# More Than React: Investigating The Role of Emoji Reaction in GitHub Pull
Requests

Teyon Son  Tao Xiao  Dong Wang  Raula Gaikovina Kula  Takashi Ishio  Kenichi
Matsumoto
Nara Institute of Science and Technology  Japan
Email: {son.teyon.sr7  tao.xiao.ts2  wang.dong.vt8  raula-k  ishio
<EMAIL_ADDRESS>
###### Abstract

Context: Open source software development has become more social and
collaborative, especially with the rise of social coding platforms like
GitHub. Since 2016, GitHub started to support more informal methods such as
emoji reactions, with the goal to reduce commenting noise when reviewing any
code changes to a repository. Interestingly, preliminary results indicate that
emojis do not always reduce commenting noise (i.e., eight out of 20 emoji
reactions), providing evidence that developers use emojis with ulterior
intentions. From a reviewing context, the extent to which emoji reactions
facilitate for a more efficient review process is unknown.
Objective: In this registered report, we introduce the study protocols to
investigate ulterior intentions and usages of emoji reactions, apart from
reducing commenting noise during the discussions in GitHub pull requests
(PRs). As part of the report, we first perform a preliminary analysis to
whether emoji reactions can reduce commenting noise in PRs and then introduce
the execution plan for the study.
Method: We will use a mixed-methods approach in this study, i.e., quantitative
and qualitative, with three hypotheses to test.

##  I Introduction

In the past few years, open source software development has become more social
and collaborative. Known as social coding,open source development promotes
formal and informal collaboration by empowering the exchange of knowledge
between developers [9]. GitHub, one of the most popular social coding
platforms, attracts more than 72 million developers collaborating across 233
million repositories.111https://github.com/search Every day, thousands of
people engage in conversations about code, design, bugs, and new ideas on
GitHub. To promote collaboration, GitHub implements a vast number of social
features (i.e., follow, fork, and stars).

Since 2016, GitHub introduced a new social function called “reaction” for
developers to quickly express their feelings in issue reports and PRs.
Especially for discussing a Pull Requests(PRs), we find
that222https://tinyurl.com/3rpdr6dp:

> “In many cases, especially on popular projects, the result is a long thread
> full of emoji and not much content, which makes it difficult to have a
> discussion. With reactions, you can now reduce the noise in these threads” -
> GitHub

In the context of code review, we assume that a thread full of emoji may also
contribute to the existing forms of confusion for reviewers during the code
review process. For instance, Ebert et al. [10] pointed out that confusion
delays the merge decision decreases review quality, and results in additional
discussions. Hirao et al. [16] found that patches can receive both positive
and negative scores due to the disagreement between reviewers, which leads to
conflicts in the review process.

The Figure 1 depicts two typical cases where the emoji reactions occur. Figure
1(a) shows the case where the reaction does reduce unnecessary commenting in
the thread, hence may lead to less confusion and conflicts. The example
illustrates how Author B reduces the commenting by simply reacting with a
quick expression of approval through THUMBS UP *X. In contrast, as shown in
Figure 1(b), there exists a case where the emoji usage has an ulterior
intention and does not reduce comments in the discussion thread. In detail,
Contributor D uses three positive emoji reactions (THUMBS UP *X, HOORAY *X,
and HEART *X) to represent the appreciation to this PR. Then later goes on to
provide detailed comments on the PR. We posit that the intention of the emoji
was to express appreciation for the PR, and did not reduce the amount of
commenting in the thread discussions. As part of our preliminary study, we
also found other cases where the emoji did not always reduce the commenting in
the discussion. Under a closer manual inspection of 20 emoji reactions, we
find that there are eight cases where the emoji reactions did not reduce
commenting noise.

Therefore, in this registered report, we present our study protocol to
investigate ulterior intentions and usages of emoji reactions, apart from
reducing commenting noise during the discussions. Specifically, we would like
to (i) investigate the effect of emoji reaction related factors on the pull
request process (i.e., review time), (ii) investigate whether the first time
pull request is more likely to receive reactions, (iii) analyze the
relationship between the reaction and the intention of comment, and (iv)
explore consistency between sentiments of an emoji reaction and the sentiment
of the comment. To enable other researchers to extend our study, we plan to
make the study data publicly available.

(a) Example of emoji reaction reduce commenting noise. (b) Example of emoji
reaction does not reduce commenting noise.

Figure 1:  Examples of emoji reactions used in GitHub.

##  II Preliminary Study

The goal of our preliminary study is to explore the extent to which emoji
reaction usage reduces commenting noise of a PR. We selected the Eclipse
projects as our case study since it is a mature thriving open source project
with a number of contributors that actively submit and merge PRs.

TABLE I: Preliminary Dataset Summary Statistics

| # Repositories | # PR | # PR Comments
---|---|---|---
With reactions | 203 | (48%) | 6,867 | (8%) | 9,256 | (4%)
Without reactions | 217 | (52%) | 76,202 | (92%) | 249,354 | (96%)
Total | 420 | (100%) | 83,069 | (100%) | 258,610 | (100%)

###  II-A Data Collection

We collected a list of 683 Eclipse repositories by using the official API of
GitHub [2]. Since we focus on the reactions that are used in PRs,we excluded
the repositories that do not have any PRs. We obtained 420 repositories that
have 83,069 PRs. Then, we extracted PRs with reactions using GitHub GraphQL
API [1], along with the information of reaction types, reaction time, the
developer who posts the reaction. We excluded those reactions or PRs that are
posted by bots since we investigate the reactions used by developers. Based on
our manual check, we performed the following two exclusions: (i) exclude
developer name by using a regular expression matching(‘github.app’) (ii)
exclude the dependabot, a popular bot used to automatically notify developers
of dependency upgrades.333https://dependabot.com/ In the end, we obtained
9,256 comments having emoji reactions across 203 Eclipse repositories, as
shown in Table I.

###  II-B Data Analysis

Figure 2: Proportion of PRs with reactions by year in Eclipse repositories.

With our preliminary dataset, we conducted three exploratory analyses related
to emoji reaction usage. First, we investigate the prevalence of reaction
usage yearly since this feature was initially introduced in 2016. To do so, we
measured the proportion of PRs that have at least one emoji reaction. Second,
we investigate what are the common reactions for developers to express during
the PR process. To do so, we grouped the eight existing emoji reactions into
four categories:

  * •

Positive \- is the single usage or the combination usage of THUMBS UP *X,
LAUGH *X, HOORAY *X, HEART *X, and ROCKET *Xreactions.

  * •

Negative \- is the single usage or the combination usage of THUMBS DOWN *X and
CONFUSED *X.

  * •

Neutral \- is the usage of EYES *Xreaction.

  * •

Mixed \- is the combination usage of the four categories mentioned above.

For example, if a PR comment is reacted by the THUMBS UP *X and the THUMBS
DOWN *X, we then classify this case as Mixed. Third, we further investigate
whether or not the reaction is used to reduce commenting noise during the
review process. To do so, we randomly select 20 PR comment samples from the
preliminary dataset and did a manual classification (i.e., reduce commenting
noise or not) among the first four authors. After the emoji reactions were
posted, if there are no additional comments related to the existing topic by
the developers who react, we classify such case as Reduce Noise. Otherwise, we
classify the case as Not Reduce Noise. For example, in Figure 1(a), the Author
B reacted with THUMBS UP to the suggestion provided by the Contributor A and
added three commits without any additional comments. This case is labeled as
Reduce Noise.

Positive emoji reactions are widely used in PRs. Two preliminary results are
summarized. First, we find that around 8% to 10% of PRs have at least one
reaction in Eclipse repositories between 2016 and 2020. Figure 2 shows the
proportion of PRs with reactions yearly. Second, we observe that most of the
reactions in PR comments are Positive (i.e., the single or combination usage
of *X, *X, *X, *X, and *X), accounting for 98.1%. Upon a further inspection,
among these Positive reactions, the single usage of *X almost reached 86.5%.
On the other hand, the usage of Negative, Neutral, and Mixed only accounts for
1.84%. Table II shows the distribution of the sentiment of emoji reaction
usage, indicating that the positive reaction is the most prevalent.

Emoji reactions do not always reduce the commenting noise. Table III shows the
frequency of samples where whether reactions reduce commenting noise from our
manual classification. The table shows that eight samples (40%) are classified
as Not Reduce Noise. In these samples, after the emoji reactions were posted,
developers post additional comments to express and discuss issues on the PR.

Summary: Preliminary results show that around 8% to 10% of PRs have reactions
in Eclipse repositories. We find cases where the emoji did not always reduce
commenting noise in the discussion. Under a closer manual inspection of 20
emoji reactions, we find that there are eight cases where the emoji reactions
did not reduce commenting noise.

TABLE II: Distribution of Sentiments of Emoji Reactions. Emoji reaction sentiments | # PR Comments |
---|---|---
Positive | 9,084 | (98.10%) |
Negative | 67 | (0.72%) |
Neutral | 74 | (0.79%) |
Mixed | 31 | (0.33%) |
Total | 9,256 | (100%) |
TABLE III: The frequency count of manual samples where whether the emoji reactions reduced the commenting noise. Category | Count |
---|---|---
Reducing commenting noise | 12 |
Not Reducing commenting noise | 8 |

##  III Study Protocols

In this section, we present the design of our study. This section consists of
our research questions with their motivations.

###  III-A Research Questions

Inspired by the motivating examples and the preliminary study, we formulate
four research questions to guide our study:

  * •

RQ1: Does the emoji reaction used in the review discussion correlate with
review time?
Prior studies [3, 19] have widely analyzed the impact of technical and non-
technical factors on the review process (e.g., review outcome, review time).
However, little is known about whether or not the emoji reaction can be
correlated with review time. It is possible that emoji reaction may shorten
the review time, as it could reduce the noise during the review discussions.
Thus, our motivation for the first research question is to explore the
correlation between the emoji reaction used in the review discussion and
review time.

  * •

RQ2: Does a PR submitted by a first-time contributor receive more emoji
reactions?
As shown in Figure 1(b), we find that the emoji reaction might be used to
express appreciation for submitting a PR. Our motivation for this research
question is to understand if contributors that have never submitted to the
project before receive more emoji reactions. Furthermore, answering this
research question will provide insights into a potent ulterior motive for a
emoji reaction.

Our assumption is that:

> H1: PRs submitted by first-time contributor receive more emoji reactions.
> Existing contributors express positive feelings to attract newcomers to the
> project.

  * •

RQ3: What is the relationship between the intention of comments and their
emoji reaction?
Our preliminary study findings show that emoji reactions do not always reduce
the commenting noise. Hence, our motivation for the third research question is
to explore the relationship between the intention of comments and their
reactions.

Our assumption is that:

> H2: Most emojis are uniformly distributed across the different intentions.
> Specific intentions may explain the ulterior purpose of reacting with an
> emoji reaction.

  * •

RQ4: Is emoji reaction consistent with comment sentiment?

We found that specific sentiments of the emoji (i.e., THUMBS UP *X) are widely
used in PRs from our preliminary study. Our motivation for this research
question is to investigate whether there is any inconsistency between
sentiments of the comments and sentiments of the emoji reactions. Furthermore,
we plan to manually check the reasons why inconsistency happened. We believe
answering RQ4 would help newcomers better understand the emoji usage in the PR
discussion.

Our assumption is that:

> H3: The sentiment of emoji reactions are uniformly distributed across the
> same comment sentiments. Specific sentiments may explain the ulterior
> purpose of reacting with an emoji. This may be useful in understanding what
> information is needed in code review.

##  IV Data collection

To generalize the results of the study, we plan to expand on our dataset from
active software development repositories shared by Hata et al. [13]. Each
repository in this dataset has more than 500 commits and at least 100 commits
during the most active two-year period. In total, this dataset contains 25,925
repositories from seven languages (i.e., C, C++, Java, JavaScript, Python,
PHP, and Ruby). We will use the GraphQL API [1] to obtain PRs created before
March 13rd 2016 where GraphQL was introduced. The whole dataset will be used
for all four research questions.

##  V Execution Plan

In this section, we present the execution plan of our experiment. We will use
a mixed method consisting of both quantitative and qualitative analysis to
answer our research questions.

###  V-A Research Method for RQ1:

For the first research question, we plan to use a quantitative method. To
investigate the effect of emoji reaction related factors on the pull request
process (i.e., review time), we plan to perform a statistical analysis using a
non-linear regression model. This model allows us to capture the relationship
between the independent variable and the dependent variable. The goal of our
statistical analysis is not to predict the review time but to understand the
associations between the emoji reaction and the review time.

For the independent variables, similar to the prior studies [19, 23], we will
select the following confounding factors as our independent variables:

  * •

PR size: The total numbers of added and deleted lines of code changed by a PR.

  * •

Change file size: The number of files what were changed by a PR.

  * •

Purpose: The purpose of a PR, i.e., bug, document, feature.

  * •

# Comments: The total number of comments in a PR discussion thread.

  * •

# Author Comments: The total number of comments by the author in a PR
discussion thread.

  * •

# Reviewer Comments: The total number of comments by the reviewers in a PR
discussion thread.

  * •

Patch author experience: The number of prior PRs that were submitted by the PR
author.

  * •

Reviewers: The number of developers who posted a comment to a review
discussion.

  * •

Commit size: The number of commits in a PR.

Since we investigate the effect of the emoji reaction, we plan to compute
additional independent variables that are related to emoji reaction:

  * •

With emoji reaction: Whether or not a PR includes any emoji reaction (binary).

  * •

The number of emoji reactions: The count of emoji reaction in a PR.

For the dependent variable (i.e., review time), we measure the time interval
in hours from the time when the first comment was posted until the time when
the last comment was posted. For the model construction, we will adopt the
steps that are similar to the prior studies, including (i) Estimating budget
for degrees of freedom, (ii) Normality adjustment, (iii) Correlation and
redundancy analysis, (iv) Allocating degrees of freedom, and (v) Fitting
statistical models.

(a) Analysis Plan: We will analyze the constructed regression models in the
following three steps: (i) Assessing model stability. To evaluate the
performance of our models, we will report the adjusted $R^{2}$ [12]. We will
also use the bootstrap validation approach to estimate the optimism of the
adjusted $R^{2}$. (ii) Estimating the power of explanatory variables. Similar
to prior work [23], we plan to test the significant correlation of independent
variables with p-value and employ Wald statistics to measure the impact of
each independent variable. (iii) Examining relationship. Finally, we will
examine and plot the direction of the relationship between each independent
variable (i.e., especially emoji reaction related variables) and the dependent
variable.

###  V-B Research Method for RQ2:

For RQ2, we plan to use a quantitative method.  To do so, we will construct
two groups of pull requests to compare against: first-time contributors and
non-first time contributors (control group). For the first-time contributor
group, we will identify all pull requests that are submitted by first-time
contributors from our dataset. For the non-first time contributor group, to
construct a balanced control group, we will randomly select the equal number
of pull requests that are submitted by non-first time contributors. We will
then divide the pull requests into ones having emoji reactions and the other
ones without emoji reactions, respectively.

(a) Analysis Plan: We will present a pivot chart to show the frequency of pull
requests having emoji reactions or without emoji reactions by first-time
contributors and non-first time contributors. The plot x-axis will represent
two groups of first-time contributors and non-first time contributors.
Furthermore, each group will be divided into two parts: pull requests with
emoji reaction and pull reactions without emoji reactions. The plot y-axis
will represent the frequency count of pull requests.

(b) Significant Testing: To select a suitable statistical test, we will adopt
the Shapiro-Wilk test with alpha = 0.05. In the case when the p-value is
greater than 0.05, we will perform a two-tailed independent t-test with alpha
0.05. Otherwise, we will adopt a two-tailed Mann Whitney U test [21] with
alpha = 0.05 to validate.

In addition, we will investigate the effect size. In case when the data is
normally distributed, we will use Hedges g effect size [14]. Effect size is
analyzed as follows: (1) $|d|$ < 0.2 as Negligible, (2) 0.2 $\leq$ $|d|$ <0.5
as Small, (3) 0.5 $\leq$ $|d|$ <0.8 as Medium, or (4) 0.8 $\leq$ $|d|$ as
Large. If the data are not normally distributed, we will apply Cliff’s delta
(Romano et al, 2006) to measure effect size. Effect size is analyzed as
follows: (1) $|\delta|$ < 0.147 as Negligible, (2) 0.147 $\leq$ $|\delta|$
<0.33 as Small, (3) 0.33 $\leq$ $|\delta|$ <0.474 as Medium, or (4) 0.474
$\leq$ $|\delta|$ as Large.

###  V-C Research Method for RQ3:

For RQ3, we plan to use a quantitative method to classify the intentions of
the comments.  To categorize the intentions of the comments, we will use a
taxonomy of intention proposed by Huang [17]. They manually categorized 5,408
sentences from issue reports of four projects in GitHub to generalize the
linguistic pattern for category identification.

The taxonomy of intention category is described below:

  * •

Information Giving (IG): Share knowledge and experience with other people, or
inform other people about new plans/updates (e.g., “The typeahead from
Bootstrap v2 was removed.”).

  * •

Information Seeking (IS): Attempt to obtain information or help from other
people (e.g., “Are there any developers working on it?”).

  * •

Feature Request (FR): Require to improve existing features or implement new
features (e.g., “Please add a titled panel component to Twitter Bootstrap.”).

  * •

Solution Proposal (SP): Share possible solutions for discovered problems
(e.g., “I fixed this for UI Kit using the following CSS.”).

  * •

Problem Discovery (PD): Report bugs, or describe unexpected behaviors (e.g.,
“the firstletter issue was causing a crash.”).

  * •

Aspect Evaluation (AE): Express opinions or evaluations on a specific aspect
(e.g., “I think BS3’s new theme looks good, it’s a little flat style.”).

  * •

Meaningless (ML): Sentences with little meaning or importance (e.g., “Thanks
for the feedback!”).

To facilitate the automation, they proposed a convolution neural network based
classifier with high accuracy. For RQ3, we will use this classifier to
automatically label the intention of the comments. To evaluate the robustness
of this classifier in our dataset, we will first use the proposed classifier
to automatically classify the intentions of the randomly sampled 30 comments.
Then, we will manually check whether the labeled intentions of these 30
comments are correct or not. The result of this sanity check will be presented
as a percentage of the false positive, under 10% being considerable.

(a) Analysis Plan: To analyze the relationship between the intention of
comments and their emoji reaction, we will use the association rule mining
technique. To show the diversity of different intentions from the
classification, we will draw a histogram plot. To show the results of the
relationship, a table will be drawn with descriptive statistics,including the
criteria support and confidence.

(b) Significant Testing: To inspect whether or not the classified intentions
of comments are normally distributed, we will adopt the Shapiro-Wilk test with
alpha = 0.05, which is widely used for the normality test. In addition, to
inspect whether the intentions of comments are significantly different, we
will use Kruskal-Wallis non-parametric statistical test [4].

###  V-D Research Method for RQ4:

For RQ4, we plan to use a qualitative and quantitative method. First of all,
we will use a quantitative method to investigate whether there is any
inconsistency between sentiments of emoji reaction and sentiments of the
comments. We determine the sentiments of the emoji based on the definition we
discussed in section 2. Hence, Emoji sentiment can be categorized into the
following types: Positive, Negative, Neutral, and Mixed. To extract the
sentiment of the first responses, we plan to use SentiStrength-SE [18], the
state-of-the-art sentiment analysis tool for software engineering text.
Similar to the tool we plan to use for RQ3, the input is reacted comments, and
the output is the sentiment score of the given comment. The sentiment score
varies from -5 (very Negative) to 5 (very Positive). Based on the above
definition, we consider it as inconsistent if the sentiments of the comment
are different from the sentiments of the emoji.

Then, we will conduct qualitative analysis to explore the possible reasons for
inconsistency between sentiments of emoji and sentiments of the comments. To
do so, we will apply the open coding approach [5] to classify the reasons for
inconsistency. To discover as a complete list of reasons as possible, we
strive for theoretical saturation [11]. Similar to prior work [15], we set our
saturation criterion to 50, i.e., we continue to code randomly selected
comments until no new reasons have been discovered for 50 consecutive
comments. Furthermore, we perform the kappa agreement score [22] to evaluate
the classification quality. Similar to Hata et al. [13], the agreement of the
coding guide will be performed using a kappa agreement. Kappa result is
interpreted as follows: values $\leq$ 0 as indicating no agreement and
0.01–0.20 as none to slight, 0.21–0.40 as fair, 0.41–0.60 as moderate,
0.61–0.80 as substantial, and 0.81–1.00 as almost perfect agreement. The
agreement scores larger than 0.81 (i.e., almost perfect) are considered for
the manual analysis. Based on the prior experience, we estimate the size of
the samples to range from 200-300 samples.

(a) Analysis Plan: Similar to RQ3, we will depict a histogram plot to show the
distribution of emoji types by sentiments of the comment. To show the results
of inconsistency reasons, we will draw a histogram plot to show the frequency.
As part of our result presentation, we will paste the real examples during the
analysis to describe the reason taxonomy.

(b) Significant Testing: Similar to RQ3, we will use Shapiro-Wilk test to
inspect whether or not the sentiments of emoji usage are normally distributed
and use Kruskal-Wallis non-parametric statistical test to validate the
significant difference.

##  VI Implications

We summarize our implications with the following take-away messages for the
key stakeholders:

  * •

Researchers: Answering RQ1 will help researchers understand the impact of
emoji reactions, hence may contribute to existing knowledge on the code review
process. As informal communication such as the usage of emojis becomes
prevalent, it is a need to understand its role in keeping an efficient code
review process. We believe that emojis may also help remove toxic and other
forms of anti-patterns [8] in the code review process. In terms of the
intention of the emoji reactions, our study will complement all related works
on emoji usage [6, 7].

  * •

Contributors: In terms of practitioners, we envision our study to assist
projects to attract and maintain existing and potential contributors to the
project. Especially for RQ2, the results of the study may provide some
insights into how to attract newcomers and also how to provide a more friendly
and welcoming environment. Furthermore, answering RQ3 and RQ4 will provide
some insights for contributors into undersanding the consistency of emoji
reactions. As emoji usage becomes popular [20], the results of the study
should provide guidelines on how to represent the common intentions when an
emoji reaction is warranted.

##  VII Threats to validity

We identified three key threats to our study. First, the Eclipse projects that
were used in our preliminary study may not be representative of all types of
GitHub projects. To increase generalizability, we will extend our study to
include a sample of random GitHub projects [13]. Our second threat is
concerning the qualitative aspect of the study, as this is bias to human error
in the classification. This is because the interpretation of emoji usage may
not be trivial. To mitigate this, we employ the Kappa method to have multiple
co-authors for agreement of each code. For instance, if positive emoji is used
in an ironic context, it does not mean positive. This usage of emojis may
influence our results. Third, our quantitative analysis and data collection
may include some false positives, such as bot reactions and comments.
Currently, we manually exclude these bots for the preliminary study. To
mitigate this, we plan to carefully identify and systematically remove bots
based on official documentation.

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11institutetext: Indian Institute of Science Education and Research,
Berhampur, 760010, India

# Azimuthal anisotropy measurement of multi-strange
hadrons in Au+Au collision at $\sqrt{s_{NN}}$ = 27 and 54.4 GeV
at STAR

Prabhupada Dixit (for the STAR collaboration) 11<EMAIL_ADDRESS>
###### Abstract

In these proceedings, we present the second order ($v_{2}$) and the third
order ($v_{3}$) azimuthal anisotropy measurements of strange and multi-strange
hadrons such as $K^{0}_{S}$, $\Lambda$, $\bar{\Lambda}$, $\phi$, $\Xi^{-}$,
$\bar{\Xi}^{+}$, $\Omega^{-}$, $\bar{\Omega}^{+}$ in Au+Au collisions at
$\sqrt{s_{NN}}$ = 27 and 54.4 GeV at STAR. The centrality dependence of
$v_{2}$ and $v_{3}$ as well as the number of constituent quark scaling (NCQ
scaling) for all the particles mentioned above are discussed. The ratio of
$v_{2}$ of $\phi$ mesons to that of $\bar{p}$ is presented at $\sqrt{s_{NN}}$
= 54.4 GeV which gives a hint of mass ordering violation at low $p_{T}$.

##  1 Introduction

Relativistic heavy-ion collision is a unique tool to study matter at the
extreme conditions of temperature and density. The collective expansion of the
medium formed in heavy-ion collision is called flow. Depending on the initial
spatial configuration of the colliding system and fluctuations, there can be
the presence of different orders of anisotropy in the azimuthal distributions,
known as anisotropic flow. In non-central nucleus-nucleus collisions, the
overlap region of two colliding nuclei looks like the shape of an almond and
in this region the pressure gradient is different along different axis of the
system. The difference in pressure gradient leads to momentum anisotropy which
is the dominant cause for the development of elliptic flow. But the event-by-
event fluctuations of the nucleons in the colliding nuclei can give rise to
other types of flow such as triangular flow, rectangular flow, etc. All these
flow coefficients can be studied by using Fourier series expansion of the
azimuthal distribution of particles given by

$E\frac{d^{3}N}{dp^{3}}=\frac{1}{2\pi}\frac{d^{2}N}{p_{T}dp_{T}dy}\left[1+\sum_{n}2v_{n}\cos
n(\phi-\Psi_{R})\right].$ (1)

Here $v_{2}$ and $v_{3}$ represent elliptic flow coefficient and triangular
flow coefficient respectively. The $n^{th}$ order flow coefficient is given by

$v_{n}=\langle\cos n(\phi-\psi_{n})\rangle.$ (2)

Here $\phi$ is the azimuthal angle of the particles and $\psi_{n}$ is the
$n^{th}$ order event plane angle. The simultaneous measurement of $v_{2}$ and
$v_{3}$ is important to constrain the shear viscosity over entropy density
ratio ($\eta/s$) Ref1 . In this study, we have used strange and multi-strange
hadrons because of their small hadronic interaction cross-section and early
freeze-out Ref2 , the measured flow coefficients remain least affected in the
hadronic phase.

##  2 Analysis method

The $n^{th}$ order event plane is given by

$\psi_{n}=\frac{1}{n}\tan^{-1}\Big{(}\frac{\sum_{i}w_{i}\sin(n\phi_{i})}{\sum_{i}w_{i}\cos(n\phi_{i})}\Big{)},$
(3)

where $\phi_{i}$ is the azimuthal angle of the particle track and $w_{i}$ is
the weight factor equal to $p_{T}$ $\times$ $\phi$-weight. The factor
$\phi$-weight is the correction factor for azimuthal acceptance of the
detector. Due to the finite resolution of the event plane, the $v_{n}$ needs
to be corrected by event plane resolution given by

$R_{n}=\sqrt{\langle\cos n(\psi_{A}-\psi_{B})\rangle}.$ (4)

Here $\psi_{A}$ and $\psi_{B}$ are two sub-event planes in -1$<\eta<$-0.05 and
0.05 $<\eta<$1 regions respectively. In Eq. 2, the particle’s $\phi$ and the
event plane $\psi_{n}$ are taken from opposite $\eta$ region.

Invariant mass method Ref3  is used to calculate $v_{n}$. A detailed procedure
to apply this method can be found in Ref. Ref8 .

##  3 Results and discussion

Figure 1 shows $v_{2}$ and $v_{3}$ plotted as a function of $p_{T}$ for
$K^{0}_{S}$, $\Lambda$, $\phi$, $\Xi^{-}$, $\Omega^{-}$ at $\sqrt{s_{NN}}$ =
54.4 GeV in 0-80% centrality. In low $p_{T}$ both $v_{2}$ and $v_{3}$ show a
mass ordering which is due to the radial flow of the system Ref4 , but at
$p_{T}$ above 2 GeV/c a separation between baryons and mesons is observed
which can be explained by the quark recombination model of hadronization Ref5
.

Figure 1:  Panel-a shows $v_{2}$ and panel-b shows $v_{3}$ as a function of
$p_{T}$ for $K^{0}_{S}$, $\Lambda$, $\phi$, $\Xi^{-}$, $\Omega^{-}$ in 0-80%
centrality. The vertical lines represent the statistical error bars and the
shaded boxes represent the systematic error bars.

A centrality dependence study for both $v_{2}$ and $v_{3}$ is shown in Fig. 2
and Fig. 3 in three different centrality classes, 0-10%, 10-40% and 40-80%.
The elliptic flow $v_{2}$ shows a strong centrality dependence because of
initial spatial anisotropy dominance, whereas $v_{3}$ does not because it
arises mostly due to event-by-event fluctuations in the initial state.

Figure 2: $v_{2}$ is plotted as a function of $p_{T}$ for 0-10%, 10-40% and
40-80% centrality classes. The vertical lines represent the statistical error
bars and the shaded boxes represent the systematic error bars. Figure 3:
$v_{3}$ is plotted as a function of $p_{T}$ for 0-10%, 10-40% and 40-80%
centrality classes. The vertical lines represent the statistical error bars
and the shaded boxes represent the systematic error bars.

The NCQ scaled $v_{2}$ is studied as a function of NCQ scaled transverse
kinetic energy for 0-80% centrality at $\sqrt{s_{NN}}$ = 27 and 54.4 GeV. The
NCQ scaled $v_{2}$ for all the particles shown in Fig. 4 is found to fall on a
single curve which suggests that the collectivity is developed during the QGP
stage of the collision. The same result is found for $v_{3}$ also as shown in
Fig. 5.

Figure 4: NCQ scaled $v_{2}$ is plotted as a function of NCQ scaled transverse
kinetic energy for 0-80% centrality for $\sqrt{s_{NN}}$ = 27 (left) and 54.4
GeV (right).  Figure 5: NCQ scaled $v_{3}$ is plotted as a function of NCQ
scaled transverse kinetic energy for 0-80% centrality for $\sqrt{s_{NN}}$ =
54.4 GeV. The vertical lines represent the statistical error bars and the
shaded boxes represent the systematic error bars.

The ratio of $\phi$ meson $v_{2}$ to $\bar{p}$ $v_{2}$ is calculated as a
function of $p_{T}$ as shown in Fig. 6. Since the mass of the $\phi$ meson
(1.019 GeV/$c^{2}$) is greater than that of $\bar{p}$ (0.938 GeV/$c^{2}$),
according to mass ordering at low $p_{T}$ the ratio should have been less than
one. But we observe a hint of mass ordering violation at low $p_{T}$ which can
be explained by larger hadronic rescattering effect on $\bar{p}$ compared to
$\phi$ meson Ref6 . A similar result is found in previous STAR measurement in
minimum biased Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV Ref7 .

Figure 6: The left panel shows the ratio of $\phi$ meson $v_{2}$ to $\bar{p}$
$v_{2}$ in Au+Au collision at $\sqrt{s_{NN}}$ = 54.4 GeV in 10-40% and 40-80%
centralities. The right panel shows the comparison of the ratio with 200 GeV
data points for minimum bias events. The vertical lines represent the
statistical error bars and the shaded boxes represent the systematic error
bars.

##  4 Summary

Measurements of $v_{2}$ and $v_{3}$ as a function of $p_{T}$ of strange and
multi-strange hadrons, $K^{0}_{S}$, $\Lambda$, $\bar{\Lambda}$, $\phi$,
$\Xi^{-}$, $\bar{\Xi}^{+}$, $\Omega^{-}$, $\bar{\Omega}^{+}$ is presented in
Au+Au collisions at midrapidity for 0-80% centrality. A mass ordering is
observed in the low $p_{T}$. At high $p_{T}$ ($>$ 2 GeV/c), we observe a
baryon to meson separation. A strong centrality dependence is observed for
$v_{2}$, but the centrality dependence is absent for $v_{3}$. The NCQ scaling
is found to hold for all the particles at $\sqrt{s_{NN}}$ = 27 and 54.4 GeV.
The ratio of $v_{2}$ of $\phi$ meson to that of $\bar{p}$ gives a hint of mass
ordering violation at low $p_{T}$.

## References

  * (1) Chun Shen et al., J. Phys. G: Nucl. Part. Phys. 38, 124045 (2011)
  * (2) A. Shor, Phys. Rev. Lett. 54, 1122 (1985)
  * (3) N. Borghini and J.-Y. Ollitrault, Phys. Rev. C 70, 064905 (2004)
  * (4) D. Teaney, J. Lauret, and E. V. Shuryak, Phys. Rev. Lett. 86, 4783 (2001)
  * (5) D. Molnar and S. A. Voloshin, Phys. Rev. Lett. 91, 092301 (2003)
  * (6) T. Hirano et al., Phys. Rev. C 77, 044909 (2008)
  * (7) L. Adamczyk et al. (STAR), Phys. Rev. Lett. 116, 062301 (2016)
  * (8) P. Dixit, arXiv:2103.04646 [hep-ex]


# Directional effects of antiferromagnetic ordering on the electronic
structure in NdSb

Yevhen Kushnirenko  Division of Materials Science and Engineering, Ames
Laboratory, Ames, Iowa 50011, USA  Department of Physics and Astronomy, Iowa
State University, Ames, Iowa 50011, USA  Brinda Kuthanazhi  Division of
Materials Science and Engineering, Ames Laboratory, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  Lin-Lin Wang  Division of Materials Science and Engineering, Ames
Laboratory, Ames, Iowa 50011, USA  Benjamin Schrunk  Division of Materials
Science and Engineering, Ames Laboratory, Ames, Iowa 50011, USA  Evan O’Leary
Division of Materials Science and Engineering, Ames Laboratory, Ames, Iowa
50011, USA  Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA  Andrew Eaton  Division of Materials Science and Engineering,
Ames Laboratory, Ames, Iowa 50011, USA  Department of Physics and Astronomy,
Iowa State University, Ames, Iowa 50011, USA  P. C. Canfield  Division of
Materials Science and Engineering, Ames Laboratory, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  Adam Kaminski  Corresponding author<EMAIL_ADDRESS>Division of
Materials Science and Engineering, Ames Laboratory, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA

###### Abstract

The recent discovery of unconventional surface state pairs, which give rise to
Fermi arcs and spin textures, in antiferromagnetically ordered NdBi raised the
interest in rare-earth monopnictides. Several scenarios of antiferromagnetic
order have been suggested to explain the origin of these states with some of
them being consistent with the presence of non-trivial topologies. In this
study, we use angle-resolved photoemission spectroscopy (ARPES) and density-
functional-theory (DFT) calculations to investigate the electronic structure
of NdSb. We found the presence of distinct domains that have different
electronic structure at the surface. These domains correspond to different
orientations of magnetic moments in the AFM state with respect to the surface.
We demonstrated remarkable agreement between DFT calculations and ARPES that
capture all essential changes in the band structure caused by transition to a
magnetically ordered state.

††preprint: APS/123-QED

##  I Introduction

Many rare-earth monopnictides order antiferromagnetically at low temperatures
due to moment bearing rare earth ions [1, 2, 3, 4, 5, 6, 7]. In the last
decade, some materials of this family were predicted to host topological Weyl
states [8, 9, 10], while, in some other materials, topological surface Dirac
states are expected to be already present in the paramagnetic state [11, 12]
due to band inversion. These predictions are supported by numerous
experimental studies [13, 14, 15, 16, 17, 7, 18, 19]. In a recent ARPES study
[20], reported emergence of unconventional surface states (SS) below AFM
transition temperature. These new states undergo Kaminski-Canfield (K-C)
splitting that leads to formation of Fermi arcs.

Similarly to other rare-earth monopnictides, NdSb and NdBi have a rock salt
crystal structure and undergo AFM transition at low temperatures. This
transition was observed in neutron diffraction [21, 22, 23, 24], magnetization
[25, 26, 27], resistivity, and specific heat [28] measurements. The transition
temperatures reported in these studies for NdSb and NdBi are 15 K and 24 K,
respectively. In contrast to some other rare-earth monopnictides [29, 30],
NdSb and NdBi have only one AFM phase with the magnetic structure shown in
Fig. 1(a). Two recent ARPES studies [31, 32] reported changes in the
electronic structure of NdSb upon AFM transition. However, there was a
substantial difference in the band dispersion present below AFM transition.
One study [32] showed the development of two additional features at
$\Gamma$-point, one of which forms a small round pocket on the Fermi surface
(FS). In contrast another study [31] showed the development of surface state
dispersion at $\sim$0.2 $\AA^{-1}$ away from the $\Gamma$-point. These surface
states form arcs and elliptical pockets of the Fermi surface. Similar surface
state dispersions were also observed in NdBi [20].

Here, we show the existence of different variations of band structures in NdSb
below Neel temperature using laser ARPES. These variations coexist in the same
sample at different surface locations. Also, we present DFT calculations of
the band structure in the AFM ordered NdSb. By comparing the experimental
results with DFT calculations, we show that observed variants of band
structures are attributed to domains that have different directions of AFM1
ordering. In addition, we show that magnetically ordered NdBi can have band
structure different from the one reported before. This indicates that NdBi can
have different domains as well.

##  II Experimental details

Single crystals of NdSb crystals were grown out of Sn flux using an initial
concentration of Nd4Sb4Sn96. The elements were weighed out into a fritted
alumina crucible set (Canfield Crucible Set) [33, 34] and sealed in a fused
silica tube under partial pressure of argon. The prepared ampules were heated
up to 1100∘ C over 4 hours and held there for 5 hours. This was followed by a
slow cooling to the decanting temperature of 800∘ C over 100 hours and
decanting of the excess flux using a centrifuge [35]. The cubic crystals
obtained were stored and handled in a glovebox under Nitrogen atmosphere.

Most of the ARPES data was collected using vacuum ultraviolet (VUV) laser
ARPES spectrometer that consists of a Scienta DA30 electron analyzer,
picosecond Ti:Sapphire oscillator and fourth-harmonic generator [36]. Data
from the laser based ARPES were collected with 6.7 eV photon energy. Angular
resolution was set at $\sim$ 0.1∘ and 1∘, along and perpendicular to the
direction of the analyser slit respectively, and the energy resolution was set
at 2 meV. The VUV laser beam was set to vertical polarization. The diameter of
the photon beam on the sample was $\sim 15\,\mu$m. Supplementary He-lamp-based
ARPES measurements were carried out using R8000 analyzer and GammaData helium
discharge lamp with custom focusing optics. The diameter of the photon beam on
the sample was $\sim 1$mm.

##  III Results and Discussion

Fig. 1(b) shows a two-dimensional projection of bulk Fermi surface of
paramagnetic NdSb calculated using density functional theory (DFT). As well as
other rare-earth monopnictides[6, 7, 12, 17, 20, 31], it has several hole
pockets at the center of the Brillouin zone (BZ) and ellipsoidal electron
pockets at its corners. The experimental Fermi surface (FS) map in Fig. 1(c)
was measured in the PM phase using 21.2 eV light from He-lamp. It is in good
agreement with DFT calculations. The observed broad FS features are the result
of a strong three-dimensionality of the band structure that is projected onto
$k_{z}$, $k_{y}$ plane and limited $k_{z}$-selectivity of the ARPES
experiment.

Figure 1: NdSb crystal- and band-structure. (a) Schematic magnetic structure
of AFM1 ordering. Green and blue squares represent (100/010) and (001) planes,
respectively. (b) DFT-calculated non-magnetic Fermi surface. (c) Fermi surface
map measured in the paramagnetic state (T = 17.5 K). (d) Three Fermi surface
maps measured at different locations on one sample in the AFM state (T = 5.5
K).

For further analysis, we measured detailed datasets around the center of the
BZ using laser-based ARPES. Fig. 1(d) shows three FS maps measured at
$T<T_{N}$ at different locations on the same sample surface cleave. All these
maps differ from the map measured in the PM state (Fig. 1(c)) and from each
other. As it will be shown later, the first map in Fig. 1(d) corresponds to
the case of AFM1 ordering with magnetic moments oriented orthogonal to the
sample surface (001 surface in Fig. 1(a)), while two other maps correspond to
the case of AFM1 ordering with magnetic moments oriented parallel to the
sample surface (100/010 surface in Fig. 1(a)) along the vertical and
horizontal directions.

In Fig. 2, we analyze the effects of magnetic ordering on the electronic band
structure at (001) surface. Parts of the DFT calculated FS near the Brillouin
zone center for AFM1 (001), and PM phases are shown in Fig. 2(a) and (b),
respectively. In the magnetically ordered state, we observe the suppression of
the inner bulk pocket and the appearance of two sharp contours. This
suppression is a result of the opening of a hybridization gap (for details,
see Appendix B). The inner contour is formed by electron-like surface-state
dispersion (see Fig. 2(c)), and the other contour is formed by a folded band
and W-shaped surface-state dispersion. To analyze the development of these new
features and compare them with the calculations, we measured datasets at
several temperatures in the magnetically ordered and PM state from A-domain.
In Fig. 2(d) and (e), we show the temperature evolution of the FS and the band
dispersions along a high-symmetry direction. At $T=$ 6.5K, the bottom of the
electron-like and the W-shaped dispersion are observed in the experimental
data as it is predicted by DFT calculations.

Figure 2: Band structure at the (001) cleaving plane. (a) and (b) DFT-
calculated part of the Fermi surface near $\Gamma$-point for AFM and PM phases
of NdSb, respectively. (c) DFT-calculated band dispersions along the cut
marked with a line in (a). (d) Temperature dependence of the Fermi surface
from the A-domain. (e) Temperature dependence of band dispersions along the
cut marked with a line in (d) and corresponding second derivative plots. (f)
$E_{F}$ MDCs from spectra in (e). All the experimental data in this plot were
measured using photon energy 1.67 eV. The arrows in (c) mark SS dispersions

With the temperature increasing, both SS dispersions move to higher binding
energies. At the same time, the W-shaped dispersion gradually changes its
shape towards a regular parabola. As a result, the FS pockets increase in
size. The inner FS pocket appears as a solid circle with a sharp perimeter.
This happens because, besides the SS dispersion, which should form just a
contour of the FS, there are bulk states above the SS parabola, as is seen
from the DFT calculations in Fig. 2(c). In the experimental data, the SS
parabola is the most pronounced in the 13.5 K spectrum near the Fermi level.
The momentum distribution curve obtained from this spectrum at the Fermi
energy ($E_{F}$) is shown in Fig. 2(f). Two sharp peaks associated with the
surface states can be seen in this curve. Finally, in the paramagnetic state
at T = 15.5 K, both SS dispersions disappear. Also, from the plots in Fig.
2(e) and MDCs in Fig. 2(f), one can see that in the AFM1 state, a
hybridization gap is opening inside the bulk states. These new SS are directly
linked to this gap: the electron-like dispersion is the upper boundary of the
gap, and the W-shaped dispersion exists inside the gap and partially merges
with its lower boundary.

For further analysis, we measured a set of spectra along $\Gamma-X$ direction
at T= 10.5 K using light with different photon energy. These spectra are shown
in Fig. 3, together with the corresponding second-derivative plots. This
allowed us to access parts of the BZ with different $k_{z}$ and distinguished
SS from bulk states. Two SS dispersions mentioned above indeed do not shift
with photon energy. The only difference between these spectra is the
suppression of some parts of them at lower photon energy. This suppression is
likely a result of matrix elements. Interestingly, in 6.79 and 6.85 eV
spectra, we observe a Dirac-like feature. However, the DFT (Fig. 3(c))
calculations do not predict a Dirac cone there. Two other hole-like SS
dispersions predicted by DFT calculations can also be observed in the
experiment. They correspond to the top of the band located at 300 meV and the
another one at 420 meV in the experimental data. We can also observe two more
relatively sharp dispersions (marked with red arrows in Fig. 3). These
dispersions shift with photon energy and are associated with bulk states.

Figure 3: Photon energy dependence. (a) Spectra measured along $\Gamma$-X
direction using photon energy from 6.05 to 6.85 eV. (b) Corresponding plots of
second derivatives for the spectra from (a). (c) DFT-calculated surface state
dispersion. The arrows in (b) and (c) mark corresponding SS dispersions. The
arrows in (a) mark bulk dispersions.

In Fig. 4, we analyze the effects of (100/010) ordering on the surface state
band dispersions. The DFT calculated FS and dispersions along the $\Gamma-X$
and $\Gamma-Y$ directions are shown in Fig. 4(a)-(c). These calculations
predict the existence of several additional surface states $\Gamma-X$ in the
magnetically ordered phase, but the most interesting for us are the two
outermost surface states, which form Fermi arcs and elliptical electron
pockets on the FS. Such states were already observed in the sibling compound
NdBi [20]. The calculations successfully reproduce the experimental FS from
domains B and C, shown in Fig 1(d). Both maps demonstrate that the SS Fermi
arcs and elliptical pockets are present only along one direction, within a
given domain. For further analysis, we plot experimental dispersions along two
orthogonal cuts for both maps in Fig. 4(d-g). The cuts along the $\Gamma-X$
direction show the presence of SS, namely pair of K-C split bands near the Ef
(panels d and f), which are responsible for the formation of Fermi arcs and
elliptical pockets. At the same time, we observe no signs of the surface state
near the $\Gamma$-point or suppression of the bulk states in this region
similar to those observed in the A-domain data. The SS are seemingly absent
along $\Gamma$-Y direction (Fig. 4(e) and (g)). Taking into account that
datasets from all domains were measured under identical experimental
conditions (experiment geometry, light polarization, and photon energy), we
can conclude that the absence of SS states in the experimental data in one
direction is an intrinsic property of this material and not caused by the
matrix elements.

Figure 4: Band structure in the case of (100/010) cleavage plane. (a) DFT-
calculated Fermi surface for AFM1 state. (b) and (c) DFT-calculated surface
state dispersions along $\Gamma-X$ and $\Gamma-Y$ in AFM1 state, respectively.
(d) and (e) Experimental band structure along $\Gamma-X$ and $\Gamma-Y$
directions obtained from the B-domain dataset respectively. (f) and (g) The
same as (d) and (e), respectively, but for the C-domain. (h) Cuts parallel to
$\Gamma-X$ taken at $k_{x}=$ 0.065 $\AA^{-1}$ and -0.056 $\AA^{-1}$,
respectively. (i) Temperature dependence of $\Gamma-X$ spectrum from (d). The
arrows demonstrate the correspondence between particular SS features in the
DFT calculations and the experimental data

The temperature evolution of the spectrum from Fig. 4(d) is shown in Fig.
4(i). The splitting of the SS decreases with increased temperature, and the SS
vanishes above $T_{n}$. This result agrees with the previous studies of NdSb,
and NdBi [31, 20]. This behavior is similar to the behavior of the SS on the
(001) surface (Fig. 2(e)), where the distance between the surface state bands
also decreases with increasing temperature. Moreover, the absolute positions
of the bottoms of these bands are the same for both domain types. Their
positions estimated for (100) surface at T = 10.5 K (181 nm plot of Fig. 3)
are 66$\pm$4 meV and 104$\pm$2 meV. Their positions estimated for (001)
surface at T = 10.5 K (Fig. 4(g)) are 68$\pm$2 meV and 105$\pm$2 meV. This
similarity is not surprising since, in both cases, SS are formed inside the
same hybridization gap.

In the paramagnetic state at T $\sim$ 15.5 K, SS dispersions disappear. The
high-temperature spectra in Fig. 2(e) and Fig. 4(g) look identical, which
indicates that both domain types are indeed formed in the same material.
However, after several cycles of heating the sample to $T>T_{N}$ and cooling
it down, we found the same domains at their original locations (for the
details, see Appendix A). Thus, most likely, the domain type can be
predetermined by some factors, e.g., pinned by the strain present in the
crystal.

The spectra in Fig. 4(d) and (f) exhibit one more SS dispersion (marked with
green arrows), which disappears in the paramagnetic state. Under closer
inspection, one can see that this dispersion is split into two branches. This
can be better seen in the cuts taken slightly off the high-symmetry direction
where this splitting is large (see Fig. 4(h)). As well as the other SS
dispersions, this splitting is predicted by DFT calculations. The
corresponding features can be found in the calculated FS (Fig.4 (a)).

Elliptical SS pockets and arcs with four-fold symmetry were reported in NdBi
in previous ARPES work [20, 31]. In analogy to NdSb, these states can be
associated with AFM1 (100/010) domains [37]. However, no features associated
with (001) domains have been observed in NdBi. In order to prove the presence
of (001) domains, we performed more ARPES measurements on NdBi. The results of
these measurements are shown in Fig. 5. These spectra do not demonstrate the
SS features observed in the previous studies, but they show a hybridization
gap at the $\Gamma$ point and the development of several additional SS
dispersions near the $\Gamma$ point. This makes this band structure similar to
the band structure of the (001) domain of NdSb. Also, as well in NdSb, the
additional SS, which are located in the gap, move up with the temperature
decreasing. Thus, we can associate this band structure with AFM(001) ordered
domain in NdBi.

Figure 5: AFM1 ordering in NdBi ((001) surface). Temperature dependence of
band structure along $\Gamma-X$ direction. TN = 24 K

##  IV Conclusions

We investigated the evolution of the electron structure in NdSb and NdBi upon
antiferromagnetic transition using ARPES and demonstrated the presence of
domains with different configurations of surface states. This result agrees
with the earlier neutron diffraction study [23], which also reported the
presence of three domain types in NdSb.

The development of a domain with magnetic moments oriented orthogonal to the
sample surface causes the formation of a hybridization gap at the $\Gamma$
point and several additional surface-state dispersions near the $\Gamma$
point. The development of a domain with magnetic moments oriented parallel to
the sample surface causes the formation of a different set of surface states,
including an unconventional surface-state pair [20] that forms an arc and an
elliptical pocket on the Fermi surface. Except for a Dirac-like feature in
(001) domain which remains unexplained, the observed electron structure of all
three domains in NdSb is in exceptional agreement with our DFT calculations
for AFM1 ordered phase.

###### Acknowledgements.

This work was supported by the U.S. Department of Energy, Office of Basic
Energy Sciences, Division of Materials Science and Engineering. Ames
Laboratory is operated for the U.S. Department of Energy by Iowa State
University under Contract No. DE-AC02-07CH11358. Crystal growth,
characterization and were supported by Center for the Advancement of
Topological Semimetals (CATS), an Energy Frontier Research Center funded by
the U.S. DOE, Office of Basic Energy Sciences.

Figure 6: NdSb: additional data. (a) Domain map measured using ARPES. Red,
green, and blue colors represent parts of the sample where one domain type.
Cyan, magenta, yellow, and wight colors represent parts of the sample where we
observed the superposition of signals from several domains.

##  Appendix A Reproducibility

Most of the data presented in the main part of the manuscript were collected
from one sample. The order in which data were collected is shown in Table I.
We see that after heating the sample up and cooling it down, the band
structure measured from the part of the sample where the B-domain was, did not
change. Moreover, the band structure measured from the part of the sample
where the A-domain was, did not change after three cycles of temperature
variation. Also, it shows that under the experimental conditions, the aging of
the sample surface is negligible: the map measured from the A-domain soon
after cleaving the sample (Fig. 1(c)) looks identical to the similar map
measured 24 hours later (Fig. 3(d)).

The mapping of another sample (Fig. 6(a)) has shown that it also has parts
where each one of the three domain types is dominant, as well as parts where
we observe the superposition of signals from several domains.

Time | Type | Domain | Temperature | Plot
---|---|---|---|---
(hours) |  |  |  |
0.9 | FS map | B | 5.7K | 1(c), 4(e,f)
2.6 | FS map | A | 5.7K | 1(c), 4(c,d)
4.3 | FS map | C | 5.7K | 1(c)
6.4 | spectrum | A | 6.5K | 2(e)
6.5 | spectrum | A | 9K | 2(e)
6.6 | spectrum | A | 10.5K | 2(e)
6.7 | spectrum | A | 12K | 2(e)
6.7 | spectrum | A | 13.5K | 2(e)
6.8 | spectrum | A | 15.5K | 2(e)
7.0 | spectrum | B | 10.5K | 4(i)
7.1 | spectrum | B | 13.5K | 4(i)
7.2 | spectrum | B | 15.5K | 4(i)
34.6 | FS map | A | 13K | 2(d)
36.4 | FS map | A | 11K | 2(d)
37.7 | FS map | A | 15.6K | 2(d)
38.5 | FS map | A | 7K | 2(d)
Table 1: The order in which data were collected.

##  Appendix B DFT calculations

All density functional theory [38, 39] (DFT) calculations with spin-orbit
coupling (SOC) were performed with the PBE [40] exchange-correlation
functional using a plane-wave basis set and projector augmented wave method
[41], as implemented in the Vienna Ab-initio Simulation Package (VASP) [42,
43]. Using maximally localized Wannier functions [44, 45], tight-binding
models were constructed to reproduce closely the band structure including SOC
within $E_{F}\pm$1eV with Nd s-d-f and and Sb p orbitals. The surface spectral
function and 2D Fermi surface (FS) were calculated with the surface Green’s
function methods [46, 47] as implemented in WannierTools [48]. In the DFT
calculations, we used a kinetic energy cutoff of 254 eV, a Gaussian smearing
of 0.05 eV, a $\Gamma$-centered Monkhorst-Pack [49] (11×11×8) k-point mesh for
the 1q tetragonal unit cell and a (8×8×8) k-point mesh for the 2q cubic unit
cell. For band structure calculations, we have used the experimental lattice
parameters of 6.336 $\AA$. To account for the strongly localized Nd 4f
orbitals in NdSb, an onsite Hubbard-like [50] U=6.3 eV and J=0.7 eV have been
used. Our DFT+U+SOC calculation on NdSb 1q and 2q gives a similar spin moment
of 2.7 $\mu_{B}$ and an orbital moment of 5.7 $\mu_{B}$ in the opposite
direction, resulting in a total magnetic moment of 3.0 $\mu_{B}$ on Nd.

Figure 7: DFT-calculated surface spectral function of NdSb AFM 1q on (a) (001)
and (c) (100)/(010) surfaces. The corresponding surface only contributions are
plotted in (b) and (d), respectively, to highlight the surface bands. Notably
on (001), the hybridization gap and the associated surface states around the
$\Gamma$ point as discussed in Fig.2 has an overall bulk band projection
background, in contrast to the unconventional surface state pairs away from
the $\Gamma$ point on (100)/(010) residing in the hybridization gap without
such a background.

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# Characterizations of $e^{\star}$-open sets and nearby open sets on Infra
topological spaces

G<EMAIL_ADDRESS>and D. Jaya
<EMAIL_ADDRESS>3331,2Department of Mathematics, Vel
Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology (Deemed
to be University), Avadi, Chennai-600062, India.

###### Abstract

The study of infra-topological spaces focuses on characterizations of
$e^{\star}$-open sets and nearby open sets in infra-topological spaces. The
$e^{\star}$-open sets, a variation of open sets, are explored for their unique
properties and relationships within the infra-topological framework.
Additionally, nearby open sets, which capture the notion of points being close
to each other, are investigated to provide a comprehensive understanding of
the topological structure. The research aims to contribute to the broader
field of topology by extending traditional concepts to infra-topological
spaces, offering new perspectives on openness and proximity. The findings not
only deepen our understanding of mathematical structures but also open avenues
for applications in various scientific and engineering disciplines.

Keywords and phrases: $e$-open set, $e^{\star}$-open sets, infra topological
spaces

AMS (2000) subject classification:54A05,54A10

##  1 Introduction

A seminal work by S. Mashhour et al. [mas] laid the foundation for supra
topological spaces, wherein they delved into the intricacies of $s$-continuous
functions and $s^{*}$-continuous functions. The groundbreaking exploration of
supra topological spaces marked a significant contribution to the field.

Building upon this foundation, Adel. M. Al. Odhari [ade] extended the
theoretical landscape by introducing and studying infra topological spaces.
Odhari’s work included a comprehensive examination of open sets within infra
topological spaces, shedding light on their fundamental properties.

E. Ekici, in a series of noteworthy contributions [eki1, eki2, eki3, eki4,
eki5], introduced and extensively investigated the properties of $e$ and
$e^{\star}$ along with nearby open sets in the context of general topological
spaces. Ekici’s research provided valuable insights into the behavior of these
sets, enriching the understanding of topological structures.

Motivated by these advancements, our present work focuses on extending the
study of $e$ and $e^{\star}$ open sets to infra topological structures. We
embark on an in-depth investigation, exploring the properties and
characteristics of these sets within the specific framework of infra
topological spaces. To bolster our findings, we present concrete examples that
not only validate our assumptions but also serve as illustrative instances of
the nuanced interplay between infra topological structures and the introduced
open sets. Through this research, we aim to contribute to the ongoing dialogue
surrounding the interplay of different topological spaces and the behavior of
distinct open sets within them.

##  2 Preliminaries

###### Definition 2.1.

Let $X$ be any arbitrary set. An infra topological space ($\mathcal{ITS}$) on
$X$ is a collection infra topology ($\mathcal{IT}$) of subsets of $X$ such
that the following axioms are satisfying:

  1. (i)

$\emptyset,~{}X\in\mathcal{IT},.$

  2. (ii)

The intersection of the elements of any subcollection of $\mathcal{IT}$ in
$X.$

i.e, If $O_{i}\in\mathcal{IT},~{}1\subseteq i\subseteq n,~{}\cap
O_{i}\in\mathcal{IT}.$

Terminology, the ordered pair $(X,\mathcal{IT})$ is called $\mathcal{ITS}$. we
simply say $X$ is a $\mathcal{ITS}$.