# A Sticker is Worth a Thousand Words: Characterizing the Use
of Stickers in WhatsApp Political Groups in Brazil

Philipe Melo 1, 2, João M. M. Couto 1, Daniel Kansaon 1, Vitor Mafra 1,
Julio C. S. Reis 2, Fabricio Benevenuto 1

###### Abstract

With the increasing use of smartphones, instant messaging platforms turned
into important communication tools. According to WhatsApp, more than 100
billion messages are sent each day on the app. Communication on these
platforms has allowed individuals to express themselves in other types of
media, rather than simple text, including audio, videos, images, and stickers.
Particularly, stickers are a new multimedia format that emerged with messaging
apps, promoting new forms of interactions among users, especially in the
Brazilian context, transcending their role as a mere form of humor to become a
key element in political strategy. In this regard, we investigate how stickers
are being used, unveiling unique characteristics that these media bring to
WhatsApp chats and the political use of this new media format. To achieve
that, we collected a large sample of messages from WhatsApp public political
discussion groups in Brazil and analyzed the sticker messages shared in this
context111Warning! This paper contains images and terms that may be
potentially offensive..

## Introduction

“A picture is worth a thousand words” is a popular expression that means that
complex and sometimes multiple ideas can be transmitted by a single still
image, which may convey its essence more effectively than a verbal
description. Similarly, stickers are a form of visual media, popular in
instant messaging services, through which users can express emotion, nonverbal
language, and complex ideas (Wang 2016). With the increasing importance of
online communication in our daily lives, especially after the COVID-19
pandemic, users seek new and more efficient ways to express themselves through
computers and smartphones, emphasizing the need for meaningful online
emotional connections (Liu and Tong 2020). In this scenario, sometimes a
simple text-based communication may not be enough to express every feeling we
might want to convey. Even with emoji characters in text, this may prevent
users from fully understanding the contextual subtlety and accuracy of the
conversation. Users, however, may overcome such limitations with the usage of
stickers (Lee et al. 2016), embodying some forms of interactions, gestures,
and symbols with this emerging multimedia messaging format.

When it comes to WhatsApp, they introduced stickers on the platform referring
to them as something to “help you share your feelings in a way that you cannot
always express with words”222https://blog.whatsapp.com/introducing-stickers.
WhatsApp allows users to create stickers of anything and for any occasion, and
them to be quickly incorporated into various contexts on the platform,
especially for political discussion, which is a popular topic on WhatsApp
(Resende et al. 2019b). This becomes even more relevant considering that
WhatsApp has recently gained attention due to campaigns that abuse this
environment by quickly spreading misinformation on its network (Melo et al.
2019b) and playing a central role in the dissemination of fake news during
political events (Bursztyn and Birnbaum 2019; Resende et al. 2019a).

In Brazil, stickers had a significant increase in popularity, transcending
their role as a mere form of humor to become a key element in political
strategy. Throughout the 2022 presidential Brazilian election, stickers
evolved into essential tools for disseminating political content across
diverse groups with the goal of proliferating, capturing a broad audience, and
circulating in groups that extended beyond the realm of fervent political
activism (Couto 2022). Such was their widespread, the Brazilian Electoral
Court responded by creating a sticker pack, aiming to guide people in fact-
checking news (TSE 2022). Moreover, since users are able to create their own
stickers on WhatsApp, this permissive model is also being abused by users to
create offensive and hateful stickers threatening other group members and
creating an unsafe environment for the users. Recently, even the tool
introduced by WhatsApp to automatically create stickers using AI was accused
of generating xenophobic stickers within the app (Bhuiyan 2023).

Although stickers became very popular among users, we have observed this media
being abused for a range of issues within WhatsApp, from misinformation
campaigns to hateful harassing content. However, we still lack a systematic
analysis of their usage on a large scale within the messaging platforms
ecosystem due to the closed architecture of instant messaging platforms. In
this study, we put forth a systematic characterization of sticker usage on
WhatsApp, focusing specifically on their role within political public groups
during the 2022 Brazilian elections. Our analysis aims to address the
following research questions:

  * •

RQ1: Do users use stickers the same way they use other media formats on
WhatsApp?;

  * •

RQ2: What are the characteristics of stickers shared on WhatsApp political
groups in Brazil?

  * •

RQ3: How users are abusing of stickers to spread offensive content through
WhatsApp?

To answer these questions, we collect a large-scale set of messages sent to a
thousand political public groups on WhatsApp during five months of the
Brazilian election period in 2022 (i.e., June to October). In total, we
collected more than 6M messages, from which 650k are sticker messages. We also
process distinct stickers, counting 57,031 unique media. In our initial
efforts, we uncovered key insights indicating intriguing avenues for
exploration in this content. Despite the greater prevalence of image and text
messages, stickers are also extremely popular on WhatsApp. Our initial results
suggest that stickers not only possess a distinct structure compared to other
media formats but also reflect varied usage patterns among users. Furthermore,
we discovered evidence of offensive and provocative stickers shared on public
groups on WhatsApp, especially during the electoral period when public groups
from opposing political parties suffered several attacks from users who
invaded the group and spread hate speech.

## Background and Related Work

The use of mobile instant messaging applications became popular has been
changing the way users communicate online. Compared to the traditional short
messaging service (SMS), instant messaging offers a more social,
conversational, and informal communication environment (Church and de Oliveira
2013). Therefore, smartphone users felt closer to their correspondents through
the exchange of messages (Zhou, Hentschel, and Kumar 2017). This change in the
way people connect also came with more complex ways of expressing, other than
emoticons and emojis, influencing the emergency of stickers.

Stickers are usually described as emoticons in the form of colored images
(Chen, Freeman, and Balakrishnan 2019), as both can be used as visual reaction
messages or determine the feeling of the interlocutor during the conversation,
but differ from in-line emoticons and emojis in diversity, complexity, and
usage (Lee et al. 2016), providing a richer communication (Wang 2016). Also,
they differ from simple static images as they can be short animated movies,
similar to a GIF. Stickers are also closely related to memes (Wang et al.
2019), which often present a combination of a picture and a statement,
typically with sarcastic or humorous intention (Davison 2012). Usually, there
is a template-based image that people modify, edit, and publish their version
while keeping some key aspects so that the meme template is still
recognizable. Both memes and stickers can be created based on personal
experiences, but inspiration can be obtained from popular cultural products
such as television shows and video games (Ge 2020). Additionally, users
publish memes on the Web to create an incentive for others to share by
replicating the original (Ling et al. 2021).

The origin of these stickers dates back to 2011 with LINE in 2011, a Korean
instant massaging application that got popular in Japan, mixing cartoons and
“emojis”(Russell 2013). This tool allowed users to express social intention,
identities, context, and social-cultural differences in a more specific way,
inspiring other applications to adopt stickers as well. In 2013, Facebook
added stickers to its platform, bringing the idea to the western market, but
still with a limited set of precreated images. This media was adopted by
WhatsApp in 2018, nine years after the creation of this app, granting its
users the opportunity to create their stickers on the platform. Due to its
broad variety of images, stickers allowed users to be more self-represented
while also increasing sympathy and social presence (Lee et al. 2016),
therefore becoming a relevant tool to express opinions and feelings through
the app.

While there are some efforts dedicated to the study of memes (Ling et al.
2021; Zannettou et al. 2018), few are focused on stickers. These mostly
attempt to understand why and how people use stickers on messaging platforms.
In (Lee et al. 2016) the authors conducted a qualitative approach with semi-
structured interviews to understand sticker usage patterns, observing
individual and social characteristics of emoticon usage to reinterpret the
meaning of emoticons in instant messages. They suggest that, in addition to
using stickers to express emotions, users may also have other strategic and
functional purposes. Zhou et al. (Zhou, Hentschel, and Kumar 2017) also
present a qualitative study of mobile messaging, focusing on the rapid
proliferation of emojis and stickers and the decrease in dependence on text
within the platform.

Given the importance of stickers in messaging services and their usage as
reaction or responses, some studies go in the direction of predicting a
sticker as a reply to a message in a conversation. In (Gao et al. 2020), the
authors trained a deep learning model to respond to messages with the most
suitable sticker for the situation in conversation, and (Laddha et al. 2020)
explores the sticker recommendation problem in two steps, the first one being
predicting the message that the user is likely to send in the chat and the
second one being substituting the predicted message with an appropriate
sticker. Furthermore, the study presented in (Al-Maroof et al. 2020) explores
the social and cognitive factors that affect the use of stickers on WhatsApp,
creating a unified model for the usage and acceptance of stickers in social
media messaging.

Stickers can help users define their intentions or feelings in messages, but
their complexity and placement constraints can also result in
miscommunication. For instance, (Cha et al. 2018) explores the sticker
misinterpretations in instant messaging platforms, showing how people perceive
emotion in stickers as well as how sticker miscommunication occurs in actual
chat contexts. Most of the issues occurred due to stickers’ ambiguous
(multiple) facial/bodily expressions and different perceptions of dynamism in
gestures. In real chats, there were also contextual misinterpretations, that
senders and receivers differently interpreted the representation of stickers
and the corresponding textual messages. Another facet of miscommunication is
studied by (Tang et al. 2021). They analyze the outcomes of sticker usage in
authentic online communications, finding differences between the sender’s
intention and the receiver’s interpretation on 34.7% of the stickers, although
they say that these disparities did not adversely affect communication.

It is interesting to observe this miscommunication in sticker-based
conversations, especially in the WhatsApp ecosystem, in which misinformation
is also an issue on the platform (Resende et al. 2019b; Bursztyn and Birnbaum
2019). The app is pointed as a pivot in the spreading of misinformation
campaigns during political elections in Brazil and India (Melo et al. 2019b).
Studies on WhatsApp show that the contents of text (Resende et al. 2019a;
Machado et al. 2019), image (Resende et al. 2019b; Garimella and Eckles 2020)
and audio (Maros, Almeida, and Vasconcelos 2021; El-Masri, Riedl, and Woolley
2022) are used to transmit false stories on the platform, leading to the use
of systems to monitor these contents(Melo et al. 2019a), evidencing a
multimodal trait of misinformation within WhatsApp.

Memes are not only used for entertainment, but also carry more in-depth
meanings, such as political opinions (Koetse 2015). In their work, (Zannettou
et al. 2018) found a substantial number of politics-related memes on both
mainstream and fringe Web communities, supporting media reports that memes
might be used to promote or harm politicians. They also observed that Internet
memes are increasingly used to sway and manipulate public opinion, and this
prompts the need to study their propagation, evolution, and influence across
the Web. Given the parallels between memes and stickers, including that the
latter has an even higher emotional link with users, and knowing the important
role of WhatsApp in political scenarios, we also need to understand the
implications of the usage of this multimedia format within the messaging
platforms and its relationship with partisan context.

While many qualitative studies investigate the intention of users in sending
stickers, providing meaningful characteristics of this medium and key insights
about what they are, we lack a more systematic characterization of this new
media format, about how they appear within chat structures, their political
implications, and, moreover, the offensive aspect with which they can threaten
vulnerable groups. In this work, we fill this research gap by providing a
large-scale analysis of the use of stickers in WhatsApp political groups,
shedding light on the contribution that stickers have in the architecture of
our current digital communication.

## Methodology

Next, we briefly describe out methodology to gather stickers from WhatsApp and
process them, as well as some potential limitations of our strategy.

### Data Collection

WhatsApp has more than 2 billion users and more than 100 billion messages are
exchanged every day333https://twitter.com/WhatsApp/status/1364714386078621703.
Given the closed architecture of the app, this data is mostly covered by end-
to-end encryption in private chats. However, there is a large number of public
WhatsApp groups shared online from which data can be collected. Through an
invite in the form of a URL, any user can join a group and participate in
public chats with hundreds of people, whose topics include, but are not
limited to, politics, sports, work, and entertainment. Therefore, an effective
and popular practice, then, is to get access to this kind of data by joining
these public groups to collect messages shared within them (Garimella and
Tyson 2018; Resende et al. 2019b; Melo et al. 2019a; Bursztyn and Birnbaum
2019; Reis et al. 2020). In this work, we follow a similar methodology by
selecting public groups dedicated to political discussions shared on social
networks in Brazil and collecting messages exchanged within these chats during
the four months of the Brazilian electoral campaign period, dating back from
June of 2022 and up to September of the same year. For each message, we
collect the following data: (i) the user ID, (ii) the ID of the group in which
the message was posted, (iii) the timestamp, (iv) whether the message was
labeled “forwarded” or not, (v) the text of the message, (vi) the media type
of the message (i.e. sticker, image, audio, video or document) and, when
available, (vii) the actual attached multimedia file. In Table 1 we show the
summary of the data collected. In total, we collected more than 6M WhatsApp
messages, of which 659K were stickers sent in 1,063 different groups by more
than 64k users.

Table 1: Summary of data collected from WhatsApp.

| Number of
---
Messages
| Number of
---
Stickers
| Unique
---
Stickers
| Number of
---
Groups
| Number
---
of Users
6,039,760 | 659,576 (10.9%) | 57,031 | 1,063 | 64,227

### Processing Sticker Data

After collecting the data, we filter messages by media type, selecting all
stickers sent in the chats. For each individual sticker, we also processed a
visual hash for the image file related to it444Note that, for animated
stickers, we convert them to a static one using the first frame so that it can
be processed as an image. For this step, we use the pHash algorithm, which
takes as input the sticker file and whose output is a 64bit-integer hash,
calculated using a discrete cosine transform that extracts a specific
“fingerprint” to each file (Zauner 2010). As perceptual hashes are constructed
in a way that small variations in the pixels of the input image will result in
similar hashes (opposing to crypt hashes, in which a single byte difference
will drastically change the final hash), this process allows us to visually
compare two similar images (Zannettou et al. 2018). By processing all 659K
sticker messages, we found 57,031 unique stickers (i.e., with exactly the same
hash). For each sticker, we also extract an attribute of its dominant color by
grabbing the main color from the palette of that
image555https://pypi.org/project/colorthief/. After that, we represent each
sticker file as a combination of its hash and color.

We use this representation of the images to cluster them together and obtain
insights that emerge from the clusters. A similar methodology was used to
study memes from Web communities (Zannettou et al. 2018), in which annotated
images were mapped to their corresponding clusters. Using the DBSCAN
algorithm, we can provide an investigation on the popularity and diversity of
stickers on WhatsApp and how users make use of this form of media.

Besides the visual appearance of stickers, we also assess them by the context
in which they are used. For that, we grouped stickers based on the group
network connection in which they were sent by building a graph of the stickers
sent in the same group. We also labeled the political groups whose data were
collected as right- or left-wing alignments to investigate the relationship
between the graph topology and the political alignment of their nodes.

Finally, we analyzed the abusive use of stickers on WhatsApp. We use the NSFW
Yahoo model(Mahadeokar and Pesavento 2016) to measure information of how
“explicit” each sticker is. We also use the SafeSearch detection with Google
Vision API666https://cloud.google.com/vision/docs/detecting-safe-search to get
complementary data about potentially harmful content being posted on WhatsApp
through the stickers.

Data Limitations. It is known that, at least for WhatsApp, most of the
conversations occur in private chats. Therefore, this data collection may be
not representative of the entire platform group dynamic, especially
considering that we observe only public groups whose topics are dedicated to
political discussion. Nevertheless, this methodology collects a large volume
of data from the app Since political groups are particularly inclined to
massively share content on WhatsApp, they can bring meaningful insights for
the discussion. It is important to highlight, however, that results refer only
to content circulating on the public layer of the platform and within the
context of politics, so they may not hold true for the entire network. Despite
this limitation, our efforts aim to shed light on important aspects that can
support others towards understanding how the WhatsApp platform is being
abused, since some of these aspects would be otherwise hidden behind the
closed nature of the platform’s architecture.

Ethical considerations. Unlike other approaches, this work does not involve
direct experiments with human subjects. Regarding the data analyzed, we
collected messages only from publicly accessible groups on WhatsApp and in a
public political context. In addition, we look exclusively at stickers sent
within those chats, and all sensitive information (i.e. usernames, phone
numbers, and further text messages) has been previously anonymized to ensure
the privacy of users. Last, we are providing a large-scale study regarding the
use of stickers in WhatsApp political groups presenting the results in
aggregated way.

## Stickers as a distinctive form of media

Multimedia messages are very popular on WhatsApp (Resende et al. 2019b). Even
though text messages still make up the majority of the data, users often share
different kinds of media using images, videos, audio, and stickers. Although
images and stickers are very similar (both are visual media displayed in the
middle of the chat) some key changes distinguish them. Stickers may show an
animated image on chat, they are smaller than actual images, they are stored
in a different format by the app and they are usually displayed directly to
users on chat (as emojis) while images need do be downloaded. These
characteristics, however, refer to the structure of the sticker media and not
about their usage. We do not know whether the thought process employed by
users when sending a sticker is similar or not to other kinds of media. In
this section, we evaluate some aspects of the stickers to investigate how
members of political groups use this medium.

In Figure 1, we compare stickers with images and texts regarding the volume of
media sent per group and per user. In the cumulative distribution function
(CDF) of the messages per group (Fig. 1(a)), we see that sticker messages were
less popular in the groups than images. About 30% of the groups did not send
any stickers in the chat; images were absent in 20% of them. Moreover, 60% of
groups sent a maximum of 100 stickers, while 40% of the groups sent less than
that. Still, we observe that some groups have more than 10k stickers and
images. In the distribution of messages per user (Fig. 1(b)), we have similar
curves, with stickers being the least common medium. Here, even more users do
not send any stickers (75%), but we also have users who individually sent
around 10k stickers. Hence, we can observe that, as expected in a messaging
app, text format is much more used than other types of media. However, it is
interesting to see that users prefer to send images than stickers on the
groups, even though both media are widely spread on WhatsApp, especially when
considering that stickers are generally easier to send, as they do not need to
select a file from the gallery and are more incorporated into the WhatsApp
interface.

(a) Groups

(b) Users

Figure 1: Cumulative Distribution Function (CDF) of stickers sent per groups
and users compared to image and text.

(a) Shares

(b) Forwards

Figure 2: Cumulative Distribution Function (CDF) of total shares and
forwarding per sticker and image medias.

As we processed and merged stickers, we measured the number of shares each
sticker has within our dataset. In Figure 2(a) we evaluate the distribution of
total shares per media by comparing images and stickers. Even though we
observed more image messages in the data, when looking at the total shares per
unique media, we note that stickers are more shared than images. About half of
the stickers appeared more than once, many of which were shared more than a
thousand times. In contrast, about 75% of images have only a single
appearance.

Another attribute we obtained for each message is whether it was forwarded or
not. Forwarding is an important tool in WhatsApp communication, in which users
share content with their contact (Melo et al. 2024). When available, this info
allows us to determine how many messages were forwarded for each piece of
media in our dataset. By analyzing this quantity, we found that images are
much more frequently forwarded than stickers. Almost none of the stickers
collected were found in forwarded messages, meaning that nearly all of them
were sent directly from the associated users to a group. On the other hand,
for images, we observe that 40% have been shared as a forward at least once.
We can see that stickers not only have a different structure when compared to
images, but they also present a different usage by the users. In general,
there are fewer stickers than other media in groups, and users also send fewer
stickers. However, they are individually shared more often and directly by
user within chat. A possible explanation for this is that stickers can have a
collectible tone (Zhou, Hentschel, and Kumar 2017), in which users prefer to
save and keep a collection to have at their disposal a range of options to use
in a specific situation or as a self-disclosure reaction in the chat (Wang
2016) instead of just forward a piece of content as they do for images. This
implies that a sticker can be reproduced more than once in different
scenarios, while images are usually more concise and portray a specific piece
of information.

## Stickers Characterization

In this section, we present different forms of grouping stickers, including by
visual resemblance using the pHash and by context through building the graph
of stickers according to the groups in which they were posted.

Figure 3: UMAP visualization of all stickers from WhatsApp dataset.

A visual representation of all stickers collected is shown in Figure 3. We
create a depiction by using both the pHash binary vector and each sticker’s
dominant color. Then, we employ dimensionality reduction via UMAP to plot the
sticker representations in a 2D space. Each point is associated with a single
sticker’s representation and its color is the actual sticker’s dominant color.
It is visible that there is a concentration of red and also green and blue
stickers, which are related to the political context we monitored, and many
stickers within the data are directly related to political campaigns for the
2022 Brazilian elections. The leading left-wing party in Brazil employs red
colors associated with the party’s branding, while the leading nationalist
right-wing party mostly uses green, blue, and yellow, colors derived from the
Brazilian flag. The stickers in our dataset reflect the partisan nature of the
branding of the competing parties. A thorough manual investigation of the
yellow chunk of stickers revealed presence of many “emoji” stickers, which are
faces drawn in emoji style representing existing emoji characters. The grayish
area in the plot encompasses many different sticker styles and we do not could
point to a single characteristic of them.

### Clustering by Visual Similarity

To explore the similarity of these stickers, we leverage features of images to
cluster them together and investigate whether there exist patterns and
characteristics to be observed within stickers on WhatsApp. A similar
methodology was employed for the study of memes in Web communities (Zannettou
et al. 2018), annotating and mapping them to clusters. Since the perceptual
hash (pHash) offers a comparable fingerprint of stickers, we can use it to
cluster our stickers through the density-based spatial clustering of
applications with noise (DBSCAN) algorithm, merging them in visually similar
groups of content shared on WhatsApp. Using DBSCAN enables the investigation
of the popularity and diversity of stickers found on WhatsApp and insights
into how users may use this form of media.

Among the ensuing clusters, we find pHash-grouped stickers that represent
variations of a base image, emoji, or meme. Some of the resulting clusters
demonstrate a key aspect of stickers: their similar use as a meme and also as
emojis, as shown in Figures 4 and 5. Emojis are text characters that refer
directly to pictorial representations of feelings (often in the form of round
yellow faces) and other objects (Zhou, Hentschel, and Kumar 2017). There are
clusters with elaborated versions of those characters, occasionally even as
animated stickers. These are usually employed as reactions to other messages,
in a similar fashion emojis appear in text messages (Wang et al. 2019). The
clusters associated with meme templates put together analogous images that
share a unique base template, but are differentiated by slight changes in text
or detail to instantiate a joke or a funny situation, but are still recognized
as the same meme (Ge 2020). This pattern is also observed in memes, in which
popular standardized comical images are widely spread across the Internet.
This behavior of creating, remixing, transforming, and editing variations of
an image is a common practice in meme distribution on the Web (Lee et al.
2016; Ling et al. 2021; Davison 2012)

Figure 4: Example of a cluster of stickers representing emojis.

Figure 5: Example of the cluster with meme sticker template with small
variations.

It is interesting to note that within certain clusters, very visually similar
stickers may also portray opposing ideas. In the context of the public
political groups encompassed in this work, many clusters are related to the
poll-leading politicians in the Brazilian 2022 elections and contain
advertising/provocative material associated with their campaigns. Here, we can
find examples of corresponding sticker images that carry drastically opposing
partisan content. Figure 6 offers an example of a cluster that grouped two
visually analogous stickers but which are shared in different contexts. The
stickers portray an edited image of opposing candidates of the left and right-
wing parties in a criminal mugshot, suggesting that stickers are coopted by
adversary groups and used with opposing semantic but adopting with the same
visual aesthetics.

These results alert us to the implications of grouping stickers exclusively
based on their visual appearance. Small details oftentimes imply drastic
changes in the semantic value of the sticker.

Figure 6: Examples of visually similar stickers used by opposing political
leanings.

### Political Alignment of Stickers

As exemplified in the previous section, even though stickers can be very
visually similar, they might contain drastically different ideas and thus be
used in very distinct scenarios. To evaluate the usage of stickers from this
perspective, we built a network of stickers based on the groups in which they
were shared. Figure 7 offers a graph visualization of this network assembled
with the most popular stickers in our WhatsApp data. There, each node is a
sticker, and they are linked with an edge if both appear in the same group, we
also applied a community detection algorithm based on modularity optimization
(Blondel et al. 2008) to extract the communities of stickers in this network.
For visualization clarity, we used only stickers shared more than 100 times in
the observed groups and edges with at least 10 groups in common.

Figure 7: Sticker network.

Even with a very dense network (density measured at 0.698), we could identify
two main larger communities and a third smaller one. These two main
communities reflect the polarized nature of our data and have very analogous
stickers from both political leaning. There were two major candidates for the
Brazilian 2022 presidential elections (Bolsonaro, current right-wing Brazilian
president, and Lula, former leftist president of Brazil). The majority of the
stickers found on the monitored public groups, therefore, represent these
sides of this dispute. Some stickers are symbols, numbers, and colors
associated with each political party, while others depict their candidate if
choice as a superhero or sporting items associated with the elected president
of Brazil. Numerous stickers aim to criticize or enrage members of the
opposing political leaning. Together, these sticker archetypes form a rather
unique and connected political community. The smaller third community,
however, presents a more diverse type of content, including some with sexually
explicit nature.

To evaluate the partisan leaning of stickers, we label each monitored group as
left-wing, right-wing, or undefined, following the political stance declared
by the group’s name and, if necessary, by their description. The annotation
shows that most groups support the right-wing candidate Bolsonaro, with 666
groups, whereas just 126 groups support a left-wing agenda. The remaining 617
groups were tagged as undefined as it was not possible to determine their
alignment (e.g. groups of support of a different political leaning or for
general political debate).

Next, we examine each sticker to determine its political leaning based on the
groups that posted them. Then, taking into account the label of these groups,
we calculate the proportion associated with each political alignment. Finally,
we assigned a political alignment label for the stickers according to the side
of the political spectrum that had the highest portion of messages. This label
indicates if the sticker is mostly used in the context of right- or left-wing
groups. Through this process, we characterize the political nature of stickers
shared through Brazilian WhatsApp groups. Note that certain stickers might be
equally used by both sides of the political spectrum, or also predominantly by
groups that are not clearly defined politically. To prevent mislabeling in
such cases, we introduced a third category called “undefined”, used for
stickers that do not have a 60% majority on either side. This approach allows
us to better identify stickers that are closely linked to political
discussions. Through this annotation, we found that 18k stickers are biased
towards usage in right-wing groups and 6k towards the left, as shown in Table
2. Note that the difference could be explained by the fact that, in Brazil,
the right-wing movement is seen to be actively mobilized on the WhatsApp
platform (Resende et al. 2019b; Bursztyn and Birnbaum 2019).

Table 2: Political alignment of groups and sticker context. | Right | Left | Undefined
---|---|---|---
Total Groups | 666 | 126 | 617
Total Stickers | 18,158 | 6,704 | 32,169

Given the potential for stickers to convey strong partisan sentiments, our
investigation delves into the hypothesis that political stickers serve as
tools for provocation or even direct attacks against opposing political
groups. The public nature of the monitored groups allows individuals from
adversarial political positions to join, establishing a channel for launching
targeted assaults against individuals aligned with opposing ideologies.
Leveraging the ease of sticker sharing with just a single click, these
initiatives are particularly conducive to flooding group chats with divisive
content. To examine this phenomenon, we explore whether stickers predominantly
associated with a specific political alignment also make appearances within
groups representing the opposing political spectrum.

Our analysis reveals that the majority of politically annotated stickers were,
in fact, exclusively utilized within their corresponding ideological circles.
However, we also identified instances where political stickers transcended
partisan boundaries, being shared in groups with opposing political
affiliations. Employing our criteria, we identified 869 partisan stickers,
with a notable majority (794) exhibiting right-wing bias and being
disseminated within leftist groups. In contrast, only 75 left-wing biased
stickers were observed in right-wing groups.

Figure 8 showcases examples of these “politically inconsistent” stickers,
which promote a partisan candidate in rival environments. The presence of such
stickers in politically opposing groups suggests a deliberate act of
confrontation by users, trying to cause reactions and distress among
individuals holding contrasting political views. For instance, Figures 8,8,8,8
depict stickers featuring the right-wing presidential candidate Bolsonaro,
perceived as provocative within leftist groups. One sticker even includes the
text “Leftists will infarct with hate”, tailored to instigate distress.
Similarly, leftist stickers depicted in Figures 8 and 8 endorse Lula and are
incongruous with right-wing ideologies. Of particular concern are images such
as those in Figure 8, depicting Bolsonaro wielding a firearm, and Figure 8,
featuring heavy boots crushing a red star with the words “This is our fight.
Communism, not here!” These images evoke menacing connotations, symbolizing a
dominance of right-wing ideologies over leftist counterparts, thereby
exacerbating tensions within a polarized online political discourse.

Figure 8: Stickers used to “provoke” opposing political groups on WhatsApp.

Our findings reveal that clustering stickers based on their image pHashes,
which denotes visual similarity, enables the identification of image sets
exhibiting even minor variations. However, this may not capture the subtle
political context carried with them and may also inadvertently put together
stickers with opposite political messages. On the other hand, examining
stickers through the perspective of the groups in which they are shared, and
thus their political alignment, exhibits a strong association with the
contextual environment rather than solely relying on visual resemblances of
the images. This highlights the importance of considering the broader socio-
political context in understanding sticker usage patterns.

Additionally, we observed that stickers often demonstrate remarkably partisan
alignments and are frequently utilized as tools for provocation and abuse
within political public groups on WhatsApp. This highlights the pivotal role
of stickers as a media for expressing and perpetuating political ideologies
and tensions within this network.

Moreover, the asymmetric distribution of stickers aligned with right-wing
ideologies may contribute to the reinforcement of echo chambers and
polarization within WhatsApp groups. Users who predominantly encounter content
aligned with their own political beliefs may become further entrenched in
their viewpoints, hindering constructive dialogue and fostering division
within online communities. Also, the mechanism of sharing biased stickers
within WhatsApp raises questions about the dynamics of political engagement
and manipulation on social media platforms. The deliberate dissemination of
partisan content, particularly in the form of stickers designed to provoke or
attack opposing political groups, shows a strategic usage of digital media for
political propaganda and agitation.

In conclusion, the prevalence of right-wing biased stickers in our dataset
shed light on the complex interplay between digital media, political
discourse, and ideological polarization on WhatsApp in Brazil. Understanding
this ecosystem is essential for promoting informed and inclusive dialogue
within online communities and mitigating the risks of manipulation and
division.

## Abusive Stickers on WhatsApp

While political attacks are one domain through which stickers can be abusive
(Kansaon et al. 2024), there are also stickers disseminating offensive and
inappropriate content on WhatsApp. In this section, we delve into these
pathways, shedding light on the various forms of abusive behavior facilitated
by stickers on the platform.

On WhatsApp, users can freely create their own stickers based on any image
they want. However, this model is not unanimous among all messaging apps. Most
of them have only predetermined and limited sets of stickers that users can
use during conversations. Facebook and its messaging system “Messenger” does
not allow all users to add their own creations as stickers (Facebook 2022).
The mobile messaging app LINE even sells millions of curated sticker packs per
month, and stickers have long been one of LINE’s key revenue drivers (Russell
2016). Despite Meta restricting stickers on its other platforms and selling
stickers can be a highly profitable economic model, for WhatsApp particularly,
Meta chose to let users freely create and share their own stickers. However,
this permissive model comes with some challenges, as it opens the door for the
dissemination of offensive and inappropriate content.

WhatsApp’s terms of service, however, determine that stickers created must be
legal, authorized, and acceptable images. Furthermore, users are not allowed
to use WhatsApp services in ways “that are obscene, defamatory, threatening,
intimidating, harassing, hateful, racially or ethnically offensive, or
instigate or encourage conduct that would be illegal, such as promoting
violent crimes”777https://www.whatsapp.com/legal/terms-of-service. Although
WhatsApp’s TOS does not allow for offensive stickers, we find clear examples
of stickers that do not comply with these rules. Figure 9 shows unacceptable
stickers found within our WhatsApp dataset that are highly offensive towards
specific groups and violate the terms of use of WhatsApp. Sticker˜9 is an
example of a homophobic image against LGBTQIA+ people; Sticker˜9 depicts a
swastika, a symbol amply associated with the former Nazi party; Sticker 9
depicts a black male holding a knife and the phrase “around blacks never
relax”, which is a recurring racist remark on the Web (Froese 2015).
Similarly, Sticker 9 portrays a derogatory illustration of a Jewish man based
on antisemitic characterizations of Jewish people, which is also a widely
known hate symbol (ADL 2022). These examples demonstrate that stickers,
uploaded to and freely shared between users on WhatsApp, can be used as media
for harassment and targeted attacks toward vulnerable and marginalized people.

Figure 9: Example of offensive stickers sent by users that violate terms of
use of WhatsApp.

In our data, we also found a considerable presence of stickers with sexually
explicit content. To assess this, we apply the convolutional neural network
model for Not Suitable for Work (NSFW) proposed by Yahoo Inc. (Mahadeokar and
Pesavento 2016) to identify adult themed stickers. Images with a score greater
than 80% are labeled as explicit. Figure 10(a) presents the volume of NSFW
stickers. Compared to images, stickers are five times more likely to depict
explicit content (0.5% of images and 2.3% of stickers are NSFW). In total, we
discovered 33,335 messages containing NSFW stickers, accounting for 5.5% of
all sticker messages. Notably, in Figure 10(b), we see that approximately 10%
of users from the monitored groups shared a NSFW sticker, with some
individuals posting hundreds of NSFW content. There is even a peculiar user
who shared alone more than 3000 NSFW stickers. These results corroborate with
the idea of abusive behavior on WhatsApp, especially considering the nature of
the public political groups monitored are usually not NSFW (some of them even
state rules in the description forbidding adult content on that chat).

(a) NSFW Score

(b) NSFW per Users

Figure 10: Distribution of NSFW stickers sent. Table 3: “Not Safe” stickers
categories on WhatsApp.

| Unique Stickers | Number of Shares
---|---|---
Yahoo_NSFW | 1,322 (2.3%) | 33,335 (5.05%)
Adult | 1,423 (2.5%) | 30,828 (4.6%)
Violence | 416 (0.7%) | 4,583 (0.7%)
Racy | 4,503 (7.9%) | 55,476 (8.4%)
Medical | 735 (1.3%) | 14,300 (2.1%)
Spoof | 17,834 (31.2%) | 150,351 (22.7%)

Furthermore, we evaluate other categories of NSFW content through the Safe
Search detection from Google’s Vision API888https://cloud.google.com/vision.
It detects content within an image across five categories (adult, spoof,
medical, violence, and racy) and returns the likelihood for each one of them.
According to the API, Adult content encompasses elements such as nudity,
pornographic images, cartoons, and activities of a sexual nature. The medical
category accounts for the likelihood that the content is associated with
medical imagery (e.g. wounds, surgeries, blood). Violence offers an estimation
of the probability that the image portrays a violent action. Spoof is the
likelihood that a modification was made to the image’s canonical version to
make it appear funny or offensive. Content deemed racy may include (but is not
limited to) skimpy clothing and strategically covered nudity, lewd,
provocative poses, and extreme close-ups of sensitive body areas. In Table 3
we aggregated the results of Safe Search detection on our data collection of
stickers considering the likely and very likely labels for each category.

These findings underscore the urgent need for greater moderation and
enforcement of content guidelines within WhatsApp’s sticker ecosystem. Unlike
images, which users can choose not to automatically download, stickers are
directly displayed to users in WhatsApp chats, making them susceptible to
abuse by malicious actors. While stickers can enhance user engagement and
communication (Wang 2016), their misuse brings significant risks, including
promoting hate speech, perpetuating harmful stereotypes, and creating unsafe
environments within online communities.

## Conclusion

Stickers are an emerging media format that has attained massive popularity on
instant message platforms. With digital communication more present in our
daily lives than ever, users demand more effective ways to express feelings
and specific concepts within virtual chats. In this context, stickers
represent a welcome and rapidly adopted media to extend the expressiveness of
online conversation. Despite their widespread popularity, our understanding of
sticker usage remains limited, primarily due to the closed and private nature
of messaging app architectures. The opaque nature of these platforms hinders
the comprehensive analysis of sticker and their implications for online
discourse.

In this work, we analyze 659K stickers messages from WhatsApp political public
groups comprising a total of 57,031 unique stickers shared during four months
of the 2022 presidential Brazilian elections and offer insights into how
stickers are employed in this specific domain. Our findings reveal some
characteristics that suggest stickers are indeed a distinctive media format
but share similarities with others: stickers are less frequently sent compared
to other media types, such as images within groups. However, a single sticker
usually presents higher total shares, which means they exhibit a higher rate
of recurrence within conversations, similar to an emoji. Moreover, stickers
are rarely forwarded, suggesting that users often save and keep their
collection of preferred stickers for future use instead of using sharing tools
of the system. This behavior highlights the collectible nature of stickers, in
which individuals accumulate a personalized repertoire of stickers to deploy
on various occasions like a meme.

Furthermore, our analysis investigates two distinct methodologies for
categorizing stickers, each offering unique insights into their usage
patterns. Initially, we employ a visual similarity approach, clustering
stickers based on perceptual hashing – pHash. This method reveals parallels of
stickers with the usage of emojis and memes, showing how stickers can present
minor variations while keeping the same essence of a base image template.
However, we also uncover instances where visually similar stickers convey
entirely different meanings, emphasizing the importance of considering other
aspects when grouping images by perceptual hashes. Particularly in political
groups, where misinformation dissemination is a concerning issue, it is
important to avoid merging distinct ideas within the same category, especially
when minor alterations in images could propagate misleading narratives.

Additionally, our second approach includes building a network of stickers
based on their co-occurrence in public groups. This network analysis resulted
in two prominent communities of stickers, mirroring the polarized partisan
landscape of our political WhatsApp dataset. Subsequently, by labeling
stickers according to their political alignment, we find a strong association
between stickers and political affiliations. Notably, we observe instances
where political stickers are shared across ideologically opposing groups,
often serving as tools for provocation and inciting discord.

Moreover, our findings reveal the urgent need for better moderation mechanisms
to curb the dissemination of offensive stickers and safeguard users from
encountering inappropriate content on WhatsApp. Our analysis of the dataset
reveals a notable prevalence of potentially offensive sticker messages,
including content perpetuating homophobia, featuring hate symbols like the
swastika, perpetrating racist stereotypes, and depicting derogatory
illustrations of marginalized communities. We also discovered that stickers
are five times more likely to depict explicit (NSFW) content compared to
images; which represents 2.3% of unique stickers and 5% of all sticker
messages containing sexually explicit content. Furthermore, we identified a
substantial volume of sticker messages featuring violent (0.7%), medical
(2.1%), and racy (8.4%) content, which may be highly sensitive for certain
audiences. Given the visual immediacy of stickers within chat interfaces, the
unrestricted access to public groups, and the ease with which users can create
and disseminate stickers from virtually any image, there are significant risks
of offensive content spreading unchecked due to lack of moderation.

In conclusion, stickers have transformed communication on instant messaging
platforms by offering users a nuanced way to express emotions and ideas. While
they bring novelty to online chats, their impact on platform dynamics is
largely unexplored. Despite WhatsApp’s predominantly private and encrypted
nature, our research provides valuable insights into how stickers are utilized
within public political groups, bringing more transparency to the platform. We
discover key differences that make a distinction between stickers and other
more traditional formats. and analyzed the implications of the usage of
stickers within political groups and their partisan nature. Finally, we
identify a substantial abuse of stickers with offensive content on WhatsApp,
which requires actions to bring more safety to users.

As future work, we aim to investigate the use of stickers across different
scenarios and subjects within WhatsApp. We also intend to further explore the
prevalence and impact of hate stickers, seeking to identify patterns and
strategies to mitigate their dissemination on the platform. Additionally, we
plan to expand our analysis to other messaging platforms such as Telegram,
broadening our understanding of sticker usage in different contexts and
environments. By continuing to explore and address the complexities of
stickers, we can contribute to building a safer and more informed online
environment for users worldwide.

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# Quantum phases of spin-1 system on 3/4 and 3/5 skewed ladders

Sambunath Das<EMAIL_ADDRESS>Solid State and Structural Chemistry
Unit, Indian Institute of Science, Bangalore 560012, India  Dayasindhu Dey
<EMAIL_ADDRESS>Solid State and Structural Chemistry Unit, Indian
Institute of Science, Bangalore 560012, India  S. Ramasesha
<EMAIL_ADDRESS>Solid State and Structural Chemistry Unit, Indian
Institute of Science, Bangalore 560012, India  Manoranjan Kumar
<EMAIL_ADDRESS>S. N. Bose National Centre for Basic Sciences,
Block - JD, Sector - III, Salt Lake, Kolkata 700106, India

###### Abstract

We study the quantum phase transitions of frustrated antiferromagnetic
Heisenberg spin-1 systems on the 3/4 and 3/5 skewed two leg ladder geometries.
These systems can be viewed as arising by periodically removing rung bonds
from a zigzag ladder. We find that in large systems, the ground state (gs) of
the 3/4 ladder switches from a singlet to a magnetic state for $J_{1}\geq
1.82$; the gs spin corresponds to ferromagnetic alignment of effective $S=2$
objects on each unit cell. The gs of antiferromagnetic exchange Heisenberg
spin-1 system on a 3/5 skewed ladder is highly frustrated and has spiral spin
arrangements. The amplitude of the spin density wave in the 3/5 ladder is
significantly larger compared to that in the magnetic state of the 3/4 ladder.
The gs of the system switches between singlet state and low spin magnetic
states multiple times on tuning $J_{1}$ in a finite size system. The switching
pattern is nonmonotonic as a function of $J_{1}$, and depends on the system
size. It appears to be the consequence of higher $J_{1}$ favoring higher spin
magnetic state and the finite system favoring a standing spin wave. For some
specific parameter values, the magnetic gs in the 3/5 system is doubly
degenerate in two different mirror symmetry subspaces. This degeneracy leads
to spontaneous spin parity and mirror symmetry breaking giving rise to spin
current in the gs of the system.

##  I Introduction

Spin chains and ladders show strong quantum fluctuations due to spatial
confinement and are extensively studied to explore the intriguing magnetic
properties of these systems. The excitations and ground state (gs) properties
of these systems depend on the magnitude of the spins at the lattice sites
(half-odd-integer or integer) Haldane (1983a); *haldane83b; Affleck  _et al._
(1987, 1988) and topology of the exchange interactions Majumdar and Ghosh
(1969a); *ckm69b; White and Affleck (1996); Kumar  _et al._ (2015); Soos  _et
al._ (2016); Hamada  _et al._ (1988); Chubukov (1991); Chitra  _et al._
(1995); Itoi and Qin (2001); Heidrich-Meisner  _et al._ (2006, 2007); Kecke
_et al._ (2007); Vekua  _et al._ (2007); Mahdavifar (2008); Dmitriev and
Krivnov (2008); Hikihara  _et al._ (2008); Sudan  _et al._ (2009); Heidrich-
Meisner  _et al._ (2009); Sirker (2010); Kumar  _et al._ (2010); Kumar and
Soos (2012); Parvej and Kumar (2017). A Heisenberg antiferromagnetic(HAF)
spin-1/2 chain with only nearest neighbor exchange interaction, $J_{1}$, shows
a quasi-long range order in the gs and a gapless spectrum Mikeska and Kolezhuk
(2004), whereas a spin-1/2 HAF chain with nearest neighbor and next nearest
neighbor exchange interactions $J_{1}$ and $J_{2}$, respectively, can show
quasi-long range or short range order and gapless or gapped spectrum depending
on the ratio $J_{2}/J_{1}$ Majumdar and Ghosh (1969a); *ckm69b; White and
Affleck (1996); Kumar  _et al._ (2015); Soos  _et al._ (2016). The HAF spin-1
chains, on the other hand, with only nearest neighbor-exchange interaction
have short range order in the gs and gapped spectrum as pointed out by Haldane
Haldane (1983a); *haldane83b, and gs can be represented as a valance bond
solid (VBS) Affleck  _et al._ (1987, 1988); Schollwöck  _et al._ (1996) which
is of the same universality class as the Affleck, Kennedy, Lieb and Tasaki
(AKLT) states Affleck  _et al._ (1987, 1988). The AKLT state has inspired many
numerical techniques like matrix product states Östlund and Rommer (1995);
Verstraete  _et al._ (2008); Schollwöck (2011), Tensor Network methods Orús
(2014) and projected entangled pair states (PEPS) methods Murg  _et al._
(2007); Verstraete  _et al._ (2008). AKLT states can also be represented as
cluster states which can be used in measurement-based quantum computation
Verstraete and Cirac (2004); Wei  _et al._ (2011).

The first experimental realization of the spin-1 Haldane system was in the
well known transition metal chain compound Ni(C2H8N2)2NO2ClO4 (NENP) Renard
_et al._ (1987, 1988); Katsumata  _et al._ (1989). The HAF spin-1 chain
exhibits topological phase with, spin-1/2 edge modes leading to four fold
degenerate gs in thermodynamic limit. In spin-1 chain, the gs correlation
length, $\xi=6.05$ lattice constant and a large spin gap to the excited state,
$\Delta_{\rm{ST}}\sim 0.41J_{1}$ Dey  _et al._ (2016); White and Huse (1993).
The gs of a two leg HAF spin-1 normal ladder is a plaquette-singlet solid
state (PSSS) where two spin-1/2 singlet dimer are sitting at each rung and
there is no overlap to the VBS state in the large rung exchange limit Todo
_et al._ (2001). On a zigzag ladder, the gs of a spin-1 HAF model is the
Haldane phase in the weak rung interaction limit, while a double Haldane state
is the gs in the large rung interaction limit Hikihara  _et al._ (2002);
Chepiga  _et al._ (2016). Another class of spin ladders that have been studied
in recent times is the skewed ladder which contains slanted rung bonds which
periodically displace rung bonds as well as periodically missing rung bonds of
the original ladder (Fig. 1). This leads to fused equivalent or inequivalent
cyclic rings. An example of such a skewed ladder is the 5/7 skewed ladder with
alternately fused five- and seven-membered rings while 5/5 skewed system
consists of equivalent five membered rings fused together. The 5/7 system has
been studied in both the spin-1/2 Thomas  _et al._ (2012); Giri  _et al._
(2017); Dey  _et al._ (2020) and spin-1 cases Das  _et al._ (2020). Similarly,
the 3/4, 3/5 and 5/5 spin-1/2 skewed ladders have also been studied earlier
and it was shown that the gs of hetero skewed ladders (polygons of unequal
number of vertices fused together), such as 5/7 and 3/4 are magnetic Giri  _et
al._ (2017). In the large rung exchange limit the gs wavefunction of 3/4
skewed ladder can be represented as a product of rung singlet dimers and one
ferromagnetically interacting spin-1/2 object per triangle Giri  _et al._
(2017). The gs of a spin-1/2 system on a 3/5 geometry is a low spin magnetic
state for intermediate values of $J_{1}$, but evolves to a gapless
antiferromagnetic state for large $J_{1}$ Giri  _et al._ (2017). As outlined
above, the spin-1/2 and spin-1 systems exhibit fundamentally different
behavior in case of the HAF spin chains. This has prompted us to study the
spin-1 system on these ladders, to explore the quantum phase transitions in
these systems as the ratio of the rung exchange to the ladder exchange
strengths is varied.

Figure 1:  Schematic diagram of (a) zigzag ladder, (b) 3/4 skewed ladder and
(c) 3/5 skewed ladder. The unit cell of each structure is shown inside the box
in red.

In this paper, we focus on the gs properties of spin-1 objects arranged on 3/4
and 3/5 skewed ladders interacting via competing rung and leg
antiferromagnetic exchange interactions of strength $J_{1}$ and $J_{2}$
respectively. The structure of zigzag, 3/4 and 3/5 skewed ladders are shown in
Fig. 1(a)–1(c). In section II we discuss the model Hamiltonian and numerical
methods used in the paper. The results for 3/4 and 3/5 skewed ladders are
discussed in section III. In section III.1, we show that the gs of the 3/4
system switches from a singlet to a magnetic state for $J_{1}\geq 1.82$ with
magnetization per unit cell ‘$\langle m\rangle$’ taking a value of $2$. In
section III.2, we show that the gs of 3/5 ladder exhibits short range spin
correlations and a spiral spin density wave, the spin of the gs depends both
on size of the system and $J_{1}$ for intermediate $J_{1}$ values; the gs is
magnetic with one unpaired spin per unit cell in the large $J_{1}$ limit. For
some specific $J_{1}$ values, the gs is degenerate and exhibits vector chiral
phase due to simultaneous breaking of spin inversion and reflection
symmetries. Section IV provides summary of results and conclusions.

##  II Model and method

We consider the HAF spin-1 model on 3/4 and 3/5 skewed ladders shown in Fig.
1(b) and (c) respectively; all exchange interactions between the neighboring
spins are antiferromagnetic in nature. The site numberings are shown in Figs.
1(b) and 1(c). The exchange interaction along rung and leg bonds are
represented as $J_{1}$ and $J_{2}$ respectively. The exchange interaction
$J_{2}$ is set to 1, to define the energy scale in all the studies. The model
Hamiltonian of 3/4 skewed ladder can be written as

$\displaystyle H_{3/4}$ $\displaystyle=$ $\displaystyle
J_{1}\sum_{k=1}^{N/3}\vec{S}_{3k-1}\cdot\left(\vec{S}_{3k-2}+\vec{S}_{3k}\right)$
(1) $\displaystyle\qquad+J_{2}\sum_{k=1}^{N-2}\vec{S}_{k}\cdot\vec{S}_{k+2}$

where the system consists of $N=6n$ sites, with $n$ being the number of unit
cells and an open boundary condition is assumed (Fig. 1(b)). The first term
denotes the rung exchange, the second denotes the exchange interactions along
the legs. The model Hamiltonian of 3/5 skewed ladder can be written as

$\displaystyle H_{3/5}$ $\displaystyle=$ $\displaystyle
J_{1}\sum_{i=1}^{n}\vec{S}_{4i-2}\cdot\left(\vec{S}_{4i-3}+\vec{S}_{4i-1}\right)$
(2)
$\displaystyle\qquad+J_{1}\,\vec{S}_{4n+2}\cdot\vec{S}_{4n}+J_{2}\sum_{i=1}^{4n}\vec{S}_{i}\cdot\vec{S}_{i+2}.$

Where $n$ is the number of unit cells and an open boundary condition is
assumed. In case of periodic boundary condition, the Hamiltonian is modified
accordingly and sites $4i-2$ and $4i$ are the inversion centres of the system.
Both $H_{3/4}$ and $H_{3/5}$ conserve total spin $S^{2}$, and $z$-component of
the total spin $S^{z}$.

We use exact diagonalization (ED) technique for solving these ladders with up
to 16 spins with periodic boundary condition (PBC). For larger system sizes,
we use the now well known finite density matrix renormalization group (DMRG)
method White (1992, 1993); Schollwöck (2005); Hallberg (2006) which retains
the size of the Hamiltonian matrix at all system sizes. This is achieved by a
systematic truncation of the irrelevant degrees of freedom by selecting ‘$m$’
states with largest eigenvalues of the density matrix to span the Fock space
of a part of the full system. The chosen value of $m$ is up to $500$ in our
studies and it keeps the truncation error of the density below $\sim
10^{-10}$. We also carry out 6-10 finite sweeps for convergence. For some
values of $J_{1}$ and for large system sizes we have employed $m=1600$ and 12
finite DMRG sweeps to improve convergence. The largest system size used in
this paper is $N=98$ or 24 unit cells with open boundary condition.

##  III Results and discussions

In this section we present the results for both 3/4 and 3/5 skewed ladders.
The gs of the 3/4 skewed ladder shows a transition from a singlet to a
magnetic phase, whereas the gs of 3/5 ladder has spiral spin arrangement and
switches between different magnetic and singlet states upon tuning $J_{1}$. In
the decoupled limit the gs of both systems show Haldane phase, whereas, in
large $J_{1}$ limit the gs exhibits strong rung trimer formation. In strong
rung coupling limit (very large $J_{1}$) the gs of the system has an effective
spin-1 on each triangle with those triangles interacting ferromagnetically. To
analyze the magnetic transitions, various quantities like gs energies
$E_{gs}$, energy gaps to low-lying excited states $\Gamma_{l}$, bond orders
$b_{i,j}$ between sites $i$ and $j$ and local spin densities $\rho_{i}$ at
site $i$ are analyzed, as a function of $J_{1}$. The excitation gaps
$\Gamma_{l}$ are defined as

$\Gamma_{l}=E_{0}(S^{z}=l)-E_{0}(S^{z}=0)$ (3)

where $E_{0}(S^{z}=l)$ and $E_{0}(S^{z}=0)$ are the energies of the lowest
states in the $S^{z}=l$ and $S^{z}=0$ manifold, respectively, and $l$ is an
integer. The model Hamiltonians in Eqs. 1 and 2 are isotropic and therefore,
conserve total spin $S$ and its $z$-component $S^{z}$. For a magnetic gs with
spin $S$, $E_{0}(S^{z}=S)$ is degenerate with $E_{0}(S^{z}=m)$ where $-S\leq
m\leq S$ and the state is $(2S+1)$ fold degenerate. Thus $S$ is the gs spin if
the lowest energy level in every $S^{z}\leq S$ are the same or $\Gamma_{l}=0$
for all $l\leq S$. The bond order
$b_{ij}=-\langle\vec{S}_{i}\cdot\vec{S}_{j}\rangle$, spin density
$\rho_{i}=\langle{S}_{i}^{z}\rangle$ and correlation function
$C(r)=\langle\vec{S}_{i}\cdot\vec{S}_{i+r}\rangle$ are the gs expectation
values calculated to characterize the gs with $S^{z}=S$. Due to symmetry there
are only two unique sites and three unique bonds per unit cell in the 3/4
ladder, whereas there are three unique sites and four bonds in the 3/5 ladder.
Therefore, it is sufficient to obtain all the above quantities for these
unique sites and bonds.

###  III.1 3/4 Skewed ladder

The magnetic gs of the 3/4 skewed ladder is obtained using excitation gaps
$\Gamma_{l}$, defined in Eq. (3). In this system the gs rapidly evolves from a
singlet state to a magnetic state with spin $S=2n$ where $n$ is the number of
unit cells in the system. This transition seems continuous in systems with
open boundary condition (OBC) with system size $N=50$. In this system (Fig.
2(a)) the gs spin, $S_{G}$, changes gradually beyond $J_{1}=1.67$ and near
$J_{1}\sim 1.75$, $S_{G}$ increases rapidly and beyond $J_{1}=1.82$, $S_{G}$
achieves the highest gs spin of $S_{G}=16$. The transition region shrinks
dramatically to $1.65<J_{1}<1.67$ for $N=18$ with periodic boundary condition
(PBC) and for both PBC and OBC case the transition region decreases with
system size. The scaling of the total spin $S_{G}$ of the gs is shown as
function of unit cell in two parameter regimes in Fig. 2(b). The gs remains a
singlet for $J_{1}<1.6$, whereas, for larger $J_{1}\geq 1.82$, $S_{G}$ vs $N$
shows a linear variation with slope 1/3. The linear variation for $S_{G}$ vs
$N$ for $J_{1}=2$ is shown in Fig. 2(b). The finite size effect of $S_{G}$ is
shown in $S_{G}$ vs $J_{1}$ curve for three system sizes $N=14$, 26 and $50$
in Fig. 2(c). We notice that the $S_{G}$ increases in a stepwise fashion for
small system size but shows a sharp increase as the system size increases.

Figure 2: (a) The lowest excitation gaps $\Gamma_{l}$ are shown as function of
$J_{1}$ for 3/4 ladder of 50 sites with open boundary condition. For
$J_{1}<1.6$, all the $\Gamma_{l}$ are non-zero, whereas, for $1.6<J_{1}<1.82$,
the gs transitions to a magnetic state. For $J_{1}\geq 1.82$, the gs spin
equals twice the number of unit cells in the system. (b) The $S_{G}$ is shown
as a function of system sizes for $J_{1}=1.0$ and $2.0$. Each data point
corresponds to integer number of unit cells. (c) $S_{G}$ vs. $J_{1}$ for
different system sizes.

In small $J_{1}$ limit, spin correlation $C(r)$ behavior is similar to that of
a HAF spin-1 chain with short correlation length. In Fig. 3(a) and 3(b) total
correlation function $C(R,R+2r)$ is shown for $J_{1}=1.0$ and 2.0
respectively. There are two unique sites one at the base of the triangle and
other at the apex of the triangle. We consider reference sites on the middle
triangle of the ladder with site labels 49, 50 and 51 of a $N=98$ site system.
Site 49 and 51 are at the base of the triangle whereas site 50 is at the apex
of the triangle, and correlations from these points $C(49,49+2r)$ and
$C(51,51+2r)$ are the correlations between the reference sites on the lower
leg of the ladder with all the sites on the lower leg, whereas $C(50,50+2r)$
are the correlations between the reference site on the upper leg of the ladder
with all the other sites on the upper leg. The $C(R,R+2r)$ for $R=49$ and 51
sites are similar as both these sites belong to the base of the triangle and
are equivalent by symmetry. In Fig. 3(a) $C(R,R+2r)$ for $J_{1}=1.0$ seems to
show oscillatory behavior with exponentially decreasing amplitudes. This seems
to indicate existence of spin wave packets in the singlet state. The
$C(R,R+2r)$ in magnetic regime $J_{1}=2.0$ are shown Fig 3(b), and spins are
antiferromagnetically aligned and have long range correlations. In this
regime, in the bulk of the system, the correlations are oscillating with
almost constant amplitude. The correlation between the reference spins in the
base of the triangle is always ferromagnetic with spins at the base of the
triangle and antiferromagnetic with the spin at the apex of the triangle. This
is true on both legs. We have also computed the spin densities in the gs of
the ladder as a function of $J_{1}$. The spin densities vanish at all the
sites in the singlet gs. We note from Fig. 4, that the apex and the base sites
have opposite spin densities in the magnetic gs. The spin densities at all the
sites are nearly constant. Indeed, the spin-spin correlations in the magnetic
state are well approximated by the product of the spin densities at the
corresponding sites.

Figure 3:  The spin-spin correlations between the spins in the lower leg (R =
49 and R=51), spins in upper leg (R = 50) for a 3/4 ladder with N=98 spins
with OBC with (a) $J_{1}=1.0$, the singlet regime and (b) $J_{1}=2.0$, the
magnetic regime.

In large $J_{1}$ limit the spin densities show that the gs behavior can be
thought of as each triangle having effective spin-1, which are interacting
ferromagnetically. The spin density of an isolated triangle with large $J_{1}$
interaction is also $0.75$ on the basal sites and $-0.5$ at the apex which is
close to the values found in the gs at large $J_{1}$.

Figure 4: (a)The base and the apical spins are shown in blue and red; three
different types of bonds namely, the rung bonds ($b_{r}$), the basal bonds
($b_{b}$) and the apical bonds ($b_{a}$) are shown for a unit cell (b) Spin
densities $\rho_{b}$ and $\rho_{a}$ for the base sites and the apical sites,
respectively, as a function of $J_{1}$ (c) bond orders $b_{r}$, $b_{b}$ and
$b_{a}$ as a function of $J_{1}$ are shown.

In 3/4 system there are only three distinct bond orders corresponding to the
rung bond $b_{r}$, bond between the basal sites $b_{b}$ and bond between the
apical and base sites $b_{a}$. We notice that both types of bonds on the legs
($b_{a}$ and $b_{b}$) show decreasing bond order as $J_{1}$ increases and the
basal bond order shows jump at $J_{1}=1.65$. Whereas, the magnitude of rung
bond order increases continuously with $J_{1}$ and jumps suddenly at
$J_{1}=1.65$. The jump in the rung bond order is due to transition of the gs
from a singlet to a magnetic state. The saturation value of the three bonds
for large $J_{1}$ limit is $+1.5$, $+1.0$ and $-1.0$ respectively. The value
$b_{r}=+1.5$ corresponds to two spin-1/2 objects separately forming bonds and
total bond order is twice of a singlet bond order $+3/4$. The $b_{b}=+1$ and
$b_{a}=-1$ corresponding to the ferromagnetic bond formation between axial and
basal sites and intermediate spin state between the basal sites. In small
$J_{1}$ limit, the $b_{r}$ rung bond is weak and $b_{r}$ is close to zero.
Whereas, $b_{b}$ and $b_{a}$, for small values of $J_{1}$, start with value
close to the bond order of a spin-1 chain $\sim 1.40$, they decrease with
increasing $J_{1}$; $b_{b}$ becomes ferromagnetic and $b_{a}$ remains
antiferromagnetic.

To understand the spin density and bond order behavior in large $J_{1}$ limit,
we analyze a system with three spins on a triangle, and exchange interaction
of the two sides and base are $J_{1}$ and $J_{2}=1$, respectively. The gs is
in spin-1 states and there are six possible spin configurations. For
$J_{1}\geq 2$ we calculate the gs wavefunction of the system in the $S=1$ and
$S^{z}=1$ manifold as

$\displaystyle\mid\Psi$ $\displaystyle($ $\displaystyle S=1,S^{z}=1)\rangle$
(4) $\displaystyle=$ $\displaystyle\sqrt{\frac{3}{5}}\left[\mid
1,-1,1\rangle+\frac{1}{2}(\mid 0,0,1\rangle+\mid 1,0,0\rangle)\right.$
$\displaystyle\left.-\frac{1}{6}(\mid-1,1,1\rangle+\mid
1,1,-1\rangle)-\frac{1}{3}\mid 0,1,0\rangle\right].$

We notice that the large contribution ($60\%$) is from the state $\mid
1,-1,1\rangle$ and a smaller ($15\%$) contribution is from the linear
combination $1/2(\mid{0,0,1}\rangle+\mid 1,0,0\rangle)$ spin configuration.
The sites 1 and 3 are symmetric and total contribution of spin density comes
from these two configurations, and $60\%$ contribution to $\langle
S^{z}\rangle$ arises from $\mid 1,-1,1\rangle$ while a contribution of $15\%$
to $\langle S^{z}\rangle$ arises from $(\mid 0,0,1\rangle+\mid 1,0,0\rangle)$,
resulting in a total spin density of $0.75$. The configuration $\mid
1,-1,1\rangle$ contributes $-0.6$ to $\langle S^{z}\rangle$ at site $2$ while
the states $(\mid-1,1,1\rangle+\mid 1,1,-1\rangle)$ and $\mid 1,-1,1\rangle$
contribute $+0.1$ to $\langle S^{z}\rangle$ at site $2$. This results in total
spin density at site 1 and 2 of $0.75$ and $-0.5$ respectively in large
$J_{1}$ ($J_{1}\geq 2.0$) limit. The contribution to the $z$ component of all
three bond orders mostly arises from the configuration $\mid 1,-1,1\rangle$
and the bonds $1-2$ and $2-3$ are singlet in nature whereas the bond $1-3$ is
ferromagnetic.

###  III.2 3/5 Skewed ladder

The 3/5 ladder has 4 spins per unit cell as shown schematically in Fig. 1(c)
and in each unit cell three sites form a triangle, whereas $4^{\rm{th}}$ site
is connected to the apex of the triangle. There is one rung bond at each odd-
numbered site and two rung bonds at alternate even numbered sites, $(4k-2)$,
$k=1$, 2, $\cdots$. With periodic boundary condition, the sites $2k$ at the
apices of triangles and pentagons are inversion centres. The model Hamiltonian
of this system is shown in Eq. (2)

The gs of the 3/5 system is singlet in low $J_{1}$ limit and toggles between
singlet and different magnetic states as $J_{1}$ is increased. We find that
the evolution of the spin of the gs depends not only on $J_{1}$ but also on
the system size. We show this dependence for four representative sizes of the
systems with open boundary condition. For the system with 6 unit cells (Fig.
5) ($N=26$), the gs switches from a singlet to a triplet at $J_{1}=0.84$ and
it then switches back to a singlet at $J_{1}=1.7$. The triplet then becomes a
gs from $J_{1}=2.15$ to $J_{1}=3.0$. Beyond $J_{1}=3.0$ the gs is a quintet
until $J_{1}=6.5$, a septet ($S=3$) from $J_{1}=6.5$ to $7.0$ a nonet ($S=4$)
from $J_{1}=7.0$ to $8.5$ and $S=5$ beyond $J_{1}=8.5$. The highest spin in
the $N=26$ systems with 6 unit cells can be $S=6$, corresponding to
ferromagnetic arrangement of spins at sites $4k$. In large systems these
transitions occur at different values of $J_{1}$ (Fig. 5).

Figure 5: The lowest excitation energy gaps in different $S^{z}$ sectors,
$\Gamma_{l}$ for different $l=S^{z}$ manifolds as a function of the rung
exchange interaction $J_{1}$ for systems of (a) $N=26$ spins (6 unit cells),
(b) $N=50$ spins (12 unit cells), (c) $N=74$ spins (18 unit cells), and (c)
$N=98$ (24 unit cells) with open boundary condition. For larger system sizes,
the gs switches between magnetic and singlet states as $J_{1}$ is increased.

To understand this seemingly strange behavior, we have computed spin densities
and spin-spin correlations for the gs with different spins (corresponding to
different $J_{1}$ values) for the $N=98$ system. We see from Fig. 6 that in
the interior of the system, higher spin densities are found at $4k$ sites.
Besides the spin densities of a given type of sites show a wavelike behavior.
Thus, the gs spin is dictated by two factors. Large $J_{1}$ favors a high spin
gs, while the wavy nature of the spin density favors a standing wave as
interference effects will be lowest in this state and so will be the spin
fluctuations which tend to increase the energy of a state. Hence, the gs shows
the unusual spin state changes as $J_{1}$ is increased. This also explains the
nonmonotonic dependence on the switching values of $J_{1}$ for different
system sizes. When $J_{1}$ is very large ($J_{2}/J_{1}\to 0$), we expect the
gs spin of the system to be $4n$ where $n$ is the number of unit cells in the
system. This unusual behavior is also reflected in the spin-spin correlation
function (Fig. 7), where the wavelength of the correlation function decreases
with $J_{1}$ and the magnitude of the correlation between spins at sites $4k$
has larger amplitude than the rest of the correlations.

Figure 6: Spin densities of a 3/5 ladder of $N=98$ spins (OBC). (a) spin
densities for $J_{1}=1.0$ in the triplet gs are shown with continuous red,
blue, green and orange curve (the eye guide) for 1, 2, 3 and 4 sites in every
unit cells. (b) spin densities for $J_{1}=2.5$ are shown in the quintet gs. We
have chosen $S^{z}=S_{G}$ in both the cases. Figure 7: Spin-spin correlations
between spins at same type of sites in each unit cell of a 3/5 ladder of
$N=98$ spins (OBC) for (a) $J_{1}=1.0$, (b) $J_{1}=1.3$, (c) $J_{1}=1.7$, and
(d) $J_{1}=2.5$. Starting from the middle of the ladder $R=49$, 50, 51, 52 are
the reference sites.

In case of our 3/5 ladder system, there are two mirror planes one which is
passing through the centre of the five membered ring along the upper leg and
perpendicular to the lower leg, whereas second mirror plane ‘$\sigma$’ passes
through the apex of triangle and bisects the base triangle. The system also
has spin inversion symmetry $P$ which exists in the $S^{z}=0$ subspace and
corresponds to invariance of the Hamiltonian when all the spins in the system
are rotated by $\pi$ around the $y$-axis. This is equivalent to invariance of
Hamiltonian under spin inversion. This symmetry divides the $S^{z}=0$ subspace
into an even (‘$+$’) and an odd (‘$-$’) subspace with the even (odd) subspace
spanning states with even (odd) spins. Thus, the even subspace consists of
basis functions with even integer total spin ($S$) and the odd subspace
consists of basis functions with odd integer total spin($S$). The symmetry
group of the 3/5 skewed ladder consists of $E$, $P$, $\sigma$ and $\sigma P$
and all these elements commute with each other and form an abelian group. The
irreducible representations of this group is represented by $A^{+}$ , $A^{-}$
, $B^{+}$ and $B^{-}$. $A(B)$ corresponds to even (odd) space under $\sigma$
and $+(-)$ corresponds to even(odd) space under $P$. The spin inversion
symmetry is broken when the lowest energy states with odd and even total spins
are degenerate and reflection symmetry is broken when lowest energy levels in
both $A$ and $B$ spaces are degenerate. Both the symmetries are broken when
the lowest energy states in $A^{+}$ ($A^{-}$) and $B^{-}$ ($B^{+}$) spaces are
degenerate. We show the degeneracy of singlet and triplet states at
$J_{1}=0.8075$ and triplet and quintet at $J_{1}=1.2183$ in Table 1. For these
values of $J_{1}$, the reflection symmetry is also broken as lowest energy
states in spaces odd and even under reflections are also degenerate for these
$J_{1}$ values. Hence we observe spontaneous spin current. The $z$-component
of the spin current is defined as

$\displaystyle\kappa^{z}(j,k)$ $\displaystyle=$
$\displaystyle-i\langle\Psi_{G}(-)\mid(\vec{S}_{j}\times\vec{S}_{k})^{z}\mid\Psi_{G}(+)\rangle$
(5) $\displaystyle=$
$\displaystyle\frac{1}{2}\langle\Psi_{G}(-)\mid(S_{j}^{+}S_{k}^{-}-S_{j}^{-}S_{k}^{+})\mid\Psi_{G}(+)\rangle$

which is the eigen value of the spin current operator
$(\vec{S}_{j}\times\vec{S}_{k})^{z}$ expressed as a matrix in the degenerate
$\mid\Psi_{G}(+)\rangle$ and $\mid\Psi_{G}(-)\rangle$ basis where the function
$\mid\Psi_{G}(+)\rangle$ $\left(\mid\Psi_{G}(-)\rangle\right)$ is the gs in
the even (odd) subspace for reflection and even (odd) subspace for spin
inversion. In Fig. 8, we show the spin currents for $J_{1}=0.8705$ and
$1.2183$. The current in triangle is counter clockwise, whereas it is
clockwise in the pentagons and weak in upper leg of pentagons. Interestingly
the spin current retains the qualitative features for both the $J_{1}$ values.

Table 1: Lowest energy levels for different $S_{z}$ values of $N=16$ $J_{1}$ | $E(S^{z}=0)$ | $E(S^{z}=1)$ | $E(S^{z}=2)$
---|---|---|---
0.8705 | $-23.0172$ | $-23.0172$ |
| $-23.0172$ |  |
1.2183 | $-23.7554$ | $-23.7554$ | $-23.7554$
| $-23.7554$ | $-23.7554$ |
Figure 8: The spin current in a 3/5 skewed ladder of $N=16$ spins with
periodic boundary condition at (a) $J_{1}=0.8705$ and (b) $J_{1}=1.2183$. The
direction of the spin current is indicated by arrowheads and the magnitude is
given by the numbers adjacent to the arrows.

##  IV Summary and conclusions

In this paper we study the gs properties of spin-1 Heisenberg
antiferromagnetic model on 3/4 and 3/5 skewed ladder geometries shown in Fig.
1(b) and (c). The 3/4 spin ladder system goes from a singlet state to
partially magnetized state. For $J_{1}\geq 1.82$ the magnetization per unit
cell is $\langle m\rangle=2$ with each triangle in the system contributing
$\langle m\rangle=1$ to the magnetization in the gs. The gs of this system for
$J_{1}<1.6$ is a singlet and shows short range correlations. In this system
there are two unique sites and three unique bonds in the six site unit cell.
In the weak $J_{1}$ limit, the gs is a singlet and spin densities are zero and
bond orders along the leg are close to that of the valance bond state of a
spin-1 chain. The bond order in the upper leg which connect the apex or base
of two different triangles decreases gradually as $J_{1}$ is increased while
bond-order for base of the triangle increases rapidly. In large $J_{1}$ limit
or in magnetic gs the spin densities for base and apex is $0.75$ and $-0.5$
respectively, and the effective spin per triangle is 1. The bond order along
the rung and base of the triangle are $1.5$ or equivalent two spin-1/2 dimers
and $-1$ for bond between a basal site and an apical site, indicating a
ferromagnetic interaction. In the large $J_{1}$ limit, the effective spin one
on each triangle interact ferromagnetically. In the spin-1/2 skewed ladder
system the ferromagnetic arrangement of spin is an example of the McConnell
mechanism McConnell (1963); Giri  _et al._ (2017) with antiferromagnetic
exchange (here $J_{2}$) between sites with positive and negative spin
densities leading to a ferromagnetic interaction between the unpaired spins.
The effective interaction between the spins in two neighboring triangles can
be written as $J_{\rm{eff}}=2J_{2}\rho_{a}\rho_{b}$ where $\rho_{a}$ and
$\rho_{b}$ are the spin densities at $a$ and $b$ type sites. We note that
ferromagnetic mechanism for spin-1 system seems similar to spin-1/2 system.

The gs properties of the 3/5 ladder seems more complicated unlike the 3/4
system. The gs is a singlet for $J_{1}<0.84$ and shows spin density wave gs
for larger $J_{1}$ values. What is interesting is that the gs spin $S_{G}$
varies between $0$, $1$ and $2$ before attaining the saturation value of $n$
(the number of unit cells) for very large $J_{1}$. The switching of the gs
depends on system size and there is no apparent systematics. Analysis of spin
densities and spin-spin correlations seem to indicate that there is a
competition between the standing spin density wave of a lower spin state and
the higher exchange stability of the successively higher spin state. This
seems to render a qualitative explanation of the seemingly unsystematic
switching in the gs spin of the system as a function of system size. This
system also shows vector chiral phase due to simultaneous breaking of the
reflection symmetry and spin inversion symmetry. The broken symmetry phase is
characterized by nonzero spin currents in the system.

In summary, we have studied the gs properties of an antiferromagentic
Heisenberg spin-1 system on 3/4 and 3/5 skewed ladders. Both systems
transition from singlet to a partially polarised gs on tuning $J_{1}$. The 3/5
system shows switching of the spin state between singlet and different
magnetic states due to a competition between $J_{1}$ which favors a high spin
state and standing spin density wave which favors a lower spin state. For two
different parameter values the gs of the system has doubly degenerate gs which
leads to breaking of spin parity and reflection symmetry in the system
resulting in spontaneous spin current. Although such systems have not yet been
experimentally realized, we believe that they can be realized in molecular
magnets based on transition metal compounds.

## Author’s contributions

SD and DD contributed equally to this work.

###### Acknowledgements.

MK thanks SERB for financial support through project file No. CRG/2020/000754.
SR thanks Indian National Science Academy and DST-SERB for supporting this
work.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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