels. This paper proposes an adaptive
data encoding scheme at the transmitter by joining network coding and machine
learning (NCML) for retransmission of lost packets. Our proposed NCML extracts
the important features from historical feedback signals received by the
transmitter to train a classifier. The constructed classifier is then used to
predict states of transmitted data packets at different receivers based on
their corrupted feedback signals for effective data mixing. We have conducted
extensive simulations to collaborate the efficiency of our proposed approach.
The simulation results show that our machine learning algorithm can be trained
efficiently and accurately. The simulation results show that on average the
proposed NCML can correctly classify 90$\%$ of the states of transmitted data
packets at different receivers. It achieves significant bandwidth gain
compared with the ARQ and NC based schemes in different transmission terrains,
power levels, and the distances between the transmitter and receivers.

###### Index Terms:

Machine learning, network coding, wireless broadcasting.

##  I Introduction

Reliable broadcast is a mechanism for disseminating identical information from
one source to many receivers. It is widely used in many applications ranging
from satellite communications to wireless mobile ad hoc and sensor networks.
For instance, a cluster head may want to reliably broadcast information to
hundreds of sensors in its cluster for new software update. In such a
scenario, every time the source node sends a data packet to the receivers, a
feedback signal is used to inform the source whether it receives the data
packet successfully or not via acknowledgement (ACK) and negative-
acknowledgement (NAK), respectively. When the communication channels are
lossy, ACK or NAK data can be corrupted at the source. As a result, the source
node might retransmit redundant data, e.g., packets have been received
successfully at the receivers. These unnecessary retransmissions not only
reduce transmission bandwidth efficiency but also quickly drain the limited
energy of the sensors.

Currently, Forward Error Correction (FEC) in conjunction with Automatic
Repeat-reQuest (ARQ) is used to cope with packet errors or losses [1]. Using
FEC, the transmitter adds redundant information to transmitted data packets
which allows the receivers to detect and correct some of corrupted bits of the
received packets. When number of corrupted bits is greater than the correction
capability of the FEC codes, transmitter needs to retransmit the corrupted
data packet. These approaches work well in small size networks with good
channel condition. However, in large networks with the existence of high
interference, these approaches are bandwidth inefficient due to redundant
retransmission. That will significantly reduce the quality of service of the
network.

Recently, network coding [2] has been shown to be a promising technique for
improving the network bandwidth efficiency. Using NC, the transmitter combines
the lost data packets across multiple receivers for retransmission[3, 4]. In
this way, with a single retransmission of network coded packet, multiple
receivers can recover their lost data simultaneously, resulting in significant
bandwidth efficiency improvement. However, the existing NC-based approaches
are usually assumed that the transmitter instantly knows states of transmitted
data packets at the receivers [5, 6] to determine which packets should be
combined for retransmissions. However, such an assumption does not hold in
practical transmissions where both transmission and feedback links are subject
to errors. Some approaches, e.g., [7, 8], applied the framework of Markov
Decision Process at the transmitter to determine optimal packet combination
via statistics of link conditions. These approaches, however, suffer the curse
of dimensionality when number of receivers and transmitted data packets
increases.

In this paper, we propose a new approach by combining machine learning with
network coding for efficient bandwidth usage. In our approach, the transmitter
exploits the historical feedback signals from receivers to train a classifier
which is then used to predict the states of new transmitted data packets. The
transmitter then uses the predicted outcomes of the classifier to determine
its optimal data mixing for retransmission. The proposed machine learning
based network coding scheme (NC-ML) is efficient and data-driven without
assumption on the knowledge of the transmission channel statistics. Our
simulation results show that NC-ML can achieve very high classification
accuracy of the states of transmitted data packets and its accuracy improves
over time when more data is collected for training our classifier.

The rest of the paper is organized as follows. In Section II, we discuss some
recently published network coding and machine learning papers in wireless
networks. In Section III, we describe our system model and background on
network coding and supervised machine learning. We then describe in detail our
proposed NC-ML approach in IV. In Section V, we present our extensive
simulation results and discussion. Finally, we conclude our paper in Section
VI.

Figure 1: A typical setting for reliable wireless broadcast in which a sender
sends a stream of packets to multiple receivers and the receivers send
feedback to the sender to notify the status of received packets.

##  II  Related Work

Network coding has been shown to be an effective technique for sending data
both for multicast and broadcast scenario, particularly in wireless broadcast
networks [9, 10, 11, 3, 7]. In [9], the authors consider the problem of
wireless broadcasting in ad-hoc network with the use of network coding that
helps to reduce the energy consumption significantly. Katti et al. [11] and
Nguyen et al. [3] show that network coding can improve the channel efficiency
both in theory and practical networks. In addition, the wireless network
coding schemes are applied in scheduling for video and multimedia streaming
over the erroneous network to increase the multimedia transmission quality[7,
6].

Recently, many studies have been conducted on machine learning used in the
dynamic wireless networking environment to improve the transmission
throughput. [12] lists comprehensive review on how machine learning techniques
can be used to provide the solution for maximizing the resource utilization
and prolonging the network lifespan of wireless sensor networks. In another
study [13], the authors illustrate a data-driven approach to link adaptation,
where a machine learning classifier is employed to determine the optimal
modulation and coding scheme for multiple input-multiple output (MIMO)
systems. Kotobi et al. [14] propose the data-mining cognitive radio using
learning techniques for decision making and improving the performance of a
wireless network. By mining the dataset, the algorithms create more efficient
cognitive radio networks. In addition, [15] introduces frequency pattern
mining to efficiently allocate wireless spectrum between licensed and
unlicensed users. In [16], the authors present a review of using machine
learning algorithms for wireless sensor networks to adapt the environment
changes over time, from that, to increase the network efficiency and save the
limited resources of the networks.

Differently, in this paper, we propose a new method by utilizing machine
learning algorithms to predict states of transmitted data packets and
adaptively use network coding for retransmission. Our proposed approach is
efficient and accurate without any assumption on the knowledge of transmission
channel statistics. We will describe in detail our system model and the
proposed approach in the next sections.

##  III System Model

We consider reliable wireless broadcasting for multiple receivers in lossy
networks. In this setting, the transmitter wishes to reliably broadcast data
to several receivers with different channel conditions. Our system model and
related background are described in the next subsections.

Figure 2: A network coding model for wireless broadcast with feedback: (i) S
broadcasts packet a to R1 and R2, R2 does not receive it correctly, (ii) S
broadcasts packet b to R1 and R2, R1 does not receive it correctly, (iii) S
broadcasts the combined packet $a\oplus b$ so that R1, R2 can decode lost
packets.

###  III-A Reliable Wireless Broadcasting over Erroneous Links

We consider data broadcasting in lossy wireless networks where the transmitter
wants to reliably broadcast data to several receivers as illustrated in Figure
1. Every transmitted packet from the transmitter to receiver $R_{i}$ is
subject to an error probability $p_{i}$. To ensure reliable data transmission,
feedback signals, i.e., ACK/NAK, are sent from receivers to inform the
transmitter the states of the transmitted packet. In our paper, we assume that
the feedback links are also subject to errors. Whenever the transmitter
receives an NAK signal, it will retransmit the lost data packet to the
signaling receiver until it receives the packet successfully. When all
receivers obtain the transmitted packet successfully, the transmitter will
proceed to the next data packet. The process is repeated until all data
packets are successfully delivered.

###  III-B Wireless Network Coding

Network coding has been used to improve bandwidth efficiency of lossy wireless
networks [5]. To illustrate how NC is used to improve the network bandwidth,
we provide a simple example in Figure 2. In this example, a transmitter, e.g.,
an access point or a base station, wishes to reliably broadcast a stream of
data packets to two receiver R1 and R2. For the sake of simplicity, we assume
that the transmitter has two packets a and b for transmission. Due to
erroneous transmission links, we assume that packet $a$ is received correctly
at R1 while lost or received incorrectly at R2. In the second transmission,
packet $b$ is lost at R1 while successful at R2.

In the NC-based retransmission, the transmitter won’t retransmit lost data
immediately. Instead, the transmitter will mark lost packets at different
receivers and consider data combination for retransmission. Particularly, the
transmitter broads a combined packet $c=a\oplus b$ to all receivers. Upon
receiving the coded data packet successfully, both R1 and R2 can recover their
wanted data by using $a\oplus(a\oplus b)$ and $b\oplus(a\oplus b)$,
respectively. In this example, by using NC, the number of transmissions is
reduced from 4 to 3, thus, resulting in about $33\%$ bandwidth gain.

Figure 3: A flowchart of a supervised machine learning model

###  III-C Machine Learning for Classification

Machine Learning has been widely used to solve several challenges in many
fields, including wireless sensor networks. In this subsection, we present
some of the classification techniques which are used to identify the
transmitted feedback from the receiver. For the sake of completeness, we
provide a brief introduction of supervised machine learning for classification
here while its detail can be found in [17] .

Figure 3 illustrates a typical workflow of a supervised learning algorithm for
classification. In the first step, raw training data is cleaned and fed into
feature extraction to select only useful features from all available ones.
Next, labels or known class and extracted features are then passed to the
training phase where machine learning algorithms are used to identify a good
model which map inputs to desired outputs. The evaluation phase provides
feedback to the feature extraction and learning phases for adjustment to
improve model accuracy.The training process is repeated until a desired
accuracy level achieved. Once a model is constructed, it is then used to
predict label of new data. We will utilize the framework of supervised
learning to train our classifier to predict the state of feedback signals in
the next section.

##  IV The Proposed Machine Learning Based Network Coding

Figure 4: Inefficiency of error-prone feedback: (a) the sender correctly
retransmit the error packet; (b) the sender sends duplicate packet 2 due to
the error feedback.

###  IV-A The Impact of Error-Prone Feedbacks

To provide reliable wireless broadcast in lossy networks, feedback signals are
used to request retransmission of a lost data packet. Despite of using higher
redundancy, the feedback signals are also subject to error. Network bandwidth
efficiency will decrease significantly when feedback signals are
misclassified. Figure 4 illustrates an example of inefficient bandwidth usage
when feedback signal is incorrectly estimated.

In Figure 4(a), packet 2 is corrupted during transmission, thus, an NAK is
sent by the receiver over the feedback channel to inform the transmitter for
retransmission. Upon receiving of the feedback signal, the transmitter
retransmits packet 2. However, due to the error in the feedback channels,
feedback signals are subject to error as well. Without any mechanism to detect
the states of a transmitted data packet at the receivers, the transmitter
might retransmit redundant data packet, resulting in inefficient bandwidth
usage. Figure 4(b) shows an example of impact of error feedback on
transmission bandwidth. In this case, packet 2 is actually received
successfully at the receiver; however, due to feedback error, the transmitter
still retransmits packet 2. This inefficiency of retransmission can render
more overhead in a broadcast channel in which one error on particular feedback
can initiate a retransmission to all other receivers in the networks. The
erroneous feedback not only decreases bandwidth efficiency but also quickly
drains out limited power of wireless receivers.

###  IV-B The Proposed Machine Learning Based Network Coding (NCML)

In this subsection, we will describe our proposed machine learning based
network coding (NCML) approach for reliable data broadcast in lossy wireless
networks. Our NCML illustrated in Figure 5 consists of three main steps:

  * •

Collecting Feedback Data: In the first step, historical feedback data is
collected based on realistic simulation of data broadcast networks. In
particular, we simulate data transmission in different settings including
transmission range, transmit power, proximity environment, modulation, etc. to
gather our feedback signal data.

    * –

Transmission Environments: To simulate transmission different environments, we
adopt the link models proposed in [18]

$\displaystyle PL$ $\displaystyle=[A+10(a-bh_{b}+c/h_{b})log_{10}(d/d_{0})]+$
(1)
$\displaystyle[10x\sigma_{\gamma}log_{10}(d/d_{0})+y\mu_{\sigma}+yz\sigma_{\sigma}],$

where $A=20log_{10}(4\pi d_{0}/\lambda)$ denotes the fixed free-space path
loss at reference distance $d_{0}$ and wave length $\lambda$; $a,b,$ $c$ and
$\sigma_{\gamma}$ are data-derived constants for each terrain category;
$h_{b}$ denotes transmitter antenna height in meters; $d$ is distance between
transmitter and receiver; $x$, $y$, and $z$ denote a zero-mean Gaussian
variable of unit standard deviation $N[0,1]$; $\mu_{\sigma}$ and
$\sigma_{\sigma}$ are are both data-derived constants for each terrain.
Typically, the first part of Equation (1) denotes median the path loss at
distance $d$ depending on a specific terrain, base antenna height and
transmission distance while the second part represents a zero-mean random
variation about that median.

    * –

Data Generation: Simulation program is run several times to collect data of
different transmission configurations. The states of feedback signals are
labeled based on their decoded data (i.e., ACK or NAK). At the training step,
we discard all corrupted feedback signals which cannot be decoded.

  * •

Supervised Learning Classifier for Erroneous Feedbacks: In the first step, the
transmitter constructs a classifier for predicting the states of data packet
received at different receivers based on the historical feedback data. It
includes the following components:

    * –

Historical Feedback Data: This component collects and stores the past data of
feedback signals from different receivers. The states of feedback signals are
labeled based on only the ACK/NAK signals which don’t have any errors. In
other words, only successfully received ACKs/NAKs at the transmitter are used
for labeling the states of the transmitted data packets.

    * –

Feature Extraction and Selection: Obviously, there are many features of the
communication links between the transmitter and receivers can be utilized for
constructing our classifier. However, some of these features are correlated.
This step is to extract and select features of feedback signals for training
classifier in the next step. Table I shows list of our extracted features and
their units used in our training phase.

    * –

Training Classifier: Our classifier is constructed by using features extracted
from the Feature Extraction and Selection step. The collected dataset is first
partitioned into training and validation datasets. The training dataset is
used to construct our classifier and the validation dataset is used to
evaluate the constructed classifier. This training step is carried out
iteratively on different combination of the features. The classifier with
minimum classification error on the validation dataset is selected as our
final classifier. Algorithm 1 describes pseudocode of the training process.

0: Historical feedback dataset with features and labels

0: The best classifier (i.e., minimum classification error) Initialization :
$E\leftarrow$ collected feedback dataset; $\Sigma\leftarrow\\{\text{set of
machine learning techniques}\\}$; $\Delta\leftarrow\\{\text{set of all
feedback signal features}\\}$; $n\leftarrow|\Delta|$;
$P_{0}\leftarrow\emptyset$

1: for $T\in\Sigma$ do

2: best $\leftarrow$ CHOOSE$\\_$ATTRIBUTE{$\Delta,E$, method = $T$}

3: $P\leftarrow$ TRAIN$\\_$CLASSIFIER(best, data = $E_{training}$, method =
$T$)

4: if (EVALUATE($P$, data = $E_{Validation}$) $>$ EVALUATE($P_{0}$, data =
$E_{Validation}$)) then

5: $P_{0}\leftarrow P$

6: end if

7: end for

8: return $P_{0}$

Algorithm 1 Pseudocode of NCML Classifier

TABLE I: Feedback Signal Parameters No. | Features |  Description
---|---|---
1 | distance (m) |  Estimated distance between the transmitter and receivers
2 | noise (dBm) |  Noise level
3 | terrains |  Types of transmission environments (terrains)
4 | snr (dBm) |  Signal to noise ratio of feedback signal
5 | rx (dBm) |  Received power of feedback signal at the transmitter
6 | mod |  Types of modulation
  * •

Machine Learning Based Network Coding Retransmission: In the third step, NCML
performs network coding based data mixing for retransmissions. Our
transmission includes two phases: transmission and retransmission phases. In
the transmission phase, the transmitter keeps sending data packets to the
receivers and collects feedback signals. These feedback signals are fed to our
constructed classifier to determine the states of the transmitted data packets
at different receivers. The output of the classifier is used to create a
packet state map of the transmitted data packets at different receivers for
all data packets transmitted in transmission phase. After the transmission
phase, the transmitter will consider lost packets for retransmission. Based on
the constructed packet state map, the transmitter will determine which lost
packets are combined together for retransmission.

Figure 5: Block diagram of NCML

##  V Simulation Results and Discussion

###  V-A Simulation Setting and Transmission Schemes

We set up a basic simulation setting of a WiMAX based wireless broadcast
scenario in which there are one transmitter and multiple receivers. We assume
that data is always available for transmission at the transmitter.
Transmission is divided into transmission and retransmission phases. In the
transmission phase, data packets are transmitted to all receivers and feedback
signals are collected. In our simulation, both transmission and feedback
signals are subject to errors due to interference and fading in the wireless
networks. When a data packet is lost at a receiver, the transmitter will
retransmit it until it is received successfully. In our simulation, we
consider the following types of terrains:

TABLE II: Parameters of Simulation Environments MODEL PARAMETER | TERRAIN CATEGORY
---|---
| Terrain 1
---
(Hilly/Moderate-to-
Heavy Tree Density)
| Terrain 2
---
(Hilly/Light/Tree Density
or Flat/Moderate-to-
Heavy Tree Density)
| Terrain 3
---
(Flat/Light Tree
Density)
a | 4.6 | 4 | 3.6
|  b (in m-1)
---
0.0075 | 0.0065 | 0.005
| c (in m)
---
12.6 | 17.1 | 20
$\sigma_{\gamma}$ | 0.57 | 0.75 | 0.59
$\mu_{\sigma}$ | 10.6 | 9.6 | 8.2
$\sigma_{\sigma}$ | 2.3 | 3 | 1.6

  * •

Path Loss Model: We simulate suburban areas with different terrain categories
[18] in Table II. In our simulation, we also vary the distance between the
transmitter and receivers as well. This feature will be exploited in training
our model.

  * •

Transmission Schemes: In our simulation, we consider different transmission
schemes for reliable data delivery.

    * –

Naive re-transmission ARQ (ARQ) Scheme: In this scheme, if an NAK is received
from a receiver, the transmitter retransmits the lost packet until the
receiver receives it successfully. We emphasize that in this scheme, the state
of the transmitted data packet is purely based on the recovery of feedback
signal received at the transmitter. The process is repeated to all receivers
until all data packets are delivered successfully to all receivers. This
simulates a typical existing reliable data transmission protocol.

    * –

ARQ with Machine learning (ARQ-ML) Scheme: In ARQ-ML, the states of a
transmitted data packet at different receivers are determined by the
constructed classifier. The feedback signals from the receivers are fed
through our constructed classifier and its output will be used to determine if
the data packet is retransmitted to a receiver or not. The process is repeated
until all data packets are delivered successfully to all receivers.

    * –

Network Coding (NC) Scheme: In this scheme, the transmitter implements NC for
retransmission. Basically, the transmission is divided into two phases. In the
first phase, the transmitter sends out all data packets to the receivers and
collect feedback signals from the receivers. In the second phase, the
transmitter will consider lost data packets for retransmission. The
combination is performed adaptively based on the states of retransmission at
different receivers. In this scheme, the states of transmitted data packets
are based on the recovery of feedback signals from the receivers.

    * –

Machine Learning Based Network Coding (NC-ML) Scheme: Similar to NC schemes,
the NC-ML scheme also divides the data transmission into transmission and
retransmission phases where the transmitter sends out data packets in the
first phase and collect states of transmitted data packets at different
receivers. The second phase is for retransmission of lost data packets. The
difference between NC-ML compared to the NC is that in NC-ML, the states of
data packets are predicted by using our constructed classifier based on the
feedback signals from the transmitter.

  * •

Performance Metric: To compare different transmission schemes, we use the
network effective throughput defined as the average number of transmission per
data packet per receiver. Mathematically, we define our effective network
throughput by

$\eta=\lim_{N\rightarrow\infty}\frac{\sum_{i=1}^{N}n_{i}}{NMK},$ (2)

where $N$ denotes number of simulation trials, $M$ is number data packets for
transmission, $K$ and $n_{i}$ are number of receivers and number of
transmission of trial $i$, respectively. The transmission scheme with smallest
$\eta$ is the best approach.

Figure 6: Accuracy of machine learning algorithm versus the training size.

###  V-B Results and Discussion

We first evaluate the performance of our machine learning classifier with
different sizes of training dataset in Figure 6. As expected, the accuracy of
our classifier increases proportionally to the size of the training dataset.
This is because when number of training examples increases, the training
dataset retains almost all important information of the population which then
translates into the classifier, resulting in higher accuracy.

Figure 7: Comparison of different machine learning models

Next, we compare the classification accuracy of different machine learning
algorithms given the same size of training dataset. As shown in Figure 7, the
Neural Networks obtained the best performance with an average classification
accuracy of $90\%$ while Naive Bayes has the worst performance. This can be
explained by the flexibility thanks to multiple layers of the neural networks
which can be trained a classifier to fit to non-linear data. However, the high
accuracy of the Neural Networks also comes with the longer training time.

Figure 8: Bandwidth efficiency vs. transmission distance

In addition, we evaluate the network bandwidth efficiency of different schemes
in Figure 8. Particularly, we compare the average number of transmissions per
packet of different schemes versus transmission distances. In this simulation,
we consider two receivers and both of them have the same distances to the
transmitter. Those distances vary from 200 meters to 500 meters while the
transmit power is unchanged at 20 (dBm). As expected, the number of
transmissions per packet increases with the distance. This is because the
received powers at the receivers decrease when distance increases. As a
result, more number of packets will be lost during transmission requiring more
number of retransmissions. We also observe that ARQ and NC-ML achieve the
worst and best performance, respectively. We further observe that in the
regime of short distance, i.e., from 200 to 300 meters, the machine learning
based schemes, i.e., ML and NC-ML, obtain similar performance compared with
their counterparts, i.e., ARQ and NC, respectively. This is because in the
range of short distance, most of feedback signals are not corrupted.
Therefore, there is not much gain by using machine learning in predicting the
states of data packets in the regime. However, the performance gain increases
when transmission range is greater than 300 meters. This is because more
feedback signals got corrupted and machine learning based schemes bring
benefit by predicting correctly the actual states of transmitted data packets
at the receivers. As a result, they reduce the number of retransmissions.

Figure 9: Bandwidth efficiency vs. transmit power

Figure 9 compares the bandwidth efficiency of different transmission schemes
versus transmit power. In this simulation, we keep the distance between the
transmitter and receivers at 500 meters and used the Terrain Type 1 as our
transmission environment. As we can see, NC-ML achieves the best performance
thanks to the machine learning algorithm that correctly predicts states of
transmitted data packets. Consequently, it reduces the number of
retransmissions compared with other schemes.

Figure 10: Bandwidth efficiency vs. error probability of forward channels
Figure 11: Bandwidth efficiency vs. different types of terrains

Next, we evaluate the impact of forward channel conditions on the bandwidth
efficiency of different schemes. In this simulation, we assumed the two
receivers have the same packet error rate $p$ and varied it from 2$\%$ to
20$\%$. As $p$ increases, more packets will be corrupted at the receivers, as
such, the number of transmission per packet increases. ARQ requires more
number of retransmissions due to large number of lost packets. In addition,
ARQ suffers due to failing to estimate the actual states of the transmitted
data packets. As a result, it consumes more bandwidth for retransmitting
several unnecessary redundant packets. On the other hand, NC-ML exploits the
advantages of both machine learning algorithm for predicting states of
transmitted data packets and network coding for mixing data for
retransmission, it significantly improves network bandwidth efficiency.

Finally, we compare network bandwidth efficiency of different schemes in
different transmission environments in Figure 11. Particularly, we considered
three types of terrains as specified in [18] in our simulation. As expected
the number of transmission per packet is largest in terrain type 1 while
smallest in terrain type 3. This is because terrain type 1 models for hilly
and high tree densities while terrain type 3 models for flat terrain with
light tree densities. In all terrain categories, NC-ML achieves the best
performance thanks to the combination of machine learning algorithm and
network coding for retransmission.

##  VI Conclusion

In this paper, we have proposed a new approach by using machine learning
algorithms in conjunction with network coding for efficient data transmission
in lossy wireless networks. Our proposed machine learning algorithms are
efficient and accurate by exploiting important features of network channels
via historical feedback signals at the transmitter. Our simulation results
show that the transmitter can accurately classify more than 90$\%$ states of
transmitted data packets at the receivers. In combination with network coding
for data retransmission, the proposed scheme NC-ML significantly increases the
bandwidth efficiency compared with the existing approaches. Our future work
will implement a testbed of the proposed approach.

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YGHP-24-02

aainstitutetext: Research and Education Center for Natural Sciences, Keio
University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japanbbinstitutetext:
Department of Physics, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa
223-8521, Japanccinstitutetext: Department of Physics, Yamagata University,
Kojirakawa-machi 1-4-12, Yamagata, Yamagata 990-8560, Japanddinstitutetext:
International Institute for Sustainability with Knotted Chiral Meta
Matter(SKCM2), Hiroshima University, 1-3-2 Kagamiyama, Higashi-Hiroshima,
Hiroshima 739-8511, Japan

#  Topological solitons stabilized by a background gauge field and soliton-
anti-soliton asymmetry

Yuki Amari<EMAIL_ADDRESS>c,a,d Minoru Eto<EMAIL_ADDRESS>b,a,d Muneto Nitta<EMAIL_ADDRESS>
###### Abstract

We study topological lumps supported by the second homotopy group
$\pi_{2}(S^{2})\simeq{\mathbb{Z}}$ in a gauged $O(3)$ model without any
potential term coupled with a (non)dynamical $U(1)$ gauge field. It is known
that gauged-lumps are stable with an easy-plane potential term but are
unstable to expand if the model has no potential term. In this paper, we find
that these gauged lumps without a potential term can be made stable by putting
them in a uniform magnetic field, irrespective of whether the gauge field is
dynamical or not. In the case of the non-dynamical gauge field, only either of
lumps or anti-lumps stably exists depending on the sign of the background
magnetic field, and the other is unstable to shrink to be singular. We also
construct coaxial multiple lumps whose size and mass exhibit a behaviour of
droplets. In the case of the dynamical gauge field, both the lumps and anti-
lumps stably exist with different masses; the lighter (heavier) one
corresponds to the (un)stable one in the case of the nondynamical gauge field.
We find that a lump behaves as a superconducting ring and traps magnetic field
in its inside, with the total magnetic field reduced from the background
magnetic field.

##  1 Introduction

Solitons are non-perturbative excitations playing significant roles in physics
and mathematics. In particular, topological solitons are protected by topology
and thus are quite stable, ubiquitously appearing in quantum field theory
Rajaraman:1987 ; Manton:2004tk ; Shnir:2005vvi ; Eto:2006pg ;
Vachaspati:2006zz ; Weinberg:2006rq ; Shifman:2009zz ; Dunajski:2010zz ;
Weinberg:2012pjx ; Shnir:2018yzp ; Eto:2013hoa , cosmology Kibble:1976sj ;
Vilenkin:1984ib ; Hindmarsh:1994re ; Vilenkin:2000jqa ; Vachaspati:2015cma
and condensed matter systems Mermin:1979zz ; Volovik:2003fe ; Pismen ;
Svistunov:2015 ; Kawaguchi:2012ii . Topological solitons are classified into
defects and textures, the both of which are characterized by homotopy groups
in a different fashion. The former contains domain walls, vortices and
monopoles, while the latter does sine-Gordon solitons, baby(2D) Skyrmions, and
Skyrmions. However, topology is not sufficient for the stability of solitons;
energetically they may prefer to shrink or expand without suitable settings.
The Derrick’s scaling argument offers a simple criterion of the stability
under scaling of solitons of finite energy Derrick:1964ww . For instance, in
$d=3+1$ dimensions, Skyrmions Skyrme:1961vq ; Skyrme:1962vh  and Hopfions
Gladikowski:1996mb ; Faddeev:1996zj ; Battye:1998pe  are unstable to shrink
without a four derivative Skyrme term. In $d=2+1$ dimensions, topological
lumps (equivalently sigma model instantons in 2+0 dimensions) are scale
invariant Polyakov:1975yp , while they shrink in the presence of a potential
term, which can be prevented by introducing a four-derivative Skyrme term as
baby Skyrmions Bogolubskaya:1989ha ; Bogolyubskaya:1989fz ; Piette:1994ug ;
Weidig:1998ii  or time dependence as Q-lumps Leese:1991hr ; Abraham:1991ki .
Vortices can be made finite energy by introducing gauge field, otherwise they
have infinite energy.

Here, we pursue a yet another possibility for the stabilization of topological
solitons in terms of a background field. Topological solitons unstable to
shrink or expand can be prevented from collapsing by a background field. In
fact, such a stabilization mechanism is well known in condensed matter physics
as far as background fields are not dynamical, while our target is the case of
dynamical fields with backgrounds; In condensed matter physics, a typical
example with nondynamical background field is given by magnetic Skyrmions
Bogdanov:1989 ; Bogdanov:1995 ; Rossler:2006 ; doi:10.1126/science.1166767 ;
doi:10.1038/nature09124 ; Han:2010by ; Barton-Singer:2018dlh ; Ross:2020hsw ;
Amari:2023gqv , that is 2D Skyrmions in chiral magnets, which are stable even
without a four derivative Skyrme term. Chiral magnets are described by the
$\mathbb{C}P^{1}$ model with a so-called Dzyaloshinskii-Moriya (DM)
interaction Dzyaloshinskii ; Moriya:1960zz  which prevents Skyrmions from
shrinking. This interaction can be reformulated as a background $SU(2)$ gauge
field (a constant $SU(2)$ magnetic field) Schroers:2019hhe ; Amari:2023gqv .
See e.g. Refs. Ross:2022vsa ; Amari:2023gqv  for a Derrick’s scaling argument.
One of characteristic features is that only Skyrmions are stable while anti-
Skyrmions are unstable to shrink. Magnetic Skyrmions have been generalized to
${\mathbb{C}}P^{2}$ Skyrmions Akagi:2021dpk ; Amari:2022boe . Ultracold atomic
gases with spin-orbit couplings or synthetic gauge fields Lin:2009vfu ;
Goldman:2013xka  offer various examples such as 2D Skyrmions
PhysRevA.84.011607  and 3D Skyrmions Kawakami:2012zw . As mentioned above, in
all condensed matter examples, a gauge field is nondynamical and thus is a
constant background field.

Let us illustrate our idea with taking a concrete example of topological lumps
(or sigma model instantons in $d=2+0$) supported by the second homotopy group
$\pi_{2}(S^{2})\simeq{\mathbb{Z}}$ in an $O(3)$ model or equivalently
${\mathbb{C}}P^{1}$ model in $d=2+1$ dimensions Polyakov:1975yp . Topological
lumps are scale invariant solutions in the ${\mathbb{C}}P^{1}$ model without a
potential. The structure of topological lumps becomes completely different if
we gauge an $SO(2)\simeq U(1)$ subgroup of $O(3)$. The $U(1)$ gauged
${\mathbb{C}}P^{1}$ model with an easy-plane potential term is known to admit
stable lumps Schroers:1995he ; Schroers:1996zy ; Baptista:2004rk ;
Nitta:2011um , where a lump is decomposed into a pair of a vortex and anti-
vortex with fractional lump charges (sometimes called merons). However, if
there is no potential term, lumps are unstable to expand and eventually
diluted.

In this paper, we show that these gauged-lumps without potentials can be made
stable if we put them in a uniform magnetic field background. We separately
consider the cases that the $U(1)$ gauge field is nondynamical and dynamical,
and construct topological lump solutions in the presence of a uniform magnetic
field background. In the both cases, we apply the Derrick’s scaling argument,
numerically solve the equations of motion (EOM) for the axisymmetric lumps,
and confirm the Derrick’s scaling argument within numerical accuracy. One of
interesting results peculiar to our models with the background magnetic field
is asymmetry between solitons and anti-solitons; In the case of the non-
dynamical gauge field, either of lumps or anti-lumps stably exists depending
on the sign of the background magnetic field, and the other is unstable to
shrink to a singular configuration. This situation is quite analogous to
magnetic Skyrmions in chiral magnets. We find that the stable (anti-)lumps
have no size modulus reflecting the fact that the model is not scale invariant
due to the mass scale $\sqrt{eB}$ with the background magnetic field $B$ and
$U(1)$ gauge coupling constant $e$. We numerically find that the lump size is
of order $1/\sqrt{eB}$ and asymptotic tails of profile functions decay to the
vacuum exponentially, unlike ungauged BPS lumps whose profiles decay
polynomially. The mass of the single lump is smaller than that of a BPS lump.
We also construct coaxial lumps with higher winding number $k$ and find that
their size and mass are proportional to $\sqrt{k}$ up to a constant shift,
implying that the lumps behave as droplets. The higher lumps exhibit ring-like
shape and their interiors are empty. On the other hand, in the case of the
dynamical gauge field, the above properties of the stable lumps for the non-
dynamical gauge field remain qualitatively correct. However, unlike the
nondynamical case, both the lumps and anti-lumps stably exist as regular
solutions with different masses; the lighter one corresponds to the regular
one in the case of the nondynamical gauge field, while the heavier one does to
the unstable (singular) one. The dynamical gauge field deforms the magnetic
field around the lumps. We find that a lump behaves as a superconducting ring
and traps magnetic field in its inside, with the total magnetic field reduced
from the background magnetic field because of superconductiviy expelling it.
The higher lumps appear as ring-like shape but their interiors are not empty
and are filled by trapped magnetic fields which are almost constant.

A motivation to consider the problem of this paper is originated from QCD in
the presence of strong magnetic fields. QCD phase structure with a finite
chemical potential in the presence of strong magnetic fields gather a lot of
attention due to its relevance in neutron stars and heavy-ion collisions
Kharzeev:2015kna ; Miransky:2015ava ; Andersen:2014xxa ; Yamamoto:2021oys ;
Cao:2021rwx ; Iwasaki:2021nrz . When a magnetic field and/or chemical
potential is large enough, a neutral pion $\pi^{0}$ domain wall or soliton has
a negative tension due to the anomalous term Son:2007ny  when it is
perpendicular to the magnetic field Eto:2022lhu . Thus, domain walls are
spontaneously created in such a region, and the ground state is a stack of the
domain walls, called a chiral soliton lattice (CSL) Eto:2012qd ;
Brauner:2016pko . However, it has been recently found that a large part of CSL
with stronger magnetic field and larger baryon density should be replaced by a
new phase, the domain-wall Skyrmion phase Eto:2023lyo ; Eto:2023wul  (see also
Ref. Eto:2023tuu  for a counterpart under rapid rotation), where topological
lumps supported by $\pi_{2}(S^{2})\simeq{\mathbb{Z}}$ have negative energy and
spontaneously appear inside a soliton Nitta:2012wi ; Nitta:2012rq ;
Gudnason:2014nba . To conclude this result, the effective theory on a single
soliton was used, which is nothing but a $d=2+1$ dimensional $U(1)$ gauged
${\mathbb{C}}P^{1}$ model without a potential in a constant magnetic field
background. In the previous studies Eto:2023lyo ; Eto:2023wul ; Eto:2023tuu ,
we have used conventional BPS lumps by neglecting the electromagnetic
interaction as an approximation. However, they should be replaced by gauged
lumps constructed in this paper that may improve the phase boundary between
the CSL and domain-wall Skyrmion phase.

This paper is organized as follows. In Sec. 2, we construct gauged lumps with
nondynamical gauge field in a constant magnetic field background. In Sec. 3,
we consider dynamical gauge field and find that gauged lump solutions remain
stable. Sec. 4 is devoted to a summary and discussion.

##  2 Gauged lumps with nondynamical gauge field in a constant magnetic field
background

In this section, we consider nondynamical constant background gauge field. In
Sec. 2.1, we briefly explain the gauged $\mathbb{C}P^{1}$ model. Then we
explain the Derrick’s theorem in Sec. 2.2, and show the numerical solutions
for the single lump in Sec. 2.3 and for the higher charged lumps in Sec. 2.4.

###  2.1 The $U(1)$ gauged $\mathbb{C}P^{1}$ model

We consider a $U(1)$ gauged $O(3)$ model or equivalently $\mathbb{C}P^{1}$
model in $2+1$ dimensions. The Lagrangian is given by

$\displaystyle{\cal L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+D_{\mu}\vec{n}\cdot
D^{\mu}\vec{n}\,,$ (2.1)

where $\vec{n}$ is a three-component real column vector of scalar fields with
the $S^{2}$ constraint

$\displaystyle\vec{n}=\begin{pmatrix}n_{1}\\\ n_{2}\\\
n_{3}\end{pmatrix}\,,\quad\vec{n}^{2}=v^{2}\,,$ (2.2)

where $v$ is a radius of $S^{2}$ whose mass dimension is $\dfrac{1}{2}$. We
gauge the $SO(2)$ symmetry about the $n_{3}$-axis as

$\displaystyle n_{1}+in_{2}\to
e^{-i\theta(x)}\left(n_{1}+in_{2}\right)\,,\quad n_{3}\to n_{3}\,,$ (2.3)

with the covariant derivatives

$\displaystyle D_{\mu}n_{1}=\partial_{\mu}n_{1}+eA_{\mu}n_{2}\,,\quad
D_{\mu}n_{2}=\partial_{\mu}n_{2}-eA_{\mu}n_{1}\,,\quad
D_{\mu}n_{3}=\partial_{\mu}n_{3}\,.$ (2.4)

Here, $A_{\mu}$ is the $U(1)$ gauge field and $e$ is the $U(1)$ gauge coupling
constant. This can be also expressed as
$D_{\mu}(n_{1}+in_{2})=(\partial_{\mu}-ieA_{\mu})(n_{1}+in_{2})$ with
$A_{\mu}\to A_{\mu}-\partial_{\mu}\theta/e$. The mass dimensions are
summarised as $[\vec{n}]=[v]=\dfrac{1}{2}$, $[A_{\mu}]=\dfrac{1}{2}$, and
$[e]=\dfrac{1}{2}$. The symmetry of the model is $U(1)\times\mathbb{Z}_{2}$
where the $\mathbb{Z}_{2}$ is $n_{3}\to-n_{3}$.

In the rest of this section we will consider a constant magnetic field
background $F_{12}=B$, where the gauge field is not dynamical. In order to
distinguish the nondynamical background gauge field from a dynamical one, we
will use ${\cal A}_{\mu}$ for the former. Hereafter, we will take

$\displaystyle{\cal A}_{0}=0\,,\quad{\cal A}_{1}=-\frac{By}{2}\,,\quad{\cal
A}_{2}=\frac{Bx}{2}\,.$ (2.5)

###  2.2 Derrick’s theorem

We will study the topological lump solutions for $B\neq 0$ below. The static
energy of the scalar field in the temporal gauge ${\cal A}_{0}=0$ is given by

$\displaystyle E[\vec{n}]=\int d^{2}x\,|D_{i}\vec{n}|^{2}=\int
d^{2}x\left[|\partial_{i}\vec{n}|^{2}+2e{\cal
A}_{i}\varepsilon_{ab}\partial_{i}n_{a}n_{b}+e^{2}{\cal
A}_{i}^{2}(n_{1}^{2}+n_{2}^{2})\right]\,.$ (2.6)

Here we remove infinite energy of the constant magnetic field, so that
$E[\vec{n}]$ can be finite. We now want to apply the Derrick’s scaling
argument to $E[\vec{n}]$. For this purpose, we divide the energy into three
pieces as

$E=E_{2}+E_{1}+E_{0}$ (2.7)

with

$\displaystyle E_{2}$ $\displaystyle=$ $\displaystyle\int
d^{2}x\,|\partial_{i}\vec{n}|^{2}\,,$ (2.8) $\displaystyle E_{1}$
$\displaystyle=$ $\displaystyle\int d^{2}x~{}2e{\cal
A}_{i}\varepsilon_{ab}(\partial_{i}n_{a})n_{b}\,,$ (2.9) $\displaystyle E_{0}$
$\displaystyle=$ $\displaystyle\int d^{2}x~{}e^{2}{\cal
A}_{i}^{2}(n_{1}^{2}+n_{2}^{2})\,.$ (2.10)

where the indices on $E$ represent the number of the partial derivative. The
indices $a$ and $i$ runs from 1 to 2. Note that $E_{0}$ and $E_{2}$ are
positive semidefinite while $E_{1}$ can be either positive, negative or zero.

Let us define the scaling of $\vec{n}(x)$ as usual

$\displaystyle\vec{n}^{(\lambda)}(x)=\vec{n}(\lambda
x)\,,\quad\partial_{i}\vec{n}^{(\lambda)}(x)=\lambda\partial_{i}^{\prime}\vec{n}(\lambda
x)\,,$ (2.11)

where $\partial_{i}^{\prime}$ stands for $\partial/\partial(\lambda x^{i})$.
On the other hand, the scaling of the gauge field differs from the dynamical
one because we deal with the constant magnetic field and the gauge field is
not dynamical. We define the scaling of the background gauge field as follows

$\displaystyle{\cal A}_{i}^{(\lambda)}(x)=\lambda^{-1}{\cal A}_{i}(\lambda
x)\,.$ (2.12)

Plugging Eq. (2.5) into this, we indeed see ${\cal A}_{i}^{(\lambda)}(x)={\cal
A}_{i}(x)$, and therefore the constant background magnetic field is not
affected by the rescaling.

The $\lambda$ dependence of the energy functional reads

$\displaystyle e(\lambda)$ $\displaystyle=$ $\displaystyle
E\left[\vec{n}^{(\lambda)}(x)\right]$ (2.13) $\displaystyle=$
$\displaystyle\int d^{2}x\bigg{[}\left(\frac{\partial n_{a}(\lambda
x)}{\partial x^{i}}\right)^{2}+2e\left(\lambda^{-1}{\cal A}_{i}(\lambda
x)\right)\varepsilon_{ab}\frac{\partial n_{a}(\lambda x)}{\partial
x^{i}}n_{b}(\lambda x)$ $\displaystyle+e^{2}\left(\lambda^{-1}{\cal
A}_{i}(\lambda x)\right)^{2}(n_{1}(\lambda x)^{2}+n_{2}(\lambda
x)^{2})\bigg{]}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}x^{\prime}}{\lambda^{2}}\bigg{[}\left(\lambda\frac{\partial
n_{a}(x^{\prime})}{\partial
x^{\prime}{}^{i}}\right)^{2}+2e\left(\lambda^{-1}{\cal
A}_{i}(x^{\prime})\right)\varepsilon_{ab}\left(\lambda\frac{\partial
n_{a}(x^{\prime})}{\partial x^{\prime}{}^{i}}\right)n_{b}(x^{\prime})$
$\displaystyle+e^{2}\left(\lambda^{-1}{\cal
A}_{i}(x^{\prime})\right)^{2}(n_{1}(x^{\prime})^{2}+n_{2}(x^{\prime})^{2})\bigg{]}$
$\displaystyle=$ $\displaystyle E_{2}+\lambda^{-2}E_{1}+\lambda^{-4}E_{0}\,,$

where we have used $x^{\prime}{}^{j}=\lambda x^{j}$. In order to see if this
functional has a stationary point or not, we differentiate this with respect
to $\lambda$ to obtain

$\displaystyle\frac{de(\lambda)}{d\lambda}=-2\lambda^{-3}E_{1}-4\lambda^{-5}E_{0}\,.$
(2.14)

If $\vec{n}(x)$ is a static solution, this must be zero at $\lambda=1$,
provided

$\displaystyle\delta\equiv E_{1}+2E_{0}=0\,.$ (2.15)

As mentioned above $E_{0}\geq 0$ whereas $E_{1}$ can be positive or negative.
Hence, the system can successfully evade the Derrick’s no-go theorem only for
$E_{1}<0$.

###  2.3 $k=\pm 1$ lump solution and the selecting rule

Under the presence of nonzero background magnetic field $B\neq 0$, the charged
field cannot be condensed in the vacuum. Therefore, the vacuum configuration
should be $n_{3}=\pm v$ with $n_{1}+in_{2}=0$. Hence, the $\mathbb{Z}_{2}$
symmetry is spontaneously broken. We refer $n_{3}=+v$ to the up vacuum while
$n_{3}=-v$ to the down vacuum.

Let us construct lump and anti-lump solutions. To this end, we make an axially
symmetric Ansatz, given by

$\displaystyle n_{1}=v\frac{x}{r}\sin\Theta(r)\,,\quad
n_{2}=v\frac{y}{r}\sin\Theta(r)\,,\quad n_{3}=v\cos\Theta(r)\,,$ (2.16)

with $r=\sqrt{x^{2}+y^{2}}$. The lump with the winding number $k=+1$ satisfies
the boundary condition $\Theta(0)=0$ and $\Theta(\infty)=\pi$. Equivalently,
this can be expressed as $n_{3}(0)=+v$ and $n_{3}(\infty)=-v$. Thus, this lump
with $k=+1$ lives in the down vacuum $n_{3}=-v$. Eq. (2.16) can also describe
an anti-lump with $k=-1$ with the different boundary condition:
$\Theta(0)=\pi$ and $\Theta(\infty)=0$ or equivalently $n_{3}(0)=-v$ and
$n_{3}(\infty)=+v$. Hence, the anti-lump lives in the up vacuum $n_{3}=+v$.
The lump in the down vacuum and the anti-lump in the up vacuum can be
exchanged by $\Theta\to\pi-\Theta$ or $n_{3}\to-n_{3}$.

Plugging this into Eq. (2.1), we find the Lagrangian written in terms of
$\Theta$ as

$\displaystyle{\cal
L}=-\frac{B^{2}}{2}-v^{2}\Theta^{\prime}{}^{2}-\frac{v^{2}(2-eBr^{2})^{2}}{4r^{2}}\sin^{2}\Theta\,,$
(2.17)

and the corresponding EOM reads

$\displaystyle\Theta^{\prime\prime}+\frac{\Theta^{\prime}}{r}-\frac{(2-eBr^{2})^{2}}{4r^{2}}\sin\Theta\cos\Theta=0\,,$
(2.18)

where $v$ disappears from EOM. Here, it is important to point out that EOM is
not invariant under flipping the sign of $eB$. Note that the sign of $eB$ is
also important in the Derrick’s scaling condition. Having the Ansatz (2.16),
we can express $E_{0}$ and $E_{1}$ with respect to $\Theta(r)$ as

$\displaystyle E_{0}$ $\displaystyle=$ $\displaystyle\frac{\pi
e^{2}B^{2}v^{2}}{2}\int^{\infty}_{0}dr\,r^{3}\sin^{2}\Theta\,,$ (2.19)
$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle-2\pi
eBv^{2}\int^{\infty}_{0}dr\,r\sin^{2}\Theta\,.$ (2.20)

Both $E_{0}$ and $E_{1}$ can be nonzero for $B\neq 0$, thus the above scaling
argument is meaningful only for the lump under a nonzero magnetic field. While
$E_{0}\geq 0$ is obvious from this expression, the sign of $E_{1}$ is the same
as $-eB$. Since it has to be negative for Eq. (2.15) to be satisfied, we
should choose $eB>0$ for the Ansatz (2.16). More specifically, the lump in the
down vacuum and the anti-lump in the up vacuum can only exist for $eB>0$.

We can take another Ansatz with $n_{2}$ being replaced by
$n_{2}=-v\dfrac{y}{r}\sin\Theta$ in Eq. (2.16) while $n_{1}$ and $n_{3}$ are
unchanged. We call this transformation ($n_{2}\to-n_{2}$ up to $U(1)$ gauge
transformation) the topological charge (TC) conjugation. This should be
distinguished from the previous transformation $n_{3}\to-n_{3}$ that flips not
only the winding number but also the vacuum. In contrast, the only lump charge
is flipped by the TC conjugation. Indeed, the solution satisfying the boundary
condition $\Theta(0)=0$ and $\Theta(\infty)=\pi$ (equivalently $n_{3}(0)=+v$
and $n_{3}(\infty)=-v$) lives in the down vacuum and has the winding number
$k=-1$, whereas that satisfying $\Theta(0)=\pi$ and $\Theta(\infty)=0$
(equivalently $n_{3}(0)=-v$ and $n_{3}(\infty)=+v$) lives in the up vacuum and
has the winding number $k=+1$. Note that the TC conjugation is identical to
the normal charge conjugation $n_{1}+in_{2}\to n_{1}-in_{2}$. The charge
conjugation is also same as $e\to-e$. Therefore, Eqs. (2.17)–(2.20) remain
correct with replacement $eB\to-eB$. Hence, we now need $eB<0$ for realizing
$E_{1}<0$. More specifically, the anti-lump in the down vacuum and the lump in
the up vacuum can only exist for $eB<0$.

#### The selecting rule and soliton-anti-soliton asymmetry

This results in an interesting phenomenon. To clarify the lump solutions, we
need to specify signs for the following three quantities: the first is the
sign for the vacuum ($n_{3}=\pm v$), the second is that for the magnetic field
($eB>0$ or $eB<0$), and the third is that for the topological charge ($k>0$ or
$k<0$). If the background magnetic field is absent, we can put a lump or an
anti-lump either in the up or down vacuum. However, when $B\neq 0$, there is a
selecting rule that the lumps with positive topological charges ($k>0$) can
exist only when $\text{sign}(n_{3})\cdot\text{sign}(eB)<0$ is satisfied.
Similarly, the anti-lumps with a negative topological charge ($k<0$) can exist
only when $\text{sign}(n_{3})\cdot\text{sign}(eB)>0$ holds. Hence, once the
vacuum is chosen, one can select either lumps or anti-lumps by applying
magnetic field. We call this phenomenon a violation of the TC conjugation, see
Fig. 1.

Figure 1: The selecting rule. The cones stand for $\vec{n}$. The cone upward
is $n_{3}=+v$ and that downward is $n_{3}=-v$. The TC conjugation exchanges
two diagonal entries, and two off-diagonal entries.

Solutions for $B\neq 0$ can be only available by a numerical computation.
However, asymptotic behavior can be analytically understood as follows. Since
EOM includes $eB$ (its mass dimension is $2$), a solution will not have a size
modulus. Indeed, the asymptotic behavior should obey an exponential law rather
than a power law. This can be easily verified by analyzing the linearized EOM
for the fluctuation $\vartheta(r)=\pi-\Theta(r)$ at $r\gg 1/\sqrt{eB}$ as

$\displaystyle-\vartheta^{\prime\prime}+\frac{e^{2}B^{2}r^{2}}{4}\vartheta=0\,,\quad\vartheta\propto\exp\left(-\frac{eB}{4}r^{2}\right)\,,$
(2.21)

with the expected decay length $1/\sqrt{eB}$.

Let us show the numerical solution. To this end, we first change the variable
as $\rho=\sqrt{eB}\,r$. Then, EOM to be solved uniquely becomes

$\displaystyle\Theta^{\prime\prime}+\frac{\Theta^{\prime}}{\rho}-\frac{(2-\rho^{2})^{2}}{4\rho^{2}}\sin\Theta\cos\Theta=0\,,$
(2.22)

where $\Theta^{\prime}=\dfrac{d\Theta}{d\rho}$. The energy density and the
mass are expressed as

$\displaystyle{\cal
E}=\frac{B^{2}}{\beta}\left[\Theta^{\prime}{}^{2}+\frac{\left(2-\rho^{2}\right)^{2}}{4\rho^{2}}\sin^{2}\Theta\right]\,,\quad
M=2\pi v^{2}\int\rho d\rho\,{\cal E}\,,$ (2.23)

respectively. Here, we have introduced the dimensionless parameter for later
convenience

$\displaystyle\beta\equiv\frac{B}{ev^{2}}\,.$ (2.24)

A numerical solution is shown in Fig. 2.

Figure 2: The numerical solution $\Theta$ and energy density of the single
lump for $eB>0$.

Let us discuss features of this solution. First, we verify the Derrick’s
constraint in Eq. (2.15). With respect to the dimensionless coordinate, we
have

$\displaystyle E_{0}$ $\displaystyle=$ $\displaystyle 2\pi
v^{2}\int^{\infty}_{0}d\rho\,\frac{\rho^{3}}{4}\sin^{2}\Theta\,,$ (2.25)
$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle-2\pi
v^{2}\int^{\infty}_{0}d\rho\,\rho\sin^{2}\Theta\,,$ (2.26)

and we numerically obtain $\delta/(2\pi v^{2})=3.32494\times 10^{-7}$, thus
Eq. (2.15) is satisfactory met within numerical accuracy. The mass of the lump
is numerically evaluated as $M/(2\pi v^{2})=3.62188$. Note that this does not
depend on $eB$. We also numerically determine the size
$r_{0}=\dfrac{0.544784}{\sqrt{eB}}$ of the lump from the condition
$n_{3}(r_{0})=0$ $\left(\Theta(r_{0})=\dfrac{\pi}{2}\right)$. Hence, the
magnetic flux inside the lump can be evaluated as $\Phi=\pi r_{0}^{2}B\simeq
0.15\times\dfrac{2\pi}{e}$.

We can compare our numerical solution with the BPS lump at $B=0$, see Appendix
A for some details. The mass of BPS lump is $M_{\text{BPS}}/(2\pi v^{2})=4$,
therefore our numerical solution for $B\neq 0$ is slightly lighter than the
BPS lump at $B=0$. Furthermore, rough estimation of the fixed size of a BPS
lump under the constant magnetic field $B$ was given as
$\tilde{r}_{0}=\sqrt{\dfrac{2}{eB}}$ due to the magnetic flux quantization
condition that the magnetic flux inside $\tilde{r}_{0}$ is
$\tilde{\Phi}=\dfrac{2\pi}{e}$ Eto:2023lyo . Our numerical solution improves
this, and the actual size is about 40% of $\tilde{r}_{0}$, and the magnetic
flux inside the lump is about 15% of $\tilde{\Phi}$.

###  2.4 Coaxial multiple lumps

We here quickly show the axially symmetric lumps with the winding number
$k>1$. To this end, we slightly change the Ansatz

$\displaystyle n_{1}=v\frac{{\rm Re}\,z^{k}}{r^{k}}\sin\Theta(r)\,,\quad
n_{2}=v\frac{{\rm Im}\,z^{k}}{r^{k}}\sin\Theta(r)\,,\quad
n_{3}=v\cos\Theta(r)\,,$ (2.27)

with $z\equiv x+iy$. The EOM reads

$\displaystyle\Theta^{\prime\prime}+\frac{\Theta^{\prime}}{r}-\frac{(2k-eBr^{2})^{2}}{4r^{2}}\sin\Theta\cos\Theta=0\,,$
(2.28)

and $E_{0}$ and $E_{1}$ are expressed as

$\displaystyle E_{0}$ $\displaystyle=$ $\displaystyle 2\pi
v^{2}\int^{\infty}_{0}d\rho\,\frac{\rho^{3}}{4}\sin^{2}\Theta\,,$ (2.29)
$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle-2k\pi
v^{2}\int^{\infty}_{0}d\rho\,\rho\sin^{2}\Theta\,.$ (2.30)

Here $E_{0}$ is independent of $k$ while $E_{1}$ is proportional to $k$. The
numerical solutions for $k=1,2,3,4,5,6$ are shown in Fig. 3.

Figure 3: The numerical solutions $\Theta$ and energy densities for
$k=1,2,3,4,5$ with $eB>0$.

We numerically verify if the Derrick’s condition is satisfied and find
$\delta/(2\pi v^{2})=-6.83875\times 10^{-7}$, $-1.17859\times 10^{-6}$,
$-1.58112\times 10^{-6}$, $-1.63619\times 10^{-6}$, $-2.50128\times 10^{-6}$
for $k=2,3,4,5,6$, respectively.

The lumps get fat as $k$ increased. We numerically measure the radius $r_{0}$
for $k$ as shown in Fig. 4 and find that it can be well approximated by
$\dfrac{r_{0}}{\sqrt{eB}}=a\sqrt{k}+b$ with $a=1.77$ and $b=-1.16$. Similarly,
the mass is found as $\dfrac{M}{2\pi v^{2}}=c\sqrt{k}+d$ with $c=4.86$ and
$d=-1.18$. This suggests that the lumps are well described by a droplet model.
Note that the lumps with $k>1$ are empty in the sense that the energy
densities at the cores are negligibly small.

Figure 4: The numerical solutions $\Theta$ and energy densities for
$k=1,2,3,4,5,6$ with $eB>0$.

##  3 Gauged lumps with dynamical gauge field in a magnetic field background

In this section, we consider a dynamical gauge field with a uniform background
magnetic field. In Sec. 3.1, we introduce the dynamical gauge field. Then we
explain the Derrick’s theorem in Sec. 3.2, and show the numerical solutions
for the gauged lumps in Sec. 3.3. The anti-lumps and the soliton-anti-soliton
asymmetry are explained in Sec. 3.4.

###  3.1 The model with dynamical gauge field

Next we deal with $A_{\mu}$ as a dynamical field. Since we are interested in
the lumps under the constant magnetic field, we first need to decompose the
gauge field into two parts as

$A_{\mu}={\cal A}_{\mu}+a_{\mu}\,,$ (3.1)

where ${\cal A}_{\mu}$ is the nondynamical background gauge field given in Eq.
(2.5) and $a_{\mu}$ is the dynamical gauge field.

###  3.2 Derrick’s theorem

We modify the Derrick’s scaling argument in the previous section to the case
with a dynamical gauge field. The static and finite energies of the scalar
field in the temporal gauge $A_{0}=0$ are given by

$\displaystyle E[\vec{n},a_{\mu}]$ $\displaystyle=$ $\displaystyle\int
d^{2}x\,\left(\frac{F_{12}^{2}}{2}-\frac{B^{2}}{2}+|D_{i}\vec{n}|^{2}\right)$
(3.2) $\displaystyle=$ $\displaystyle\int
d^{2}x\bigg{[}\frac{(B+f_{12})^{2}}{2}-\frac{B^{2}}{2}$
$\displaystyle+|\partial_{i}\vec{n}|^{2}+2e({\cal
A}_{i}+a_{i})\varepsilon_{ab}\partial_{i}n_{a}n_{b}+e^{2}({\cal
A}_{i}+a_{i})^{2}(n_{1}^{2}+n_{2}^{2})\bigg{]}\,,$

respectively. We have subtracted $B^{2}/2$, and so that the energy remains
finite. Let us decompose this into four pats as follows:

$\displaystyle E[\vec{n},a_{\mu}]=E_{4}+E_{2}+E_{1}+E_{0}\,,$ (3.3)

with

$\displaystyle E_{4}$ $\displaystyle=$ $\displaystyle\int
d^{2}x\,\frac{f_{12}^{2}}{2}\,,$ (3.4) $\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\int
d^{2}x\,\left[Bf_{12}+|\partial_{i}\vec{n}|^{2}+2ea_{i}\varepsilon_{ab}\partial_{i}n_{a}n_{b}+e^{2}a_{i}^{2}(n_{1}^{2}+n_{2}^{2})\right]\,,$
(3.5) $\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle\int
d^{2}x\,\left[2e{\cal A}_{i}\varepsilon_{ab}\partial_{i}n_{a}n_{b}+2e^{2}{\cal
A}_{i}a_{i}(n_{1}^{2}+n_{2}^{2})\right]\,,$ (3.6) $\displaystyle E_{0}$
$\displaystyle=$ $\displaystyle\int d^{2}x\,e^{2}{\cal
A}_{i}^{2}(n_{1}^{2}+n_{2}^{2})\,,$ (3.7)

and $f_{12}=\partial_{1}a_{2}-\partial_{2}a_{1}$.

Now we apply the Derrick’s scaling argument to these as before. To this end,
we use the same scaling laws Eqs. (2.11) and (2.12) for $\vec{n}$ and ${\cal
A}_{i}$, respectively. On the other hand, we adopt the standard scaling law
for the dynamical gauge field $a_{\mu}$ as

$\displaystyle a_{i}^{(\lambda)}(x)=\lambda\,a_{i}(\lambda x)\,.$ (3.8)

The $\lambda$ dependence of the energy functional reads

$\displaystyle
e(\lambda)=E[\vec{n}^{(\lambda)},a_{\mu}^{(\lambda)}]=\lambda^{2}E_{4}+\lambda^{0}E_{2}+\lambda^{-2}E_{1}+\lambda^{-4}E_{0}\,.$
(3.9)

In order to see if this functional has a stationary point or not, we
differentiate this with respect to $\lambda$ as

$\displaystyle\frac{de(\lambda)}{d\lambda}=2\lambda
E_{4}-2\lambda^{-3}E_{1}-4\lambda^{-5}E_{0}\,.$ (3.10)

If $\\{\vec{n}(x),a_{\mu}\\}$ is a static solution, this must be zero at
$\lambda=1$, provided

$\displaystyle\delta^{\prime}\equiv-E_{4}+E_{1}+2E_{0}=0\,.$ (3.11)

As before, $E_{1}$ can be either positive or negative whereas $E_{0}$ and
$E_{4}$ are positive.

###  3.3 Multiple gauged lumps in the down vacuum with $eB>0$

For a single lump, we make the following Ansatz

$\displaystyle a_{0}=0\,,\quad a_{1}=-\frac{Ba(r)}{2}y\,,\quad
a_{2}=\frac{Ba(r)}{2}x\,,$ (3.12)

where $a_{\mu}$ is the dynamical gauge field as given in Eq. (3.1). Together
with the background gauge field in Eq. (2.5), the full gauge field is given by

$\displaystyle A_{0}=0\,,\quad A_{1}=-\frac{B(1+a(r))}{2}y\,,\quad
A_{2}=\frac{B(1+a(r))}{2}x\,.$ (3.13)

The mass dimension of $a(r)$ is zero. The magnetic field is given by

$\displaystyle F_{12}=B(1+a)+\frac{Bra^{\prime}}{2}\,.$ (3.14)

We impose the profile function $a(r)$ to approach $0$ as $r\to\infty$, so that
the magnetic field asymptotically behaves as $F_{12}\to B$.

Plugging Eqs. (2.27) and (3.13) into Eq. (2.1), we find the reduced Lagrangian
for $\Theta$ and $a$

$\displaystyle{\cal
L}=-\frac{B^{2}}{2}\left(1+a+\frac{ra^{\prime}}{2}\right)^{2}-v^{2}\Theta^{\prime}{}^{2}-v^{2}\frac{\left(2k-eBr^{2}(1+a)\right)^{2}}{4r^{2}}\sin^{2}\Theta\,.$
(3.15)

The corresponding EOMs are given by

$\displaystyle\Theta^{\prime\prime}+\frac{\Theta^{\prime}}{r}-\frac{\left(2k-eBr^{2}(1+a)\right)^{2}}{4r^{2}}\sin\Theta\cos\Theta=0\,,$
(3.16) $\displaystyle
a^{\prime\prime}+\frac{3a^{\prime}}{r}+\frac{2ev^{2}\left(2k-eBr^{2}(1+a)\right)}{Br^{2}}\sin^{2}\Theta=0\,.$
(3.17)

Up to here in this subsection, the prime stands for a derivative in terms of
the physical coordinate $r$.

For numerical analysis, let us rewrite these with respect to the dimensionless
coordinate $\rho=\sqrt{eB}\,r$. Then we have

$\displaystyle\Theta^{\prime\prime}+\frac{\Theta^{\prime}}{\rho}-\frac{\left(2k-\rho^{2}(1+a)\right)^{2}}{4\rho^{2}}\sin\Theta\cos\Theta=0\,,$
(3.18) $\displaystyle
a^{\prime\prime}+\frac{3a^{\prime}}{\rho}+\frac{2\left(2k-\rho^{2}(1+a)\right)}{\beta\rho^{2}}\sin^{2}\Theta=0\,.$
(3.19)

Now the prime indicates a derivative in terms of $\rho$. The EOMs include the
dimensionless parameter $\beta$ defined in Eq. (2.24). If we assume $a=0$, Eq.
(3.18) is identical to Eq. (2.22). On the other hand, Eq. (3.19) with $a=0$
cannot be satisfied except for $\Theta=0$ or $\pi$. Eq. (3.19) with $a=0$ is
approximately satisfied for a large $B$ (or small $e$) such as $B\gg ev^{2}$.
This is a reasonable condition for us to ignore the back reaction to the
background gauge field.

Figure 5: The numerical solution of $k=1$ lump. $\Theta$ (top-left) and $a$
(top-right). The back reaction $f_{12}$ to the background magnetic field $B$
(bottom-left) and energy density excited by the single lump. We take
$\beta=\\{1/4,\,1/2,\,1,\,2\\}$.

In what follows, we will concentrate on the lumps with $k>0$ in the down
vacuum with $eB>0$. Namely, the boundary condition for $\Theta$ is
$\Theta(0)=0$ and $\Theta(\infty)=\pi$. The numerical solutions of $k=1$ with
different $\beta=\frac{1}{4}$, $\frac{1}{2}$, $1$, and $2$ are shown in Fig.
5. The anti-lumps with $k<0$ in the negative vacuum with $eB>0$ will
separately be explained in Sec. 3.4. We first realize that back reaction to
$\Theta$ is negligibly small irrespective of $\beta$, see the top-left panel.
On the other hand, as can be seen in the top-right panel, the dynamical gauge
field $a_{\mu}$ sensitively reacts to $\beta$. We also show the back reactions
of the magnetic flux density and energy density

$\displaystyle f_{12}$ $\displaystyle=$ $\displaystyle
B\left(a+\frac{\rho}{2}a^{\prime}\right)\,,$ (3.20) $\displaystyle{\cal E}$
$\displaystyle=$
$\displaystyle\frac{B^{2}}{\beta}\left[\frac{\beta}{2}\left\\{\left(1+a+\frac{\rho}{2}a^{\prime}\right)^{2}-\frac{1}{2}\right\\}+\Theta^{\prime}{}^{2}+\frac{(2k-\rho^{2}(1+a))^{2}}{4\rho^{2}}\sin^{2}\Theta\right]\,,$
(3.21)

respectively, in the second row of Fig. 5. Note that the net magnetic flux is
$F_{12}=B+f_{12}$ and we show $f_{12}$ in Fig. 5. The magnetic flux is
condensed at the center of the lump. It gets stronger than the background
value $B$ inside the lump core, and it reduces and changes its sign as going
away from the lump, and asymptotically decays at spatial infinity. As
expected, the greater $\beta$ is, the smaller deformation of the profile $a$
is. Namely, the back reaction tends to be small for the strong background
magnetic field $B\gg ev^{2}$. This behavior can be explained as follows. Since
$|n_{1}+in_{2}|\neq 0$ at the edge of lump, the lump behaves as a
superconducting ring and traps magnetic field in its inside, with the total
magnetic field reduced from the background magnetic field because of
superconductiviy expelling it.

Let us quantitatively evaluate the numerical solutions. First, we measure a
net change of the magnetic flux due to the single lump,

$\displaystyle\delta\Phi=\int d^{2}x\,f_{12}=\frac{2\pi}{e}\int\rho
d\rho\,\left(a+\frac{\rho}{2}a^{\prime}\right)\,.$ (3.22)

This is shown for $\beta=\\{\frac{1}{4},\,\frac{1}{2},\,1,\,2\\}$ in Table 1.
We find that $\delta\Phi$ is always negative, and thus the net background
magnetic field is weakened under the presence of lump regardless of $\beta$.

Second, we measure the mass of the lump by

$\displaystyle M=2\pi v^{2}\int\rho
d\rho\,\left[\frac{\beta}{2}\left\\{\left(1+a+\frac{\rho}{2}a^{\prime}\right)^{2}-1\right\\}+\left\\{\Theta^{\prime}{}^{2}+\frac{(2k-\rho^{2}(1+a))^{2}}{4\rho^{2}}\sin^{2}\Theta\right\\}\right]\,.$
(3.23)

This is also summarised in Table 1.

Third, we calculate $E_{4}$, $E_{1}$, and $E_{0}$ by

$\displaystyle E_{4}$ $\displaystyle=$ $\displaystyle\frac{2\pi B}{e}\int\rho
d\rho\,\frac{1}{2}\left(a+\frac{\rho}{2}a^{\prime}\right)^{2}\,,$ (3.24)
$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle\frac{2\pi
B}{e\beta}\int\rho d\rho\,\left(\frac{\rho^{2}a}{2}-k\right)\sin^{2}\Theta\,,$
(3.25) $\displaystyle E_{0}$ $\displaystyle=$ $\displaystyle\frac{2\pi
B}{e\beta}\int\rho d\rho\,\frac{\rho^{2}}{4}\sin^{2}\Theta\,,$ (3.26)

and verify the scaling-stability condition $\delta^{\prime}=0$. The results
are summarize in Table. 1 and they are satisfactory small.

$k=1$ | $\beta=1/4$ | $\beta=1/2$ | $\beta=1$ | $\beta=2$ | $\beta=\infty$
---|---|---|---|---|---
$\delta\Phi/(2\pi/e)$ | $-1.11$ | $-0.400$ | $-0.132$ | $-0.0402$ | $0$
$M/(2\pi v^{2})$ | $2.90$ | $3.14$ | $3.33$ | $3.45$ | $3.62$
$\delta^{\prime}/(2\pi B/e)$ | $-5.61\times 10^{-4}$ | $-7.18\times 10^{-4}$ | $-2.43\times 10^{-4}$ | $-8.84\times 10^{-5}$ | —
Table 1: The quantitative properties of the $k=1$ lump for $\beta=1/2$, $1$,
and $2$. We numerically evaluate changes of the net magnetic flux, the mass,
and the Derrick’s scaling condition $\delta^{\prime}=0$.

Finally, we show the lumps with higher topological charges $k=1,2,3,4,5,6$ for
$\beta=1$ in Fig. 6. The profiles of $\Theta$ are qualitatively the same as
those for the constant magnetic field shown in Fig. 3. The gauge field $a$ and
magnetic flux density $f_{12}$ show plateaus inside the lump cores. The lumps
are no longer empty and are filled by the magnetic field. These behaviors in
$f_{12}$ and ${\cal E}$ are similar to those of well-known Abrikosov-Nielsen-
Olsen vortices.

Figure 6: The axially symmetric lumps for $k=1,2,3,4,5,6$ for $\beta=1$.
$\Theta$ and $a$ are shown in the top row, and $f_{12}$ and ${\cal E}$ are
shown in the bottom row.

We also numerically solved EOMs for $\beta=1/4,\,1/2$ and $\beta=2$, and the
results are qualitatively the same as those for $\beta=1$. As an example to
see the similarity, we show the relation between the lump width $r_{0}$ and
the winding number $k$ for $\beta=\\{1/4,\,1/2,\,1,\,2\\}$ in Fig. 7. We find
it is well approximated by $\dfrac{r_{0}(k)}{\sqrt{eB}}=\sqrt{ak}+b$ as
before. Namely, the lumps with magnetic fluxes behave as droplets.

Figure 7: The relation between the lump size $r_{0}$ and the winding number
$k$ for $\beta=\\{1/4,\,1/2,\,1,\,2\\}$.

###  3.4  Gauged anti-lumps in the down vacuum with $eB>0$

Let us construct anti-lumps with the negative winding number $k<0$ which do
not exist in the case of nondynamical gauge field. For comparison, we take the
same boundary condition as that taken in Sec. 3.3. Namely, we consider the
anti-lumps in the negative vacuum and the background magnetic field is fixed
by $eB>0$.

We first remind that, in the treatment that the $U(1)$ gauge field is non-
dynamical in Sec. 2, the Derrick’s scaling condition (2.15) strictly prohibits
the anti-lumps existing in the down vacuum with $eB>0$. On the other hand, in
the treatment that the $U(1)$ gauge field is dynamical, the scaling condition
is relaxed as Eq. (3.11) due to the additional contribution by $E_{4}$.
Therefore, there would exist a room for the anti-lumps to exist in the model
with dynamical gauge fields.

To verify this, we numerically solve EOMs in Eqs. (3.18) and (3.19) with $k<0$
and the boundary condition $\Theta(0)=0$ and $\Theta(\infty)=\pi$. Indeed, we
numerically found non-singular anti-lump solutions.

Figure 8: The axially symmetric lumps for $k=-1,-2,-3$ for $\beta=1/4$ in the
down vacuum. $\Theta$ and $a$ are shown in the top row, and $f_{12}$ and
${\cal E}$ are shown in the bottom row.

As an example, we show the numerical solutions with $k=-1,\,-2,\,-3$ for
$\beta=1/4$ in Fig. 8. The anti-lumps trap the negative magnetic field in the
cores which is opposite to the background one $B$($>0$). Unlike the case with
$k>0$, the back reaction $f_{12}$ is everywhere negative.

The asymmetry between the lumps and anti-lumps in the model with the dynamical
gauge field is more modest than that in the model without the dynamical field.
However, the asymmetry does not disappear and remains as obviously seen in the
EOMs in Eqs. (3.18) and (3.19). This can be also verified by our numerical
solutions. Fig. 9 shows the energy densities (more explicitly, we show ${\cal
E}\rho\beta/B^{2}$) of the $k=-1$ and $k=+1$ lumps under the same down vacuum
with $\beta=1/4$. We find that $k=-1$ lump has higher energy around its core
and slightly lower at the outer bump. We numerically evaluate the mass of
$k=-1$, and found $\dfrac{M}{2\pi v^{2}}=2.97$ which is higher than
$\dfrac{M}{2\pi v^{2}}=2.90$ for $k=+1$, as shown in Table 1.

Figure 9: The comparison of energy densities of $k=-1$ and $k=+1$ for
$\beta=1/4$.

In conclusion, both the lumps and anti-lumps exist in the down vacuum with
$eB>0$, but the former is energetically favored than the latter. Intuitively,
this reflects the fact that putting the soliton with the magnetic field
opposite to the vacuum costs an energy than the soliton with the same magnetic
field as the vacuum. The mass difference becomes larger as $\beta$ is
increased. In the limit $\beta\to\infty$ where the gauge field is non-
dynamical, the anti-lumps tend to be singular and to be excluded from the
system.

The same can be said for the up vacuum with $eB>0$ or $eB<0$ by appropriately
exchanging the lump and anti-lump.

##  4 Summary and discussion

In this paper, we have worked out stable lumps in a gauged $O(3)$ model
without any potential term coupled with a (non)dynamical $U(1)$ gauge field.
We have found that the gauged-lumps without a potential term can be made
stable by putting them in a uniform magnetic field, irrespective of whether
the gauge field is dynamical or not. In the case of the non-dynamical gauge
field, we have found that only either of lumps or anti-lumps stably exists
depending on the sign of the background magnetic field, and the other is
unstable to shrink to be singular. The case of nondynamical gauge field
resemble magnetic Skyrmions in chiral magnets. We have also constructed
coaxial lumps with higher winding number $k$, whose size and mass are
proportional to $\sqrt{k}$ up to a constant shift, implying that they behave
as droplets. On the other hand, in the case of the dynamical gauge field, we
have found that both the lumps and anti-lumps stably exist with different
masses; the lighter (heavier) one corresponds to the (un)stable one in the
case of the nondynamical gauge field. The lumps behave as superconducting
rings, trapping magnetic fields in their interiors. The total magnetic fluxes
are reduced from the background magnetic field because of superconductivity.

In this paper, we have discussed only axially symmetric configurations for
multiple lump states. The next step should be to investigate separated
configurations and study interaction among them. Since the profile of a single
lump exponentially decays, the interaction would be exponentially suppressed
like vortices in superconductors. In particular, we expect a repulsive force
between two lumps and the existence of a lattice as the case of magnetic
Skyrmions Rossler:2006 ; doi:10.1126/science.1166767 ; doi:10.1038/nature09124
; Han:2010by ; Ross:2020hsw 111 However, the interaction between magnetic
Skyrmions was polynomial like Ross:2020hsw . . This is quite important for an
application to domain-wall Skyrmions in QCD under strong magnetic field.

One natural generalization of the current work is the ${\mathbb{C}}P^{N-1}$
model. From a theoretical point of view, a naive question is if one $U(1)$
gauging is enough or $U(1)^{N-1}$ gauging is necessary for the stability of
${\mathbb{C}}P^{N-1}$ lumps. From a physical point of view, such a $U(1)$
gauged ${\mathbb{C}}P^{2}$ model appears on a chiral soliton with three
flavors (up, down and strange quarks) in which case ${\mathbb{C}}P^{2}$ lumps
are $SU(3)$ Skyrmions from the bulk point of view. The $U(1)$ gauged
${\mathbb{C}}P^{2}$ model also appears Vinci:2012mc ; Hirono:2012ki ;
Chatterjee:2015lbf  on the worldsheet of a non-Abelian vortex in dense QCD
Balachandran:2005ev ; Nakano:2007dr ; Eto:2009kg ; Eto:2009bh ; Eto:2009tr ;
Eto:2013hoa  in which case lumps are sigma model instantons viewed as Yang-
Mills instantons from the bulk Eto:2004rz .

Our model would have an impact on production of topological solitons. During a
phase transition accompanied with a spontaneous symmetry breaking, topological
solitons are in general created by the Kibble-Zurek-mechanism. Usually the
numbers of solitons and anti-solitons created in this way are the same. If we
consider such a phase transition in our model, the background magnetic field
works as a bias leading to an asymmetry between the lumps or anti-lumps.
Hence, this suggests a new impact on solitogenesis and might be useful for
baryogenesis in early Universe.

###### Acknowledgements.

YA is grateful for the hospitality at the Department of Physics of Yamagata
University, where this work was initiated. This work is supported in part by
JSPS KAKENHI [Grants No. JP23KJ1881 (YA), No. JP22H01221 (ME and MN)] and the
WPI program “Sustainability with Knotted Chiral Meta Matter (SKCM2)” at
Hiroshima University (ME and MN).

##  Appendix A A BPS lump for $B=0$

Here let us quickly recall the analytic solution for $B=0$. The EOM reduces to
one for a usual $\mathbb{C}P^{1}$ lump, and its analytic solution is available
as

$\displaystyle\frac{n_{3}(r)}{v}=\cos\Theta(r)=\frac{a^{2}-r^{2}}{a^{2}+r^{2}}\,.$
(A.1)

Here, $a$ is so called size modulus. As usual, we define the size $r_{0}$ of
lump by $n_{3}(r_{0})=0$. Clearly, we have $r_{0}=a$. This solution satisfies
the boundary condition $\cos\Theta(0)=0$ and $\cos\Theta(\infty)=\pi$
($n_{3}(0)=v$ and $n_{3}(\infty)=-v$). The asymptotic behavior at $r\to\infty$
is power law as $\frac{n_{3}}{v}=-1+\frac{2a^{2}}{r^{2}}+\cdots$.

Let $C$ be a closed curve on which $n_{3}=0$, and $D$ be the interior of $C$.
The $U(1)$ is spontaneously and maximally broken around $|n_{1}+in_{2}|=v$.
Thus, the closed curve $C$ is a superconducting ring, and there is a
persistent current along it. Let us write $n_{1}+in_{2}=ve^{i\psi}$ on $C$.
The configuration of the gauge field along $C$ is determined by minimizing the
gradient energy $|D_{\alpha}(n_{1}+in_{2})|^{2}=0$, yielding
$\partial_{\alpha}\psi=eA_{\alpha}$. Then, we have a flux (and area)
quantization on $D$:

$\displaystyle
BS_{D}=\int_{D}d^{2}x\,B=\oint_{C}dx^{i}A_{i}=\frac{1}{e}\oint_{C}dx^{i}\partial_{i}\psi=\frac{2\pi}{e}\;\;$
(A.2)

with the area $S_{D}$ of $D$. This gives a constraint among the lump moduli.
For a single BPS lump with $n_{3}=v(a^{2}-r^{2})/(a^{2}+r^{2})$. Thus, the
size of $D$ bounded by $n_{3}=0$ is $r=a$, and the flux quantization implies a
quantization of the size $a=\sqrt{2/eB}$.

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