# A Dataset of Coordinated Cryptocurrency-Related Social Media Campaigns

Karolis Zilius1, Tasos Spiliotopoulos2, Aad van Moorsel2

###### Abstract

The rise in adoption of cryptoassets has brought many new and inexperienced
investors in the cryptocurrency space. These investors can be
disproportionally influenced by information they receive online, and
particularly from social media. This paper presents a dataset of crypto-
related bounty events and the users that participate in them. These events
coordinate social media campaigns to create artificial ”hype” around a crypto
project in order to influence the price of its token. The dataset consists of
information about 15.8K cross-media bounty events, 185K participants, 10M
forum comments and 82M social media URLs collected from the Bounties(Altcoins)
subforum of the BitcoinTalk online forum from May 2014 to December 2022. We
describe the data collection and the data processing methods employed and we
present a basic characterization of the dataset. Furthermore, we discuss
potential research opportunities afforded by the dataset across many
disciplines and we highlight potential novel insights into how the
cryptocurrency industry operates and how it interacts with its audience.

## Introduction

The last years have seen the rise of Decentralized Finance (DeFi), a form of
finance that does not rely on central financial intermediaries, such as banks
and exchanges, to offer traditional financial services, and instead utilizes
smart contracts on distributed ledgers (blockchains). The cryptographically
secured digital tokens that are stored on these blockchains, called
cryptoassets, include cryptocurrencies (like Bitcoin and Ethereum), utility
tokens, security tokens and Non-Fungible Tokens (NFTs) (Karim et al. 2022). A
simple application of this is that people can effectively invest in a company
or project by acquiring and trading digital tokens of that company without
intermediaries. Because these cryptoassets are freely traded, there is a lot
of room for price speculation and manipulation (Nizzoli et al. 2020).
Importantly, compared to traditional investing, this decentralized investing
can be conducted by lay investors (Abramova et al. 2021) who are often
influenced disproportionally by information they receive from social media
(Glenski, Saldanha, and Volkova 2019; Jahani et al. 2018). This provides a
strong incentive for such crypto projects to create artificial ”hype” in order
to convince more social media users to buy their token, or in order to
manipulate the price of a token in a certain way. In particular, they provide
payments in the form of ”bounties” to users willing to engage with the project
on social media in a positive way, such as following, commenting, retweeting
and creating content.

This paper presents a dataset of crypto-related bounty events that coordinate
social media campaigns to create artificial hype around a crypto project, as
well as the users that engage in these campaigns. Our dataset was crawled from
the Bounties(Altcoins) subforum of the Bitcointalk.org online forum. While a
lot of research has studied inauthentic, manipulative, deceptive and collusive
content online, this type of bounty hunting phenomenon is unique due to the
particularities of the cryptoassets context. In particular, the sentiment and
the volume of information online have been found to strongly influence market
prices and user investing decisions in traditional finance (Bollen, Mao, and
Zeng 2011), and even more so in relation to cryptoassets (Domingo, Piñeiro-
Chousa, and Ángeles López-Cabarcos 2020; Xie, Chen, and Hu 2020). In the
context of cryptoassets, these coordinated campaigns aim to achieve this in
two ways; directly, by influencing social media users to buy an asset based on
artificial hype, and indirectly, by influencing trading algorithms that make
investing decisions based on sentiment mined from social media (Garcia and
Schweitzer 2015). What makes this practice particularly concerning is that
bounty hunters are typically paid in the token that they are promoting. This
gives them an incentive to ”dump” (i.e., sell) the asset when the campaign
ends, and since they have inside knowledge of exactly when the campaign will
end, this practice can be considered equivalent to a ”pump-and-dump” scheme
(Hamrick et al. 2021) that will dramatically crash the price of the token
causing the investors outside the campaign to lose the funds they invested.

While this type of bounty hunting phenomenon is unique, it builds on known
practices that have been identified and studied earlier in the literature.
These practices have been tailored specifically to the crypto space and are of
varying ethical and legal positioning and justification. Bounty programs, in
general, have been commonly used to manage rewards traditionally aimed at
incentivizing positive and productive behavior, such as finding bugs in
software code (Ding, De Jesus, and Janssen 2019), providing worthy answers in
question-answer platforms (Parnin et al. 2012) and verifying the bias and
safety of AI systems (Brundage et al. 2020). The practices and ethics of
advertising, in general (Schauster, Ferrucci, and Neill 2016), and social
media promotion (Zeng, Kohno, and Roesner 2021), in particular, have been
debated by researchers. Related work has highlighted particularly questionable
practices, such as astroturfing (i.e., hidden coordinated information
campaigns that mimic genuine user behavior by incentivizing agents to spread
information online) for consumers (Kauppila and Soliman 2022; Lee,
Tamilarasan, and Caverlee 2013) and citizens (Schoch et al. 2022). Dutta and
Chakraborty (2021) have studied blackmarket services that are employed by
”collusive” users to inflate the popularity of their online account and get
appraisals for their content, while spam has been a topic of intensive
research over the years (e.g., Benevenuto et al. 2010; Yardi et al. 2009).

From the perspective of finance, researchers have examined the inner workings,
the nuances and the relationships between cryptocurrency pump-and-dump schemes
(Hamrick et al. 2021), market price manipulation (Nizzoli et al. 2020), and
Initial Coin Offerings (ICOs) (Ante, Sandner, and Fiedler 2018). From a legal
and regulatory perspective, researchers are examining the ways and the extent
to which social media content creators comply with the guidelines of the US
Federal Trade Commission (FTC) which requires them to disclose their
endorsements in order to prevent deception and harm to users (Mathur,
Narayanan, and Chetty 2018). Specifically in the crypto space, scholars are
examining the practices of crypto users and businesses in order to determine
the status of cryptoassets as securities (i.e., investment contracts),
something that would also place restrictions on discussions around
cryptoassets and would require disclosure of any connections or endorsements
(Henderson and Raskin 2019). Finally, such practices are in violation of the
Terms of Service (ToS) of online platforms; for example Twitter’s platform
manipulation and spam policy111https://help.twitter.com/en/rules-and-
policies/platform-manipulation prohibits artificially amplifying content and
makes specific mentions to coordinated activities for doing so. This mix of
characteristics of other practices that is present in crypto-related social
media bounty hunting muddles the way that campaign organisers, participants
and even targets of the campaigns perceive the ethical and legal implications.

This paper presents a dataset collected from the
Bounties(Altcoins)222https://bitcointalk.org/index.php?board=238.0 subforum
(message board) of the Bitcointalk.org333https://bitcointalk.org/ forum
describing crypto-related bounty events, the participants, the actions taken
and the rewards received from 13-May-2014 when the first thread was started to
31-Dec-2022. We collected and analyzed information about 15.8K unique cross-
media bounty events, 185K bounty hunters (i.e., users that participated in
such events), 10.01M comments, 18M spreadsheet lines, 56K images, 42.1M Tweets
and 33.9M links to Facebook posts, among other information posted on the
subforum. We further enhanced the dataset to make it accessible to a broader
set of researchers and enable future research in this area by linking
information collected across the forum threads and aggregating it to calculate
useful metadata. This resulted in a rich set of descriptive metadata; for
example, we provide 31 data fields describing each Bounty event and 17 data
fields describing each participant.

While previous research has focused on detecting coordinated social media
campaigns and attempting to infer the actors and the details of deceptive
actions from their social media traces (e.g., Peng et al. 2017; Keller et al.
2020), our dataset provides comprehensive information on these campaigns and
their organisation from their source, including the structure of incentives,
exact instructions and timing of actions, and a full account of all
participants. We expect that this unprecedented detail afforded by our dataset
will be invaluable to researchers studying online coordinated deceptive
practices.

We make the entire dataset, along with a smaller sample and supplementary
material, publicly available at the following link: (Dataset URL:
https://zenodo.org/record/7539178).

## The Bitcointalk.org Forum and the Bounties(Altcoins) Subforum

The BitcoinTalk forum was created by the developer of Bitcoin in 2009, the
same year that Bitcoin was
developed444https://bitcointalk.org/index.php?topic=5. While originally
focusing on Bitcoin itself, the discussions on the forum gradually expanded to
include alternative cryptocurrencies (altcoins). The forum currently has more
than 3.5M registered users and 61.5M posts on 1.3M
topics555https://bitcointalk.org/index.php?action=stats. As the first and very
popular community created around cryptocurrencies, the forum had an original
intention of furthering adoption and supporting users. Areas of discussion
include technical aspects of distributed ledgers, such as development and
mining, economics, politics, as well as supporting users in using the
technology, trading, and recovering from scams. As cryptocurrencies became
more widespread and less technically-inclined users turned to the forum for
information and advice, researchers utilized the BitcoinTalk forum to study
crypto-related collective intelligence (Jahani et al. 2018), speculation (Xie,
Chen, and Hu 2020) and scams (Vasek and Moore 2019).

The main BitcoinTalk forum comprises multiple subforums, each focusing on a
separate aspect of cryptoassets. Each subforum contains many discussion
threads initiated by different users. The earliest entries of the
Bounties(Altcoins) subforum date back to 2014, although it didn’t receive much
attention until 2017. The subforum experienced a massive growth in popularity
during 2017 – 2018, which aligns with the first steep increase in crypto
adoption. More than 15,000 threads have been created since and it has evolved
to the most popular subforum of BitcoinTalk666see note 5.

Bounties(Altcoins) serves as a place to organize airdrops (cryptocurrency
giveaways) and alert the public about upcoming projects or ICOs, however it is
dominated by bounty events. A bounty event involves the distribution of
cryptocurrency or monetary rewards for the accomplishment of event-specific
objectives. These rewards are typically distributed in the token being
promoted, although it is not uncommon to use stablecoins (e.g., USDT) and even
US dollars transferred via a service like Paypal. Each bounty event can have
one or more campaigns which are allocated a percentage of the total reward
pool. The vast majority of the campaigns require participants to use their
social media accounts to spread awareness about the project, however event
organizers may also be interested in producing quality content with campaigns
such as translation (e.g., of a whitepaper or website content), copywriting,
or visual content creation such as infographics or visual art.

Over time, certain norms have arisen to make it easier for users to
participate in a bounty event. Although for some events the procedure differs
slightly, most events follow a three-part structure:

  1. 1.

A user shows their interest in participating by posting a “proof of
registration” comment and including information about themselves (e.g.,
Telegram username, campaigns joined, crypto wallet address) which may differ
slightly from event to event.

  2. 2.

The participant completes the tasks defined in the campaign rules section. The
event can last from a few weeks to a few months or even up to a year,
therefore the tasks can be completed more than once (usually once a week).

  3. 3.

The participant provides evidence of engagement with the project by posting
“proof of participation” comments that include some required information
(e.g., social media handle, social media interaction links) which is usually
similar for all events. However, a small subset of events uses other means for
providing evidence, such as Google forms.

Users that employ social media bot accounts are identified by the event
organisers and are removed from the event. Participants are rewarded for their
contribution at the end of an event. Each campaign has different amounts of
stakes that are distributed to users which is proportional to the difficulty
of the task, the quality of content, or the value provided by the user (e.g.,
a user with more Twitter followers will be allocated more stakes for the same
task). The total reward can be calculated with the formula

$User\>Rewards=\frac{Campaign\>Prize\>Pool}{Distributed\>Stakes}*User\>Stakes$

## Data Collection

In this section we describe our data-scraping methods. The process consisted
of crawling subforum pages, user pages, thread pages, images, and Google
spreadsheets. To achieve this, a Python crawler was developed which used HTML
requests library777https://github.com/psf/requests to fetch images and
information from forum pages,
BeautifulSoup888https://github.com/waylan/beautifulsoup to parse HTML and
gspread library999https://docs.gspread.org/en/v5.7.0/ to access Google Sheets
API endpoints.

#### Subforum and thread crawling

The forum comprises multiple subforums, which contain a collection of threads
divided into pages. Each subforum has a unique identifier; Bounties(Altcoins)
ID is 238. Each page in the subforum contains up to 40 threads and has a
unique identifier which starts at 0 for the first page and is increased by 40
for every subsequent page. Additionally, each subforum page contains a
navigation strip, which can be utilized to retrieve the last page ID. This
allowed us to generate a unique URL for each page by inserting the identifiers
into https://bitcointalk.org/index.php?board=238.page_id. Crawling all pages
resulted in retrieval of information from 15K unique events.

#### Comments crawling

Each forum thread has a unique identifier retrieved during subforum crawling.
Similarly to subforum page, each thread has a comment page ID which starts at
0 for the first page and is increased by 20 for each subsequent page and a
navigation section which can be used to derive the last comment page ID. This
allowed us to generate unique URLs for each comment page by inserting unique
identifiers into https://bitcointalk.org/index.php?topic=thread_id.page_id.
10M comments were obtained after crawling each page.

#### User profile crawling

Forum users are given a unique identification number which was retrieved
during comment crawling. Inserting it into
https://bitcointalk.org/index.php?action=profile;u=user_id allowed us to
access each user’s profile page. Information on 185K users was retrieved after
crawling every page.

#### Image crawling

Some events in the forum used images to convey information, which had to be
retrieved, converted to text and inserted back into the main post for more
accurate processing. The image URLs could be found in HTML `<img>` elements of
the main forum post. In total, 64K unique image URLs were detected out of
which 56K were downloaded successfully and 8K were no longer being hosted.

#### Spreadsheet crawling

Some events used Google spreadsheets to track user progress and keep user
information in one place. Spreadsheet URLs were collected from main posts
using regular expressions to match spreadsheets
domain101010https:///www.docs.google.com/spreadsheets and cleaned to extract
unique spreadsheet identifiers. Using this information, all rows were
retrieved from Google sheets by querying Google Sheets API with gsread. This
resulted in the retrieval of more than 20M distinct rows from Google
spreadsheets.

## Data Processing

In this section we provide an overview of how the collected data were
processed and labeled and we explain the structure of the dataset.

### Forum Threads

The primary focus of the dataset is bounty events. This section explains the
methods used to extract information from the main post as well as how such
events were distinguished from other sorts of events in the subforum.

#### Categorization

Information from 15,870 unique events was fetched from the first thread
creation on 13-May-2014 to 31-Dec-2022. Each thread was assigned one or more
categories based on the title, main post comment and user replies. The
category of Bounty was assigned to threads that:

  1. 1.

Had seven or more out of 14 common Bounty event traits. Nine of the traits
involved matching against a single word (Stake, Rewards, Campaign, General
rules, Token, Bounty, Pool, Spreadsheet, Whitepaper). Five traits had multiple
synonyms (Rules: 3, Reward allocation: 19, Proof: 2, Signature: 2 and Social
media campaigns: 22); or

  2. 2.

Had six out of the 14 common bounty event terms and more than 100 replies or
the terms included Bounty and Whitepaper; or

  3. 3.

More than 50% of comments were classified as registration / participation (see
next section for details) and the event was closed (the main post, title or
both removed) or had more than 200 replies.

Events containing the keyword bounty in the title but not meeting any of the
requirements above were assigned a category of Bounty(low quality). These
categories had to be separated because titles were created by users and some
threads that contained the bounty keyword in the title had nothing to do with
bounty hunting.

Figure 1: Number of unique participations and registrations per week (multiple
comments of the same type by an individual in the same event are counted as
one).

Other categories included: Moved (event moved to another subforum), ICO
(Initial Coin Offering), Closed (removed content), Announcement, and Other.
7,177 events were identified as Bounty events, 3,432 threads were identified
as Bounty (Low Quality) and 5,261 events were assigned one of the other
categories. While just fewer than half of the threads were categorized as
Bounty events, 96.4% of the collected comments were posted in Bounty events
and 1.2% to Bounty(Low Quality) events, demonstrating the importance of Bounty
events in the subforum.

#### Content processing

Information about forum events was gathered from several places. Data which
are included in every thread were collected from the subforum page which is
well structured. This information includes thread title, thread author,
username and unique ID, number of replies and views, and information about
last comment: date, author username and ID. The thread page also contained
structured information that was common between all events such as event
creation date and main post in HTML format. Further analysis was only carried
out on events assigned a category of Bounty.

All HTML `<a>` elements of the main post were inspected for links. By matching
domain names with regular expressions, URLs were divided into the following
categories: spreadsheet, image – embedded images that were converted to text,
forum – bitcointalk.org domain, social media and other – the rest of the URLs.

The remaining event information was obtained from text. To overcome a
BeautifulSoup parser limitation when converting HTML to text and to preserve
post structure, a new line character was inserted after each element that was
not `<br>` or inline. The text was split on new lines to aid the analysis
process.

The Reward pool size section of the event was identified by matching keywords
(e.g., ”Bounty pool”, ”Bounty details”). Subsequent lines were checked for
three common reward pool patterns (e.g., ”$100 worth of ABC”, ”ABC 500”, ”200$
in ABC token” ).

Token names were detected in title, reward pool and text (near phrases such as
”Token” or ”Coin”). Each of these strings was tested against a set of
conditions: two characters or longer, no words in a list of 27 phrases (e.g.,
“NFT”, “USDT”, “APY”), only contain characters A-Z and $, not a word in a USA
or UK dictionary. Strings that passed all checks and were most frequent are
included in the dataset.

Reward allocation was determined by matching phrases (e.g., ”Reward
allocation”, ”Budget”). Each subsequent line was inspected for `%` sign and
campaign name. If the percentages of one or more campaigns added up to a 100%,
then a list of campaign titles and reward allocation percentages was recorded
in the database.

Bounty events would often have sections for event general rules, and rules and
rewards for each campaign. Retrieving these sections comprised two steps;
detecting in which line the section began and where it ended:

  1. 1.

The general rules section was detected by matching the text for relevant
phrases (e.g., ”Bounty rules”). For each campaign this was achieved by
matching a phrase (e.g., ”Twitter campaign”) followed by the keywords “Rules”
and “Rewards”.

  2. 2.

Once the index of a line where the section began was found, the subsequent
lines were checked for an ordered or unordered list. If found, this
information outlined where the section began and finished.

### Thread Comments

#### Categorization

The dataset contains five comment types: mixed (participation and registration
in one), participation, registration, author (comments by the thread author)
and other. Comment category was assigned with a 4-step process:

  1. 1.

Check if userid of comment author and main post author match. If yes, assign
author category, else next step.

  2. 2.

Check how many out of six common proof of registration traits (headline
“#proof of authentication”, forum username, forum profile URL, telegram
username, campaigns joined, wallet address) the comment has.

  3. 3.

Check how many out of six common proof of participation traits (headline
“week” or “day”, social media URLs, social media campaign names, social media
username, numbered list, keywords such as ”Like” and ”Retweet”) the comment
has.

  4. 4.

If a comment passes a registration threshold of 4 traits (3 if there are fewer
than 5 lines in the comment) or has # symbol with one of 75 synonyms for
“proof of authentication” then the threshold for participation comment becomes
2 (it must contain social media links). If both thresholds are passed, the
comment is assigned a category of mixed else, it becomes registration. If
registration threshold is not passed, the comment is considered as
participation with threshold of 3 (must contain social media links or
keywords: “week”, “day”). If this threshold is passed the comment is assigned
a category of participation, else it is assigned to other.

One limitation of this process is detecting registration comments which
include little information, however it is very efficient at identifying and
distinguishing between registration and participation comments. We report the
results of a manual check of a random sample of 100 comments of each category
for different threshold values in the Supplementary Material. We also provide
the regular expressions used to match social media URLs.

#### Content processing

Each comment shares some mutual information retrieved from the forum: comment
id, author username and id, post time. Additionally, the thread id where the
comment was found is included in the dataset. Comments of type other and
author were not processed and contain the comment text saved in HTML format.
This choice was made to preserve the structure of the comment since it would
be lost if the comment was a reply (i.e., contained one or more `<div
class="quote">` elements).

On the other hand, participation, registration and mixed comments were
processed extensively. Mixed comments include a Boolean check and get analyzed
by both algorithms. Most of the information was extracted with conditional
statements and regular expressions utilizing library
re111111https://docs.python.org/3/library/re.html.

Registration comments include information such as campaigns a user registered
for, Telegram username, Twitter username, forum rank and post count (specified
by user), line number in spreadsheets and crypto wallet address. Additionally,
the dataset also contains other information which was rare and did not get
assigned any label.

Participation comments include information about user social media username
(with social media platform), campaigns participated in (based on
participation links found) and all proof of interactions which include
TweetIDs and social media links (Twitter (without id), Facebook, Instagram,
Telegram, Reddit, YouTube, Medium, LinkedIn, Discord, TikTok, Steemit, Image
sharing, Other)

#### Google spreadsheets

Each Google spreadsheet contains one or more sheets, which are processed
independently. The process comprises two parts:

  1. 1.

Many spreadsheets included banners of varying height, so we had to detect
which row contains column names by iterating through the first 15 rows and
checking which one has the most out of ten common column names (e.g.,
”timestamp”, ”twitter followers”, ”post count”).

  2. 2.

Label consequent rows based on column names.

Unidentified columns labels and data are also included in the dataset.
Identified columns include: timestamp, proof of registration post URL, forum
rank, forum profile URL, forum username, wallet address, email address,
stakes, twitter followers, twitter audit URL, post count, social media
username, social media profile URL.

### Images to Text

Using Tesseract OCR121212https://github.com/tesseract-ocr/tesserac, locally
stored images were converted to text and inserted into respective HTML `<img>`
elements. The process encountered a few limitations when dealing with
intricate graphics pictures. Occasionally when dealing with such images, there
were misinterpreted letters, random symbols, or parts of the text missing. The
dataset contains the text generated from locally retrieved images and
corresponding image URLs.

Data category | Entries collected
---|---
Total threads in subforum | 15,870
Bounty events | 7,177
Total comments in subforum | 10,024,001
Total comments in Bounty events | 9,655,442
Bounty comments (participation) | 6,926,577
Bounty comments (registration) | 1,670,871
Bounty comments (mixed) | 233,268
Bounty comments (author) | 107,650
Bounty comments (other) | 1,183,612
Social media links | 82,854,734
Images (converted to text) | 56,529
User profiles | 185,709
Google spreadsheets | 24,887
Lines in Google spreadsheets | 20,182,746
Labeled Google spreadsheets entries | 18,082,209
Table 1: Statistics of collected dataset.

### Structure of the Processed Dataset

Our data collection and analysis process resulted in a PostgreSQL database
with twelve tables. We exported the tables to twelve tsv files and added some
additional annotations. The final dataset contains four tsv files of
unprocessed information retrieved from the subforum, google spreadsheets, and
image hosting services. To accommodate the largest audience possible and to
aid further analysis by researchers, we also release eight processed tsv files
resulting from the cleaning, merging and annotation conducted in the
processing stage. In total, the files contain 159 fields (events – 31 fields,
users – 17, comments_participation - 29, comments registration – 21,
comments_author, comments_other and comments_raw – 7, rewards_and_rules – 4,
spreadsheets – 18, spreadsheets_raw – 5, images_raw – 3, threads_raw – 10).
More extensive description of each file and field can be found in the
Supplementary Material that come with the dataset. Table 1 shows some
statistics of the collected data.

## Exploratory Analysis

#### Subforum activity over time

Figure 1 shows forum activity in terms of unique user registrations and
participations. If a user created more than one participation/registration
comments in the same thread, they were only counted as one
participation/registration. To preserve plot space, the visualization starts
from 2017 as there are only ten threads that were created before that time.
From the illustration we can observe that bounty events first started gaining
traction in the second half of 2017 and peaked during the first half of 2018,
which aligns with the timeline of the first massive bull market
run131313https://coinmarketcap.com/charts/ and corresponding increase in
crypto adoption. During this period we can observe up to 30,000 unique
participations every week. Since then, the forum has stayed active with some
ups and downs in participation over time. After a substantial slump in 2019
and 2020 where activity fell to almost one tenth of the peak, activity started
to slowly recover in 2021 and falling again from 2022.

#### Size and activity of Bounty events

Most Bounty events are substantially large to make a difference in promoting a
project; 4,402 (61.4%) events had more than 100 unique participants and 837
(11.7%) had more than 1,000 unique participants. We found five outlier events
where more than 5,000 forum users participated (Figure 2).

The number of activities that participants engaged in varied substantially
across Bounty events. In the early days of the subforum the norms hadn’t
evolved yet and the comments did not have the structure that could help us
crawl them successfully and categorize them as registration or participation
comments. Still, a substantial number of events show relatively little
activity with limited participation from users. On the other hand, we also
identified a large number of events with several thousand participation
comments; each one typically being a weekly comment including many activities
across different campaigns (Figure 3).

Figure 2: Number of unique participants per Bounty event. Note that for
visualization purposes, the bins in the horizontal axis are not equal. Figure
3: Number of participation comments per Bounty event. Note that the y axis is
logarithmic.

#### Duration of events

Figure 4 shows the duration of Bounty events. The duration is determined by
calculating the time passed from the first post in the thread (which announces
the beginning of the event) to the last (which typically ends the event). An
event typically requires participants to post every week. However, sometimes,
the event can get paused and the thread blocked for a few days. For this
reason, we considered any period of more than three weeks between two
consecutive comments as the end of an event.

We can observe that 350 (5%) events last less than a day and 30% end in the
first month, which leads us to believe that the majority of Bounty events end
successfully. A small number of events (92) kept operating for more than 300
days. These projects may have a long-term objective to remain in the
cryptocurrency market and invest in a continuous bounty program.

Figure 4: Number of consecutive days a Bounty event stayed active.

#### Bounty campaign popularity

We discuss which campaigns are most likely to attract participating users and
how many links grouped by campaign type have been collected from user
comments.

Figure 5 illustrates how many unique participations for each campaign type
were recorded in the subforum each week. Twitter and Facebook are the most
utilized platforms across all time by a substantial margin. Telegram and
Instagram started being utilized considerably in recent years.

In Figure 6 we can observe campaign popularity from another angle. Twitter and
Facebook are still the most popular platforms with 42.1M and 33.9M URLs
collected respectively, while all other platforms account for less than 9% of
all links collected. No Telegram links were collected as these tasks did not
require users to share a proof of participation link. Campaigns that involved
more complex or time-consuming tasks, such as creating YouTube videos or
Medium blog posts are underrepresented compared to their relative importance.

#### Reward allocation

Bounty events typically involve more than one campaign, necessitating the
distribution of the reward pool across campaigns. Each campaign gets allocated
a percentage of the total pool based on the project owners’ preferences and
the goal of the event. We were able to determine reward allocation that added
up to 100% for only 2,705 of the 7,170 Bounty events, which we analyze below.
Figure 7 shows the average reward allocation for each campaign type as a
percentage of the full available reward pool. Campaigns that were encountered
fewer than five times are grouped under the category “rare campaigns”.

The campaign with the most funds allocated on average is Signature. Signature
is an exclusive campaign to the forum that requires users to update their
profile with a specific banner that is visible on all their posts, effectively
acting as advertisements for the Bounty event. This suggests that event
organizers may expect that forum users will consider campaigns with signatures
to be more professional, trustworthy or popular, and thus will be more likely
to join them.

The second most rewarded campaign type is Content creation. This campaign type
includes activities such as .gif and meme creation, infographics, and
copywriting. Together with the Rare category, which also attracts a
substantial level of rewards, this suggests that Bounty events are
particularly interested in unconventional, but also sophisticated and time-
consuming activities of project promotion.

Twitter and Facebook are the third and fourth most rewarded campaigns, which
is expected since they make up 91% of all links collected from proof of
participation comments and are key channels for spreading information about
Bounty events online.

Figure 5: Number of unique participations (multiple participation comments
from the same user in the same thread are counted as one) for different
campaign types over time. The legend is sorted from most to least popular
campaigns, top to bottom. Figure 6: Number of links collected from proof of
participation comments for different types of campaigns. Figure 7: Average
percentage of reward pool that gets allocated to different campaign types in
Bounty events.

## Research Opportunities Using the Dataset

The BitcoinTalk forum has a long history and has had significant influence in
the cryptocurrency space. As the largest subforum, and one that encourages
users to engage with social media, create content and participate in
discussions, the Bounties (Altcoins) dataset can provide useful insights to
researchers of various interests. Here we discuss some unique research
opportunities arising from our dataset.

#### Financial analysis

Market price manipulation has long been of interest to researchers, both in
the stock market and in the crypto space (e.g., Gandal et al. 2021). Further
studies can use our dataset to study the effect of coordinated social media
campaigns (Schou et al. 2022) by examining the price movement of the tokens
included in our dataset. This can show whether the bounty campaigns have been
successful and whether they were intended to con investors out of their
investments by inflating the price and then ”dumping” the token (Mirtaheri et
al. 2021; Nizzoli et al. 2020). Further research can also attempt to correlate
the activity in the forum or the prices of the promoted tokens with the
movements of the wider economy, Bitcoin or other cryptoassets (Gandal et al.
2021) to examine how these affect online campaigns.

#### Social media and network analysis

A lot of research has examined how and why people use multiple media
(Spiliotopoulos and Oakley 2020; Alhabash and Ma 2017). Our dataset, which
includes the details of multiple social media accounts for each user, can be
valuable for social media researchers studying cross-media user posting
practices. Researchers can also follow the social media links from the dataset
to gain a better understanding of the effect of these campaigns. For example,
how much of the discussion around a crypto project is organic and how much is
driven by bounties? How many followers of a crypto project are genuine and how
many are bounty hunters? Social network researchers can use the bounty hunter
profiles provided in the dataset to determine details about the networks
(Himelboim et al. 2017) of bounty hunters; How many are connected and follow
each other within and across media? What cliques are they part of? Are their
followers genuine or are they just following one another and effectively
promoting the project to the converted?

#### Natural Language Processing

Researchers can follow the social media links and download text content from
the Tweets, Facebook posts, Reddit discussions and blog posts to study the
specific language used by bounty hunters. Natural Language Processing (NLP)
techniques (Hirschberg and Manning 2015) can identify the topics discussed and
the linguistic characteristics of this content, in order to detect it and
protect social media users. Sentiment analysis techniques (Valle-Cruz et al.
2022) can be used to quantify the effect of changes in online sentiment
created by bounty campaigns on the crypto market (Xie, Chen, and Hu 2020).

#### Policy and regulation

Legal scholars and policy makers can use the dataset to reach a more nuanced
understanding of the crypto space and the relationship between social media or
online communities and cryptocurrency investing. Examination of the structure
and organization of Bounty events and the practices of crypto bounty hunters
will inform the evaluation of the current regulatory state and protect naive
investors. This is particularly important as the calls for regulation of the
crypto space become increasingly insistent (Henderson and Raskin 2019).

## Broader Impact of the Work and Ethical Considerations

Overall, there are numerous benefits that can be derived from our dataset. We
bring awareness to a large number of Bounty events and campaigns that can
potentially lead inexperienced crypto investors to make poor financial
decisions. Our dataset can be a useful resource for researchers that can study
the details and structure of the campaigns, gain insights into how the
cryptocurrency industry operates and how it interacts with its audience, and
develop ways to protect users and inform policy makers.

Of course, these benefits need to be considered together with the potential
risks of collecting the data and sharing the dataset, especially risks related
to unanticipated secondary use (Salganik 2019). With regards to the data
collection stage, while our dataset includes information and links from a
range of social media and other online platforms, we only collected data from
the BitcoinTalk forum. This data collection is in line with the Terms and
Conditions of the forum, and, in fact, the forum specifically encourages data
scraping from its
boards141414https://bitcointalk.org/index.php?topic=5208423.0. The collected
raw data were held in secure password-protected devices and cloud accounts.

A cross-media user dataset presents additional potential misuses compared to a
dataset from a single social media platform, such as more extensive profiling
and tracking, cyberstalking, and identity theft. In order to minimize the
risks associated with sharing the dataset, we removed email addresses.
However, we decided to keep other account information, such as social media
account IDs and crypto wallet addresses. It was clear to us that these
accounts were created for the purpose of participating in Bounty events and
there is minimal overlap with forum users’ personal social media accounts or
other personal information. The removal of personal information and the use of
non-personal (or ”throwaway”) social media accounts by the forum users keeps
the risks associated with sharing the dataset relatively low. It is possible
that a forum user in the future may decide to delete their account from the
forum and their account information will remain in our dataset, but we
consider that to be in line with the reasonable privacy expectations of public
forum users, especially since a significant amount of content scraping takes
place by other forum users in public151515see previous note. We do not foresee
that the release of our dataset can put any of the forum users in any
additional danger or risk, as the activities taking place during these Bounty
events are currently within the law. Furthermore, while we did not employ any
social media APIs to collect the data and, thus, we are not technically
subject to any of their Terms and Conditions, we still decided to not share
social media content (e.g., tweets) and to only share social media URLs and
Tweet IDs. Finally, this work received ethical approval from our institution.

#### Compliance with FAIR principles

Our dataset adheres to the four FAIR data
principles161616https://force11.org/info/the-fair-data-principles/. Our
dataset is Findable as it is assigned a unique and persistent Digital Object
Identifier (DOI): `10.5281/zenodo.7539178`. The dataset is indexed in a
searchable resource and contains rich metadata that describe it. The data and
metadata are retrievable by their identifier in an open and free manner,
making our dataset Accessible. The dataset is released in tsv format, making
it Interoperable. It is extensively documented with metadata describing every
data field in a text file that accompanies the data. We also provide further
details of the regular expressions used for some of the data processing we
conducted in a supplementary file and we also share files with the raw
collected data in order to enable unanticipated future uses. This makes our
dataset Re-usable.

## Acknowledgments

The reported research is supported by UKRI under Grant title: AGENCY: Assuring
Citizen Agency in a World with Complex Online Harms. Grant reference:
EP/W032481/2. Further support was received by the EPSRC Centre for Digital
Citizens. Grant reference: EP/T022582/1.

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# The Effect of Intrinsic Quantum Fluctuations on the Phase Diagram of
Anisotropic Dipolar Magnets

Tomer Dollberg  Juan Carlos Andresen  Moshe Schechter  Department of Physics,
Ben-Gurion University of the Negev, Beer Sheva 84105, Israel

###### Abstract

The rare-earth material $\mathrm{LiHoF_{4}}$ is believed to be an experimental
realization of the celebrated (dipolar) Ising model, and upon the inclusion of
a transverse field $B_{x}$, an archetypal quantum Ising model. Moreover, by
substituting the magnetic Ho ions by non-magnetic Y ions, disorder can be
introduced into the system giving rise to a dipolar disordered magnet and at
high disorders to a spin-glass. Indeed, this material has been scrutinized
experimentally, numerically and theoretically over many decades with the aim
of understanding various collective magnetic phenomena. One of the to-date
open questions is the discrepancy between the experimental and theoretical
$B_{x}-T$ phase diagram at low-fields and high temperatures. Here we propose a
mechanism, backed by numerical results, that highlights the importance of
quantum fluctuations induced by the off-diagonal dipolar terms, in determining
the critical temperature of anisotropic dipolar magnets in the presence and in
the absence of a transverse field. We thus show that the description as a
simple Ising system is insufficient to quantitatively describe the full phase
diagram of $\mathrm{LiHoF_{4}}$, for the pure as well as for the dilute
system.

Introduction.—Anisotropic dipolar magnets, realized in both single molecule
magnets and rare earth magnetic insulators, are at the forefront of quantum
research. The large anisotropy barrier allows their use as nano-magnets, with
possible applications in the operation of qubits and memory bits at reduced
sizes [1, 2, 3]. In lattice form, anisotropic dipolar magnets typically have
very small exchange interactions, allowing for efficient induction of quantum
fluctuations by applied transverse fields. Thus, anisotropic dipolar magnets
are perceived as experimental models for the transverse field Ising model
(TFIM). These unique characteristics motivated intense study of quantum
phenomena in these materials, including quantum phase transitions [4, 5, 6,
7], quantum annealing [8, 6, 9], domain wall dynamics [10, 11], and high-Q
nonlinear dynamics [12].

One of the most studied anisotropic dipolar magnets is $\mathrm{LiHoF_{4}}$
[8, 9, 13, 5]. Below the Curie temperature of
$T_{c}=$1.53\text{\,}\mathrm{K}$$ $\mathrm{LiHoF_{4}}$ orders
ferromagnetically due to the dipolar interaction between $\mathrm{Ho^{3+}}$
ions combined with its lattice structure [14]. By the inclusion of an external
transverse field the transition temperature is suppressed, until eventually it
is converted to a quantum phase transition (QPT) at
$B_{x}\approx$4.9\text{\,}\mathrm{T}$$ [4]. Additionally, disorder can be
introduced by randomly substituting some of the magnetic $\mathrm{Ho^{3+}}$
ions with nonmagnetic $\mathrm{Y^{3+}}$ ions, resulting in
$\mathrm{LiHo_{x}Y_{1-x}F_{4}}$, which presents a rich phase diagram—including
a spin-glass phase at low concentrations ($x\lesssim 0.25-0.3$) [15, 16,
17]—the result of the interplay of interactions, disorder, and quantum
fluctuations [18, 19, 20, 21, 22, 23, 24]

The $B_{x}-T$ phase diagram of $\mathrm{LiHoF_{4}}$ is indeed in qualitative
agreement with that of the transverse-field Ising model, but a quantitatively
correct description has proven enduringly elusive, specifically at the high-
temperature, low-field regime, where thermal, rather than quantum,
fluctuations are dominant [25, 23, 26].

The significance of off-diagonal terms of the dipolar interaction is well
appreciated in the presence of disorder and a transverse field, as they break
$\mathbb{Z}_{2}$ symmetry and transform spatial disorder to an effective
random longitudinal field, making this material one of the few magnetic
realizations of the random field Ising model [19, 20, 21, 22, 23]. In this
paper we establish the importance of offdiagonal dipolar (ODD) interaction
terms to the quantitative description of the phase diagram of
$\mathrm{LiHo_{x}Y_{1-x}F_{4}}$ at $0.4<x\leq 1$, which may thus provide
insights to open questions in the field. We show that even for the pure
system, and in the absence of a transverse field, the classical Ising model,
which does not take these terms into account, provides an insufficient
description of the system. The reason being that ODD interactions give rise to
quantum fluctuations which markedly affect the phase diagram. These
fluctuations are induced when ODD terms exert internal transverse fields that
lower the energy of the $\mathrm{Ho^{3+}}$ ions on which they are exerted. We
argue such fields are more prevalent in the paramagnetic (PM) phase than in
the ferromagnetic (FM) phase, thus favoring the former.

Figure 1: The full phase diagram of $\mathrm{LiHoF_{4}}$ as a function of
temperature and applied transverse field. (a) A compilation of previous
numerical works. Open squares are quantum Monte Carlo (QMC) results [27]. The
dotted line is 1/z calculation with $J_{\text{ex}}=$3.13\text{\,}\mathrm{mK}$$
and the dot-dashed line is the same calculation with
$J_{\text{ex}}=$1.16\text{\,}\mathrm{mK}$$ [28]. The solid line is a mean-
field calculation [29] which uses the latter exchange value. Triangles
represent results from several different experiments [4, 29, 26]. An apparent
trade-off is observed between theoretical predictions that match the
experimental results at low temperatures but completely fail at the low-field
regime, and ones that give correct zero-field $T_{c}$ but fail to predict the
correct $T_{c}(B_{x})$ dependence and give a poor match at the intermediate
$B_{x}$ region. (b) Results of this paper overlaid on top of previous
theoretical and experimental results. The green Xs are the numerical results
of this work with off-diagonal dipolar terms excluded and blue plus signs are
for numerical results where they are included. Both use
$J_{\text{ex}}=$1.16\text{\,}\mathrm{mK}$$.

Results from previous studies, using various Monte Carlo techniques [27, 25]
and mean-field analyses [4, 28], show a persistent discrepancy with
experimental results for the $B_{x}-T$ phase diagram [4, 23, 26, 29]. Namely,
when the theoretical results are fitted to the experimental results, either
the zero-field critical temperature or the low-temperature–high-field regime
can be made compatible with experiment, but not both. If the former is chosen,
then even at small fields the $T_{c}(B_{x})$ dependence is not theoretically
well reproduced, and if the latter, then the critical temperatures at low and
intermediate transverse fields are significantly overestimated. The fitting is
done by tuning the single free parameter representing the nearest-neighbor
antiferromagnetic interaction strength, where higher values correspond to
lower critical temperatures. This apparent trade-off can be clearly seen in
Fig. 1(a).

Employing classical Monte Carlo (MC) simulations with variable single-spin
magnetic moments, we find that the inclusion of ODD terms in the effective
Hamiltonian allows for fitting $T_{c}$ at zero field using the same exchange
parameter that accurately fits the data at low temperatures and high
transverse fields. At the same time, we find better agreement with
experimental results for the long unexplained weak dependence of $T_{c}$ on
the transverse field at small fields, and the linear dependence of $T_{c}$ on
$\mathrm{Ho^{3+}}$ concentration in the absence of a transverse field.

Theoretical considerations.—Off-diagonal terms of the dipolar interaction have
been known to give rise to many interesting phenomena in the case of the
diluted $\mathrm{LiHo_{x}Y_{1-x}F_{4}}$ in presence of an external transverse
field, where they do not cancel by symmetry [20, 21, 30, 22, 19]. We argue
that similar effects, arising from internal transverse fields exerted by the
single ion expectation values $\expectationvalue{J^{z}_{i}}$ on the $x$
angular momentum component $J^{x}_{j}$ through terms of the form
$V^{zx}_{ij}\expectationvalue{J^{z}_{i}}J^{x}_{j}$, make a significant impact
on the phase diagram even in the undiluted case. The reason is that these ODD
terms have a distinctly different contribution in the FM phase, where they are
more likely to cancel by symmetry, than in the paramagnetic phase, where they
are less likely to do so. For example, a pair of spins that lie along the
$a$-axis of the crystal will exert a transverse field on spins located between
them, above or below the axis connecting the two spins, if the two spins have
opposite orientations. Said field acts to lower the energy of the spin on
which it acts regardless of its state, thereby energetically favoring the
anti-aligned configuration of its two neighboring spins. See illustrations in
Fig. 2. This interaction thus constitutes a disorder-enhancing mechanism which
acts to decrease the critical temperature. It requires the existence of three
spins and correlation between two of them—an important aspect which will be
discussed further below.

Figure 2: Crystal and magnetic structures of $\mathrm{LiHo_{x}Y_{1-x}F_{4}}$.
(a) Crystal structure of $\mathrm{LiHoF_{4}}$. Only the $\mathrm{F^{-}}$ ions
nearest to the central $\mathrm{Ho^{3+}}$ site are shown. (b) Correlated
spins, more abundant in the FM phase, induce a field with vanishing transverse
component on their common intermediate neighbor. (c) Spins in opposite
orientations, more abundant in the PM phase, induce a nonzero transverse
magnetic field on the third spin. (d) When one of the $\mathrm{Ho^{3+}}$ ions
is replaced by an $\mathrm{Y^{3+}}$ ion, the remaining Ho induces a transverse
field on its neighbor regardless of its orientation. Thick green arrows
indicate dipole moments, small colored arrows show the magnetic field
generated by the two magnetic dipoles and a narrow black arrow qualitatively
indicates the magnitude of the transverse component of the magnetic field
exerted on the middle ion. The full effect captured in the simulation is a
result of the transverse field not just on the middle ion, but on all other
ions in the system. For further details on this point see Supplemental
Material [31].

Another effect of the transverse fields is to decrease the absolute value of
$\expectationvalue{J^{z}}$ for the two lowest single-ion electronic energy
states, by mixing them with the higher electronic states. This also
contributes to the reduction of $T_{c}$ just by reducing the dominant $zz$
dipolar term proportional to $\expectationvalue{J^{z}}^{2}$. This mechanism of
the correlation induced enhancement of the transverse field, by its nature, is
not likely to be captured in any sort of mean-field-like analysis as it
depends on the spatial fluctuations of the states. We posit that including the
ODD terms is necessary to explain the previously mentioned discrepancies
between theory and experiment.

Numerical details.—In order to examine the effect of ODD terms on the phase
diagram of $\mathrm{LiHo_{x}Y_{1-x}F_{4}}$ we perform Monte Carlo simulations
using an effective Hamiltonian derived building upon the work of Chakraborty
et al. [27], but keeping the ODD terms. In this way we get an effective
Hamiltonian,

$\begin{split}H_{\text{eff}}&=\sum_{i}V_{C}(\hat{\boldsymbol{J}}_{i})-g_{L}\mu_{B}\sum_{i}\hat{\boldsymbol{B}}_{i}\cdot\hat{\boldsymbol{J}}_{i}\end{split}$
(1)

where

$\displaystyle\hat{B}^{x}_{i}$ $\displaystyle=B_{x}-g_{L}\mu_{B}\sum_{j\neq
i}V^{zx}_{ij}\hat{J}^{z}_{j}$ $\displaystyle\hat{B}^{y}_{i}$
$\displaystyle=-g_{L}\mu_{B}\sum_{j\neq i}V^{zy}_{ij}\hat{J}^{z}_{j}$ (2)
$\displaystyle\hat{B}^{z}_{i}$
$\displaystyle=-\frac{1}{2}g_{L}\mu_{B}\sum_{j\neq
i}V^{zz}_{ij}\hat{J}^{z}_{j}-\frac{J_{\text{ex}}}{2g_{L}\mu_{B}}\sum_{j\in
NN}\hat{J}^{z}_{j}$

act as effective internal fields when taking their expectation values, thereby
transforming Eq. (1) to an effective Hamiltonian for the single spins $i$. The
$V_{C}(\boldsymbol{J}_{i})$ term is a crystal field potential which imposes an
Ising easy axis along the $c$ axis of the crystal, with a first excited state
at $\sim$10\text{\,}\mathrm{K}$$ above the ground-state doublet [28].
$V_{ij}^{\mu\nu}$ is the magnetic dipole interaction, $J_{\text{ex}}$ is the
nearest-neighbor exchange interaction coupling constant,
$\mu_{B}=$0.6717\text{\,}\mathrm{K}\text{\,}{\mathrm{T}}^{-1}$$ is the Bohr
magneton and $g_{L}=\frac{5}{4}$ is a Landé g-factor. $\boldsymbol{J}_{i}$ are
angular momentum operators of the $\mathrm{Ho^{3+}}$ ions. See further details
on the derivation of the effective Hamiltonian in the Supplemental Material
111See Supplemental Material at [URL will be inserted by publisher] for
further details on derivations and numerical methods and results..

Since the $\mathrm{Ho^{3+}}$ ions retain their Ising character up to
transverse fields well above the critical transverse field [30] we model a
single-ion as a 2-state Ising system under an applied field exerted by all
other ions in the system, as well as the external field. This applied field
not only shifts the energies of the two states, but also modifies the single
ion magnetic moments $\expectationvalue{J^{z}}$ associated with them. These
are the magnetic moments which in turn exert magnetic fields on other ions.
Therefore, each site has two possible states, $\ket{\uparrow}$ and
$\ket{\downarrow}$, and each of these has two quantities of interest
associated with it: local energy $\bra{\uparrow(\downarrow)}H_{\text{single-
site}}\ket{\uparrow(\downarrow)}$ and magnetic moment
$\bra{\uparrow(\downarrow)}J^{z}\ket{\uparrow(\downarrow)}$ where

$H_{\text{single-
site}}=V_{C}(\hat{\boldsymbol{J}})-g_{L}\mu_{B}\boldsymbol{B}\cdot\hat{\boldsymbol{J}}.$
(3)

The method of employment of the effective Hamiltonian within the Monte Carlo
simulations is detailed in the ”Numerical Methods” section of the Supplemental
Material [31].

Results.—Figure 1(b) shows the $B_{x}-T$ phase diagram of
$\mathrm{LiHoF_{4}}$. Our results, with ODD terms included and excluded, both
use the exchange parameter $J_{\text{ex}}=$1.16\text{\,}\mathrm{mK}$$
suggested in Refs. [28, 29] which corresponds to the fitting at low
temperatures and high transverse fields. It is easy to see that the simulation
with ODD terms excluded corresponds to the mean-field calculation, while the
results with ODD terms included are in close agreement with the experimental
results, for zero and small transverse fields.

Thus, the inclusion of the ODD terms results in good agreement with experiment
at zero transverse field without the need to choose $J_{\text{ex}}$ that is in
clear disagreement with experiment at lower temperatures and higher transverse
fields. For finite but small transverse fields we find that the decrease in
$T_{c}$ due to the ODD terms is maintained. We emphasize that our simulation
is a classical MC simulation, but one that allows for varying magnetic moments
due to the influence of transverse fields. Therefore, the $T_{c}(B_{x})$
dependence in our results is a consequence of the renormalization of
individual magnetic moments due to the quantum coupling of each of the Ising
doublet states to higher excited electronic states, as opposed to quantum
fluctuations between the two Ising doublet states. Hence, our model is
expected to be valid at low fields $B_{x}\lesssim$1\text{\,}\mathrm{T}$$ where
quantum fluctuations are small [30].

Figure 3: Transition temperature, $T_{c}$, vs. $\mathrm{Ho^{3+}}$
concentration, $x$, phase diagram from different sources: experimental from
Ref. [29] (triangles), numerical from Refs. [17, 38] (squares) and this work
with ODD terms included (circles) and ODD terms excluded (diamonds). A dotted
line shows the mean-field prediction (linear). Numerical results from this
work are scaled so that they agree with the experimental
$T_{c}(x=1)=$1.53\text{\,}\mathrm{K}$$. The results with ODD terms included
use $J_{\text{ex}}=$1.16\text{\,}\mathrm{mK}$$, while the results with ODD
terms excluded use $J_{\text{ex}}=$3.91\text{\,}\mathrm{mK}$$, as suggested,
e.g. in Ref. [25].

Another facet of the incomplete quantitative understanding of this material
has to do with the phase diagram of $\mathrm{LiHo_{x}Y_{1-x}F_{4}}$ as a
function of the $\mathrm{Ho^{3+}}$ concentration $x$ at zero applied field. At
moderate to high concentrations ($x\gtrsim 0.4$) experiments show a linear
dependence of $T_{c}$ on $x$, in agreement with the mean-field prediction [29,
15, 37], whereas the available numerical work seems to indicate a steeper
decline of $T_{c}$ as $x$ is reduced [38, 17].

We note here that the inclusion of the ODD terms in the effective low energy
Hamiltonian of the system leads to an effective 3-spin interaction,
proportional to the anti-correlation of two spins and the existence of the
third. We thus expect this term to depend strongly on Ho concentration,
allowing for its distinction from the excess antiferromagnetic exchange used
to parameterize the system, and for better agreement with the experimental
$x-T$ phase diagram. In Fig. 3 we present our results for $T_{c}$ as function
of Ho concentration $x$ in the presence of ODD terms and exchange parameter
$J_{\text{ex}}=$1.16\text{\,}\mathrm{mK}$$, and in the absence of ODD terms
and $J_{\text{ex}}=$3.91\text{\,}\mathrm{mK}$$. Indeed, the results with ODD
terms included show milder reduction of $T_{c}$ with decreased concentration,
in better agreement with the experimental findings of $T_{c}(x)=xT_{c}(x=1)$.

An additional microscopic indication of the effect of ODD terms can be
obtained by inspecting the distribution of local transverse fields. Figure 4
shows the distribution of $B_{x}$ at the end of the simulation, when the
system has reached thermodynamic equilibrium, for simulations where ODD terms
are included and where they are excluded, yet still considered at the end of
the simulation for the calculation of the effective transverse fields. It is
clear that, at a given temperature, when ODD terms are included the
distribution of $B_{x}$ becomes wider. This is expected, since when ODD terms
are included, configurations that maximize internal transverse fields become
more energetically favorable and are thus more abundant at any given
temperature.

Figure 4: Distribution of local $B_{x}$ at thermodynamic equilibrium for
system size $L=7$ with zero external transverse field and at $x=1$. The
temperatures used, around $T=$1.66\text{\,}\mathrm{K}$$, are below $T_{c}$
when ODD terms are excluded and above $T_{c}$ when ODD terms are included. The
black vertical lines at the end of each bar are standard errors. For each
distribution a solid smooth line is plotted as a guide to the eye, obtained by
convolution of the bin values with a Gaussian function. Where ODD terms are
excluded, they are nevertheless considered at the end of the simulation for
obtaining these effective transverse fields. One can clearly see that when ODD
terms are included, the distribution of $B_{x}$ is wider than when they are
excluded—the peak at $B_{x}=0$ is lower, compensated by higher values at the
exterior. This is an indication of the ODD induced mechanism at work.
Configurations maximizing internal transverse fields become more energetically
favorable and thus for any given temperature they are more common.

Discussion.—We have shown here that the description of anisotropic dipolar
systems by the Ising model, and in the presence of a transverse field by the
transverse field Ising model, is essentially insufficient. Even for the pure
system off-diagonal dipolar terms induce an effective three spin interaction,
enhancing paramagnetic fluctuations and lowering the critical temperature. We
have analyzed the effect of the ODD terms on the relation between critical
temperature and both transverse field and dilution, in that addressing
unanswered puzzles regarding discrepancies between theory and experiment. Our
results at small fields are obtained with the same exchange parameter used to
fit the phase transition at low temperatures and high transverse fields, and
produce improved fitting to experimental data at finite transverse fields and
as a function of Ho concentration. Thus, our results point to the need to
include the quantum fluctuations induced by the off-diagonal terms in any
theoretical consideration, classical and quantum, of anisotropic dipolar
systems. Examples are classical and quantum annealing protocols, and a
comprehensive quantum modeling of the system required to establish its full
phase diagram. For the latter, a full consideration of the hyperfine
interactions may be needed, as these may hinder spin flips also at
temperatures of the order of the zero field $T_{c}$, and may contribute to the
slow reduction of $T_{c}$ as function of applied field at low fields.

###### Acknowledgements.

We would like to thank Dror Orgad and Markus Müller for useful discussions.
M.S. acknowledges support from the Israel Science Foundation (Grant No.
2300/19).

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Supplemental Material: The Effect of Intrinsic Quantum Fluctuations on the
Phase Diagram of Anisotropic Dipolar Magnets

##  I Effective Hamiltonian with offdiagonal dipolar (ODD) terms included

Pure $\mathrm{LiHoF_{4}}$ forms a tetragonal structure with lattice constants
$a=5.175\ \mbox{\AA}$ and $c=10.75\ \mbox{\AA}$, as shown in Fig. 2a of the
main text. There are four $\mathrm{Ho^{3+}}$ ions per unit cell which form a
lattice with a basis with coordinates $(0,0,\frac{1}{2})$,
$(0,\frac{1}{2},\frac{3}{4})$, $(\frac{1}{2},\frac{1}{2},0)$ and
$(\frac{1}{2},0,\frac{1}{4})$ [23]. The complete Hamiltonian of
$\mathrm{LiHoF_{4}}$ in a transverse magnetic field is given by [27, 23]

$H=\sum_{i}V_{C}(\boldsymbol{J}_{i})-g_{L}\mu_{B}\sum_{i}B_{x}J_{i}^{x}+\frac{1}{2}(g_{L}\mu_{B})^{2}\sum_{i\neq
j}V_{ij}^{\mu\nu}J_{i}^{\mu}J_{j}^{\nu}+J_{\text{ex}}\sum_{\left\langle
i,j\right\rangle}\boldsymbol{J}_{i}\cdot\boldsymbol{J}_{j}+A\sum_{i}(\boldsymbol{I}_{i}\cdot\boldsymbol{J}_{i})$
(S.1)

where $V_{ij}^{\mu\nu}$ is the magnetic dipole interaction,
$V_{ij}^{\mu\nu}=\frac{\delta^{\mu\nu}|\vec{r}_{ij}|^{2}-3(\vec{r}_{ij})^{\mu}(\vec{r}_{ij})^{\nu}}{|\vec{r}_{ij}|^{5}}$.
$J_{\text{ex}}$ is the nearest-neighbor exchange interaction coupling
constant. $\mu_{B}=$0.6717\text{\,}\mathrm{K}\text{\,}{\mathrm{T}}^{-1}$$ is
the Bohr magneton and $g_{L}=\frac{5}{4}$ is a Landé g-factor.
$\boldsymbol{J}_{i}$ are angular momentum operators of the $\mathrm{Ho^{3+}}$
ions. $A$ is the hyperfine interaction strength, and $\boldsymbol{I}_{i}$ is
the nuclear spin operator, where the total $\mathrm{Ho}$ nuclear spin is
$I=\frac{7}{2}$. The $\mathrm{Ho^{3+}}$ ions may be randomly substituted by
nonmagnetic $\mathrm{Y^{3+}}$ ions to form $\mathrm{LiHo_{x}Y_{1-x}F_{4}}$.
The crystal field term $V_{C}(\boldsymbol{J}_{i})$ imposes an Ising easy axis
along the $c$ axis of the crystal, with a first excited state at
$\sim$10\text{\,}\mathrm{K}$$ above the ground-state doublet [28].

Following an approach analogous to that of Chakraborty et al. [27] we recast
the full Hamiltonian (S.1) as an effective transverse-field Ising model
Hamiltonian, but keep the neglected ODD terms. Of the diagonal dipolar terms,
we keep only the $zz$ interactions which have been established as the most
dominant, but we also keep the off-diagonal interaction terms. The dipolar
interaction is invariant under both $i\leftrightarrow j$ and
$\mu\leftrightarrow\nu$. We also use the fact that
$\commutator{J_{i}^{x}}{J_{j}^{z}}=0$ and
$\commutator{J_{i}^{y}}{J_{j}^{z}}=0$ when $i\neq j$. In accordance with
previous results we also keep only the $zz$ term among the three exchange
interaction terms. Additionally, we neglect the hyperfine interaction, as it
was found, at least within mean field approximation, not to cause a
significant difference in the $B_{x}-T$ phase diagram in the vicinity of the
classical phase transition [4]. The result is the effective Hamiltonian, given
in Eq. (1) in the main text, which acts on the full 17-dimensional Hilbert
space of the electronic angular momentum. The 2-state Ising space is created
anew at each MC operation by determining the two lowest energy states and
choosing one of them by the MC rules.

##  II Numerical Methods

In order to construct approximate many-body states of the full system, we
diagonalize the single-site Hamiltonian for each of the sites, given some
arbitrary initial set of magnetic moments. Using the magnetic moments obtained
for each spin, the fields (The Effect of Intrinsic Quantum Fluctuations on the
Phase Diagram of Anisotropic Dipolar Magnets) and ergo single-site
Hamiltonians (3) are updated, and the process is repeated until convergence.
Convergence is assumed when the absolute difference between the assigned
magnetic moment and the magnetic moment dictated by the local field, averaged
over all sites, is smaller than $\epsilon_{\text{tol}}=5\times
10^{-3}\left[\mu_{B}\right]$. Within this process, each spin is assumed to be
either ”up” or ”down”, as set by the MC simulation, and only the magnitude of
its magnetic moment is adjusted. The process is performed following each MC
spin-flip. In effect we neglect quantum many-body effects such as
entanglement, and instead consider each ion separately. Nevertheless, the
single ion is treated exactly by diagonalization of its Hamiltonian in a
manner that is self-consistent with all other ions. This method is somewhat
similar to the approach described in Ref. [32] under the name inhomogeneous
mean-field (iMF), with one important difference. We do not replace the $J^{z}$
operators in the Hamiltonian with their thermal averages but with their
quantum mechanical expectation values. Thus we allow the MC simulation to
sample thermal fluctuations which are required for the described mechanism to
come into effect.

The determination of the single-site energy and magnetic moment is performed
as follows: The Hamiltonian (3) is diagonalized numerically and its two low
energy levels are designated $\ket{\alpha}$ and $\ket{\beta}$ such that
$E_{\alpha}<E_{\beta}$. Next, the states $\ket{\alpha}$ and $\ket{\beta}$ are
identified as $\ket{\uparrow}$ or $\ket{\downarrow}$ according to their
$\expectationvalue{J^{z}}$ in the following manner: If
$\expectationvalue{J^{z}}{\alpha}>\expectationvalue{J^{z}}{\beta}$ then
$\ket{\alpha}\equiv\ket{\uparrow}$ and $\ket{\beta}\equiv\ket{\downarrow}$ and
vise versa otherwise. Hybridization between the $\ket{\uparrow}$ and
$\ket{\downarrow}$ states is suppressed due to the hyperfine interaction [18,
30], which we approximately take into account by introducing an artificial
longitudinal field in the determination of $\expectationvalue{J^{z}}$ that
ensures the $\ket{\uparrow}$ and $\ket{\downarrow}$ are not significantly
hybridized by the transverse field: For each applied local field,
$(B_{x},B_{y},B_{z})$, the $B_{z}$ component is replaced by
$1.1\times\sqrt{B_{x}^{2}+B_{y}^{2}}$ for the purpose of obtaining the
magnetic moment. The effect of this process can be seen in Supplementary
Figure S1. In practice the energy and magnetic moment are calculated as
described on a fixed grid of $B_{x}$, $B_{y}$ and $B_{z}$ from which they are
linearly interpolated during the simulation.

Figure S1: The magnetic moment $\expectationvalue{J^{z}}$ of the ”up” state of
a single $\mathrm{Ho^{3+}}$ ion as a function of the applied $B_{z}$ magnetic
field, for an applied $B_{x}=$1\text{\,}\mathrm{T}$$ field. The solid green
line shows the ”exact” result (without the hyperfine interaction), which shows
significant hybridization when $B_{z}$ is smaller than
$B_{x}=$1\text{\,}\mathrm{T}$$. The dashed blue line shows the value used in
this work, chosen as described in the text. The inset shows how this value
changes as $B_{x}$ is varied.

Periodic boundary conditions are used, and the dipolar interaction
$V_{ij}^{\mu\nu}$ is calculated using the Ewald summation method without a
demagnetization term [16, 33].

We use the parallel tempering Monte Carlo method [34]. To determine the
transitions at a given $x$ and $B_{x}$ we use the finite-size correlation
length [35, 16],

$\xi_{L}=\frac{1}{2\sin(k_{\text{min}/2})}\left[\frac{\left[\expectationvalue{m^{2}(0)}_{T}\right]_{\text{av}}}{\left[\expectationvalue{m^{2}(\boldsymbol{k}_{\text{min}})}_{T}\right]_{\text{av}}}-1\right]^{\frac{1}{2}}$
(S.2)

where

$m(\boldsymbol{k})=\frac{1}{N}\sum_{i=1}^{N}\expectationvalue{J_{i}^{z}}_{\psi}\exp(-i\boldsymbol{k}\cdot\boldsymbol{R}_{i}).$
(S.3)

Here $\expectationvalue{.}_{T}$ refers to a thermal average,
$\left[.\right]_{\text{av}}$ to a disorder average and
$\expectationvalue{.}_{\psi}$ to a quantum mechanical expectation value.
$\boldsymbol{R}_{i}$ is the location of the site $i$ and
$\boldsymbol{k}_{\text{min}}=(\frac{2\pi}{L},0,0)$. The finite-size
correlation length divided by the linear system size $L$ has a known scaling
form,

$\frac{\xi_{L}}{L}\sim\tilde{X}\left(L^{1/\nu}(T-T_{c})\right)$ (S.4)

so that for $T=T_{c}$ it is independent of the system size $L$. Hence, if a
transition exists, curves for different system sizes should cross at the
critical temperature. We simulate systems of linear sizes $L=6,7,8$. To find
$T_{c}$ we assume the scaling function (S.4) can be approximated by a third-
order polynomial close to the critical point:
$f(x)=p_{0}+p_{1}x+p_{2}x^{2}+p_{3}x^{3}$ (where $x=L^{1/\nu}(T-T_{c})$), and
perform a global fit for the six free parameters, $p_{0}\ldots p_{3}$, $\nu$
and $T_{c}$ using the Levenberg-Marquardt algorithm. All 17 individual plots
showing the analysis are presented in Supplementary Figures S2-S5. Statistical
errors are estimated using the bootstrap method [36]. See Supplementary Tables
1-2 for numerical results with statistical errors.

Equilibration is verified by logarithmic binning of the data, i.e. the
simulation time in terms of MC sweeps is successively increased by a factor of
2, and observables are averaged over that time. When all observables of
interest in three consecutive bins agree within error bars, the simulation is
deemed equilibrated [36].

For each value of $x$ or $B_{x}$ the simulation is performed twice: Once with
the off-diagonal dipolar terms included in the Hamiltonian (1) and once with
these terms omitted so that internal transverse fields are artificially
suppressed, i.e. $B^{x}_{i}=B_{x}$ and $B^{y}_{i}=0$ for all $i$ in (The
Effect of Intrinsic Quantum Fluctuations on the Phase Diagram of Anisotropic
Dipolar Magnets). The self-consistent calculation for the magnetic moments is
performed in both cases, as a means to establish its validity.

##  III Simulation Parameters

Supplementary tables 1-2 present the parameters and the results of the Monte
Carlo simulations performed for this work.

Supplementary Table 1: Simulation parameters at $x=1$ for different transverse fields $B_{x}$ and system sizes $L$, with ODD terms included and excluded. The equilibration/measurement times are $2^{b}$ Monte Carlo sweeps. $T_{\text{min}}$ [$T_{\text{max}}$] is the lowest [highest] temperature used and $N_{T}$ is the number of temperatures. $N_{sa}$ is the number of independent runs. All simulations use $J_{\text{ex}}=$1.16\text{\,}\mathrm{mK}$$. ODD terms | $B_{x}\ [\rm{T}]$ | L | b | $T_{\text{min}}\ [\rm{K}]$ | $T_{\text{max}}\ [\rm{K}]$ | $N_{T}$ | $N_{sa}$ | $T_{c}\ [\rm{K}]$ | $\nu$
---|---|---|---|---|---|---|---|---|---
Included | 0.0 | 6,7,8 | 10 | 1.528 | 1.628 | 24 | 50 | 1.5735(4) | 0.65(1)
Excluded | 0.0 | 6,7 | 10 | 1.738 | 1.838 | 24 | 50 | 1.7868(3) | 0.59(1)
Excluded | 0.0 | 8 | 11 | 1.738 | 1.838 | 24 | 50
Included | 0.3 | 6,7,8 | 10 | 1.512 | 1.612 | 24 | 50 | 1.5668(4) | 0.61(1)
Excluded | 0.3 | 6,7,8 | 10 | 1.739 | 1.839 | 24 | 50 | 1.7800(4) | 0.60(1)
Included | 0.6 | 6,7,8 | 10 | 1.498 | 1.598 | 24 | 30 | 1.5529(5) | 0.61(1)
Excluded | 0.6 | 6,7,8 | 10 | 1.727 | 1.827 | 24 | 30 | 1.7666(5) | 0.60(2)
Included | 1.0 | 6,7,8 | 10 | 1.47 | 1.57 | 24 | 30 | 1.5226(5) | 0.64(2)
Excluded | 1.0 | 6,7,8 | 10 | 1.705 | 1.804 | 24 | 30 | 1.7502(5) | 0.61(2)
Included | 1.5 | 6,7 | 10 | 1.42 | 1.52 | 24 | 30 | 1.4713(4) | 0.56(1)
Included | 1.5 | 8 | 11 | 1.42 | 1.52 | 24 | 30
Excluded | 1.5 | 6,7 | 10 | 1.67 | 1.77 | 24 | 30 | 1.7275(5) | 0.62(2)
Excluded | 1.5 | 8 | 11 | 1.67 | 1.77 | 24 | 30
Supplementary Table 2: Simulation parameters at $B_{x}=0$ for different dilutions $x\leq 1$ and system sizes $L$, with ODD terms included and excluded. The equilibration/measurement times are $2^{b}$ Monte Carlo sweeps. $T_{\text{min}}$ [$T_{\text{max}}$] is the lowest [highest] temperature used and $N_{T}$ is the number of temperatures. $N_{sa}$ is the number of independent runs. $x$ | ODD terms | $J_{\text{ex}}\ [\rm{mK}]$ | L | b | $T_{\text{min}}\ [\rm{K}]$ | $T_{\text{max}}\ [\rm{K}]$ | $N_{T}$ | $N_{sa}$ | $T_{c}\ [\rm{K}]$ | $\nu$
---|---|---|---|---|---|---|---|---|---|---
1.0 | Included | 1.16 | 6,7,8 | 10 | 1.528 | 1.628 | 24 | 50 | 1.5735(4) | 0.65(1)
0.83 | Included | 1.16 | 6,7,8 | 10 | 1.04 | 1.54 | 24 | 200 | 1.302(2) | 0.67(4)
0.67 | Included | 1.16 | 6,7,8 | 10 | 0.78 | 1.28 | 24 | 350 | 1.044(2) | 0.66(4)
0.46 | Included | 1.16 | 6 | 11 | 0.39 | 0.89 | 24 | 1000 | 0.656(2) | 0.70(3)
0.46 | Included | 1.16 | 7,8 | 12 | 0.39 | 0.89 | 24 | 1000
1.0 | Excluded | 3.91 | 6,7,8 | 10 | 1.463 | 1.563 | 24 | 30 | 1.5177(5) | 0.59(1)
0.83 | Excluded | 3.91 | 6,7,8 | 10 | 0.991 | 1.491 | 24 | 200 | 1.241(2) | 0.68(4)
0.67 | Excluded | 3.91 | 6,7 | 10 | 0.724 | 1.224 | 24 | 350 | 0.978(2) | 0.64(4)
0.67 | Excluded | 3.91 | 8 | 11 | 0.724 | 1.224 | 24 | 350
0.46 | Excluded | 3.91 | 6 | 11 | 0.354 | 0.854 | 24 | 1000 | 0.594(2) | 0.65(2)
0.46 | Excluded | 3.91 | 7 | 12 | 0.354 | 0.854 | 24 | 1000
0.46 | Excluded | 3.91 | 8 | 13 | 0.354 | 0.854 | 24 | 1000

##  IV Finite-Size Scaling analysis

This section presents the results of the finite size scaling analysis used to
obtain $T_{c}$ from each simulation. For each simulation, the value of $B_{x}$
and $x$ is set, and then a range of temperatures around $T_{c}$ is simulated
for several system sizes. These simulations are used to obtain the averages
required to calculate the finite-size correlation length $\xi_{L}$ which is
used in the finite-size scaling method to obtain an exact estimate of $T_{c}$.
Supplementary Figures S2-S5 show the finite-size scaling results. Despite
expected corrections to scaling arising from the relatively small system
sizes, all figures show reasonable collapse to a universal curve.

Figure S2: Finite size scaling results for simulations with ODD terms included
at $x=1$ and $J_{\text{ex}}=$1.16\text{\,}\mathrm{mK}$$. Each graph is divided
in two: the left side shows the raw results with the curves for different
system sizes crossing at $T_{c}$ and the right side shows the same results
with the $T$ axis rescaled showing a collapse onto a universal curve. The
parameters for each simulation are listed within the figures.

Figure S3: Finite size scaling results for simulations with ODD terms
excluded. Each graph is divided in two: the left side shows the raw results
with the curves for different system sizes crossing at $T_{c}$ and the right
side shows the same results with the $T$ axis rescaled showing a collapse onto
a universal curve. The parameters for each simulation are listed within the
figures.

Figure S4: Finite size scaling results for simulations with ODD terms included
at $x\leq 1$ and $J_{\text{ex}}=$1.16\text{\,}\mathrm{mK}$$. Each graph is
divided in two: the left side shows the raw results with the curves for
different system sizes crossing at $T_{c}$ and the right side shows the same
results with the $T$ axis rescaled showing a collapse onto a universal curve.
The parameters for each simulation are listed within the figures.

Figure S5: Finite size scaling results for simulations with ODD terms excluded
at $x\leq 1$ and $J_{\text{ex}}=$3.91\text{\,}\mathrm{mK}$$. Each graph is
divided in two: the left side shows the raw results with the curves for
different system sizes crossing at $T_{c}$ and the right side shows the same
results with the $T$ axis rescaled showing a collapse onto a universal curve.
The parameters for each simulation are listed within the figures.

##  V Estimation of the Effective Interaction

The excess change in the energy of the system resulting from the offdiagonal
dipolar (ODD) interactions can be viewed, to some extent, as a change in the
effective longitudinal (zz) pair interactions. This effective change in the
pair interactions is specific to each pair, is dependent on the specific
spatial configuration at finite concentration, on transverse magnetic field,
and to some extent on the specific configuration of all spins in the system.
Yet, it is useful to calculate it in some specific configurations to allow
estimation of the contribution of the offdiagonal terms to the energy of the
system, and thus to $T_{c}$. To calculate the effective interaction due to the
ODD mechanism, we take as one example a ferromagnetic system of all spins,
except two, in the up state. We then calculate the energy of the full system
for four different configurations of these 2 spins, $E(\uparrow\uparrow)$,
$E(\uparrow\downarrow)$, $E(\downarrow\uparrow)$, $E(\downarrow\downarrow)$,
where the energy is calculated using $H_{\text{eff}}$ given in Eq. (1) in the
main text, with magnetic moments calculated self consistently as described in
the text. The interaction between the two spins is then given by

$J=\frac{1}{4}\left[E(\uparrow\uparrow)+E(\downarrow\downarrow)-E(\uparrow\downarrow)-E(\downarrow\uparrow)\right]$

which is a result of the direct longitudinal dipolar interaction, the
(nearest-neighbor) exchange interaction and the effective excess interaction
due to the ODD mechanism. By repeating the above calculation with and without
offdiagonal dipolar terms and subtracting one from the other, we get an
estimate of the excess interaction due to the ODD mechanism
$J_{\text{eff}}\equiv J(\text{ODD included})-J(\text{ODD excluded})$.

This effective interaction is calculated for pairs of spins along the x axis,
along the z axis and for nearest neighbors (which can be seen in Fig. 2a in
the main text). Results are obtained for a system of linear size $L=14$, where
dipolar interactions are calculated using the Ewald method. For nearest
neighbors the effective interaction is $J_{\text{eff}}=-31\ \mathrm{mK}$
(ferromagnetic), for the two closest spins along the x axis it is
$J_{\text{eff}}=14\ \mathrm{mK}$ (antiferromagnetic), and for the two closest
spins along the z axis it is $J_{\text{eff}}=11\ \mathrm{mK}$
(antiferromagnetic). These values amount to roughly 7-23% of the standard
longitudinal dipolar interactions of the respective pairs. Similar results are
obtained for an average over a random distribution of the spins.