# Addressing machine learning concept drift reveals declining vaccine
sentiment during the COVID-19 pandemic

Martin Müller
Digital Epidemiology Lab
Ecole polytechnique fédérale de Lausanne (EPFL)
1202 Geneva, Switzerland
<EMAIL_ADDRESS>
Marcel Salathé
Digital Epidemiology Lab
Ecole polytechnique fédérale de Lausanne (EPFL)
1202 Geneva, Switzerland
<EMAIL_ADDRESS>

###### Abstract

Social media analysis has become a common approach to assess public opinion on
various topics, including those about health, in near real-time. The growing
volume of social media posts has led to an increased usage of modern machine
learning methods in natural language processing. While the rapid dynamics of
social media can capture underlying trends quickly, it also poses a technical
problem: algorithms trained on annotated data in the past may underperform
when applied to contemporary data. This phenomenon, known as concept drift,
can be particularly problematic when rapid shifts occur either in the topic of
interest itself, or in the way the topic is discussed. Here, we explore the
effect of machine learning concept drift by focussing on vaccine sentiments
expressed on Twitter, a topic of central importance especially during the
COVID-19 pandemic. We show that while vaccine sentiment has declined
considerably during the COVID-19 pandemic in 2020, algorithms trained on pre-
pandemic data would have largely missed this decline due to concept drift. Our
results suggest that social media analysis systems must address concept drift
in a continuous fashion in order to avoid the risk of systematic
misclassification of data, which is particularly likely during a crisis when
the underlying data can change suddenly and rapidly.

_K_ eywords Concept drift $\cdot$ Vaccine sentiment $\cdot$ Text
classification $\cdot$ COVID-19 $\cdot$ Twitter

##  1 Introduction

Supervised and semi-supervised Machine Learning algorithms are now ubiquitous
in the analysis of social media data. At the core of these algorithms is their
ability to make sense of a vast amount of semi-structured real-time data
streams, allowing them to automatically categorize or filter new data examples
into, usually pre-defined, classes. Multi-class text classification has been
successfully used in public health surveillance, election monitoring, or
vaccine stance prediction [1, 2, 3]. In recent years such algorithms have also
been developed to mitigate the negative effects of social media, such as in
the detection of cyber-bullying, hate speech, misinformation, and automated
accounts (bots) [4, 5, 6, 7].

The microblogging service Twitter has played a central role in these efforts,
as it serves as a public medium and provides easy access to real-time data
through its public APIs, making it the primary focus of this work. Twitter is
well described as a classical example of a non-stationary system with
frequently emerging and disappearing topical clusters [8]. This poses problems
for the aforementioned applications, as the underlying data distribution is
different between training time and the time of the algorithm’s application in
the real world. This phenomenon is known as concept drift [9] and can lead to
a change in performance of the algorithm over time.

It is important to distinguish concept drift from other reasons for
performance differences between training and testing, such as random noise due
to sampling biases or differences in data preprocessing [10, 11]. A classic
example of concept drift is the change in the meaning of classes, which
requires an update of the learned class decision boundaries in the classifier.
This is sometimes also referred to as real concept drift. Often, however, an
observed performance change is a consequence of a change in the underlying
data distribution, leading to what is known as virtual drift [12, 13]. Virtual
drift can be overcome by supplemental learning, i.e. collecting training data
from the new environment. A good example are periodic seasonality effects,
which may not be fully represented in the initial training data and only
become fully visible over time. However, in practice it is usually very
difficult (if not impossible) to disentangle virtual from real concept drift,
and as a consequence they are treated as the same effect [10].

On Twitter concept drift might appear on very different time scales and at
different rates. Sudden shifts in a debate might be triggered by a quickly
evolving news cycle or a catastrophic event. Concept drift may also be a slow
process in which the way a topic is discussed gradually changes over time. A
substantial amount of work has been dedicated to detecting and overcoming
concept drift [12, 10, 14]. Three basic re-training procedures for overcoming
concept drift have been proposed: (i) a time-window approach, (ii) an
incremental model, and (iii) an ensemble model [8]. In the time-window
approach, a sliding window of recent training examples is used to train an
algorithm. In this approach, the algorithm ignores training data collected
outside of that time window. The incremental model, in contrast, uses all
previously collected training examples to re-train the model. Lastly, the
ensemble model trains a model for each time window and uses the consensus of
all previous models for future predictions. As found in [8], in the case of
hashtag prediction on Twitter data, the incremental method gave the best
results.

Although sophisticated methods have been proposed to estimate concept drift in
an unsupervised way [15, 16], in practice, a certain amount of re-annotation
for both the detection and re-training of models seems unavoidable. The
decision about which of the newly collected data to annotate points to an
exploration-exploitation dilemma, which is usually addressed in the context of
an active learning framework [17]. The Crowdbreaks platform [18] is an example
of such a framework and has been built with the goal of exploring optimal
solutions to this problem in order to overcome concept drift.

A change in the underlying data distribution might not necessarily have a
negative impact on classifier performance. It is conceivable, for example,
that a polarisation in a debate on Twitter about a topic could even lead to an
improvement in classifier performance. It is therefore important to ask how
much we should be worried about concept drift: even if model performance were
to decrease, the real impacts on our analysis or interpretation might be
negligible.

The consequences of concept drift are task-, environment-, and model-dependent
[19]. Here, we will address concept drift in the specific case of vaccine
stance classification. Vaccine stance classification on Twitter data has been
widely studied and has shown promising links to vaccination decision making
and vaccine uptake rates in different countries [1, 20]. The COVID-19 pandemic
further emphasizes its importance, as evolving concerns about vaccines may
significantly influence their effect [21, 22].

To the best of our knowledge, only one study directly addressed concept drift
in vaccine stance classification. In this study [23] on tweets posted between
September 2016 and January 2017 in Italian language, the authors did not find
a substantial improvement of their model from incremental re-training before
specific events. Re-training was performed on 60 newly annotated tweets from
seven manually selected events. The authors conclude that either their
original algorithm was already quite robust towards concept change, or that
the newly collected training data was too small to see an effect.

Here, we use FastText [24] and BERT (Bidirectional Encoder Representations
from Transformers) [25], two commonly used models in social media text
classification. Most work on the topic of concept drift was conducted using
classical machine learning models, to which also FastText belongs. These types
of models are very reliant on high-quality annotation data. More recently,
models of the transformer family, such as BERT [25], have been proposed, which
require significantly less annotation data. In what follows, we will examine
whether these two models also share different concept drift characteristics.

The goal of this work is to emulate a typical social media analysis study, in
which data is collected for a certain period of time, and a supervised machine
learning model is trained on a subset of annotated data. The model is then
published and used to predict newly collected data. First, we will try to
answer whether or not concept drift can be observed, and if so, at what rate
it occurs. Second, we will investigate the influence of the study duration and
the amount of annotation data used. Lastly, we will examine to what extent
concept drift influences the final analysis outcomes, in this case a sentiment
index.

##  2 Results

###  2.1 Observing concept drift

Throughout the $1188$ day observation period, starting on July 1st, 2017 and
ending on October 1st, 2020, a total of $57.5$M English vaccination-related
tweets were collected. A random subset of $11,893$ tweets were annotated with
respect to stance towards vaccines, which resulted in $5482$ (46%) positive,
$4270$ neutral (36%), and $2141$ negative (18%) labels (for further details
see methods section 4.2). The dataset therefore bears clear label imbalance.

Figure 1:  Training and evaluation datasets. Each $90$ day bin consists of
$400$ samples of training data (blue) and $150$ samples of evaluation data
(red). Each trained model is using the most recent $1600$ samples for
training, which is an equivalent of 4 bins or $360$ days. For illustration
purposes, the training data for the second bin $b_{1}$ is indicated as blue
with white stripes. The $b_{1}$ model is then evaluated on all future
evaluation datasets, indicated as red with white stripes.

In order to observe whether classifiers experience drift in our dataset, we
analysed the performance change of a model when predicting newly collected
labelled data. For this we used a sliding time window approach, as first
proposed in [8]. We dissected the collected $11,893$ annotations into 13 bins
of 90 days each. From each bin we sampled $550$ examples and split them into a
train ($n=400$, 72%) and evaluation ($n=150$, 27%) set (see Figure 1). Each
model was trained on a window of 4 bins of training data, which is equivalent
to $1600$ samples and a time span of $360$ days. The models are subsequently
evaluated on the evaluation set corresponding to the bin at the end of their
training window as well as on all future evaluation sets. We repeat the
process of binning, splitting, training and evaluating 50 times in order to
yield a measure of confidence to our results.

Figure 2 shows the classifier performance at training time (square symbol) and
the performance at each future evaluation dataset (circle symbol) for
classifiers trained on different training windows (color). The upper left
panel shows the results of these experiments for the FastText models. We will
first compare the initial performance in terms of F1-macro score (i.e. the
arithmetic mean of the class-level F1 scores) of the classifiers on a test
dataset which was sampled from the last bin of the corresponding training
window (square symbols). The initial performance of the first model is at
$0.42$, the subsequent models plateau at around $0.50$, followed by a peak in
fall 2019 with an abrupt decline in January 2020. This variability in the
initial performance of models points to considerable differences between
training datasets over time. The performance of the FastText models is quite
low in general, which may be a consequence of the relatively small training
dataset of $1600$ examples and the lack of hyperparameter tuning.

Figure 2:  Model performance over time. The top row shows absolute performance
(in terms of F1-macro), and the bottom row shows the relative performance
change of models compared to training time. The columns show the result for
the two different classifier types FastText and BERT. The square indicates the
performance at training time, the circles correspond to performance of that
same model on future evaluation sets (compare with Figure 1). Bands correspond
to bootstrapped $95$% confidence intervals resulting from $50$ repeats.

Comparing the performance scores on future evaluation sets (circles) between
models, we observe that the oldest models (black) generally perform worse than
newer models (yellow) and that the ordering between models is preserved at all
times. However, in order to disentangle this effect from the variability in
initial performance, we compute the relative change in performance with
respect to performance at training time (lower left panel). Starting from
zero, the first model’s performance drops quickly by around
$5\text{\,}\mathrm{\char 37\relax}10\text{\,}\mathrm{\char 37\relax}$,
followed by a rebound to initial performance in fall 2019, and ending in a
sudden drop of approximately $-20$% in early 2020. The last drop indicates a
very abrupt shift in concepts, twice as strong as during a comparable time
window in 2019. In fall 2019, changes in the data distribution allowed all
models to rebound to initial performance, with some even “over-performing” by
5% compared to training time. This is a sign that the data distribution was
particularly easy to predict.

Further investigation of the F1-scores by class reveals that concept drift is
especially impactful on the negative class, whereas the positive and neutral
classes do not experience a significant drift (see Figure S1). This could
either indicate that anti-vaccine concepts are changing faster than pro-
vaccine concepts or that the negative class is harder to learn due to label
imbalance (cf. Figure S4) and might, as a consequence, be more affected by
virtual drift. We will further investigate this difference in the next
section.

Comparing these results to the BERT models (upper right panel), the models
show higher absolute performance but they experience a similar level of
relative performance loss and similar drift patterns. This confirms that the
observations are not model-specific but are likely to be observed in state-of-
the-art semi-supervised machine learning models.

As previously stated, each model was trained on $1600$ training examples over
the previous 360 days. Experiments were conducted under fewer training
examples (Figure S2) and smaller training windows (Figure S3) for FastText. As
expected, training on fewer training examples leads to lower model
performance, but we find the same drift patterns irrespective of the training
data size. Reducing the training window while keeping the number of training
examples constant does seem to have an impact on performance or drift
patterns.

###  2.2 Explaining concept drift

Next, we will try to explain both the variance in initial performance, as well
as the different rates of drift observed. We will investigate the effects of
label imbalance, annotator agreement, and corpus variability on initial
performance of models (Figure 3a-c). Additionally, we compare corpus
similarity over time and discuss it in the context of concept drift (Figure
3d). In particular, we consider the first sampling (repeat) of the combined
training ($n=$1600$$) and first evaluation set ($n=$150$$) for each training
window. The provided measures therefore correspond to what the model “saw”
during training and in the first bin of evaluation. Figure S3 shows the
equivalent metrics when limited to only the individual 90 day bins.

Figure 3:  Properties of the combined training data and first evaluation
dataset for each trained model. A. Distribution in the number of labels per
class. B. Annotator agreement, measured by Fleiss’ Kappa. C. Corpus
variability in terms of the variance of sentence embeddings within a corpus.
Variability is shown for the full corpus as well by class. D. Normalized
cosine similarity between the mean corpus vectors (i.e. the mean of all
sentence vectors in each corpus) for all data as well as by class.

#### Label imbalance

Although the used training datasets are always of equal absolute size, they
vary in the number of examples per class over time (see Figure 3a). It is
commonly known that label imbalance can negatively impact model performance,
which is also observed here (see Figure S1). However, we note that label
imbalance was highest in the very beginning of the observation period and
continuously decreased towards a more balanced situation. Given the drop in
initial performance in 2020, we conclude that label imbalance alone does not
explain the observed variability in initial performance.

#### Annotator agreement

We measure annotator agreement by computing the Fleiss’ Kappa [26] values for
each dataset. Annotator agreement is initially low at $0.37$, then increases
to almost $0.45$ and drops again to $0.36$ in mid-2020. This overlaps very
well with the initial performance trend observed in Figure 2. Variation in
inter-annotator agreement may be a consequence of differences in annotation
quality or difficulty of the annotation task, possibly hinting at semantic
ambiguity of the text, as discussed next.

#### Corpus variability

We use the BERT-large-uncased model to generate a 1024-dimensional sentence
embedding vector (i.e. the vector of the CLS token) for each tweet text in the
datasets. Note that this BERT model has not been trained on any of our
datasets, but it is able to generate rich sentence embeddings due to having
been pre-trained on large amounts of English text. Figure 3c shows the
variance in the generated sentence embeddings across time. We note that
overall, corpus variability is highest in the beginning of our observation
period, and then decreases towards the end. Also, when considering the corpus
variability by label class, we observe that negative samples have consistently
lower variability compared to text labelled as positive. The neutral class
seems to undergo a shift from high to low variability. In general, we may
hypothesize that a lower variability points to lower separability in
embeddings space, and therefore lower model performance. This hypothesis
aligns with the observations made in terms of initial performance.

#### Corpus similarity

Similarity was measured by calculating the cosine similarity between the mean
vectors for each corpus. Low cosine similarity points to large semantic
differences between datasets, which in turn could be an indicator for concept
drift. In the top left panel (“all”), the datasets are compared with each
other. We observe that over time, corpus vectors are moving further away from
each other. The biggest difference was observed between the two datasets
furthest from each other in time (2018-08-11 and 2020-07-31). We also observe
a bright area in the middle of the heatmap, which reveals that datasets
between February 2019 and February 2020 are more similar to each other
compared to datasets before (2018) or after (May & July 2020). This aligns
well with the results in Figure 2: Most of the concept drift was observed in
2018 and following 2020, whereas models in 2019 didn’t drift by a lot. When
considering the corpus similarity by class, we can attribute most of these
effects to the neutral and negative class. We therefore show that anti-vaccine
content “drifts” faster than pro-vaccine content.

In conclusion, our observations point to the fact that the differences in
initial performance of models are likely a consequence of low annotator
agreement. The reason for this low agreement could be rooted in semantic
ambiguity, as expressed by annotator agreement and corpus variability. The
degree of concept drift on the other hand is best explained by our measure of
corpus similarity.

###  2.3 Consequences of concept drift on real-time monitoring

Lastly, but perhaps most importantly, we highlight the impact of concept drift
on the inference of the previously trained models when used for real-time
monitoring of new data. We compare the predictions of a legacy model, which
was trained in August 2018 and used for the two subsequent years, to a model
we update (re-train) every 90 days. We compute the sentiment index $s$, which
corresponds to the weekly mean of positive, neutral and negative predictions,
when mapped to the numerical values of $1$, $0$ and $-1$, respectively. Figure
4 shows these sentiment trends for both the FastText and BERT model variants.
We observe that, in the case of FastText, the sentiment predicted by the
legacy model increased slightly until 2019 and then remained static. The
updated models, however, show a downwards trend starting in mid-2019 and
dropping further in 2020. By the end of our observation period the legacy
model predicts a 0.3 points higher sentiment than the up-to-date models, while
completely missing out on the downwards trend.

Figure 4:  Impact of concept drift on the predictions made by FastText and
BERT models. Each panel shows the comparison of a model which was trained in
August 2018 (black) to a model which was continuously updated every 90 days
(colored).

BERT models show a similar but smaller error, which is in agreement with our
previous analysis. The legacy BERT model was in agreement with the updated
models at the time of the first drop in 2019, but then started to diverge. We
can therefore conclude that due to their higher overall performance, BERT
models will have less severe deviations, but are not immune to effects of
concept drift in the long run. We also note a large difference in the extent
of positive and negative spikes between the legacy and re-trained models.
Drift may therefore not only affect the mean sentiment trend but also
sensitivity on shorter time scales, which could be problematic for real-time
event or anomaly detection.

As previously stated, the sentiment trend of both the updated BERT and
FastText models show a negative trend of the vaccine sentiment. Given the
current debate surrounding novel vaccines for the Sars-CoV-2 virus, this
finding is concerning from an epidemiological perspective. Note however, that
the BERT models used for these predictions are of mediocre performance and
future studies will be needed to confirm and interpret these trends.

##  3 Discussion

In this work, we investigated the effects of concept drift in vaccination-
related Twitter data streams over a duration of three years. Using a sliding
time window approach, we emulate a social media study in which (i) data is
collected for one year, (ii) an algorithm is trained, and (iii) the algorithm
is used in real-time monitoring of new data. While this may correspond to a
common setup in social media analytics, we demonstrate here that without
taking concept drift into account, the quality of the results will decay.
Using a vaccine-related dataset from 2018–2020, we demonstrate how failing to
take concept drift into account would have largely missed a rather dramatic
decay in vaccine sentiment during the COVID-19 pandemic in 2020.

We find that overall, concept drift indeed occurred, which led to a decline in
model performance of over 20% in the course of three years. However, most of
this decline happened in only ten months. Concept drift therefore affected
model performance at different rates throughout the observation period.
Furthermore, the relative performance loss was not consistently negative but
reverted to initial levels, or even slightly above that. These findings are
consistent with the various ways real and virtual concept drift can occur.
Although BERT models yielded higher performance scores, they are not immune to
issues related to concept drift. On a relative scale, BERT models show the
same degree of drift as the much less sophisticated FastText models.

In order to better understand the reasons for these phenomena, we investigate
the properties of the used datasets. We can explain the large differences in
initial performance of models with differences in semantic ambiguity of the
text, as indicated by low inter-annotator agreement and low corpus
variability. Occurrence of concept drift could be linked to differences in
corpus similarity. In particular, we find that the negative class is
responsible for most of the decay in performance over time and also shows the
strongest signs of drift. Anti-vaccine content may therefore change topics at
an increased rate compared to both positive or neutral content.

A caveat of this study is that the results are based on classifiers of
mediocre performance. Given the fact that the negative class was most affected
by concept drift and is at the same time also the smallest class in our
dataset, it is a fair question to ask whether concept drift would disappear
given more annotation data and higher performance of models. It is conceivable
that more annotation data would lead to a better representation of the
training window. However, as results in a study on automated geo-location of
tweets show [27], concept drift will still occur also under vast amounts of
annotated data and adaptive re-training on even a relatively small corpus can
overcome this drift.

Our results do not overlap with a previous study on vaccination-related
Twitter data [23], which did not find concept drift in an observation period
between September 2016 and January 2017 in Italian language. The reason for
this could be that the time scale analysed was too small to see an effect, or
that concept drift was much smaller in that particular dataset.

It is safe to assume that the COVID-19 pandemic led to severe topical shifts
in the vaccine debate, which ultimately translated into strong concept drift
and model performance loss. Based on these results, it can be expected that
future crisis situations would lead to similarly strong concept drift, thereby
severely undermining the utility of social media monitoring tools that do not
take concept drift into account. This is especially true for applications
which are intended to be used exactly in such circumstances.

Although our work focused on the singular task of vaccine stance prediction,
we believe that these results stress the general importance of addressing
concept drift in any real-time social media monitoring project. Overcoming
concept drift is a complex task, and many algorithmic solutions have been
proposed. However, in order to succeed in practice, a tightly coordinated and
fine-tuned framework for both the annotation and retraining of models is
required. The Crowdbreaks platform [18] was built with the intention to
address this issue and provide solutions for it.

##  4 Materials and methods

###  4.1 Data collection

This study is based on Twitter data collected through the Crowdbreaks platform
[18]. Between July 1st, 2017 and October 1st, 2020 a total of $57.5$M tweets
(including $39.7$M retweets) in English language by $9.9$M unique users were
collected using the public filter stream endpoint of the Twitter API. The
tweets matched one or more of the keywords “vaccine”, “vaccination”, “vaxxer”,
“vaxxed”, “vaccinated”, “vaccinating”, “vacine”, “overvaccinate”,
“undervaccinate”, “unvaccinated”. The data can be considered complete with
respect to these keywords.

###  4.2 Annotation data

Human annotation of a subset of tweets was performed through the Crowdbreaks
platform [18]. Tweets were anonymized by replacing user mentions and URLs with
placeholders. Tweets between February 2nd 2018 and November 11th 2020 were
sampled for annotation if they contained at least 3 words. Exact duplicates
were removed. Annotators were asked the question “What is the attitude of the
author of the tweet regarding vaccines?” and given the three options
“negative”, “neutral”, and “positive”. Annotation was performed both on Amazon
Turk (mTurk) and, to a smaller extent (roughly 1% of all annotations) by
public users on the Crowdbreaks website. We yield a dataset of $44,843$
annotations (Fleiss’ kappa of $0.30$), which resulted in $11,893$ three-fold
annotated tweets. Tweets with less than two-third agreement were excluded and
conflicts were decided through majority vote.

###  4.3 Training of classifiers

In this work we leverage two different classifiers: FastText [24] and BERT
[25]. For both models, hyperparameters were first tuned on the full annotation
data to yield optimal performance and then fixed for further experiments. For
FastText we used 10 dimensions, 500 epochs, a learning rate of $0.01$, and
using 1-gram embeddings. Optimal results were yielded by lower casing texts,
converting them to ASCII and using the tags “user” and “url” for
anonymization. BERT models of the type bert-large-uncased (pretrained in
English language) were trained for $20$ epochs, training batch size of $32$,
and a learning rate $2\text{\times}{10}^{-5}$ (using 10% warmup with linear
decay to zero), as recommended in recent literature [28, 29]. FastText models
were trained on a university cluster using the Crowdbreaks text-classification
library111https://github.com/crowdbreaks/text-classification and BERT models
were trained using Google Cloud v3-8 TPUs and the COVID-Twitter-BERT
library222https://github.com/digitalepidemiologylab/covid-twitter-bert [30].
For the purpose of predictions, text was preprocessed using the respective
preprocessing approach.

##### Data availability.

All data and code can be found on our public GitHub repository
https://github.com/digitalepidemiologylab/concept_drift_paper.

##### Author contributions.

M.M. collected the data, designed the experiments and analysed the data. M.M.
and M.S. conceptualized the work and wrote the manuscript.

##### Acknowledgments.

The authors would like to acknowledge Dr. Per Egil Kummervold and Dr. Burcu
Tepekule for their valuable comments and discussions.

##### Competing interests.

The authors declare no competing interests.

##### Funding.

This work received funding through the Versatile Emerging infectious disease
Observatory (VEO) grant as a part of the European Commission’s Horizon 2020
framework programme (grant agreement ID: 874735). Compute resources (Cloud
TPUs) were provided through Google’s TensorFlow Research Cloud and the work
was supported through Google Cloud credits in the context of COVID-19-related
research.

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Supplementary Material

Addressing machine learning concept drift reveals declining vaccine sentiment
during the COVID-19 pandemic
Martin Müller1, Marcel Salathé1
1 Digital Epidemiology Lab, Ecole polytechnique fédérale de Lausanne (EPFL),
1202 Geneva, Switzerland

##  1 Supplementary figures

Figure S1:  Performance scores by class for FastText and BERT models. For an
explanation of the Figure, please refer to Figure 2 in the main text. Unlike
for the negative class, performance between FastText and BERT is comparable
for the neutral and positive class. The “negative” class shows the strongest
effects due to concept drift.

Figure S2:  Drift of FastText models depending on size of training data. The
plots of the first column are identical to the FastText plots in Figure 2. For
all experiments a training window length of 360 days was used. Initial
performance is decreasing with a decreasing number of training samples.
Overcoming concept drift is increasingly difficult, and is barely visible at
400 training samples.

Figure S3:  Drift of FastText models depending on the length of the training
window. Each model was trained on an equal number of 800 training examples,
but distributed over 180, 270 or 360 days. A shorter training window is
occasionally associated with slightly higher initial performance and slightly
faster relative performance decrease on average.

Figure S4:  This figure is equivalent to Figure 3 in the main text, except for
the different datasets that were used. In Figure 3, we show the used training
and evaluation set in the full time window. This figure shows the newly added
training and evaluation data for each 90 day bin. For a detailed description
of this figure please refer to Figure 3 in the main text.


# Strength of effective Coulomb interaction in two-dimensional transition-
metal Halides MX2 and _MX 3_ (_M_ =Ti, V, Cr, Mn, Fe, Co, Ni; _X_ =Cl, Br, I)

Y. Yekta  Department of Physics, University of Guilan, 41335-1914, Rasht, Iran
H. Hadipour<EMAIL_ADDRESS>Department of Physics, University of
Guilan, 41335-1914, Rasht, Iran  E. Şaşıoğlu<EMAIL_ADDRESS>halle.de Institute of Physics, Martin Luther University Halle-Wittenberg,
06120 Halle (Saale) Germany  C. Friedrich<EMAIL_ADDRESS>Peter
Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum
Jülich and JARA, 52425 Jülich, Germany  S. A. Jafari<EMAIL_ADDRESS>Department of Physics, Sharif University of Technology, Tehran 11155-9161,
Iran  S. Blügel  Peter Grünberg Institut and Institute for Advanced
Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany  I.
Mertig  Institute of Physics, Martin Luther University Halle-Wittenberg, 06120
Halle (Saale) Germany

(August 27, 2024)

###### Abstract

We calculate the strength of the effective onsite Coulomb interaction (Hubbard
$U$) in two-dimensional (2D) transition-metal (TM) dihalides _MX 2_ and
trihalides _MX 3_ (_M_ =Ti, V, Cr, Mn, Fe, Co, Ni; _X_ =Cl, Br, I) from first
principles using the constrained random-phase approximation. The correlated
subspaces are formed from $t_{2g}$ or $e_{g}$ bands at the Fermi energy.
Elimination of the efficient screening taking place in these narrow bands
gives rise to sizable interaction parameters $U$ between the localized
$t_{2g}$ ($e_{g}$) electrons. Due to this large Coulomb interaction, we find
$U/W>$1 (with the band width $W$) in most TM halides, making them strongly
correlated materials. Among the metallic TM halides in paramagnetic state, the
correlation strength $U/W$ reaches a maximum in Ni$X_{2}$ and Cr$X_{3}$ with
values much larger than the corresponding values in elementary TMs and other
TM compounds. Based on the Stoner model and the calculated $U$ and $J$ values,
we discuss the tendency of the electron spins to order ferromagnetically.

###### pacs:

71.27.+a, 71.15.-m, 75.10.Lp, 68.90.+g

##  I Introduction

Since the discovery of graphene Geim ; Katsnelson  two-dimensional (2D)
materials have been extensively studied due to their rich physical properties
and diverse technological applications. One of the important applications is
the use of such 2D systems in spintronics for logic and memory applications
Kawakami ; Ohno ; Ashton . In particular, 2D half-metallic magnets and spin-
gapless semiconductors are desired for reconfigurable spintronic devices,
which combines memory and logic into a single device Reconfigurable . Despite
substantial interest, most of the 2D materials are, however, not magnetic in
their pristine form. From a theoretical point of view, according to the
Mermin-Wagner theorem Mermin , long-range magnetic order is not possible in 2D
systems at finite temperatures, but this restriction is removed by magnetic
anisotropy, which enables the formation of long-range magnetic order even in
monolayers. Several standard approaches such as adsorption of atoms Sofo ;
Ersoy-0 ; Boukhvalov ; Elias ; Zhou , point defects Hanif ; Yazyev ; Ugeda ;
McCreary ; Nair , and edge engineering Rao ; Magda ; Feldner ; Li ; Bagherpour
were developed to induce ferromagnetism in graphene and other graphene-like 2D
materials. But these systematic ways are not well controlled for realistic
applications. Therefore, 2D materials with intrinsic magnetism are of great
interest for testing theories of magnetism in low dimensions as well as for
ultralow-power memory and logic device applications.

Recently, intrinsic 2D ferromagnetism has been observed in materials
containing transition-metal (TM) atoms such as in Cr2Ge2Te6 Gong , Fe3GeTe2,
CrI3 Huang ; Kyle ; Shcherbakov ; Shabbir ; Jiang , VSe2Bonilla , and MnSe2
OHara . For instance, it was observed that the CrI3 monolayer exhibits
ferromagnetic order below $T_{c}$=45 K Huang . VSe2 and MnSe2 were reported to
have itinerant ferromagnetic order even at room temperature Bonilla ; OHara .
Indeed, the synthesis of the CrI3 monolayer has led to a huge experimental and
theoretical interest. First-principles theoretical studies have predicted that
the long-range magnetic order is also possible in other 2D monolayers of TM
dihalides and trihalides (_MX 2_ and _MX 3_, _M_ = V, Cr, Mn, Fe and Ni, and
_X_ =Cl, Br and I) McGuire ; Ping ; Tomar ; Wang ; Webster ; Zhang-1 ; Zhang-2
; McGuire-2 ; Botana ; Junjie-1 ; Vadym . Besides CrI3, ferromagnetic order in
the other TM halides such as CrCl3 McGuire-0 , CrBr3 Zhaowe ; Michael , NiI2
Kurumaji , and VI3 Kong ; Son  monolayers were discovered experimentally and
confirmed theoretically Torelli ; Besbes ; Tian ; Junjie ; Junyi .

Due to the presence of narrow $t_{2g}$ or $e_{g}$ states at the Fermi level
Wang  as well as reduced screening and quantum confinement effects arising
from reduced dimensionality, correlation effects are expected to play a
crucial role in determining the electronic and magnetic properties of the 2D
TM halides. Density functional theory (DFT) based on the local-spin-density
approximation (LSDA) or generalized gradient approximation (GGA) may therefore
not be a reliable method to calculate the physical properties of TM halides.
In this respect, methods beyond DFT such as DFT+$U$ and DFT+DMFT might be
necessary. Some TM halides have been studied by employing the DFT+$U$ method
Torelli ; Botana ; Vadym , in which the effective Coulomb interaction
parameters $U$ are chosen arbitrarily or the $U$ values of the 3D TMs are
used. Only recently, a self-consistent constrained DFT method within linear
response theory has been employed to calculate Hubbard $U$ parameters for
VCl3, VI3 and CrX3 (_X_ =Cl, Br, I) Junjie ; Junyi . The obtained $U$
parameters for V and Cr 3d orbitals turn out to be close to the corresponding
values in elementary transition metals and other TM compounds Ersoy ; Yekta .

The aim of this paper is a first-principles determination of the strength of
the effective Coulomb interaction (Hubbard $U$) between localized electrons in
2D TM halides _MX 2_ and _MX 3_ (_M_ = V, Cr, Mn, Fe and Ni, and _X_ = Cl, Br
and I) by employing the constrained random-phase approximation (cRPA) approach
cRPA_1 ; cRPA_2 ; cRPA_3  within the full-potential linearized augmented
plane-wave (FLAPW) method using maximally localized Wannier functions (MLWFs)
Marzari ; Freimuth . We find the Hubbard $U$ parameters for $t_{2g}$ or
$e_{g}$ electrons in metallic TM halides (paramagnetic state) vary between 1.0
and 5.1 eV, giving correlation strengths $U/W>1$, larger than corresponding
values in elementary TMs and other TM compounds demonstrating strong
electronic correlation in the TM halides. Furthermore, based on the Stoner
model, we use the calculated $U$ and $J$ values to assess the stability of the
ferromagnetic ordering.

The rest of the paper is organized as follows: In Sec. II we briefly present
the computational method and the cRPA method. In Sec. III, we present
calculated values of Coulomb interaction parameters for _MX 2_ and _MX 3_ for
TM halides. Finally, we summarize our conclusions in Sec. IV.

##  II Computational method

We consider 2D TM halides with formulas _MX 2_ and _MX 3_ (_M_ =Ti, V, Cr, Mn,
Fe, Co, Ni; _X_ =Cl, Br, I). Fig. 1(a) and Fig. 1(b) show the side and top
view of crystal structures _MX 2_ dihalides and _MX 3_ trihalides,
respectively. The lattice of TM dihalides consists of triangular nets of TM
atoms and exhibits geometrical frustration when the magnetic moments couple
antiferromagnetically. On the other hand, the TM atoms form honeycomb nets in
_MX 3_ trihalide monolayers. The lattice parameters are taken from Refs.
Botana ; McGuire-2 ; Tomar . Simulation of $MX_{2}$ and $MX_{3}$ unit cells,
containing one and two formula units, respectively, is based on the slab model
having a 25 Å vacuum separating them. For DFT calculations we use the FLEUR
code fleur_code , which is based on the FLAPW method. For the exchange
correlation functional we use the generalized gradient approximation (GGA)
parametrized by Perdew _et al._ Perdew  (PBE). A 18$\times$18$\times$1
$\mathbf{k}$-point grid is used for all systems. A linear momentum cutoff of
$G_{\mathrm{m}ax}$ = 4.5 bohr-1 is chosen for the plane waves. The effective
Coulomb interaction parameters are calculated within the cRPA method cRPA_1 ;
cRPA_2 ; cRPA_3  implemented in the SPEX code Friedrich  with Wannier orbitals
constructed from projection onto localized muffin-tin orbitals Freimuth . A
dense 16$\times$16$\times$1 k-point grid is used for the cRPA calculations.

Figure 1: (Colors online) (a) Top and side views of the two- dimensional
crystal structure of TM dihalides _MX 2_ (b) Top and side views of the two-
dimensional crystal structure of TM trihalides _MX 3_. The blue and purple
circles exhibit $M$ and $X$ atoms respectively. The $M$ atoms in (a) form a
frustrated triangular lattice, while in (b) their honeycomb structure is a bi-
partite and hence non-frustrated lattice.

To identify the correlated subspace and construct Wannier functions properly,
non-spin-polarized projected density of states (DOS) are calculated for all
systems and MLWFs are constructed for $t_{2g}$ or $e_{g}$ orbitals. To verify
the validity of the calculated Wannier functions, in Fig. 2a(a) and Fig. 2b(b)
we present a comparison of the non-spin-polarized DFT-PBE band structures with
the corresponding Wannier-interpolated band structures obtained with $e_{g}$
and $t_{2g}$ Wannier orbitals for NiI2 and CrI3, respectively. In all cases,
the original and the Wannier-interpolated bands agree very well. The bands
around the Fermi energy, formed by Ni-$e_{g}$ (Cr-$t_{2g}$) orbitals in NiI2
(CrI3), are well separated from the rest of the bands. They can thus be
employed to define an effective two-orbital (six-orbital) low-energy
Hamiltonian. In a similar fashion, correlated subspaces can be defined for all
considered systems, of $t_{2g}$ and $e_{g}$ character for early and late TM
halides, respectively, as shown in Fig. 3. We note that the $t_{2g}$-$e_{g}$
splitting is small for some systems (in particular, for some of the
dihalides)Kvashnin . It could, therefore, be necessary to go beyond the
present minimal subspace by including, for example, the full $d$ shell or by
considering spin polarization. However, it is already obvious from Fig. 3 that
one might then encounter the problem of entangled bands, which, on the one
hand, complicates the Wannier construction and, on the other, makes the
elimination of the subspace screening Eq. (4) less straightforward. For the
present comparative study of the electron-electron interaction strength in a
large class of materials, we therefore restrict ourselves to the minimal
subspaces, all formed by isolated sets of bands, which will capture the
essential physics of these materials. Our calculations may serve as reference
point for more sophisticated studies of any of the investigated materials in
the future.

Due to the systems’ symmetry, the bands are not of pure $t_{2g}$ and $e_{g}$
character but are mixtures and also exhibit admixture from I-$p$ states. The
denomination ”$t_{2g}$” and ”$e_{g}$” thus refers to their dominant orbital
character. We describe the orbitals in more detail in the following.

In all systems, the TM atoms are each bound to six halogen atoms in octahedral
coordination. The octahedron is, however, tilted with respect to the standard
cartesian coordinate system, which has the $z$-axis perpendicular to the
layers. Furthermore, the octahedron is distorted because of the two-
dimensionality of the structure. Opposite halogens are still exactly opposite
(forming a halogen-TM-halogen bond angle of 180∘), whereas of the remaining
twelve halogen-TM-halogen bond angles six are slightly below and six are
slightly above 90∘. For example, in NiI2 (CrI3) the difference to the right
angle is 8.62∘ (1.13∘).

The octahedron is tilted in such a way that two opposite faces of the
octahedron are parallel to the layers. If we take the three Ni-I bonds in
NiI2, whose iodine ends form the corners of one of these octahedron faces, as
$x^{\prime}$-, $y^{\prime}$\- and $z^{\prime}$-axes, we can identify local
$e_{g}$ orbitals in Fig. 2b(c). The orbitals show a strong admixture of iodine
$p$ states, a delocalization effect which will be reflected in reduced
interaction parameters later-on. Still, the $d_{{z^{\prime}}^{2}}$ and
$d_{{x^{\prime}}^{2}-{y^{\prime}}^{2}}$ orbital character is readily seen. In
CrI3, the octahedron is similarly tilted, and we can consider local
$x^{\prime}$-, $y^{\prime}$-, $z^{\prime}$-axes as above. While two of the
orbitals presented in Fig. 2b(d) indeed look like $t_{2g}$ orbitals, the one
in the middle actually has the form of a $d_{z^{2}}$ orbital, which is,
however, oriented along the cartesian $z$-axis, perpendicular to the layers.
In fact, if we linearly combine the local $t_{2g}$ orbitals by
$(d_{x^{\prime}y^{\prime}}+d_{y^{\prime}z^{\prime}}+d_{x^{\prime}z^{\prime}})/\sqrt{3}$,
we obtain a $d_{z^{2}}$ orbital oriented perpendicular to the layer. This
linear combination results from the breaking of octahedral symmetry caused by
the layer structure. The other two $t_{2g}$ orbitals,
$(d_{y^{\prime}z^{\prime}}-d_{x^{\prime}y^{\prime}})/\sqrt{2}$ and
$(d_{x^{\prime}y^{\prime}}+d_{y^{\prime}z^{\prime}}-2d_{x^{\prime}z^{\prime}})/\sqrt{6}$,
can be described, respectively, as a $d_{x^{\prime}y^{\prime}}$ rotated by 45∘
around the $y$-axis and a distorted $d_{x^{\prime}z^{\prime}}$ orbital.

(a)

(b)

Figure 2: (Color online) DFT-PBE (red) and Wannier-interpolated band
structures (blue) of non-spin-polarized (a) NiI2 and (b) CrI3. (c) The
$e_{g}$-like MLWFs for Ni atoms of NiI2. (d) The $t_{2g}$-like MLWFs for Cr
atoms of CrI3.

The fully screened Coulomb interaction $\tilde{U}$ is related to the bare
Coulomb interaction $V$ by

$\tilde{U}(\boldsymbol{r},\boldsymbol{r}^{\prime},\omega)=\int
d\boldsymbol{r}^{\prime\prime}\epsilon^{-1}(\boldsymbol{r},\boldsymbol{r}^{\prime\prime},\omega)V(\boldsymbol{r}^{\prime\prime},\boldsymbol{r}^{\prime}),$
(1)

where $\epsilon(\boldsymbol{r},\boldsymbol{r}^{\prime\prime},\omega)$ is the
dielectric function. The dielectric function is related to the electron
polarizability $P$ by

$\epsilon(\boldsymbol{r},\boldsymbol{r}^{\prime},\omega)=\delta(\boldsymbol{r}-\boldsymbol{r}^{\prime})-\int
d\boldsymbol{r}^{\prime\prime}V(\boldsymbol{r},\boldsymbol{r}^{\prime\prime})P(\boldsymbol{r}^{\prime\prime},\boldsymbol{r}^{\prime},\omega),$
(2)

where the RPA polarization function
$P(\boldsymbol{r}^{\prime\prime},\boldsymbol{r}^{\prime},\omega)$ is given by

$\begin{gathered}P(\boldsymbol{r},\boldsymbol{r}^{\prime},\omega)=\\\
2\sum_{m}^{\mathrm{o}cc}\sum_{m^{\prime}}^{\mathrm{u}nocc}\varphi_{m}(\boldsymbol{r})\varphi_{m^{\prime}}^{*}(\boldsymbol{r})\varphi_{m}^{*}(\boldsymbol{r}^{\prime})\varphi_{m^{\prime}}(\boldsymbol{r}^{\prime})\\\
\times\Bigg{[}\frac{1}{\omega-\Delta_{mm^{\prime}}+i\eta}-\frac{1}{\omega+\Delta_{mm^{\prime}}-i\eta}\Bigg{]}.\end{gathered}$
(3)

Here, $\varphi_{m}(\boldsymbol{r})$ are the single-particle DFT Kohn-Sham
eigenfunctions, and $\eta$ a positive infinitesimal.
$\Delta_{mm^{\prime}}=\epsilon_{m^{\prime}}-\epsilon_{m}$ with the Kohn-Sham
eigenvalues $\epsilon_{m}$. ß

In the cRPA approach, in order to exclude the screening due to the correlated
subspace, we separate the full polarization function of Eq. (3) into two parts

$P=P_{d}+P_{r},$ (4)

where $P_{d}$ includes only the transitions ($m\rightarrow m^{\prime}$)
between the states of the correlated subspace and $P_{r}$ is the remainder.
Then, the frequency-dependent effective Coulomb interaction is given
schematically by the matrix equation

$U(\omega)=[1-VP_{r}(\omega)]^{-1}V.$ (5)

It contains, in $P_{r}$, screening processes that would not be captured by the
correlated subspace and excludes the ones that take place within the subspace.

Figure 3: (Colors online) Orbital-resolved DOS for the non-spin-polarized
$M$I2 and $M$I3 TM halides. Each panel shows the DOS projected onto $3d$
states of the _M_ atom as well as on $5p$ states of the I atom. The two
distinct groups of $d$ bands correspond to $t_{2g}$ and $e_{g}$ bands,
respectively. The former overlaps with $5p$ states in the late TM iodides.

The matrix elements of the effective Coulomb interaction in the MLWF basis are
given by

$\begin{gathered}U_{\mathbf{R}n_{1},n_{3},n_{2},n_{4}}(\omega)=\\\ \int\int
d\boldsymbol{r}d\boldsymbol{r}^{\prime}w_{n_{1}\mathbf{R}}^{*}(\boldsymbol{r})w_{n_{3}\mathbf{R}}(\boldsymbol{r})U(\boldsymbol{r},\boldsymbol{r}^{\prime},\omega)w_{n_{4}\mathbf{R}}^{*}(\boldsymbol{r}^{\prime})w_{n_{2}\mathbf{R}}(\boldsymbol{r}^{\prime}),\end{gathered}$
(6)

where $w_{n\mathbf{R}}(\boldsymbol{r})$ is the MLWF at site $\mathbf{R}$ with
orbital index $n$, and the effective Coulomb potential
$U(\boldsymbol{r},\boldsymbol{r}^{\prime},\omega)$ is calculated within the
cRPA as described above. We define the average Coulomb matrix elements $U$,
$U^{\prime}$, and $J$ in the static limit ($\omega=0$) as follows Anisimov1 ;
Anisimov2 :

$U=\frac{1}{L}\sum_{m}U_{mm;mm}\,,$ (7)
$U^{\prime}=\frac{1}{L(L-1)}\sum_{m\neq n}U_{mn;mn}\,,$ (8)
$J=\frac{1}{L(L-1)}\sum_{m\neq n}U_{mn;nm}\,,$ (9)

where $L$ is the number of localized orbitals, i.e., two for $e_{g}$ and three
for $t_{2g}$ orbitals. This parametrization of partially screened Coulomb
interactions is the so-called Hubbard-Kanamori parametrization. Similar to the
definition of $U$ ($U^{\prime}$, $J$), we can also define the so-called fully
screened interaction parameters $\tilde{U}$ ($\tilde{U^{\prime}}$,
$\tilde{J}$) as well as unscreened (bare) $V$. The bare $V$ provides
information about the localization of Wannier functions and is a useful
parameter in the interpretation of the screened Coulomb interaction
parameters.

##  III Results and discussion

In the low-energy model Hamiltonian description of correlated solids, the non-
interacting one-electron part of the effective model is defined for a system,
in which there is no spontaneous symmetry breaking, i.e., it is a paramagnetic
(non-spin-polarized) metal. The calculation of effective Coulomb interaction
parameters (Hubbard $U$) should therefore be based on such a system.

To identify the correlated subspace for all studied TM halides, we present in
Fig. 3 orbital-resolved DOS for $M$I2 and $M$I3. In all compounds except for
the semiconducting FeI2 and CoI3, the Fermi energy falls into a group of
bands, which are of $t_{2g}$ character for the early halides, $M=$ Ti to Mn
(Ti to Fe) for $M$I2 ($M$I3), and of $e_{g}$ character for the late halides,
$M=$ Co and Ni (Ni) for $M$I2 ($M$I3). These bands are assumed to form minimal
correlated subspaces in this work. On the other hand, FeI2 and CoI3 are
semiconducting with the Fermi level falling in the energy gap between the
$t_{2g}$ and $e_{g}$ bands. Depending on the type of doping, electron or hole
doping, only one type of bands will form the minimal correlated subspace. For
these systems, we will present Hubbard $U$ parameters for $t_{2g}$, $e_{g}$,
and $d$ orbitals. The first two correspond to the zero-doping limit (or very
dilute doping), since the states of the subspace are either all occupied or
all empty. As hence no screening takes place in the subspace, the partially
and fully screened parameters (e.g., $U$ and $\tilde{U}$) are identical. The
orbital-resolved DOS of $MX_{2}$ and $MX_{3}$ with $X=$ Cl and Br look very
similar, so the subspaces can be defined identically to those of the iodides.
The $U$ matrix element of the p-admixed $t_{2g}$ state differs from the pure
$t_{2g}$ state by maximally 0.1 eV among the materials. The calculated Hubbard
$U$ values should therefore be applicable to standard DFT+$U$ implementations
that are based on atomic bases, as well, not only to implementations that
employ Wannier functions.

In Table 1, we present the onsite average intra-orbital unscreened (bare)
Coulomb interaction $V$, partially (fully) screened $U$ ($\tilde{U}$), as well
as average inter-orbital $U^{\prime}$ ($\tilde{U^{\prime}}$) and exchange
parameter $J$ ($\tilde{J}$). The behavior of the bare interaction $V$ for
$t_{2g}$ orbitals across the 3$d$ TM atoms, from Ti to Fe, in $MX_{2}$ is
similar to the case of elementary TMs. The $V$ parameter increases nearly
linearly with increasing electron number, which is due to the contraction of
the wave functions with increased nuclear charge and the concomitant increased
localization of the Wannier functions. By contrast, the $e_{g}$ orbitals
exhibit the opposite trend from Fe to Ni in $MX_{2}$: The $V$ decreases. An
analysis of the shape of the Wannier orbitals reveals that the coupling to
neighboring halogen p states gets stronger, which makes the orbitals
increasingly spill into these states giving rise to a delocalization and,
therefore, to smaller $V$ parameters. The same trend is seen for $MX_{3}$ from
Co to Ni. The behavior of $V$ for the $t_{2g}$ orbitals in the trihalides is
somewhat different from that of the dihalides: While early in the series from
Ti to Co we see a similar increase as in the dihalides, the values go over a
maximum and drop off sharply for Co$X_{3}$. In all cases, we see a decrease of
the bare $V$ for the halide series $MX_{2}$ with $X=$ Cl to I, which is likely
caused by the increase of the lattice constant in this order, making the
orbitals more extended.

The effects are reflected also in the band widths $W$ of the $t_{2g}$ and
$e_{g}$ bands presented in Table 1 (also see Fig. 3), which tend to decrease
from Ti to Ni, similar to the case of elementary $3d$ TMs with the difference
that, due to reduced coordination (reduced hybridization) in 2D, the $W$ is
much smaller than the corresponding values in $3d$ TMs, making the $t_{2g}$
and $e_{g}$ peaks in the DOS sharper and more atomic-like.

Table 1: Lattice constants $a$, orbital type of correlated subspace, bandwidth $W$, onsite average intra-orbital bare $V$, partially (fully) screened $U$ ($\tilde{U}$), inter-orbital partially (fully) screened $U^{\prime}$ ($\tilde{U}^{\prime}$), partially (fully) screened exchange interaction $J$ ($\tilde{J}$), correlation strength $U/W$, and the DOS at the Fermi level $D$(EF) for $MX_{2}$ and $MX_{3}$ compounds. _MX 2/3_ | a (Å) | orbitals |  $W$(eV) |  $V$(eV) |  $U$ ($\tilde{U}$) (eV) |  $U^{\prime}$ ($\tilde{U}^{\prime}$) (eV) |  $J$($\tilde{J}$) (eV) | $U/W$ |  $D$(EF)
---|---|---|---|---|---|---|---|---|---
TiCl2 | 3.56 | $t_{2g}$ | 1.43 | 14.07 | 5.07 (0.87) | 4.22 (0.43) | 0.43 (0.23) | 3.55 | 0.45
TiBr2 | 3.73 | $t_{2g}$ | 1.31 | 13.32 | 4.26 (0.79) | 3.47 (0.41) | 0.39 (0.19) | 3.25 | 0.51
TiI2 | 4.11 | $t_{2g}$ | 1.15 | 11.70 | 3.26 (0.63) | 2.61 (0.32) | 0.33 (0.16) | 2.83 | 0.73
VCl2 | 3.62 | $t_{2g}$ | 1.20 | 15.95 | 4.60 (0.95) | 3.62 (0.63) | 0.51 (0.16) | 3.83 | 1.28
VBr2 | 3.81 | $t_{2g}$ | 1.03 | 14.96 | 3.98 (0.86) | 3.03 (0.57) | 0.49 (0.16) | 3.86 | 1.91
VI2 | 4.08 | $t_{2g}$ | 0.94 | 13.63 | 3.10 (0.44) | 2.33 (0.27) | 0.39 (0.10) | 3.30 | 2.25
CrCl2 | 3.55 | $t_{2g}$ | 1.05 | 17.32 | 4.13 (0.23) | 3.13 (0.03) | 0.54 (0.09) | 3.93 | 4.50
CrBr2 | 3.74 | $t_{2g}$ | 0.99 | 16.51 | 3.64 (0.23) | 2.68 (0.06) | 0.51 (0.08) | 3.68 | 4.87
CrI2 | 3.99 | $t_{2g}$ | 0.96 | 14.88 | 2.96 (0.26) | 2.16 (0.07) | 0.42 (0.10) | 3.08 | 3.64
MnCl2 | 3.64 | $t_{2g}$ | 0.85 | 19.10 | 3.71 (0.82) | 2.69 (0.62) | 0.56 (0.08) | 4.36 | 6.41
MnBr2 | 3.84 | $t_{2g}$ | 0.81 | 18.12 | 3.25 (0.29) | 2.29 (0.06) | 0.52 (0.07) | 4.01 | 6.02
MnI2 | 4.12 | $t_{2g}$ | 0.75 | 17.19 | 2.76 (0.26) | 1.90 (0.04) | 0.46 (0.06) | 3.68 | 4.18
FeCl2 | 3.49 | $t_{2g}$ | 1.10 | 20.11 | 3.50 (3.50) | 2.43 (2.43) | 0.56 (0.56) | 3.18 | 0.00
|  | $e_{g}$ | 0.70 | 17.00 | 3.06 (3.06) | 2.06 (2.06) | 0.50 (0.50) | 4.37 | 0.00
|  | $d$ | 2.41 | 19.96 | 5.97 (3.21) | 4.84 (2.42) | 0.57 (0.42) | 2.48 | 0.00
FeBr2 | 3.69 | $t_{2g}$ | 0.97 | 19.02 | 3.14 (3.14) | 2.11 (2.11) | 0.53 (0.53) | 3.24 | 0.00
|  | $e_{g}$ | 0.78 | 15.24 | 2.64 (2.64) | 1.76 (1.76) | 0.44 (0.44) | 3.38 | 0.00
|  | $d$ | 2.37 | 19.13 | 5.21 (3.07) | 3.86 (2.20) | 0.55 (0.42) | 2.20 | 0.00
FeI2 | 3.98 | $t_{2g}$ | 0.80 | 17.39 | 2.52 (2.52) | 1.66 (1.66) | 0.44 (0.44) | 3.15 | 0.00
|  | $e_{g}$ | 0.91 | 12.83 | 2.03 (2.03) | 1.36 (1.36) | 0.33 (0.33) | 2.23 | 0.00
|  | $d$ | 2.21 | 17.64 | 4.07 (3.05) | 2.98 (1.67) | 0.53 (0.38) | 1.84 | 0.00
CoCl2 | 3.49 | $e_{g}$ | 0.59 | 16.51 | 3.18 (0.38) | 2.20 (0.26) | 0.49 (0.06) | 5.39 | 3.26
CoBr2 | 3.73 | $e_{g}$ | 0.61 | 14.93 | 2.67 (0.19) | 1.82 (0.07) | 0.42 (0.06) | 4.38 | 3.51
CoI2 | 3.92 | $e_{g}$ | 0.64 | 12.26 | 2.16 (0.16) | 1.52 (0.03) | 0.32 (0.07) | 3.38 | 4.54
NiCl2 | 3.45 | $e_{g}$ | 0.53 | 15.17 | 3.37 (0.16) | 2.43 (0.04) | 0.47 (0.06) | 6.36 | 4.12
NiBr2 | 3.64 | $e_{g}$ | 0.59 | 13.52 | 3.09 (0.22) | 2.27 (0.07) | 0.41 (0.07) | 5.24 | 3.28
NiI2 | 3.94 | $e_{g}$ | 0.62 | 10.60 | 2.36 (0.44) | 1.78 (0.27) | 0.29 (0.09) | 3.81 | 2.76
TiCl3 | 5.88 | $t_{2g}$ | 1.16 | 14.48 | 4.62 (0.68) | 3.75 (0.24) | 0.46 (0.22) | 3.98 | 1.09
TiBr3 | 6.27 | $t_{2g}$ | 1.08 | 14.15 | 3.97 (0.51) | 3.12 (0.16) | 0.43 (0.17) | 3.68 | 0.82
TiI3 | 6.62 | $t_{2g}$ | 1.01 | 12.86 | 2.99 (0.37) | 2.26 (0.06) | 0.38 (0.15) | 2.96 | 0.61
VCl3 | 6.04 | $t_{2g}$ | 0.91 | 16.28 | 4.49 (0.31) | 3.46 (0.04) | 0.54 (0.14) | 4.93 | 1.17
VBr3 | 6.33 | $t_{2g}$ | 0.86 | 15.59 | 3.82 (0.36) | 2.84 (0.09) | 0.51 (0.14) | 4.44 | 1.86
VI3 | 6.86 | $t_{2g}$ | 0.82 | 14.47 | 2.91 (0.28) | 2.07 (0.06) | 0.44 (0.12) | 3.55 | 2.44
CrCl3 | 5.75 | $t_{2g}$ | 0.81 | 16.53 | 4.08 (0.48) | 3.07 (0.20) | 0.53 (0.17) | 5.04 | 2.72
CrBr3 | 6.34 | $t_{2g}$ | 0.76 | 16.24 | 3.50 (0.31) | 2.52 (0.12) | 0.51 (0.10) | 4.61 | 2.40
CrI3 | 6.85 | $t_{2g}$ | 0.61 | 15.29 | 2.67 (0.25) | 1.84 (0.10) | 0.44 (0.08) | 4.38 | 2.47
MnCl3 | 6.08 | $t_{2g}$ | 0.85 | 18.05 | 3.64 (0.56) | 2.63 (0.30) | 0.55 (0.13) | 4.28 | 2.15
MnBr3 | 6.39 | $t_{2g}$ | 0.79 | 17.28 | 3.15 (0.55) | 2.19 (0.31) | 0.51 (0.13) | 3.99 | 3.19
MnI3 | 6.85 | $t_{2g}$ | 0.65 | 15.73 | 2.62 (0.48) | 1.61 (0.26) | 0.44 (0.12) | 4.03 | 3.39
FeCl3 | 6.05 | $t_{2g}$ | 0.82 | 18.14 | 3.30 (0.33) | 2.32 (0.14) | 0.54 (0.09) | 4.02 | 2.52
FeBr3 | 6.43 | $t_{2g}$ | 0.78 | 17.24 | 2.85 (0.29) | 1.92 (0.11) | 0.50 (0.09) | 3.65 | 2.47
FeI3 | 6.97 | $t_{2g}$ | 0.69 | 15.29 | 2.14 (0.22) | 1.37 (0.03) | 0.41 (0.08) | 3.10 | 2.62
CoCl3 | 6.07 | $t_{2g}$ | 0.65 | 16.82 | 2.95 (2.95) | 2.05 (2.05) | 0.50 (0.50) | 4.54 | 0.00
|  | $e_{g}$ | 0.45 | 13.95 | 2.43 (2.43) | 1.67 (1.67) | 0.38 (0.38) | 5.40 | 0.00
|  | $d$ | 1.72 | 16.21 | 3.60 (2.70) | 2.67 (2.04) | 0.47 (0.35) | 2.09 | 0.00
CoBr3 | 6.30 | $t_{2g}$ | 0.93 | 14.47 | 2.45 (2.45) | 1.67 (1.67) | 0.43 (0.43) | 2.63 | 0.00
|  | $e_{g}$ | 0.78 | 12.84 | 2.11 (2.11) | 1.42 (1.42) | 0.34 (0.34) | 2.71 | 0.00
|  | $d$ | 2.45 | 14.27 | 3.08 (2.23) | 2.26 (1.71) | 0.39 (0.31) | 1.26 | 0.00
CoI3 | 6.81 | $t_{2g}$ | 1.62 | 10.34 | 1.34 (1.34) | 0.94 (0.94) | 0.24 (0.24) | 0.83 | 0.00
|  | $e_{g}$ | 1.10 | 10.92 | 1.48 (1.48) | 0.96 (0.96) | 0.26 (0.26) | 1.35 | 0.00
|  | $d$ | 3.32 | 10.65 | 2.10 (1.40) | 1.55 (1.05) | 0.24 (0.18) | 0.63 | 0.00
NiCl3 | 6.05 | $e_{g}$ | 0.63 | 11.76 | 2.01 (0.15) | 1.39 (0.04) | 0.31 (0.06) | 3.19 | 2.52
NiBr3 | 6.16 | $e_{g}$ | 0.75 | 10.55 | 1.79 (0.25) | 1.22 (0.11) | 0.28 (0.07) | 2.39 | 1.85
NiI3 | 6.64 | $e_{g}$ | 0.91 | 8.55 | 1.08 (0.24) | 0.69 (0.10) | 0.20 (0.07) | 1.19 | 1.40

To discuss the partially screened (Hubbard $U$) effective Coulomb interaction
parameters, we focus on the TM iodides. As seen in Table 1, the $U$ values for
$M$ sites in $M$I2 ($M$I3) compounds vary between 2.1$-$3.3 eV (1.1$-$3.0 eV)
and decrease with moving from Ti to Ni, which can be described by the
projected density of states in Fig. 3. Just below the $d$ states there is a
broad peak of iodine $5p$ states, which should contribute with $5p\rightarrow
d$ transitions sizably to the screening. Across the series Ti to Ni, the $5p$
states are seen to approach the $d$ states, which effectively increases the
electronic screening and, thus, acts to compensate the increase of $U$ caused
by Wannier localization, giving rise to the reduction of $U$ parameters with
increasing $3d$ electron number in both types of TM halides. The same
behaviour is observed in Br- and Cl-based TM halides (see Table. 1).

Moving upwards in the group of halogens, from I to Cl, the _M_ $-$_X_ bond
lengths decrease. As a consequence, orbitals overlap more strongly and
hybridization increases, pushing the states apart energetically. The larger
energy difference makes electronic transitions $p\rightarrow t_{2g}$ less
likely and reduces electronic screening. From this, one would expect to see a
tendency to larger Hubbard $U$ values, which is, in fact, what we observe in
Table 1.

Comparing the partially screened ($U$) with the fully screened parameters
($\tilde{U}$) gives information about the screening within the correlated
subspace. Except for the semiconductors Fe$M_{2}$ and Co$M_{3}$, this
screening is metallic. It is very efficient and becomes the dominant screening
channel. As a consequence, the fully screened parameters $\tilde{U}$ show a
behavior very different from $U$: The values are all very small for the
metallic systems and fall in the range between 0.1 and 0.5 eV, about one fifth
of the $U$ values. They do not follow a specific trend across the TM series,
nor do they follow a general ordering with respect to the different kinds of
halides (Cl, Br, I) as in the partially screened $U$ parameters. Our results
show that $t_{2g}\rightarrow t_{2g}$ and $e_{g}\rightarrow e_{g}$ transitions
contribute substantially to the screening of the fully screened Coulomb
interaction $\tilde{U}$ in metallic systems, while in the semiconducting ones
this screening vanishes, making $\tilde{U}$ identical to $U$ for the $t_{2g}$
and $e_{g}$ orbitals, whereas the $t_{2g}\rightarrow e_{g}$ transitions reduce
$\tilde{U}$ with respect to $U$ for the full $3d$ shell.

In the semiconducting cases Fe$X_{2}$ and Co$X_{3}$, the $3d$ parameters show
a behavior parallel to that of the $t_{2g}$/$e_{g}$ parameters and are larger
by about 20-70%, elucidating that transitions between $t_{2g}$ and $e_{g}$
states play an important quantitative role in the electronic screening but do
not affect the screening qualitatively.

In cubic symmetry, the Hubbard-Kanamori inter-orbital Coulomb interaction term
$U^{\prime}$ satisfies the relation $U^{\prime}=U-2J$. This relation is nearly
fulfilled in most TM halides, even though cubic symmetry is broken. The $J$
parameters vary in the range 0.20$-$0.57 eV and show a behavior very much in
parallel to the bare parameters $V$, despite the very different range of
values, which reveals a more quantitative than qualitative effect of the
electronic screening on the exchange parameters $J$.

We now compare our calculated Hubbard $U$ values with reported ones in the
literature. Besbes et al. Besbes  calculated Hubbard $U$ values for bulk CrCl3
and CrI3 and obtained $U$ values of 1.79 eV (CrCl3) and 1.15 eV (CrI3), which
are significantly smaller than our calculated Coulomb matrix elements
presented in Table 1, whereas their exchange parameters are larger (0.85 and
0.78 eV) than our values. A possible reason for the disagreement is the
different dimensionality of the systems, 3D versus 2D. 3D materials usually
offer more screening with the consequence of smaller $U$ parameters. There are
other possible reasons: Besbes et al. employ a method different to ours, a
combined cLDA/RPA scheme Solovyev . They also used a different Wannier basis.
They combined chromium $d$ and halogen $p$ states into a larger Wannier basis
for a $dp$ tight-binding description, whereas we have employed minimal
$t_{2g}$ and $e_{g}$ subspaces. And, finally, the definition of the Hubbard
$U$ parameter is different. We have calculated Hubbard-Kanamori parameters for
the $t_{2g}$ and $e_{g}$ subsets (see Eq. (7)), whereas Besbes et al. defined
the Hubbard parameters for the full atomic $p$ and $d$ shells. In this
different kind of $U$ parameter the averaging is over the full matrix instead
of just over the diagonal elements.

In the case of V-based trihalides, He et al. calculated $U$ values for VCl3
and VI3 using the self-consistent linear response method within the cLDA
approach and obtained $U$=3.35 eV (for VCl3) and 3.68 eV (for VI3) Junjie .
These values are closer to our results.

In the following, we discuss the appearance of ferromagnetism in TM halides.
The ferromagnetic state is the ground state for most of the TM trihalides
$MX_{3}$, while TM dihalides $MX_{2}$ exhibit diverse magnetic behavior
ranging from half-metallic ferromagnetism (FeCl2) to antiferromagnetism (VI2)
and from 120∘-antiferromagnetism (MnI2) to helical magnetism (NiI2). A
richness of magnetic phases that results from the triangular lattice of $M$
atoms, which frustrates the exchange coupling. Such a frustration is not
present for $MX_{3}$ compounds, where $M$ atoms reside on a bi-partite
(honeycomb) lattice. The calculation of $U$ parameters for such complex
magnetic ground states is beyond the scope of the present paper. Note that
such a calculation would require the full $3d$ shell to be included in the
correlated subspace. The present values for the minimal $t_{2g}$ and $e_{g}$
subspaces would not be immediately applicable in this case.

Among the TM trihalides, CrI3 is of particular interest for spintronic
applications as well as theories of low-dimensional magnetism because it was
one of the first materials in which ferromagnetism was detected experimentally
in the monolayer limit. It is a ferromagnetic semiconductor with a Curie
temperature of 45 K in the monolayer and 61 K in the bulk.

As the $MX_{2}$ and $MX_{3}$ halides contain partially filled $3d$ TM atoms,
we can use the simple Stoner model to discuss the appearance of ferromagnetism
in these materials. The Stoner criterion for ferromagnetism is given by
$I\cdot D$(EF)$>$ 1, where $I$ is the Stoner parameter and $D$(EF) is the DOS
at the Fermi energy in the nonmagnetic state. The Hartree-Fock solution of the
multiorbital Hubbard model gives a relationship between the Stoner parameter
$I$, Hubbard $U$, and exchange $J$ by $I=(U+6J)/5$Stollhoff . Stollhoff et al.
showed that in elementary TMs the electron correlations reduce $I$ by roughly
40$\%$. We note that the magnetism in some of the TM halides might not be
describable by the simple Stoner mechanism. For example, MnI2 has an
antiferromagnetic (120 degrees) ground state. For reasons of consistency, we
nevertheless discuss the Stoner condition for all materials and leave more
detailed analyses taking into account Hund’s rule coupling for future studies.
In Fig. 4 (a) we present the correspondingly scaled Stoner parameter $I\cdot
D$(EF) for iodine-based compounds. The experimentally observed and
theoretically predicted ferromagnetic TM halides satisfy the Stoner criterion,
and the paramagnetic state is unstable toward the formation of ferromagnetism,
which reasonably agrees with the results of spin-polarized DFT total energy
calculations and the large magnetic moments presented in Fig. 4(b). The
failure of the Stoner criterion to predict the ferromagnetism of Fe$I_{2}$ is
due to the fact that this compound is a semiconductor, while the Stoner
criterion is based on a metallic parent (non-interacting) limit.

Figure 4: (Colors online) (a) Stoner criterion for $M$I2 and $M$I3 TM halides.
(b) Calculated magnetic moments (in units of $\mu_{B}$) of TM atoms for $M$I2
and $M$I3 TM halides.

With this tendency to form magnetic ground states, one might wonder whether
and to what degree the Hubbard $U$ parameters would change if a magnetic
ground state were taken as reference system instead of the non-spin-polarized
one. For instance for the case of MnI2, we obtain a $U$ value of 3.37 eV,
which is not too different from the paramagnetic case (2.76 eV). This might be
due to the fact that the formation of the magnetic moments predominantly
affects the low-energy states, but these are just the ones that make up the
correlated subspace whose screening channels are eliminated from the effective
Hubbard $U$ interaction.

Finally, we want to briefly discuss the strength of the electronic
correlations in 2D TM halides. Qualitatively, the correlation strength is
defined as the ratio of the effective Coulomb interaction $U$ to the bandwidth
$W$ ($U/W$). In Table 1 we present $U/W$ values for all TM halides. Note that
the $U/W$ values are calculated for a non-spin-polarized (paramagnetic) state.
In the case of $M$X2 and ignoring the non-metallic system (Fe _X 2_), the
correlation strength $U/W$ tends to increase from Ti- to Ni-based materials.
There is no clear trend in the trihalides, although a strong decrease of $U/W$
is noted for the $e_{g}$ subspace in the iodides and, less pronounced, also in
the bromides. For most of the metallic TM halides, we find $U/W>$ 2 with
maxima for NiCl2 and CrCl3. We thus expect electron correlations to be strong
in these materials. They should play an important role in model Hamiltonian
studies of the TM halides.

As a consequence of $U/W>1$, one may expect rich correlation phenomena such as
magnetic order, Mott-insulating phases, etc. For instance, in contrast to
experimental resultsKong ; Son  showing insulating behavior in VI3, the
partially filled bands in PBE calculations give rise to a half-metallic
behaviorTian ; Junjie . This implies that electron-electron interactions play
a crucial role in electronic and magnetic properties of TM halides, especially
in the systems with nearly half-filled $3d$ bands. Applying the DFT+$U$ method
with $U$=3.8 eV to Mn _X_ 2 dihalides not only increases the band gap but also
results in ferromagnetic order, whereas these systems remain antiferromagnetic
when calculated without $U$ Vadym . In the case of VI3, DFT+$U$ employing
$U$=3.5 eV opens a finite band-gap of about 0.84 eV Tian , which is in
agreement with experiments, while the system is a half-metal within DFT-PBE.
VI3 is therefore commonly classified as a Mott insulator. This inconsistency
between experiment and DFT-PBE has also been found for Cr _X_ 3, manifesting a
possible Mott insulating stateMcGuire ; Zhang-1 . Experimentally, CrI3 is
insulatingHuang ; Jiang ; Kyle  even above the Curie temperature, which
suggests that the band-gap does not stem from the exchange splitting (i.e,
from magnetism) but that the strong electron-electron correlation is
responsible for the formation of the band gap in this material. There are
theoretical works that argue that the band gap of Cr-based trihalides can be
described as a mixture of Mott-Hubbard and charge-transfer typeZhang-1 . Note
that layered materials that exhibit a Mott-insulating character are very rare.
The Mott phase has been experimentally discovered in bilayer twisted
grapheneShen-x ; Seo-x ; Choi-1 , $\sqrt{13}\times\sqrt{13}$ supercell of
1T-TaS(Se)2 and 1T-NbSe2 materials (the Star of David cluster)Ma-x ; Perfetti
; Ritschel , and TM phosphorous trichalcogenidesKim1 , _M_ P$X_{3}$, where _M_
is a TM and _X_ are chalcogen elements, which were confirmed by _ab initio_
calculations Kim2 ; Lee-2 ; Choi-1 . In all mentioned monolayers, the observed
insulating phase has been discussed to originate from the presence of narrow
bands around the Fermi energy. For example, in the distorted phase of 1T-TaS2,
there is a flat band with mainly $d_{z^{2}}$ character at $E_{\mathrm{F}}$
which increases $U/W$ substantially Tresca ; Kamil ; Darancet , even though
the Coulomb interaction parameter $U$ has been calculated to be 0.4 eV only
Kamil , which is significantly smaller than the corresponding value $U$=2.27
eV in undistorted 1T-TaS2 Cococcioni . Such narrow bands, of $t_{2g}$ or
$e_{g}$ character, are also present in CrI3, VI3, and NiI2, resulting in a
large $U/W$ difference between TM halides and elementary TMs.

##  IV Summary and Outlook

We have performed systematic _ab initio_ calculations to determine the
strength of the effective Coulomb interaction (Hubbard $U$) between localized
electrons in various 2D TM Halides with formulas _MX 2_ and _MX 3_ (_M_ =Ti,
V, Cr, Mn, Fe, Co, Ni; _X_ =Cl, Br, I) employing the parameter-free cRPA
scheme. We found that in most of the metallic TM halides (in the non-magnetic
state) the Hubbard $U$ parameters for $t_{2g}$ or $e_{g}$ electrons are larger
than 3.0 eV, and the band widths $W$ are less than 1.0 eV. As a consequence,
we find that $U/W>1$. So, these materials can be classified as moderately to
strongly correlated systems. The correlation strength in TM halides is much
larger than the corresponding values in elementary TMs and TM compounds.
Furthermore, using the calculated $U$ and $J$ values we discuss the stability
of the ferromagnetic ordering within the Stoner model. The obtained Coulomb
interaction parameters are important both for a basic understanding of the
physics of TM halides and for use in model Hamiltonians applied to describe
electronic, magnetic, and optical properties of these materials.

The ferromagnetic state of $MX_{3}$ compounds can undergo thermal and quantum
fluctuations. The thermal fluctuations are relevant to all of them. This
provides a nice experimental handle to study the role of the underlying
magnetic background in transport and other properties by tuning the
temperature across the Curie temperature. This is a unique opportunity not
present in generic two-dimensional materials. For the compounds with small
magnetic moment, the quantum fluctuations are expected to play a significant
role. The effective theory of small fluctuations around a magnetically ordered
state is known as non-linear sigma model FradkinBook . They describe spin-1
bosons known as magnons. Such bosons can mediate forces between the electrons,
pretty much the same way gauge bosons (photons) mediate the Coulomb
interactions. The part of the interaction mediated by spin-1 bosons cannot be
screened TsvelikBook . Therefore our $MX_{3}$ compounds having smaller
magnetic moments are expected to display many unexpected correlation
phenomena, such as non-Fermi liquids. The present paradigm for study of
correlation physics in 2D systems, unlike high temperature superconductors or
heavy fermion systems are not buried in the bulk, but live in a true 2D layer.
They can therefore enjoy most of the control and functionalization methods
developed in the context of graphene KatsnelsonBook .

Furthermore, in analogy to graphene nanoribbons BreyFertig  where armchair
ribbons offer low-energy one-dimensional bands, the nanoribbons of TM halides
are expected to serve as a new platform for strongly correlated one-
dimensional bands. Such bands are spatially separated from the other bands,
and hence the fascinating Luttinger physics can be studied using local probes.
Also gapping out the edge modes in such ribbons Louie  in the present
correlated systems can be an interesting framework for electronic applications
that require an energy gap, albeit with substantial Mott character.

### Acknowledgements

E.S. and I.M. gratefully acknowledge funding provided by the European Union
(EFRE, Project No. ZS/2016/06/79307).

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