# Challenges in Pre-Training Graph Neural Networks for Context-Based Fake News
Detection: An Evaluation of Current Strategies and Resource
Limitations††thanks: Preprint accepted at LREC-COLING 2024

Gregor Donabauer
Information Science
University of Regensburg
Germany
<EMAIL_ADDRESS>
Udo Kruschwitz
Information Science
University of Regensburg
Germany
<EMAIL_ADDRESS>

###### Abstract

Pre-training of neural networks has recently revolutionized the field of
Natural Language Processing (NLP) and has before demonstrated its
effectiveness in computer vision. At the same time, advances around the
detection of fake news were mainly driven by the context-based paradigm, where
different types of signals (e.g. from social media) form graph-like structures
that hold contextual information apart from the news article to classify. We
propose to merge these two developments by applying pre-training of Graph
Neural Networks (GNNs) in the domain of context-based fake news detection. Our
experiments provide an evaluation of different pre-training strategies for
graph-based misinformation detection and demonstrate that transfer learning
does currently not lead to significant improvements over training a model from
scratch in the domain. We argue that a major current issue is the lack of
suitable large-scale resources that can be used for pre-training.

##  1 Introduction

Fake news spreading online nowadays poses one of the biggest threats to
democracy [1], presenting a urgent requirement for the development of robust
detection techniques [2]. Advances in recent years were mainly driven by
context-based methods, e.g. [3, 4, 5]. Compared to the content-based paradigm,
that aims to identify fake articles based on their textual content, context-
based methods include various other signals as additional information [6].
This for example comprises information from social media platforms, like posts
associated with news articles, sharing patterns, and the profiles of connected
users which are helpful cues for the task at hand [7, 8]. These different
signals are often put together in graph data structures and can then be
aggregated using Graph Neural Networks (GNNs), e.g. [9, 10]. At the same time,
pre-training of Large Language Models (LLMs) on massive text corpora has
recently transformed the field of Natural Language Processing (NLP) as this
approach equips the trained models with an understanding of general aspects of
written language, e.g. [11, 12, 13]. These advances have led to impressive
progress across the entire spectrum of NLP, also in the area of fake news
detection, e.g. [14, 15]. In a study conducted by [16], they presented a
comprehensive evaluation of pre-training techniques within the context of
GNNs. Inspired by this progress, we propose to adopt similar strategies to
learn universal attributes of heterogeneous social media context graphs
relevant for the domain of context-based fake news detection.

We present a set of experiments across two fake news detection datasets
evaluating how different pre-training strategies perform in the domain.
Towards the end we discuss current limitations of the approach and summarize
our observations and findings. In the spirit of LREC-COLING we share our
implementations and pre-trained models with the community to foster
reproducibility111https://github.com/doGregor/pretrain_gnns_FakeNewsNet.

##  2 Related Work

###  2.1 Graph-Based Fake News Detection

We start by providing a brief contextualization within existing related work
on the context-based paradigm of graph-based fake news detection. Sometimes
context refers to graph-like conversation patterns of posts related to news on
social media platforms [17, 18, 3]. However, many approaches also include user
[19, 4, 10] or publisher [20, 5] relationships as contextual information.
Recently, it has been demonstrated that taking into account such information
within heterogeneous structures can lead to further improvements [21, 22, 23].
In conclusion, graph structures, particularly heterogeneous ones, offer high
potential in modelling diverse dimensions of contextual information resulting
in notable improvements in the domain of fake news detection.

###  2.2 Pre-Training of Graph Neural Networks

While pre-training neural networks has become standard in NLP, we will briefly
summarize relevant work in graph machine learning below. The most
comprehensive evaluation of pre-training strategies was performed by [16] in
the domain of Biology and Chemistry. Generally, GNN pre-training is mainly
employed in specific domains like recommender systems [24] and biomedical
networks [25]. Some concentrate on large-scale heterogeneous graphs, using
methods like contrastive learning-based node and schema-level pre-training
[26], node masking [27], or structural clustering with attention aggregation
[28]. Others target large homogeneous graphs with node and graph-level pre-
training tasks [29], or generative approaches applicable to both homogeneous
and heterogeneous large-scale graphs [30]. In this study, we aim to bridge the
gap of applying GNN pre-training approaches within the domain of graph-based
fake news detection.

##  3 Methodology

###  3.1 Data and Graph Setup

Existing approaches to context-based fake news detection are primarily
associated with only a small number of datasets. This includes Twitter 15 and
Twitter 16 [31], PHEME [32] and WEIBO [33], which are all very limited in
terms of size and social context provided. Apart from that there is
FakeNewsNet [34] which stands out as the largest dataset in the domain and
offers a comparatively large amount of contextual features. This dataset
comprises news articles that have been fact-checked by reputable fact-checking
organizations and were found circulating on social media platforms.
Furthermore, it includes identifiers for the associated social media data,
allowing to retrieve and analyze this information through a recrawling
processes. Recent efforts to introduce even more extensive and diverse
resources were made by [35], who provide social context cues similar to
FakeNewsNet but in greater amount and even with multilingual contents.
However, the vast majority of these datasets heavily depends on contextual
information from access-restricted social media platforms, predominantly
Twitter. Recently, new restrictions posed on the Twitter
API222https://developer.twitter.com/en/docs/twitter-api are making it
exceedingly challenging for researchers to gain access to these datasets at a
reasonable cost or with a reasonable amount of effort. Due to these
limitations, we conduct our experiments using the FakeNewsNet dataset. A
useful characteristic of FakeNewsNet is that it consists of two subsets:
Politifact (483 graphs) and Gossipcop (12,214 graphs). This allows us to
conduct pre-training on one subset and perform fine-tuning on the other
subset. However, the amount of data that potentially can be used for pre-
training of GNNs is very limited compared to the amounts of data that are used
for pre-training of LLMs in NLP. The majority of methods that incorporate
social media context through graph structures, primarily target the
identification of individual nodes within large graphs. These nodes hold
contextual information alongside the news articles to classify. This is
different to the approach of [16], who focus their methodology on a large
number of small graphs, each representing an individual data point. We thus
adopt our previous approach [23] to construct the social media context graphs
as we formulated the problem as a graph classification task which makes the
setup similar to [16]. Each graph consists of three different sets of nodes:
(1) the actual news article to classify at the center of the graph, (2)
tweets/retweets/timeline tweets related to the article, and (3) user profiles
of people linked with the aforementioned posts. Each node set is disconnected
from the other sets of nodes and all initial node representations are BERT
text embeddings. These node sets are linked via multiple types of edges
([tweet, cites, article], [user, posts, tweet], [user, posts, retweet],
[retweet, cites, tweet] and [user, posts, timeline tweet]) and each edge type
can be interpreted as an individual adjacency matrix that connects two
distinct sets of nodes. Data setup and hyperparameters in our work are adopted
from our previously provided implementations
[23]333https://github.com/doGregor/Graph-FakeNewsNet.

However, one issue we need to solve is that these are heterogeneous graphs,
compared to homogeneous graphs used by [16]. We therefore modify the pre-
training strategies to be applicable to graphs with different node/edge types.

###  3.2 Pre-Training Objectives

The pre-training objectives we adopt have been proven most effective in [16]
and are twofold: we use node-level objectives and graph-level objectives. The
latter can predominantly be viewed as strategies aimed to learn holistic graph
representations useful for the fake/real news graph classification task. On
the other hand, the node-level tasks serve as regularization of the GNNs to
mitigate the risk of overfitting to the self-supervised graph-level pre-
training.

####  3.2.1 Node-Level Objectives

We use two different node-level pre-training objectives, namely node masking
and context prediction.

Node masking is similar to word masking in pre-training of LLMs. We mask a
proportion of 15% of all nodes with placeholder values at random. Given that
the initial node attributes are continuous (BERT embeddings) we simply set all
masked nodes to a uniform value of $1.0$ in the dimension of the original
features. Since we have different node types within our heterogeneous social
media graphs, we only mask nodes of type tweet/retweet/timeline tweet which
have demonstrated the biggest impact on classification performance. During
pre-training we jointly train a GNN that serves as node encoder and a fully
connected neural network for feature reconstruction.

For context prediction we first create subgraphs of the original graphs. As
can be seen in Figure 1 we remove the centre node (news node) from the
original graphs to create the subgraphs. We perform negative sampling with a
ratio of $1$ to create matching and non-matching pairs of original graphs and
subgraphs. In particular, the learning objective is a binary classification
task of whether context and actual graph belong to the same news node. During
pre-training we jointly train one GNN that serves as node encoder of the
original graph and another GNN that serves as node encoder of the context
graph. Afterwards we perform mean pooling over all tweet and user nodes and
predict whether the embeddings originate from the same graphs. As we are
classifying the whole (sub-) graphs this is slightly different to [16].

Figure 1: Formulation of pre-training on context prediction between article
graph and context graph.

####  3.2.2 Graph-Level Objective

As described previously, the graph-level objective serves the purpose of
learning holistic graph characteristics that are useful for the upcoming
fake/real news graph classification task. We compared the influence of
different graph properties on the fake news detection task which revealed that
the number of retweets associated with an article seems to have a comparably
high influence. We therefore employ the self-supervised task of predicting the
number of retweet nodes each graph holds. Since they are part of the same node
set as normal tweet nodes and timeline tweet nodes, the GNN faces a challenge
in determining the count of retweet nodes solely from the graph properties.
Specifically, it cannot read out the number of retweet nodes from a dedicated
node set, as these nodes are intertwined with the other tweet nodes (as
described in Section 3.1). The GNN must extract this understanding by
leveraging the structural information inherent in the graph, which from our
point of view makes this prediction task more challenging.

###  3.3 Experiments

Inside our node encoder we use Heterogeneous Graph Transformer (HGT) GNN
layers [36] that are based on the Transformer architecture [37] which has
proven suitable for pre-training in the domain of NLP [11] and in other
studies on pre-training of GNNs [27, 30]. The encoder consists of two stacked
HGT layers where each has 64 input dimensions. The GNN learns distinct sets of
weights for each edge type present within the heterogeneous graph and applies
them for message passing and aggregation of each node’s neighborhood to update
the initial node representations. We first apply the two stacked HGT layers,
and finally a fully connected layer that maps the learnt node representations
to the target space (graph labels during fine-tuning or node features/context
during pre-training).

During pre-training we use a batch-size of 128 and run 50 epochs for node
level pre-training tasks with a learning rate of $0.001$; 25 epochs
(Politifact) and 50 epochs (Gossipcop) respectively for self-supervised graph-
level pre-training, again with a learning rate of $0.001$. For fine-tuning we
again use a batch-size of 128 and a learning rate which amounts to $0.001$
(same value as used during pre-training in [16]).

Our implementations are based on PyTorch Geometric [38]. All experiments are
executed using a single NVIDIA RTX A6000 GPU with 48 GB graphical memory size.

We evaluate how different combinations of node-level and graph-level pre-
training tasks affect fake news detection performance after fine-tuning. Pre-
training is always performed on one full subset (Politifact or Gossipcop),
while fine-tuning is performed on 80% of the left-out subset and tested on the
remaining 20% (with 5-fold cross validation).

Furthermore, we are conducting experiments involving fine-tuning with limited
resources. This is particularly interesting due to the high cost associated
with acquiring labeled data. This additional evaluation focuses on the
utilization of models pre-trained on the Gossipcop dataset as we have a
greater amount of pre-training data. Subsequently, we perform fine-tuning
using 50 data points ($\sim$10%) from the Politifact dataset. The model’s
performance is then evaluated on the complete, remaining test set, using
5-fold cross-validation.

Pre-training task | Fine-tuning on POL | Fine-tuning on GOS
---|---|---
Node-level |  Graph-level | P | R | ACC | F1 | P | R | ACC | F1
– |  – | 0.977 | 0.978 | 0.977 | 0.977 | 0.966 | 0.979 | 0.984 | 0.972
ContextPred |  – | 0.975 | 0.976 | 0.975 | 0.975 | 0.962 | 0.977 | 0.982 | 0.969
NodeMasking |  – | 0.981 | 0.982 | 0.981 | 0.981 | 0.959 | 0.978 | 0.981 | 0.968
– |  #RT | 0.973 | 0.974 | 0.973 | 0.973 | 0.963 | 0.977 | 0.982 | 0.970
ContextPred |  #RT | 0.955 | 0.956 | 0.955 | 0.954 | 0.951 | 0.977 | 0.978 | 0.963
NodeMasking |  #RT | 0.973 | 0.974 | 0.973 | 0.973 | 0.969 | 0.978 | 0.984 | 0.973
Table 1: Precision, Recall, Accuracy and macro F1-scores for fake news detection on the Politifact (POL) and Gossipcop (GOS) datasets respectively after fine-tuning. Pre-training task | Fine-tuning POL
---|---
Node-level |  Graph-level | ACC | F1
– |  – | 0.942 | 0.942
ContextPred |  – | 0.953 | 0.952
NodeMask. |  – | 0.965 | 0.965
– |  #RT | 0.957 | 0.957
ContextPred |  #RT | 0.957 | 0.956
NodeMask. |  #RT | 0.953 | 0.952
Table 2: Accuracy and macro F1 scores for fine-tuning with only 50 labeled
samples from Politifact on models pre-trained on the Gossipcop dataset.

##  4 Results

In line with common practice in NLP, we report precision, recall, accuracy and
macro F1 scores for all setups [39, Ch. 4]. All values are the average results
obtained by 5-fold cross validation.

As illustrated in Table 1 there are only minimal fluctuations between all the
setups without any significant differences (using pairwise t-tests at $p<0.01$
and Bonferroni correction on the F1- and Accuracy scores derived from cross-
validation across all experimental setups). The most positive impact can be
seen for the pre-training strategies involving the node masking approach. For
comparison, a strong state-of-the-art baseline excluding any graph-based
approaches, achieves $0.956$ ACC and $0.952$ F1-score for Politifact as well
as $0.963$ ACC and $0.923$ F1-score for Gossipcop [15]. The best-performing
baseline using a graph-based approach achieves $0.975$ ACC and $0.975$
F1-score for Politifact as well as $0.990$ ACC and $0.986$ F1-score for
Gossipcop [22]. Thus, our results on Politifact are marginally (though not
significantly) better than the best-reported results in the literature.

However, the potential positive impact on model performance becomes more
obvious when fine-tuning on a smaller fraction of the actual labeled data. In
that case the learnt properties during pre-training lead to higher accuracy
and F1 scores (compare table 2) for all setups. Again, these differences are
not significant.

##  5 Discussion and Limitations

While our experiments do not demonstrate significant improvements by deploying
a pre-training approach, we would like to discuss this in a broader context.

Datasets: The most relevant issue relates to the resources available to
conduct our experiments. As already argued in Section 3.1, their sizes are
notably small in comparison to the extensive amount of training data employed
for pre-training in the domain of NLP. A general advantage of self-supervised
pre-training is that these strategies do not require any labeled data. At the
same time, online social media interactions that can serve as context are
available in rich amount which means that there is a theoretical solution to
this problem. However, current limitations related to the platforms’ APIs are
making it almost impossible to solve this problem at a reasonable cost or with
a reasonable amount of effort. A potential workaround could be data
augmentation by, for example, generating graphs used for pre-training at
various temporal snapshots during the evolution of user interactions with a
news article [40]. Concluding, the community working on these topics would
benefit from solving the problem of limitations in access to shareable
resources, similar to the practice in the area of linguistics [41].

Another factor influencing the overall results could be that performance on
FakeNewsNet is already very high, which is why choosing datasets that are more
challenging and thus leave more room for improvements could better underscore
the potential improvements achievable through pre-training. However, we note
that our choice of FakeNewsNet for our exploratory experiments was deliberate,
driven by multiple considerations: (1) its status as the largest dataset in
the domain, a critical factor for effective pre-training; (2) its richness in
social context cues compared to other datasets, allowing to construct
heterogeneous graphs with diverse node and edge types; and (3) its division
into two subsets, allowing a structured approach of pre-training on one subset
followed by fine-tuning on the other subset. Examples for datasets that could
be included into follow-up experiments as additional resources are the work by
[42] (which is based on [43] and introduces richer contextual cues) as well as
the ones discussed previously such as Twitter 15 and Twitter 16 [31], PHEME
[32], WEIBO [33] and FbMultiLingMisinfo [35].

Pre-training Objectives: As shown in Table 1, graph-level pre-training as well
as context prediction do not lead to improvements while these have worked well
in other domains [16]. This possibly is due to the tasks’ poor alignment with
the problem. For context prediction this issue becomes evident from only small
fluctuations in loss and accuracy throughout the pre-training phase. One
possible explanation is that, unlike in the cases of Biology and Chemistry,
where node features hold high expressiveness, our situation places greater
emphasis on the graph structure. This was demonstrated in our prior research,
where we systematically compared the contributions of node features and graph
structure to classification performance [23]. However, the ego-graph structure
of our social media graphs where all context converges onto a single central
node is not directly comparable to molecules or proteins (which often exhibit
multiple paths between nodes) and thus seems to be less fitting for this pre-
training objective’s requirements. Moreover, the graph-level task evaluated
might generally be less suitable for the problem of fake news detection as we
could not observe any positive effects on classification performance. To
address the challenges just described, one potential solution lies in
employing larger, more densely connected social media context graphs. This is
a common setup when framing fake news detection as a node classification task
(deviating from the graph classification task alignment we maintained with
[16]).

Another suggestion is to explore approaches similar to [30] who apply
generative techniques to pre-train GNNs. This could even go beyond their
proposed idea of learning to reconstruct the input attributed graph but move
towards synthetic data generation, effectively implementing data augmentation.
This direction could also serve as a potential strategy to mitigate the
problem of data scarcity described previously.

Model Size: While BERT’s hidden layers consist of 768 nodes [11] and [16] used
GNNs with 300 dimensional layers for pre-training, we experimented with a
comparatively shallow model dimension of 64. This aligns with our goal of
exploring the general effectiveness of GNN pre-training in the domain.
However, using larger networks with higher capacity will potentially lead to
better results.

##  6 Conclusion

We demonstrate how a current lack of large-scale resources leads to
limitations in pre-training GNNs in the domain of context-based fake news
detection. However, there is much potential left as we could observe
improvements in experimental setups with small amounts of labelled data.
Taking into account all of our observations, we proposed a range of potential
avenues for future research in the field.

##  7 Ethical Considerations

We recognize the importance of responsible and ethical research practices,
especially in the context of fake news detection. We provide clear
documentation of our methodologies, including the pre-training strategies
employed, as well as datasets and evaluation metrics used. We encourage open
and transparent discussions about our findings and welcome scrutiny from the
research community which is why we also share all our implementations and pre-
trained models. Our research aims to contribute to the advancement of AI in a
manner that benefits society, while also being aware of potential risks,
challenges and limitations. We acknowledge the potential biases in data,
algorithms, and models and have made efforts to mitigate these biases to the
best of our knowledge. We acknowledge that fake news detection is a complex
field with ethical implications. While our study focuses on technical aspects,
we are aware of the broader societal impact.

##  8 Acknowledgements

We would like to thank the anonymous reviewers for their constructive feedback
which has helped us improve the paper.

This work was supported by the project COURAGE: A Social Media Companion
Safeguarding and Educating Students funded by the Volkswagen Foundation, grant
number 95564.

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# Disordered Hyperuniform Quasi-1D Materials

Duyu Chen111These authors contributed equally to this work.
<EMAIL_ADDRESS>Materials Research Laboratory, University of
California, Santa Barbara, California 93106, United States  Yu
Liu11footnotemark: 1 HEDPS, Center for Applied Physics and Technology, School
of Physics and College of Engineering, Peking University, Beijing 100871,
People’s Republic of China  Yu Zheng  Department of Physics, Arizona State
University, Tempe, AZ 85287  Houlong Zhuang  Mechanical and Aerospace
Engineering, Arizona State University, Tempe, AZ 85287  Mohan Chen  HEDPS,
Center for Applied Physics and Technology, School of Physics and College of
Engineering, Peking University, Beijing 100871, People’s Republic of China
Yang Jiao  Materials Science and Engineering, Arizona State University, Tempe,
AZ 85287  Department of Physics, Arizona State University, Tempe, AZ 85287

###### Abstract

Carbon nanotubes are quasi-one-dimensional systems that possess superior
transport, mechanical, optical, and chemical properties. In this work, we
generalize the notion of disorder hyperuniformity, a recently discovered
exotic state of matter with hidden long-range order, to quasi-one-dimensional
materials. As a proof of concept, we then apply the generalized framework to
quantify the density fluctuations in amorphous carbon nanotubes containing
randomly distributed Stone-Wales defects. We demonstrate that all of these
amorphous nanotubes are hyperuniform, i.e., the infinite-wavelength density
fluctuations of these systems are completely suppressed, regardless of the
diameter, rolling axis, number of rolling sheets, and defect fraction of the
nanotubes. We find that these amorphous nanotubes are energetically more
stable than nanotubes with periodically distributed Stone-Wales defects.
Moreover, certain semiconducting defect-free carbon nanotubes become metallic
as sufficiently large amounts of defects are randomly introduced. This
structural study of amorphous nanotubes strengthens our fundamental
understanding of these systems, and suggests possible exotic physical
properties, as endowed by their disordered hyperuniformity. Our findings also
shed light on the effect of dimensionality reduction on the hyperuniformity
property of materials.

##  I Introduction

Carbon nanotubes, a class of quasi-1D materials that can be conceptually
constructed by rolling a graphene sheet, have been investigated extensively
since their discovery Iijima (1991); Iijima and Ichihashi (1993). Benefiting
from their superior physical, chemical, and mechanical properties, crystalline
carbon nanotubes are popular candidates for a variety of applications, such as
field-effect transistors Tans et al. (1998), rectifiers Collins et al. (1997),
and sensors Kong et al. (2000). However, the industrial production of carbon
nanotubes (e.g., by catalytic chemical vapor deposition) is still not
sufficiently well controlled and various defects can form during nanotube
growth Zhou and Shi (2003). Thus, it is crucial to understand how the defects
affect the physical properties and performance of carbon nanotubes. For
example, Robinson et al. demonstrated controlled introduction of oxidation
defects can enhance sensitivity of a SWNT network sensor to a variety of
chemical vapors Robinson et al. (2006). Recently, Gifford et al. investigated
the effect of the $sp^{3}$-hybridized defects on the energies of the optical
emissive features and the influence of synthetic modifications on the
resulting defect geometry toward attaining desired narrow photoluminescence
capacity Gifford et al. (2020).

A variety of single-walled carbon nanotubes (SWNTs) can be formed by rolling
up an infinitely long strip of a single graphene sheet along different
directions. Conventionally, the type of SWNTs can be specified by a rolling
vector $(n,m)$ (with $n>0$, $m\geq 0$, and $n\geq m$) in the basis of two
linearly independent vectors that connect a carbon atom in the graphene sheet
to either two of its nearest atoms with the same bond directions Sinnott and
Andrews (2001). Two most common types of SWNTs are: (i) zigzag nanotubes with
$n>0$ and $m=0$, and (ii) armchair nanotubes with $n=m$. Multi-walled
nanotubes (MWNTs) consisting of purely zigzag nanotubes, or purely armchair
nanotubes, or a mixture of both are also of great interest. A type of commonly
seen defects in covalently-bonded network materials is the Stone-Wales (SW)
topological defects Stone and Wales (1986). A SW topological defect rotates a
bond in the network by 90 degrees, transforming 4 adjacent hexagons into a
pair of pentagons and a pair of heptagons in the absence of adjacent SW
defects. Experimentally, the SW defects can be introduced into carbon
nanotubes via high-energy radiations, during the synthesis of materials, or by
applying strains Zhou and Shi (2003). In Fig. 1 we schematically shows the
generation of a defected (3,0) zigzag nanotube and a defected (3,3) armchair
nanotube with SW defects. Previous computational studies have also
investigated the effect of the orientation and concentration of SW defects on
the electronic properties of various single-walled carbon nanotubes at low
defect concentrations Crespi et al. (1997); Azadi et al. (2010); Partovi-Azar
et al. (2013). However, a comprehensive study of large-scale structures of
carbon nanotubes with SW defects across a large range of defect concentrations
is still lacking.

$\begin{array}[]{c}\\\
\includegraphics[width=212.47617pt]{fig1.png}\end{array}$

Figure 1: (Color online) Illustration of a defected (3,0) zigzag (a) nanotube
and a defected (3,3) armchair (b) nanotube formed by rolling graphene sheets
with randomly distributed Stone-Wales defects along different rolling
directions in two dimensions.

Very recently, it is found that the SW defects in a variety of 2D material
systems, including amorphous 2D silica Zheng et al. (2020), amorphous graphene
Chen et al. (2021a), defected transition metal dichalcogenides Chen et al.
(2021b), and defected pentagonal 2D materials Zheng et al. (2021), can lead to
special amorphous states Zheng et al. (2020); Chen et al. (2021a); Zheng et
al. (2021); Chen et al. (2021c), in which the density fluctuations in the
materials are largely suppressed, a property termed “hyperuniformity”.
Interestingly, it was found that the disordered hyperuniform (DHU) states of
these materials usually possess a significantly lower energy than other
disorder models, suggesting the stability of DHU states for these systems over
other disorder models. Moreover, DHU states can lead to unique electronic and
thermal transport properties, resulted from mechanisms distinct from those
that have been identified for their crystalline counterparts. A key discovery
is that the SW defects in these systems play a significant role in
establishing hyperuniformity of these 2D material systems Chen et al. (2021a).

Disordered hyperuniformity (DHU) is a recently discovered exotic state of
matter Torquato and Stillinger (2003); Torquato (2018), which lies between a
perfect crystal and liquid. DHU systems are similar to liquids or glasses in
that they are statistically isotropic and possess no Bragg peaks, and yet they
completely suppress large-scale normalized density fluctuations like crystals.
In this sense, DHU materials possess a hidden long-range order Torquato and
Stillinger (2003); Zachary and Torquato (2009); Torquato (2018) similar to
that in crystals. DHU is also manifested as the vanishing of static structure
factor $S(k)$ in the infinite-wavelength (or zero-wavenumber) limit, i.e.,

$\lim_{k\rightarrow 0}S(k)=0,$ (1)

where $k$ is the wavenumber Torquato and Stillinger (2003); Torquato (2018).
Here $S(k)$ is defined as $S(k)\equiv 1+\rho\tilde{h}(k)$, where
$\tilde{h}(k)$ is the Fourier transform of the total correlation function
$h(r)=g_{2}(r)-1$, $g_{2}(r)$ is the pair correlation function, and $\rho$ is
the number density of the system. Note that this definition implies that the
forward scattering contribution to the diffraction pattern is omitted. For
computational purposes, $S({k})$ is the angular-averaged version of $S({\bf
k})$, which can be obtained directly from the particle positions ${\bf
r}_{j}$, i.e.,

$S({\bf k})=\frac{1}{N}\left|{\sum_{j=1}^{N}\exp(i{\bf k}\cdot{\bf
r}_{j})}\right|^{2}\quad({\bf k}\neq{\bf 0}),$ (2)

where $N$ is the total number of points in the system Zachary and Torquato
(2009), and ${\bf k}$ is the wavevector. Equivalently, the local number
variance $\sigma_{N}^{2}(R)\equiv\langle N^{2}(R)\rangle-\langle
N(R)\rangle^{2}$ associated with a spherical observation window of radius $R$
grows more slowly than the window volume for DHU systems in the large-$R$
limit Torquato and Stillinger (2003); Torquato (2018), i.e.,

$\lim_{R\rightarrow\infty}\frac{\sigma^{2}_{N}(R)}{R^{d}}=0$ (3)

where $N(R)$ is the number of particles in a spherical window with radius $R$
randomly placed into the system, $R^{d}$ is proportional to the observation
window volume, and $d$ the dimensionality of the system. The small-$k$ scaling
behavior of $S(k)\sim k^{\alpha}$ dictates the large-$R$ asymptotic behavior
of $\sigma_{N}^{2}(R)$, based on which all DHU systems can be categorized into
three classes: $\sigma_{N}^{2}(R)\sim R^{d-1}$ for $\alpha>1$ (class I);
$\sigma_{N}^{2}(R)\sim R^{d-1}\ln(R)$ for $\alpha=1$ (class II); and
$\sigma_{N}^{2}(R)\sim R^{d-\alpha}$ for $0<\alpha<1$ (class III) Torquato
(2018).

Disordered hyperuniformity has been observed in a wide spectrum of physical
and material systems Torquato (2018), in both equilibrium and non-equilibrium,
and both classical and quantum mechanical varieties. Examples include the
density fluctuations in early universe Gabrielli et al. (2002), maximally
random jammed packing of hard particles Zachary et al. (2011); Jiao et al.
(2014); Chen et al. (2014), exotic classical ground states of many-body
systems Zachary and Torquato (2011); Torquato et al. (2015), jammed colloidal
systems Kurita and Weeks (2011); Dreyfus et al. (2015), driven non-equilibrium
systems Jack et al. (2015); Weijs et al. (2015), certain quantum ground states
Torquato et al. (2008); Feynman and Cohen (1956), avian photoreceptor patterns
Jiao et al. (2014), organization of immune systems Mayer et al. (2015),
amorphous silicon Hejna et al. (2013); Xie et al. (2013), a wide class of
disordered cellular materials Klatt et al. (2019), dynamic random organizing
systems Hexner and Levine (2017); Hexner et al. (2017); Weijs and Bartolo
(2017); Lei et al. (2019); Lei and Ni (2019), electron density distributions
Gerasimenko et al. (2019); Sakai et al. (2022), and vortex distribution in
superconductors Rumi et al. (2019); Sánchez et al. (2019), to name but a few.

Since nanotubes can be constructed by rolling the corresponding sheets of 2D
materials, it is natural to ask the question: Are these quasi-1D materials
containing Stone-Wales topological defects are also hyperuniform? In this
paper, we address this question by generalizing the notion of hyperuniformity
to quasi-1D materials, which can be considered as 1D projections of higher-
dimensional structures in this context, and may involve the non-trivial
situations where multiple points in the higher-dimensional structures are
mapped to the same point in the projection. In particular, we systematically
generate a diverse spectrum of amorphous carbon nanotubes by continuously
introducing Stone-Wales (SW) topological defects into the crystalline systems,
and quantify the generalized density fluctuations in these systems. We
demonstrate that all amorphous nanotubes are hyperuniform, i.e., the infinite-
wavelength density fluctuations of these systems are completely suppressed,
and the systems possess hidden long-range order, regardless of the diameter,
rolling axis, number of rolling sheets, and defect fraction of the nanotubes.
This structural study of amorphous nanotubes strengthens our fundamental
understanding of these systems, and suggests possible exotic physical
properties, as endowed by their disordered hyperuniformity.

The rest of the paper is organized as follows: In Sec. II we discuss the the
generalization of the hyperuniformity concept to quasi-1D materials. In Sec.
III we discuss the procedures to generate various types of carbon nanotubes
with randomly distributed SW defects. In Secs. IV and V, we respectively
investigate the density fluctuations of single-walled and multi-walled
nanotubes, which we use as examples to demonstrate the use of the theoretical
framework for analyzing hyperuniformity properties in quasi-1D materials. In
Sec. VI, we report the physical properties of the hyperuniform carbon
nanotubes, including their stability and density of states. In Sec. VII, we
provide concluding remarks and discuss the implications of our work in other
quasi-1D material systems.

##  II Hyperuniformity of quasi-1D materials

In this section we introduce the generalization of the hyperuniformity concept
to quasi-1D materials. These systems typically have small width in all the
directions other than the axial/propagation direction. For example, carbon
nanotubes have small finite width in the rolling direction. Since
hyperuniformity is a large-scale structural feature, only the density
fluctuations along the propagation direction should be relevant in the context
of hyperuniformity, and effectively we are looking at 1D projections of the
graphene sheets along the axial/propagation direction in the case of carbon
nanotubes.

An important issue that we need to address when looking at density
fluctuations of low-dimensional projections of higher-dimensional structures
is that multiple points in the higher-dimensional structures can be mapped to
the same point in the projections. Here we generalize the definition of the
structure factor $S({\bf k})$ to be:

$S({\bf k})=\frac{1}{G}\left|{\sum_{j=1}^{M}g_{j}\exp(i{\bf k}\cdot{\bf
x}_{j})}\right|^{2}\quad({\bf k}\neq{\bf 0}),$ (4)

where $M$ is the number of distinguishable points in the projections, the
multiplicity $g_{j}$ is defined as the number of points in the higher-
dimensional structures that are mapped to the given point ${\bf x}_{j}$ in the
projections, and $G=\sum_{j=1}^{M}g_{j}$. Accordingly, we generate the concept
of $N(R)\equiv\langle N(R;{\bf x}_{0})\rangle$ associated with
$\sigma_{N}^{2}(R)$ to be:

$N(R;{\bf x}_{0})=\sum_{j=1}^{M}g_{j}m({\bf x}_{j}-{\bf x}_{0};R),$ (5)

where $\langle\cdots\rangle$ denotes ensemble average, $m({\bf x}-{\bf
x}_{0};R)$ is the indicator function of the observation window centered at
${\bf x}_{0}$ with radius $R$ and is defined as

$m({\bf
x;R})=\left\\{\begin{array}[]{c@{\hspace{0.3cm}}c@{\hspace{0.3cm}}c}1,\hfil\hskip
8.5359pt&|{\bf x}|\leq R,\hfil\hskip 8.5359pt&\\\ 0,\hfil\hskip
8.5359pt&\textnormal{otherwise}.\hfil\hskip 8.5359pt&\end{array}\right.$ (6)

We note that using 1D observation window to look at $\sigma_{N}^{2}(R)$ of the
projection is equivalent to using observation window in the higher dimension
that encompasses the quasi-1D materials in the radial directions, but slide in
the axial/propagation direction, which is illustrated in Fig. 1. With these
generalizations of $S(k)$ and $\sigma_{N}^{2}(R)$, we can then use the
definitions of hyperuniformity specfied in Eqs. 1 and 3 for a low-dimensional
projection, and use the dimension of the projection for $d$ in Eq. 3.

##  III Generation of amorphous carbon nanotubes containing Stone-Wales
defects

To generate a defected zigzag nanotube at a given defect concentration $p$, we
first introduce prescribed number of randomly distributed SW defects
$N_{d}=N_{b}p=\frac{3}{2}Np$ into a graphene sheet with finite width $L_{y}$
in the vertical direction in Fig. 1(a), where $N_{b}=\frac{3}{2}N$ and $N$ are
the number of bonds and atoms in the sheet, respectively, and $L_{y}$ is the
length of the $(n,0)$ vector. We then relax the structure according to the
procedure described in Ref. Chen et al. (2021a), and subsequently roll up the
sheet along the vertical direction shown in Fig. 1(a). Specifically, for a
given atom at location $(x,y)$ in a graphene sheet, its coordinate
$(x^{\prime},y^{\prime},z^{\prime})$ in the resulting zigzag nanotube is given
by

$\left\\{\begin{array}[]{l}x^{\prime}=\frac{L_{y}}{2\pi}\cos(\frac{2\pi
y}{L_{y}}),\\\ y^{\prime}=\frac{L_{y}}{2\pi}\sin(\frac{2\pi y}{L_{y}}),\\\
z^{\prime}=x\end{array}\right.$ (7)

where $x$ and $y$ axis are the horizontal and vertical directions in Fig.
1(a). On the other hand, to generate a defected armchair nanotube at a given
defect concentration $p$, we first follow similar procedure described in Ref.
Chen et al. (2021a), but use a graphene sheet with finite width $L_{x}$ in the
horizontal direction in Fig. 1(a), where $L_{x}$ is the length of the $(n,n)$
vector. We note that the directions in Fig. 1(b) are rotated by 90 degrees
from those in Fig. 1(a) for clear visualization, so the horizontal direction
in Fig. 1(a) is the vertical direction in Fig. 1(b). We then roll up the sheet
along the horizontal direction shown in Fig. 1(a), or the vertical direction
in Fig. 1(b). Specifically, for a given atom at location $(x,y)$ in a graphene
sheet, its coordinate $(x^{\prime},y^{\prime},z^{\prime})$ in the resulting
armchair nanotube is given by

$\left\\{\begin{array}[]{l}x^{\prime}=\frac{L_{x}}{2\pi}\cos(\frac{2\pi
x}{L_{x}}),\\\ y^{\prime}=\frac{L_{x}}{2\pi}\sin(\frac{2\pi x}{L_{x}}),\\\
z^{\prime}=y\end{array}\right.$ (8)

where $x$ and $y$ axis are the horizontal and vertical directions in Fig.
1(a).

To obtain multi-walled carbon nanotubes with SW defects consisting of
concentric single-walled nanotubes, we simply generate the constituting
single-walled nanotubes separately according to the procedures above, and
combine the atoms together. This is possible because the aforementioned
procedure produces nanotubes that all wrap around the $z^{\prime}$ axis in
three dimensions.

##  IV Density fluctuations of single-walled nanotubes

To investigate density fluctuations of zigzag and armchair single-walled
nanotubes, we employ the procedures described in Sec. III to generate 10
configurations of (3,0), (5,0), (3,3), and (5,5) nanotubes at each of the
given defect concentrations $p=0.02$, $0.04$, $0.06$, $0.08$, $0.10$, $0.12$,
and $0.14$. We then compute their local number variance $\sigma_{N}^{2}(R)$
and structure factor $S(k)$. This setting also allows us to study the effect
of nanotube radius on the structural features of nanotubes.

As a reference, it can be easily seen that defect-free $(n,0)$-zigzag
nanotubes are mapped into two-scale 1D point patterns with $\zeta=\frac{1}{3}$
Torquato and Stillinger (2003) when projecting onto the cylinder axis of the
nanotubes, with $n$ carbon atoms superimposed onto each other at each point in
the 1D projected point pattern. The number variance of the $(n,0)$-zigzag
nanotubes $\sigma_{N}^{2}(R)$ can then be determined analytically as

$\sigma_{N}^{2}(R)=n^{2}\sigma_{N,D}^{2}(R),$ (9)

where $\sigma_{N,D}^{2}(R)$ is the number variance of the projected two-scale
1D point patterns with $\zeta=\frac{1}{3}$ Torquato and Stillinger (2003). As
a result, $\sigma_{N}^{2}(R)$ of defect-free zigzag nanotubes fluctuate around
certain constant, whose value depends on $n$, and these nanotubes are class-I
hyperuniform.

As SW defects are introduced into the nanotubes, the structures gradually
transition into amorphous ones, which are reflected in their local number
variance $\sigma_{N}^{2}(R)$. For example, we show the computed
$\sigma_{N}^{2}(R)$ for $(3,0)$-zigzag nanotubes at different defect fractions
$p$ in Fig. 2(a). At low $p$, $\sigma_{N}^{2}(R)$ exhibits “periodicity” in
the window radius $R$, suggesting the reminiscence of the crystalline order in
the systems; at large $p$, $\sigma_{N}^{2}(R)$ varies smoothly with damped
oscillations as $R$ increases, indicating the emergence of truly amorphous
states. Also, by comparing the results of (3,0)-zigzag nanotubes in Fig. 22(a)
and the results of (5,0)-zigzag nanotubes in Fig. 2(b), one can see that at
given $p$, increasing the radius (or decreasing the curvature) of the armchair
nanotube increases density fluctuations. Interestingly, the variance
$\sigma_{N}^{2}(R)$ of these zigzag nanotubes fluctuate around certain
constants as $R$ increases at all investigated $p$, indicating that these
structures are class-I hyperuniform. This is related to the fact that SW
transformation and subsequent structural relaxations only result in local
rearrangements of atoms, and do not affect the scaling of large-scale density
fluctuations in these nanotubes.

$\begin{array}[]{c}\\\
\includegraphics[width=427.11595pt]{fig2.png}\end{array}$

Figure 2: (Color online) Local number variances $\sigma_{N}^{2}(R)$ (Top) and
structure factor $S(k)$ (Bottom) of defected single-walled zigzag and armchair
nanotubes at different defect concentrations $p$. (Column 1) (3,0) zigzag.
(Column 2) (5,0) zigzag. (Column 3) (3,3) armchair. (d) (Column 4) armchair.
The results are all averaged over 10 configurations.

Similarly, defect-free $(n,n)$-armchair nanotubes are mapped into single-scale
1D point patterns Torquato and Stillinger (2003) when projecting onto the
cylinder axis of the nanotubes, with $2n$ carbon atoms superimposed onto each
other at each point in the 1D point pattern. The number variance of the
$(n,n)$-armchair nanotubes $\sigma_{N}^{2}(R)$ can then be determined
analytically as

$\sigma_{N}^{2}(R)=4n^{2}\sigma_{N,S}^{2}(R),$ (10)

where $\sigma_{N,S}^{2}(R)$ is the number variance of the projected single-
scale 1D pattern. As a result, $\sigma_{N}^{2}(R)$ of defect-free armchair
nanotubes fluctuate around certain constant, whose value depends on $n$, and
these nanotubes are class-I hyperuniform.

As SW defects are introduced into the nanotubes, the behaviors of these
armchair nanotubes are similar to those of the zigzag nanotubes. For example,
we show the computed $\sigma_{N}^{2}(R)$ for $(3,3)$-armchair nanotubes at
different defect fractions $p$ in Fig. 2(c). Importantly, at all investigated
$p$, the variance $\sigma_{N}^{2}(R)$ of these armchair nanotubes fluctuate
around certain constants, indicating that these structures are hyperuniform.
Also, by comparing the results of (5,0)-zigzag nanotubes in Fig. 2(c) and the
results of (5,5)-zigzag nanotubes in Fig. 2(d), one can see that at given $p$,
increasing the radius (or decreasing the curvature) of the armchair nanotube
increases density fluctuations.

The results of ensemble-averaged $S(k)$ of the aforementioned zigzag and
armchair nanotubes are shown in Fig. 2(e)-(h), which all decreases to zero as
$k$ goes to zero, regardless of defect concentration $p$. These results
further confirm the hyperuniformity of these nanotubes, and are consistent
with the results of $\sigma_{N}^{2}(R)$.

##  V Density fluctuations of multi-walled nanotubes

It can be easily seen that MWNTs consisting of $K$ defect-free zigzag
nanotubes with $(n_{1},0)$, $(n_{2},0)$, $\cdots$, $(n_{K},0)$ rolling
vectors, respectively, are mapped into two-scale 1D point patterns with
$\zeta=\frac{1}{3}$ Torquato and Stillinger (2003) when projecting onto the
cylinder axis of the nanotubes, with $n_{1}+n_{2}+\cdots+n_{K}$ carbon atoms
superimposed onto each other at each point in the 1D point pattern. The number
variance of the MWNTs $\sigma_{N}^{2}(R)$ can then be determined analytically
as

$\sigma_{N}^{2}(R)=(n_{1}+n_{2}+\cdots+n_{K})^{2}\sigma_{N,D}^{2}(R),$ (11)

where $\sigma_{N,D}^{2}(R)$ is the number variance of the projected two-scale
1D point patterns with $\zeta=\frac{1}{3}$ Torquato and Stillinger (2003).
Therefore, $\sigma_{N}^{2}(R)$ of defect-free MWNTs consisting of multiple
zigzag nanotubes fluctuate around certain constant, and these nanotubes are
class-I hyperuniform.

On the other hand, as SW defects are introduced into the nanotubes, the
structures gradually transition into amorphous ones, which are reflected in
their local number variance $\sigma_{N}^{2}(R)$. For example, we show the
ensemble-averaged $\sigma_{N}^{2}(R)$ for MWNTs consisting of a $(3,0)$-zigzag
nanotube and a $(5,0)$-zigzag nanotube at different defect fractions $p$ in
Fig. 3(a). At low $p$, $\sigma_{N}^{2}(R)$ exhibits “periodicity” in the
window radius $R$, indicating that the crystalline order is reminiscent in the
systems; at large $p$, the oscillations of $\sigma_{N}^{2}(R)$ become much
more damped as $R$ increases, suggesting the emergence of truly amorphous
states. Interestingly, the variance $\sigma_{N}^{2}(R)$ of these MWNTs
fluctuate around certain constants as $R$ increases at all investigated $p$,
indicating that these structures are class-I hyperuniform. These behaviors are
similar to those in the cases of the SWNTs.

$\begin{array}[]{c}\\\
\includegraphics[width=323.04477pt]{fig3.png}\end{array}$

Figure 3: (Color online) Local number variances $\sigma_{N}^{2}(R)$ and
structure factor $S(k)$ of defected multi-walled nanotubes at different defect
concentrations $p$. (a) $\sigma_{N}^{2}(R)$ of double-walled nanotubes
consisting of a (3,0) zigzag nanotube and a (5,0) zigzag nanotube. (b)
$\sigma_{N}^{2}(R)$ of double-walled nanotubes consisting of a (3,3) armchair
nanotube and a (5,5) armchair nanotube. (c) $\sigma_{N}^{2}(R)$ of double-
walled nanotubes consisting of a (5,0) zigzag nanotube and a (5,5) armchair
nanotube. (d) $S(k)$ of double-walled nanotubes consisting of a (3,0) zigzag
nanotube and a (5,0) zigzag nanotube. (e) $S(k)$ of double-walled nanotubes
consisting of a (3,3) armchair nanotube and a (5,5) armchair nanotube. The
results are all averaged over 10 configurations.

Similarly, MWNTs consisting of $K$ defect-free armchair nanotubes with
$(n_{1},n_{1})$, $(n_{2},n_{2})$, $\cdots$, $(n_{K},n_{K})$ rolling vectors,
respectively, are mapped into are mapped into single-scale 1D point patterns
Torquato and Stillinger (2003) when projecting onto the cylinder axis of the
nanotubes, with $2(n_{1}+n_{2}+\cdots+n_{K})$ carbon atoms superimposed onto
each other at each point in the 1D point pattern. As a result, the number
variance of the MWNTs $\sigma_{N}^{2}(R)$ can be theoretically determined as

$\sigma_{N}^{2}(R)=4(n_{1}+n_{2}+\cdots+n_{K})^{2}\sigma_{N,S}^{2}(R),$ (12)

where $\sigma_{N,S}^{2}(R)$ is the number variance of the projected single-
scale 1D pattern. On the other hand, as SW defects are introduced into the
nanotubes, the behaviors of these MWNTS consisting of purely armchair
nanotubes are similar to those MWNTS consisting of purely zigzag nanotubes.
For example, we show the computed $\sigma_{N}^{2}(R)$ for MWNTs consisting of
a $(3,3)$-armchair nanotube and a $(5,5)$-armchair nanotube at different
defect fractions $p$ in Fig. 3(b). Importantly, at all investigated $p$, the
variance $\sigma_{N}^{2}(R)$ of these MWNTs fluctuate around certain
constants, indicating that these structures are class-I hyperuniform.

Interestingly, MWNTs consisting of both zigzag and armchair nanotubes exhibit
different behaviors in their density fluctuations from SWNTs or MWNTs
consisting of purely zigzag or armchair nanotubes. For example, as shown in
Fig. 3(c), $\sigma_{N}^{2}(R)$ of MWNT consisting of a (5,0)-zigzag nanotube
and a (5,5)-armchair nanotube exhibit no “periodicity” in the window radius
$R$ even at low $p$, and appear close to those of a quasi-crystal Torquato
(2018). This is a direct result of the fact that the length of the smallest
repeating unit in a defect-free (5,0)-zigzag nanotube and that of the smallest
repeating unit in a defect-free (5,5)-armchair nanotube do not have an integer
common multiple, and the 1D projections of the two nanotubes collectively onto
the cylinder axis are no longer periodic. Here when placing observation window
in the process of computing $\sigma_{N}^{2}(R)$, we restrict the observation
window to fall entirely in the range of the (5,5)-armchair nanotube in the
axial direction, which is smaller than that of (5,0)-zigzag nanotube.

The results of ensemble-averaged $S(k)$ of MWNTs consisting of a
$(3,0)$-zigzag nanotube and a $(5,0)$-zigzag nanotube, and MWNTs consisting of
a $(3,3)$-armchair nanotube and a $(5,5)$-armchair nanotube, are shown in Fig.
2(d) and Fig. 2(e), respectively, which all decreases to zero as $k$ goes to
zero, regardless of defect concentration $p$. These results further confirm
the hyperuniformity of these nanotubes, and are consistent with the results of
$\sigma_{N}^{2}(R)$. We note that due to the lack of periodicity of the
simulation box in the axial direction of the MWNT consisting of a (5,0)-zigzag
nanotube and a (5,5)-armchair nanotube, $S(k)$ is not well defined for this
MWNT, and we do not compute its $S(k)$.

##  VI Stability and Density of States of Disordered Hyperuniform Carbon
Nanotubes

We use (10,0) zigzag SWNTs as examples to demonstrate the effect of different
concentrations of SW defects on the stability and electronic structure
properties of the amorphous SWNTs. Specifically, density functional theory
(DFT) Hohenberg and Kohn (1964); Kohn and Sham (1965) calculations are
performed with the Atomic-orbital Based Ab-initio Computation at UStc (ABACUS)
package Chen et al. (2010); Li et al. (2016). The norm-conserving
pseudopotential Hamann (2013); Schlipf and Gygi (2015) is employed to describe
the ion-electron interactions and the valence electron configuration of C is
$2s^{2}2p^{2}$. The generalized gradient approximation (GGA) in the form of
the Perdew-Burke-Ernzerhof (PBE) Perdew et al. (1996) is used for the
exchange–correlation functional. In order to deal with large systems of (10,0)
zigzag SWNTs, we employ numerical atomic orbitals (NAO) in the form of
double-$\zeta$ plus polarization function (DZP) orbitals as basis sets in our
calculations, whose accuracy and consistency have been verified in previous
studies Li et al. (2016); Liu et al. (2021, 2022). Specifically, we use
$2s2p1d$ NAOs for C, whose radius cutoff is set to 8 Bohr. The energy cutoff
is set to 60 Ry. Besides, oeriodic boundary conditions and a single k-point
($\Gamma$ point) are used. We employ the Gaussian smearing method with the
smearing width of 0.001 Ry. Structural optimizations are performed with the
conjugated gradient method until forces on each atom are below 0.04 eV/Å.

We first demonstrate the stability of DHU nanotubes. To this end, we compare
the results of the DHU configurations at different defect concentrations $p$
to those of the defect-free counterpart, as well as two other representative
models of nanotubes containing periodically distributed SW defects shown in
Fig. 4. Specifically, in defect model I we introduce SW defects into a unit
cell with 40 carbon atoms shown by the blue box in the top row of Fig. 4, and
then replicate the unit cell five times in both horizontal and vertical
directions to generate a graphene sheet before rolling up to form (10,0)
nanotubes. In defect model II, we introduce SW defects into a unit cell with
100 carbon atoms shown by the blue box in the bottom row of Fig. 4, and then
replicate the unit cell five times in the horizontal direction and two times
in the vertical directions to generate a graphene sheet before rolling up to
form (10,0) nanotubes. Due to the constraints associated with the construction
of defected nanotube models, we only consider three distinct defect
concentrations for defect model I SWNTs, and two concentrations for defect
model II SWNTs, which we believe are sufficiently representative.

$\begin{array}[]{c}\\\
\includegraphics[width=214.63977pt]{fig4.png}\end{array}$

Figure 4: (Color online) Two classes of defected graphene sheets with periodic
distribution of Stone-Wales defects, which are then rolled into (10,0)
nanotubes. Their unit cells are shown by the blue boxes, which are replicated
five times in the horizontal direction (propagation direction of the
nanotubes) before rolling into the nanotubes, and for clear visualization we
only show a portion in the horizontal direction. (Top) Defect model I sheets
at (a) $p=0.0167$, (b) $p=0.0333$, and (c) $p=0.05$. (Bottom) Defect model II
sheets at (d) $p=0.02$ and (e) $p=0.0333$.

In particular, we calculate the total energy $E$ (i.e., the sum of interaction
energy and electronic kinetic energy of many-body systems) of DHU carbon
nanotubes at $T=0~{}K$ with a wide spectrum of defect concentration $p$ after
structural optimizations, as well as the total energy of the two defect models
at selected $p$. Fig. 5 shows the excess energy $\Delta E$ of the
aforementioned systems compared to the defect-free carbon nanotube. Since
introducing defects always increases the energy of the system, we have $\Delta
E>0$ for all $p>0$. It can be seen that for all three different defected
nanotube models, $\Delta E$ increases approximately linearly as the
concentration $p$ increases. The DHU nanotubes appear to possess the lowest
$\Delta E$ among the three models for the same $p$ values, and possess the
smallest slope among the three. This result implies that DHU states are more
energetically stable than defect models with periodically distributed SW
defects.

Figure 5: (Color online) Excess energy $\Delta E$ (difference of the total
energy between the amorphous system and the corresponding defect-free state)
of (10, 0) zigzag DHU nanotubes at different SW defect concentrations $p$,
compared to $\Delta E$ of two other representative defect nanotube models with
periodically distributed SW defects.

We further investigate the electronic structures of DHU nanotubes to shed
light on the effects of increasing SW defects on the material properties. Fig.
6 show the DOS of (10, 0) zigzag DHU carbon nanotubes computed by PBE
functional at different $p$ as well as the DOS of defect-free carbon
nanotubes. It is well known Azadi et al. (2010); Partovi-Azar et al. (2013)
that (10,0) nanotubes possess a well-defined band gap at Fermi level, which is
captured in our calculations (see Fig. 6(a)). Increasing disorder in the DHU
system (i.e., increasing the amount of SW defects) results in two observed
effects on the computed DOS: (i) closure of the band gap at Fermi level and
(ii) broadening and flattening of the DOS, which are consistent with previous
computational study of defected nanotube that is semiconducting in its defect-
free state Crespi et al. (1997). This trend is also similar to the
observations in DHU 2D materials Zheng et al. (2020); Chen et al. (2021a).
Specifically, the band gap is closed at $p=0.0167$, suggesting the presence of
the semiconductor-to-metal transition around this defect concentration.
Moreover, as $p$ increases the DOS becomes more and more extended, converging
to a metallic characteristic. We note although the exact values of the band
gaps could be underestimated by PBE functional compared with the actual
values, the qualitative behavior of the DOS in DHU nanotubes suggests a
semiconductor-to-metal transition should be robust.

Figure 6:  (Color online) Density of states (DOS) of (10, 0) zigzag DHU carbon
nanotubes computed by PBE functional at defect concentration $p$: (a) 0.00,
(b) 0.01, (c) 0.0167, (d) 0.02, (e) 0.03, (f) 0.04, (g) 0.05, (h) 0.06.

##  VII Conclusions and Discussion

In this work, we generalized the concept of hyperuniformity to structurally
characterize quasi-one-dimensional materials, which in the context of
hyperuniformity can be viewed as one-dimensional projections of higher-
dimensional structures along the axial direction. As a proof of concept, we
systematically investigated the density fluctuations across length scales in
amorphous carbon nanotubes containing different amounts of Stone-Wales
topological defects, which can be constructed by rolling up defected graphene
sheets. We demonstrated that all amorphous nanotubes containing SW defects
studied here are hyperuniform, i.e., the normalized infinite-wavelength
density fluctuations are completely suppressed, regardless of the diameter,
rolling axis, number of rolling sheets, and defect fraction of the nanotubes.
Disordered hyperuniformity (DHU) is recently discovered exotic state of
matter. Using DFT simulations, we also showed that these amorphous carbon
nanotubes with randomly distributed Stone-Wales defects are energetically more
stable than their ordered counterparts with periodically distributed Stone-
Wales defects. We also demonstrated that the electronic bandgap closes for a
semiconducting zigzag nanotube as Stone Wales are randomly introduced into the
carbon nanotubes by determining the density of states near the Fermi level.
Our structural study of amorphous nanotubes strengthens our fundamental
understanding of these quasi-1D materials, and suggests possible exotic
physical properties, as endowed by unique disordered hyperuniformity feature.

$\begin{array}[]{c}\\\
\includegraphics[width=195.12767pt]{fig7.png}\end{array}$

Figure 7: (Color online) An anisotropic point pattern where the points are
placed randomly in the vertical direction, but are constrained to discrete
equally-distributed lattice sites in the horizontal direction with each sites
having the same number of points. The projection of this point pattern is
hyperuniform when projected along the horizontal direction, and
nonhypeurniform when projected along the vertical direction.

Our findings on the effect of the projection operation on the hyperuniformity
property of the graphene sheets may also shed light on our understanding of
the general effect of dimensionality reduction on the preservation of
(non)hyperuniformity, as projection is a common type of dimensionality
reduction operation. While we conjecture that projection should preserve
(non)hyperuniformity of isotropic higher-dimensional structures since each
dimension contributes equally to the density fluctuations in the isotropic
cases, this is definitely not always the case for anisotropic structures. This
can be seen from the following example of anisotropic point patterns, where
the points are placed randomly in the vertical direction, but are constrained
to discrete equally-distributed lattice sites in the horizontal direction with
each sites having the same number of points, as shown in Fig. 7. The
projection of this two-dimensional point pattern along the vertical direction
is a one-dimensional Poisson point pattern, which is known to be
nonhyperuniform Torquato and Stillinger (2003). On the other hand, the
projection of this two-dimensional point pattern along the horizontal
direction is a single-scale lattice in one dimension, which is known to be
class-I hyperuniform Torquato and Stillinger (2003). This example shows that
the projection of an anisotropic point could be hyperuniform when projected
one direction, but nonhyperuniform when projected along another direction.

###### Acknowledgements.

Y.L and M.C. were supported by the National Science Foundation of China under
Grant Nos. 12122401 and 12074007. Parts of the numerical simulations were
performed on the High Performance Computing Platform of CAPT. H. Z. thanks the
start-up funds from ASU. This research also used computational resources of
the Agave Research Computer Cluster of ASU and the Texas Advanced Computing
Center under Contract No. TG-DMR170070.

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