# Open-world Multi-label Text Classification with
Extremely Weak Supervision

Xintong Li1, Jinya Jiang1, Ria Dharmani1
Jayanth Srinivasa2, Gaowen Liu2, Jingbo Shang1
University of California, San Diego1 Cisco2
{xil240, j9jiang, rdharmani<EMAIL_ADDRESS>
{jasriniv<EMAIL_ADDRESS>
###### Abstract

We study open-world multi-label text classification under extremely weak
supervision (XWS), where the user only provides a brief description for
classification objectives without any labels or ground-truth label space.
Similar single-label XWS settings have been explored recently, however, these
methods cannot be easily adapted for multi-label. We observe that (1) most
documents have a dominant class covering the majority of content and (2) long-
tail labels would appear in some documents as a dominant class. Therefore, we
first utilize the user description to prompt a large language model (LLM) for
dominant keyphrases of a subset of raw documents, and then construct a
(initial) label space via clustering. We further apply a zero-shot multi-label
classifier to locate the documents with small top predicted scores, so we can
revisit their dominant keyphrases for more long-tail labels. We iterate this
process to discover a comprehensive label space and construct a multi-label
classifier as a novel method, X-MLClass. X-MLClass exhibits a remarkable
increase in ground-truth label space coverage on various datasets, for
example, a 40% improvement on the AAPD dataset over topic modeling and keyword
extraction methods. Moreover, X-MLClass achieves the best end-to-end multi-
label classification accuracy.

Open-world Multi-label Text Classification with
Extremely Weak Supervision

Xintong Li1, Jinya Jiang1, Ria Dharmani1 Jayanth Srinivasa2, Gaowen Liu2,
Jingbo Shang1 University of California, San Diego1 Cisco2 {xil240, j9jiang,
rdharmani<EMAIL_ADDRESS>{jasriniv<EMAIL_ADDRESS>

Figure 1: An overview of our X-MLClass framework. The only required
supervision from the user is a brief description of the classification
objective. During the first LLM prompting stage for keyphrases, X-MLClass
leverages this description as a part of the prompt, so it will be helpful if
the description includes some demonstrations.

##  1 Introduction

Multi-label text classification (MLTC) aims to assign one or more labels to
each input document in the corpus. Traditional methods Liu et al. (2022);
Xiong et al. (2021); Gera et al. (2022) require a complete list of class
names, which is challenging to provide beforehand given the massive number of
documents and diverse topics. This work focuses on a new problem, open-
world111Our “open-world” definition denotes the absence of any class
information during the training and testing phase, which is more challenging
than the traditional settings. MLTC under extremely weak supervision (XWS),
where the user only provides a brief description for classification objectives
without any labels or ground-truth label space. Despite the considerable
technical challenges posed by this task, its practical significance in real-
world applications cannot be underestimated. For instance, the need for
product tagging and categorization is ubiquitous in online shopping platforms.
It requires identifying multiple labels for each product without access to a
predefined label space, a challenge our model adeptly addresses.

The most related XWS problems are text clustering Zhang et al. (2023); Wang et
al. (2023b) and topic modeling Grootendorst (2022); Pham et al. (2023), where
those methods are typically only capable of assigning a single label to each
document. These single-label methods cannot be easily adapted for multi-label.

We observe that (1) most documents have a dominant class covering the majority
of content and (2) long-tail labels would appear as the dominant class in some
documents. Experiments reveal that over 90% of documents contain a dominant
class, and 100% of labels appear as the dominant class in at least one
document, as analyzed further in Section 4. Based on these observations, we
propose a novel method, X-MLClass, to discover a pragmatic label space by
iteratively adding (long-tailed) labels and construct a multi-label text
classification classifier with the assistance of LLM (i.e., llama-2-13b-chat
in our experiments), as shown in Figure 1. Our approach requires only a brief
user description about the classification objective as prompt for LLM, which
significantly reduces the cost of model training.

The first step in X-MLClass is to construct a high-quality label space. We
start with a reasonably large subset of the raw documents. For each document,
we partition it into chunks to better align with the context length of LLM and
also ensure that each chunk contains a single topic, and then prompt the LLM
to generate the most dominant keyphrases for each chunk. As previous LLM-based
text clustering work has suggested Wang et al. (2023b, a), there are very
likely some semantically redundant yet lexically different keyphrases among
the generated ones. We cluster these keyphrases, and within every cluster, we
pull together the corresponding chunks of the keyphrases closest to the
cluster center to prompt the LLM once again, generating one single label for
each cluster. After eliminating labels with high similarity scores, we
constitute an initial label space.

We then apply the state-of-the-art textual entailment-based classification
methods Pàmies et al. (2023) to construct a classifier that revisits the
documents and identifies long-tail labels. Specifically, we query every text
chunk against all the labels for the entailment score, identifying chunks with
low top predicted scores that lack a dominant class. We revisit the keyphrases
generated by these chunks to unveil more long-tail labels, selectively
choosing keyphrases with a modest presence in the entire keyphrase set but
absent in the original label space. These new keyphrases are included in the
label set, and documents are reassessed with this updated set. By repeating
these steps for a fixed number of iterations, the final label space contains a
substantial number of long-tail labels.

Extensive experiments on 5 benchmark datasets reveal the superiority of
X-MLClass outperforming all compared methods. Remarkably, compared with
baselines, X-MLClass achieves a significant enhancement of 40% and 30% in
ground-truth label space coverage on the AAPD and RCV1-V2 datasets.
Furthermore, it achieves the highest accuracy in zero-shot MLTC, surpassing
the top-ranking models on HuggingFace across all datasets.

Our contributions are summarized as:

  * •

We attack a challenging problem, open-world MLTC with XWS, where the user only
provides a brief description for classification objectives without any labels
or ground-truth label space.

  * •

We propose a novel framework X-MLClass which iteratively discovers the label
space and builds an MLTC classifier.

  * •

Compared with all traditional label generation methods, X-MLClass achieves a
significantly higher coverage score along with superior end-to-end
classification accuracy.

Our source code and dataset can be obtained here
222https://github.com/Kaylee0501/X-MLClass.

##  2 Related Work

#### Topic Modeling:

Topic modeling has been widely adopted for discovering latent thematic
structures within collections of text documents. Traditional models, such as
Latent Dirichlet Allocation (LDA) Blei et al. (2003) and Non-Negative Matrix
Factorization (NMF) Févotte and Idier (2011) represent documents as mixtures
of latent topics using bag-of-words representations. New techniques like
Top2Vec Angelov (2020) and BERTopic Grootendorst (2022) build primarily on
clustering embeddings, demonstrating the potential of embedding-based topic
modeling approaches. Another recent method, TopicGPT Pham et al. (2023), takes
a different approach by prompting large language models for topic generation,
aligning more closely with ground truth labels. However, these existing
methods typically provide a single topic for each document, which poses
challenges when extending them to multi-label scenarios.

#### Multi-label Text Classification:

Numerous approaches have been proposed to tackle the complexities of Multi-
Label Text Classification (MLTC) problems. Bhatia and Jain Bhatia et al.
(2015) employ embedding-based methods, leveraging the power of embeddings to
train individual classifiers for each label. Later, XML-CNN Liu et al. (2017)
uses a Convolutional Neural Network (CNN) to learn text representations,
demonstrating improvements in MLTC accuracy. Recent works have started to
tackle MLTC problems using a small amount of labeled data or even with no
labels at all. For example, Shen et al. (2021) achieves impressive results by
using only class names and taxonomies. Rios and Kavuluru (2018) train a neural
architecture with both true labels and their natural language descriptor.
However, these methods still require access to the ground-truth label space.

#### Open-world Single-label Text Classification:

There has been a surge in open-world models utilizing LLM prompts to derive
labels without relying on ground-truth label spaces. Notably, GOALEX Wang et
al. (2023b) generates labels for text samples based on users’ specific goals,
demonstrating a goal-driven approach. Another noteworthy model, ClusterLLM
Zhang et al. (2023), leverages API-based LLMs to guide text clustering,
resulting in improved performance. The approach of intent discovery Zhang et
al. (2022), aiming to infer latent intents from a document set, has proven
effective in generating label spaces. A newly introduced method, IDAS De Raedt
et al. (2023), prompts LLMs to succinctly summarize utterances, enhancing
intent prediction.

##  3 Problem Formulation

Given an unlabeled corpus $\mathcal{D}$ $=$ $\\{D_{1}$, $D_{2}$, $\dots$,
$D_{n}\\}$, where $D_{i}\in\mathcal{D}$ represents a document in the
collection. Our task is to (1) identify class names
$\mathcal{C}=\\{C_{j}\\}_{j=1}^{K}$, where $K$ is the _unknown_ number of
classes, and (2) build a text classifier $f(\cdot)$ to map any raw document
$D_{i}$ to its target labels $Y_{i}=\\{y_{i}^{j}\\}_{j=1}^{p}$, where
$y_{i}^{j}$ is the single label name and $p$ is the number of target labels
for $D_{i}$. The definition of "open world" denotes the absence of labeled
information during training, which is more stringent than the original
definition where new labels only appear in the test phase. Several existing
works Brunet et al. (2023); Zhu et al. (2023) adopt a similar “open world”
definition to ours.

To the best of our knowledge, this is the first work that explores open-world
multi-label text classification without the presence of a ground-truth label
space. Given the challenging nature of the problem, we assume that human
experts are willing to devote some very limited effort, i.e., _extremely weak
supervision_ , typically in the form of brief classification objective
descriptions.

##  4 Our Observations

Our two observations mentioned in Section 1 are confirmed by experiments based
on 5 benchmark datasets: AAPD Yang et al. (2018), Reuters-21578 Debole and
Sebastiani (2005), RCV1-V2 Lewis et al. (2004), DBPedia-298 Lehmann et al.
(2015), and Amazon-531 McAuley and Leskovec (2013). Specifically, we prompt a
large language model (LLM) to check if any of the ground truth labels of a
given document is dominant, i.e., covering more than 50% of the content; and
if it exists, which one is the dominant label. 333The specific prompt can be
found in Appendix A

We randomly sample 1000 examples from each dataset and calculate the
percentage of documents with a dominant class. The dominance percentages
across datasets are AAPD: 92%, RCV1-V2: 91.7%, DBPedia-298: 95.2%,
Reuters-21578: 96.4%, and Amazon-531: 87.3%. Notably, Reuters dataset exhibits
a higher proportion of dominant labels due to its mainly consisting of single-
labeled examples. Conversely, Amazon presents a lower dominance percentage,
attributed to the mix of fine-grained and coarse-grained labels, posing
challenges in determining dominance. Overall, our analysis indicates that
across all datasets, more than 90% of documents contain a dominant class.
Moving to our second observation, labels existing in less than 1% of the
dataset are identified as long-tail labels. Upon examination, instances of
long-tail labels emerging as dominant classes in at least one document are
observed, indicating that 100% of labels serve as dominant classes in some
instances. These observations highlight the potential for generating labels,
particularly long-tail labels, from raw documents and guide the design of our
framework.

##  5 Our X-MLClass Framework

X-MLClass consists of three key steps. First, every document is split into
chunks and transformed into keyphrases by prompting an LLM to construct an
initial label space. We further assign labels to each raw document $D_{i}$
using a custom keyphrase-chunk zero-shot textual entailment classifier.
Finally, we iteratively enhance the label space by incorporating additional
long-tail labels. The framework overview is depicted in Figure 1, and the
below sections provide a detailed discussion.

###  5.1 Initial Label Space Construction

The first step in X-MLClass is to construct a high-quality label space. To
balance label coverage and the computational cost of LLM, X-MLClass is applied
to a reasonably large subset of the corpus $\mathcal{D}$, denoted as
$\mathcal{D}_{sub}\subset\mathcal{D}$.

#### Dominant Keyphrase Generation:

For each document, we partition it into chunks to better align with the
context length of LLM, and then prompt for the most dominant keyphrases per
chunk. Specifically, each document $D_{i}\in\mathcal{D}_{sub}$ is segmented
into chunks $\\{S_{i}^{1},S_{i}^{2},\dots\\}$, with a predefined chunk size of
50 tokens. This choice is also made to ensure each chunk primarily contains
one label. To generate keyphrases for each chunk $S_{i}^{j}$, we employ an LLM
and provide it with an instruction based on a brief user description of the
classification objective. 444The specific prompt can be found in Appendix B
The LLM then refines keyphrases $p_{i}^{j}$ from the chunk $S_{i}^{j}$,
serving as potential class candidates for subsequent stages of our X-MLClass
model. Keyphrases generated from each chunk collectively form a set
$\mathcal{P}$.

#### Keyphrase Clustering:

As previous LLM-based text clustering work has suggested Wang et al. (2023b,
a), there are very likely some semantically redundant yet lexically different
keyphrases among the generated ones, so we employ the instruction-tuned text
embedding model, instructor-large Su et al. (2022), to generate vector
representations for all the keyphrases in $\mathcal{P}$. Traditional
clustering methods face challenges in high-dimensional spaces Aggarwal et al.
(2001); Wang et al. (2020b). To address this, we apply UMAP McInnes et al.
(2018) for dimensionality reduction, effectively balancing local and global
structures. Finally, we obtain the clusters using the Gaussian Mixture Model
(GMM) in the projected low-dimensional space, renowned for its enhanced
flexibility in capturing intricate data distributions.

#### Number of Clusters:

The number of clusters is determined by considering both the insights of human
experts regarding the magnitude of the label space and non-parametric
clustering methods such as BERTopic Grootendorst (2022), a highly effective
topic modeling method. For example, one can train BERTopic on the keyphrase
set $\mathcal{P}$ to obtain the topic number $K^{0}$, serving as the hyper-
parameter to GMM. This approach also ensures a fair comparison with baseline
methods by maintaining consistency in the number of clusters.

#### Redundant Keyphrase Removal:

Within every cluster, we focus on the three keyphrases closest to the cluster
center to synthesize one single label. Instead of directly employing the
keyphrases for label space creation, we trace back to the original chunks that
generated these keyphrases, as they likely contain similar content and
represent the same label. Concatenating these three chunks for each cluster
results in a new document. For each document, we prompt LLM with an
instruction “find one label for this document”, yielding the initial $K^{0}$
classes $\\{C_{j}\\}_{j=1}^{K^{0}}$. This strategy allows us to generate a
single label that best represents the cluster.

This initial label space may contain redundant labels. Sentence-Transformer
models Wang et al. (2020a) are used to identify distinct pairs of classes with
relatively high cosine similarity scores. The first class in each identified
pair is then removed to eliminate redundancy. For those borderline similar
label pairs, we prompt GPT-4 Achiam et al. (2023) with an instruction “Do
label pairs have similar meanings? If Yes, please output the label that we
should delete.” to help us detect distinct pairs where cosine similarity
scores alone are insufficient. This method proves effective in creating a
robust label space $\\{C_{j}\\}_{j=1}^{K^{1}}$, and while human involvement
can enhance the refinement process, it is not mandatory. Further details on
human involvement in the label space refinement are provided in appendix C.

###  5.2 Textual Entailment-based Classifier

Given a label space, we build a zero-shot textual entailment-based classifier
Yin et al. (2019). Since each chunk is designed to have only one label, state-
of-the-art zero-shot single-label text classification methods Pàmies et al.
(2023); Gera et al. (2022); He et al. (2021) are all applicable here.
Specifically, we compare every text chunk against all the labels using a
textual entailment model. For each chunk $s\in\mathcal{S}$ and each class name
$c\in\mathcal{C}$, we derive $E_{s,c}$ representing the confidence for the
chunk $s$ entailing the hypothesis “This example is constructed for $c$”, and
similarly obtain $E_{p,c}$ for each keyphrase $p\in\mathcal{P}$. Subsequently,
for each example in $\mathcal{S}$, we identify the label $c^{*}$ with the top
entailment score, denoted by $E_{s,c^{*}}>E_{s,c},\forall c\neq c^{*}$.

Finally, we find all $s$ and $p$ belong to the same document $D_{i}$ and group
them into a new set $Q$. For each instance in $Q$, we rank the label
candidates according to their entailment scores. We identify the labels that
occur most frequently with the same ranking as the predicted labels for
document $D_{i}$, progressing from the top-ranking to the lowest-ranking
order.

###  5.3 Label Space Improvement

We further identify the chunks with lower top predicted scores — these chunks
lack a dominant class in the initial label space. By ranking $E_{s,c^{*}}$ in
ascending order, we select a subset $\mathcal{S}_{sub}\subset\mathcal{S}$ with
relatively lower entailment scores, suggesting potential association with tail
classes not included in our label space. Since keyphrases generated by the LLM
may include outliers that are too specific to their corresponding documents,
we retain only keyphrases occurring more than 15 times. For each
$s\in\mathcal{S}_{sub}$, we examine all keyphrases in the corresponding $p$.
If a keyphrase is absent in the label space but occurs more than 15 times in
$\mathcal{P}$, we incorporate it into the label space $\mathcal{C}$.

Additionally, we compute the frequency of each label $c$ with the top
entailment score. Labels with lower frequency are removed from the label space
$\mathcal{C}$. The high-frequency labels, secured as a part of the label
space, are temporarily excluded from the later label space improvement
process. By iteratively training the classifier based on the updated label
space, the label set gets finalized by adding more long-tail labels. In the
concluding stages, all high-frequency labels are reintroduced, culminating in
the formation of ultimate label space.

Dataset | # Train | # Text | # Class
---|---|---|---
AAPD | 53,840 | 2,000 | 54
Reuters | 7,769 | 3,019 | 90
RCV1-V2 | 643,531 | 160,883 | 103
DBPedia | 196,665 | 49,167 | 298
Amazon | 29,487 | 19,658 | 531
Table 1: Dataset statistics.

##  6 Experiments

We assess the performance of X-MLClass through two primary criteria: label
space quality and zero-shot MLTC accuracy. Our evaluation involves a
comparison of our model’s label coverage with that of four topic modeling and
three keyword extraction methods. In terms of end-to-end classification
accuracy, we test our method with several top-ranking models available on
HuggingFace. The subsequent section provides comprehensive details on the
datasets, baseline methods, evaluation metrics, implementation specifics, and
performance analysis.

###  6.1 Datasets

We perform experiments on five benchmark datasets for multi-label text
classification across various domains: AAPD, Reuters-21578, RCV1-V2,
DBPedia-298, and Amazon-531. Detailed information about each dataset is
provided in Appendix D. Table 1 shows that the number of labels in these
datasets varies from tens to hundreds. All the methods will be applied on the
documents from the training set, and then evaluated on the test set.

###  6.2 Compared Methods

We compare our X-MLClass framework with two types of methods.

#### Label (Space) Generator:

We select four representative _topic modeling_ methods with distinct
paradigms. LDA Blei et al. (2003) and NMF Févotte and Idier (2011) extract
topics based on word frequency within documents. Topic2Vec Angelov (2020)
extends the Word2Vec model to embed topics, facilitating the exploration of
semantic relationships between documents. Meanwhile, BERTopic Grootendorst
(2022) leverages BERT embeddings and the HDBSCAN clustering algorithm to
identify topics.

The three _keyword extraction_ methods include PKEBoudin (2016),
TextRankMihalcea and Tarau (2004), and TF-IDF Ramos et al. (2003). PKE selects
keyphrase candidates based on their confidence scores, TextRank is a graph-
based ranking model that ranks keywords using a voting mechanism, and TF-IDF
scores words based on their frequency in the document and inversely
proportional to their frequency across documents.

Despite the above methods generating a single label per document, based on our
observations, they often align with the dominant label for each document.
Given their potential to cover all labels, it is reasonable to compare the
label space coverage between our task and these label generators.

#### Zero-shot Text Classification:

State-of-the-art zero-shot text classifiers typically follow textual
entailment Yin et al. (2019); Pàmies et al. (2023). Therefore, we choose three
entailment models: (1) bart-large-mnli exclusively trained on the MNLI
dataset, (2) deberta-v3-large-all trained on 33 datasets reformatted into the
universal NLI format, and (3) xlm-roberta-large-xnli fine-tuned on the XNLI
dataset. We apply these models using the HuggingFace Transformer pipeline,
with a hypothesis template “This example is {label}”.

###  6.3 Evaluation Metrics

#### Label Space Quality:

We employ an automatic evaluation metric, coverage, to meature the alignment
between the ground-truth (GT) label space and the predicted label space. A
ground-truth label is considered “covered” if it exhibits a similarity score
exceeding a predefined threshold when compared to a predicted label, or if it
receives a positive evaluation from GPT-4. We compute the similarity scores
using the all-MiniLM-L6-v2 model from HuggingFace Sentence-Transformers. If
the similarity score exceeds 0.75, the ground-truth label is considered
“covered.” For scores between 0.5 and 0.75, we prompt GPT-4 with specific
instructions (see Appendix E). Labels with scores below 0.5 are not compared
due to obvious dissimilarity.

The coverage score is computed as follows:

$\text{Coverage}=\frac{1}{N}\mathbb{G}\left(\mathbb{I}\left({C}^{\text{pred}},C^{\text{GT}}\right)\right)$

Here $N$ is the total number of topics in the GT label set,
$C^{\text{GT}}(C^{\text{pred}})$ denotes the set of ground-truth (predicted)
labels. $\mathbb{I}$ is an indicator that returns 1 if the GT label is
considered “covered”. $\mathbb{G}$ represents the bipartite graph maximum
match algorithm.

Model | AAPD | Reuters | RCV1-V2 | DBPedia | Amazon
---|---|---|---|---|---
LDA | 30.61 | 17.77 | 12.74 | 11.40 | 13.18
NMF | 28.57 | 17.77 | 12.74 | 28.18 | 15.06
Top2Vec | 32.65 | 20.00 | 28.43 | 30.87 | 14.50
BERTopic | 20.41 | 20.00 | 10.68 | 33.22 | 16.76
PKE | 14.29 | 14.44 | 25.24 | 25.84 | 15.44
TextRank | 18.37 | 11.11 | 11.65 | 14.77 | 12.43
TF-IDF | 14.29 | 15.56 | 8.74 | 26.51 | 12.24
X-MLClass | 77.56 | 37.78 | 61.17 | 67.45 | 38.04

Table 2: Label Space Coverage Comparison. Top2Vec and BERTopic generate topics
with multiple keywords. The predicted label is determined by selecting the
top-ranking keyword based on each model’s setting.

#### Classification Accuracy:

Because of the large label space, multi-label text classification typically
employs the rank-based evaluation metric precision at $k$, i.e., P@k. It
captures the percentage of true labels among top-$k$ score labels and is used
for performance comparison. P@k can be defined as:

$\textrm{P@k}=\frac{1}{N}\sum_{i=1}^{N}\frac{C_{i}^{r_{k}}\cap
L_{i}}{\min(k,|L_{i}|)}$

where $L_{i}$ and $C_{i}$ denote the true labels and predicted labels for
document $D_{i}$, $|L_{i}|$ is the number of true labels for $D_{i}$, and
$r_{k}$ is the $k$-th highest predicted label. The ground-truth label is
defined “covered” using the same coverage metric.

Classifier | Method | AAPD | Reuter | RCV1-V2 | DBPedia | Amazon
---|---|---|---|---|---|---
P@1 | P@3 | P@1 | P@3 | P@1 | P@3 | P@1 | P@3 | P@1 | P@3
bart-large- mnli | Vanilla | 0.2030 | 0.2585 | 0.0940 | 0.2547 | 0.4220 | 0.3760 | 0.6420 | 0.3657 | 0.5080 | 0.3930
X-MLClass | 0.3323 | 0.3735 | 0.1630 | 0.3617 | 0.4530 | 0.3808 | 0.6890 | 0.4000 | 0.6620 | 0.4608
deberta-v3- large-all | Vanilla | 0.3860 | 0.3700 | 0.1290 | 0.3937 | 0.4660 | 0.4152 | 0.6370 | 0.3816 | 0.5970 | 0.4068
X-MLClass | 0.3573 | 0.4009 | 0.1320 | 0.4150 | 0.4820 | 0.4307 | 0.6350 | 0.3953 | 0.6200 | 0.4650
xlm-roberta- large-xnli | Vanilla | 0.1860 | 0.2330 | 0.1450 | 0.3477 | 0.3270 | 0.3053 | 0.6500 | 0.3673 | 0.5060 | 0.3860
X-MLClass | 0.2713 | 0.3467 | 0.2040 | 0.3968 | 0.4040 | 0.3383 | 0.6610 | 0.3910 | 0.5840 | 0.4547
Table 3: Zero-Shot Multi-Label Text Classification Accuracy Comparison:
Vanilla method represents baseline classifier trained on raw documents. We
compare the Vanilla performance with X-MLClass performance.

###  6.4 Implementation Details

We use llama-2-13b-chat LLM for X-MLClass implementation. The chunk size is
uniformly set to 50 across all datasets, ensuring a consistent approach. In
configuring LDA and NMF, we align the number of topics with our approach. For
Top2Vec and BERTopic, which employ HDBSCAN as a clustering method, specifying
an exact number of topics is not feasible. However, to maintain consistency,
we ensure that these methods generate clusters neither exceeding nor falling
below 10 in comparison to our label number.

We employ a dynamic addition strategy to determine the size of
$\mathcal{D}_{sub}$ for each dataset. Starting with the empty set, we
iteratively add 1000 examples at a time, generating the initial label space
until no additional labels are generated. Consequently, $\mathcal{D}_{sub}$
contains 3000 documents for AAPD, Reuters-21578, and RCV1-V2; 8000 documents
for DBPedia-298; and 14000 documents for Amazon-531. In the label space
improvement phase, by ranking the top entailment scores in ascending order, we
select a subset of chunks $S$ with comparatively lower entailment scores. To
ensure consistency with $\mathcal{D}_{sub}$, we control the size of $S$
proportionally. For AAPD, Reuters-21578, and RCV1-V2 datasets, we select the
top 500 examples. For DBPedia-298, the subset size is set at 1,000, and for
Amazon-531, we choose 1,500 examples.

We consider labels existing in less than 1% of the dataset as long-tail
labels. Therefore, we select all keyphrases that occur in less than 1% of
$\mathcal{P}$. By calculating the semantic similarity score between these
selected keyphrases and all labels in $C$, we identify the highest score for
each keyphrase and use the median of these scores as our threshold $\gamma$.
Only labels with a semantic similarity score lower than $\gamma$ compared to
existing labels are added to the new label space. This methodology ensures a
refined and relevant augmentation of the label space. Details on the time and
computational resources required for model training and prediction are
provided in Appendix F.

###  6.5 Label Space Coverage Results

We present the coverage of the predicted label space in comparison to topic
modeling and keyword extraction baselines, as detailed in Table 2. Our method
consistently outperforms all baseline approaches. Specifically, for the AAPD,
RCV1-V2, and DBPedia-298 datasets, we achieve over 60% coverage of the ground-
truth label space, showcasing a noteworthy increase of up to 40% compared to
the baseline methods. These results demonstrate that our method excels at
predicting labels that align closely with the ground-truth label space. For
example, our model can generate long-tail labels, a task that is particularly
challenging for all baselines. However, our model exhibits comparatively lower
performance on the Reuters-21578 and Amazon-531 datasets. Regarding
Reuters-21578, the lower performance is due to a higher proportion of long-
tail labels and the use of abbreviations in ground-truth labels. For Amazon
dataset, the initially generated label space by X-MLClass is only one-third of
the ground-truth size. Even after adding more labels in the improvement stage,
the predicted label space is still less than half of the ground-truth space,
leading to lower coverage scores.

We also use Meta-Llama-3-8B-Instruct for model implementation and test on AAPD
and RCV1-V2 datasets. The coverage scores were 75.51% and 60.19%,
respectively, similar to the results obtained using llama-2-13b-chat.

###  6.6 Zero-shot Text Classification Accuracy

We present the comprehensive zero-shot performance across all methods in Table
3. The results unequivocally demonstrate that our framework consistently
outperforms nearly all baseline models. Notably, the P@3 scores of X-MLClass
surpass those of the baseline methods across all datasets. This observation
implies that training the zero-shot classifier for both the keyphrases set and
the chunk set, followed by merging the results, enhances the multi-label
performance. Specifically, our chunk-splitting procedure increases the
likelihood of finding the less dominant labels for each document, as these
labels may become dominant in smaller chunks. Similarly, our approach improves
the accurate prediction of tail labels by the classifier, contributing to the
overall MLTC performance.

Dataset | Initial | Improvement | $\Delta$
---|---|---|---
AAPD | 44.90 | 77.56 | +32.66%
Reuters-21578 | 24.44 | 37.78 | +13.34%
RCV1-V2 | 49.51 | 61.17 | +11.66%
DBPedia-298 | 55.70 | 67.45 | +11.75%
Amazon-531 | 23.35 | 38.04 | +14.69%

Table 4: Label Coverage Score Improvement Results.

###  6.7 Label Space Coverage Improvement

Table 4 shows that iteratively updating the label space leads to an
enhancement in label coverage across all datasets. Figure 2 visually
represents the normalized incremental coverage during each iteration across
datasets. Notably, the improvement is more pronounced for datasets with
smaller initial label space sizes or lower initial coverage scores. This
finding aligns with expectations, as AAPD exhibits significantly smaller label
space sizes compared to all other datasets, rendering label space improvement
easier. For the Reuters-21578 and Amazon-531 datasets, the initial coverage
score is low, meaning that many matching labels are not initially included in
the label space. This results in a higher potential for improvement by adding
these labels. Additionally, the criteria for adding new labels must align with
all existing ones in the generated label space, presenting a greater challenge
in expanding larger label spaces like DBPedia-298.

Figure 2: Coverage Improvement across Iterations.

###  6.8 Label Coverage with Human Involvement/Evaluation

We can seek expert assistance for refinement when generating the initial label
space, especially for borderline similar label pairs. Human judgment helps
determine whether to keep both labels, typically requiring inspection of only
about 30 pairs on average. Table 5 shows that the coverage score of initial
label space across all datasets is lower without human involvement, as
expected.

Our model also encounters challenges in generating labels exactly matching the
ground-truth label space. For instance, within Reuters-298, certain ground-
truth labels are abbreviations, while our model generates the full-word
version, leading to a lower semantic similarity score than the actual score.
As shown in Table 6, the ground-truth label “acq” corresponds to our predicted
label “acquisitions,” possessing identical meanings, yet their semantic
similarity score falls below 40%. In the Amazon-531 dataset, many ground-truth
labels consist of phrases, which complicates the coverage evaluation.
Achieving high scores requires precise matches, but predicting similar-meaning
phrases with different words is common, resulting in lower scores. As evident
in Table 6, “electrical_safety” and “electronics_troubleshooting” are
identical labels, but their semantic similarity scores are lower, treated as
distinct labels in our setting. Expert evaluation can assist in such cases.

Considering these factors, the actual coverage score of our predicted label
space compared to the ground-truth label space is likely higher than the
presented result in Table 2.

###  6.9 Ablation Study for Amazon-531 Dataset

The label space for the Amazon-531 dataset is significantly larger than that
of the other datasets. To address this discrepancy and enhance label coverage,
we increased the number of iterations to add more long-tailed labels. Using
the same number of iterations as for the other datasets would result in a
final label space only half the size of the ground-truth label space. As
depicted in Figure 3, increasing the number of iterations facilitates the
addition of more labels to the predicted label space, leading to an
improvement in the coverage score.

Dataset | w/o Human | w/ Human | $\Delta$
---|---|---|---
AAPD | 40.81 | 44.90 | +4.09%
Reuters-21578 | 18.89 | 24.44 | +5.55%
RCV1-V2 | 45.63 | 49.51 | +3.88%
DBPedia-298 | 46.31 | 55.70 | +9.39%
Amazon-531 | 21.28 | 23.35 | +2.07%

Table 5: Initial Coverage w/wo Human Involvement.

Ground-truth | Predicted Label
---|---
acq | acquisitions
money-fx | monetary policy
earn | earnings
plug_play_video_games | gaming_electronics
electrical_safety | electronics_troubleshooting
teether | baby_dental_care

Table 6: Matching pairs between the ground-truth labels and the predicted
labels through human evaluation.

##  7 Conclusion and Future Work

We attack a novel and challenging problem, open-world MLTC with extremely weak
supervision. In this scenario, user only provides a brief description for
classification objectives without any labels or ground-truth label space. Our
LLM-based framework, X-MLClass, is designed to overcome this challenge by
discovering a practical label space and constructing an MLTC classifier for
label prediction. Notably, it excels in identifying long-tail labels, arguably
the most challenging aspect in MLTC problems. Our experiment results show that
X-MLClass surpasses baselines in terms of ground-truth label coverage and
exhibits higher zero-shot text classification performance compared to top-
ranking models.

Despite our model’s success in generating some tail labels, a considerable
number of tail labels remain undiscovered. Future work should focus on
refining our approach to capture more long-tail labels. Subsequent studies
could explore methodologies tailored for datasets featuring significantly
larger label spaces, contributing to the broader applicability of our model.

## Limitations

Our work aims to discover the label space from extensive input text documents
and then construct a multi-label text classifier. The most formidable
challenge in this problem setting revolves around label space construction —
how can we discover the labels, especially the long-tail ones? Therefore, our
primary focus is on developing a novel method to address this challenge; we
didn’t propose any new zero-shot multi-label text classifier, since it is
beyond the scope of this paper. Given that our proposed X-MLClass starts with
a subset of documents, its efficacy may be limited for extremely long-tail
labels (e.g., those occurring less frequently than 0.0001% of the documents).
Alternatively, a considerably large subset would be required, potentially
incurring significant computational costs from LLM. While our evaluation
includes a diverse set of datasets, there is potential for further extension
to more challenging datasets with an exceptionally large label space (e.g.,
over 1000 different labels are expected).

## Ethical Considerations

This paper uses open-source datasets and models for training and evaluation,
which is reproducible. We will also release the code upon acceptance. It is
important to note that the keyphrases generated by LLMs may vary with each
run, potentially leading to minor differences in results compared to those
presented. Additionally, our evaluation toolkit uses OpenAI models, which may
impact reproducibility.

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Figure 3: Improvement of Label Coverage for Amazon-531 by increasing the
number of iterations.

##  Appendix A Prompt Templates for Dominant Label

Code 1 is the prompt we use to find the dominant label for the selected
document.

##  Appendix B Prompt Templates for Generating Keyphrases

Code 2 provides an example of the prompt used to generate keyphrases for a
selected chunk of the Amazon-531 dataset. Users can help us define the
objective with examples. For example, the coarse-grained objectives look like
“games” and “animals”, while the corresponding fine-grained objectives are
“trading_card_games” and “reptiles”.

##  Appendix C Label Space refinement with human involvement

Human experts play a crucial role in refining the label space generated by
LLM. For instance, when the cosine similarity score between two labels falls
between 0.50 and 0.75, indicating a certain degree of semantic similarity,
human intervention is preferred to determine whether these labels are
synonymous. Synonyms need to be identified and treated accordingly, with one
of them being removed from the label space. To specify, we show them with an
instruction “Do label pairs have similar meanings in the text classification
problem? If Yes, please output the label that we should delete.” However,
there is also the case that these two labels may represent concepts from
different scopes; for example, “health_care” and “health_personal_care.” In
such instances, human judgment is necessary to detect and treat them as
separate labels.

Furthermore, some predicted labels may contain multiple meanings,
necessitating human intervention to split them into distinct labels. For
instance, if a predicted label is “computer vision and machine learning,” it
is evident that the label should be divided into two separate labels. These
judgments require human expertise for accurate and context-aware decisions.

##  Appendix D Datasets Detailed Information

  * •

AAPD Yang et al. (2018) contains computer science papers. The labels are
research topics.

  * •

Reuters-21578 Debole and Sebastiani (2005) is a collection of news articles
from the Reuters financial newswire service in 1987. The labels are the news
topics.

  * •

RCV1-V2 Lewis et al. (2004) contains categorized newswire articles by Reuters
Ltd. The labels are the news topics.

  * •

DBPedia-298 Lehmann et al. (2015) are extracted from Wikipedia articles. The
labels are the article categories.

  * •

Amazon-531 McAuley and Leskovec (2013) encompasses product reviews and
associated metadata. The labels are the product tags.

##  Appendix E GPT Instructions for Verifying Matching Pairs

Code 3 is the instruction we used to verify matching pairs.

##  Appendix F Time and Computational Resources Required for Model Training
and Prediction

The number of training examples we select varies based on the scope of each
dataset. On average, we prompt LLM on approximately 18,000 chunks for each
dataset. We use a single 80GB A100 for LLM prompting, and the process consumes
less than 35GB of GPU memory. Each prompting only takes less than 1 second to
execute. Thus, on average, each dataset requires less than 4 hours for LLM
prompting, making it a manageable cost.

During the text classification and label space improvement step, the majority
of the computation cost arises from calculating the textual entailment score
for each chunk. However, the entailment model is lightweight, occupying only
around 3GB of GPU memory. When dealing with large label spaces, nearest-
neighbor techniques can be applied to trim down label candidates, reducing the
time required for calculating entailment scores. On average, we require
approximately 2 hours to complete one iteration. However, users have the
flexibility to choose the number of iterations needed to update the label
space. Given that our model already outperforms the baseline with the initial
label space, only a few iterations are necessary, making both time and cost
manageable.

⬇

sys_prompt = "You are a poetic assistant, skilled in explaining complex
programming concepts with creative flair."

user_prompt = "

Which label in the label space {true_label_array} is the dominant label that
covers more than 50% of the below content?

{documents}

Please output the dominant label only if exist or output \’NO\’ if there are
no dominant labels.

"

prompt = "{sys_prompt} \n {user_prompt}"

Code 1: Prompt to find the dominant label for the selected document

⬇

#Amazon-531 keyphrases generation template

sys_prompt = "<s>[INST] <<SYS>>

You are a helpful, respectful and honest assistant for labeling topics.

<</SYS>>."

example_prompt = "

The following is a customer review of a product bought from Amazon.

[documents]

Based on the topic information mentioned above, the coarse-grained keyphrases
are formatted as {user classification objective}, while the fine-grained
keyphrases are formatted as {user classification objective}.

"

main_prompt = "

[INST]

Based on the information about the topic above, please find two coarse-grained
and two fine-grained keyphrases for the example.

[DOCUMENTS]

Please only return the keyphrases in one line using the format below:

[/INST] [keyphrase] and [/keyphrase].

"

prompt = "{sys_prompt} \n {example_prompt} \n {main_prompt}"

Code 2: Prompt templates to generate keyphrases

⬇

sys_prompt = "You are an expert in text classification, with specialized
skills in discerning matching pairs for labels."

user_prompt = "

Given that we have established matching pairs such as

"\’Machine learning\’ and \’artificial intelligence\’",

"\’Computational Geometry\’ and \’Algebraic Geometry\’",

"\’Physics and Society\’ and \’Physics\’", //optional

"\’teether\’ and \’baby_dental_care\’", //optional

when using util.dot_score to measure semantic similarity between tokens, would
you consider {ground_truth} and {prediction} as a matching pair in a text
classification problem?

Please respond with Yes or No.

"

prompt = "{sys_prompt} \n {user_prompt}"’

Code 3: Prompt templates for verifying match pairs


# Origin of Rashba Spin-Splitting and Strain Tunability in Ferroelectric Bulk
CsPbF3

Preeti Bhumla<EMAIL_ADDRESS>Deepika Gill  Sajjan
Sheoran  Saswata Bhattacharya<EMAIL_ADDRESS>[

###### Abstract

Spin-orbit coupling (SOC) in conjunction with broken inversion symmetry acts
as a key ingredient for several intriguing quantum phenomena viz. persistent
spin textures, topological surface states and Rashba-Dresselhaus (RD) effects.
The coexistence of spontaneous polarization and the RD effect in ferroelectric
(FE) materials enables the electrical control of spin degree of freedom. In
light of this, we explore here the FE lead halide perovskite viz. CsPbF3 as a
potential candidate in the field of spintronics by employing state-of-the-art
first-principles based methodologies viz. density functional theory (DFT) with
semi-local and hybrid functional (HSE06) combined with SOC and many-body
perturbation theory (G0W0). For a deeper understanding of the observed spin-
splitting, we have analyzed the spin textures within the combined framework of
DFT and k.p model Hamiltonian. The latter confirms that there is no out of
plane spin component indicating that the Rashba splitting dominates over
Dresselhaus splitting in this system. Owing to the presence of Pb-6$p$ orbital
in conduction band, the large value of Rashba coefficient ($\alpha_{R}$) at
conduction band minimum (CBm) is noticed in comparison to that of at the
valence band maximum (VBM). We also observe that strength of Rashba spin-
splitting can be substancially tuned on application of uniaxial strain ($\pm
5\%$). More interestingly, we notice reversible spin textures by switching the
FE polarization in CsPbF3 perovskite. These findings will further pave the
path for perovskite-based spintronics devices.

+91-11-2659 1359 Indian Institute of Technology Delhi] Department of Physics,
Indian Institute of Technology Delhi, New Delhi, India

Lead halide perovskites have evolved as an excellent choice in optoelectronics
owing to their exotic properties viz. suitable optical band gap, high
absorption coefficient, low trap density and reasonable manufacturing cost 1,
2, 3, 4, 5, 6, 7, 8, 9. These alluring materials exhibit great applications as
an absorber for high power conversion efficiency solar cells 10, 11, 12. Since
last decade, consistent research efforts have led to the revolutionary growth
of power conversion efficiency to over 25% 13, 14. Additionally, the large
spin-orbit coupling (SOC) tied to the presence of heavy element Pb plays a
decisive role in determining the electronic properties of lead halide
perovskites 15, 16. Over the years, the influence of SOC is well documented on
the band structures of these perovskites 16. Intriguingly, SOC in conjunction
with broken inversion symmetry acts as a key ingredient for several exotic
phenomena such as persistent spin textures 17, 18, topological surface states
19 and Rashba-Dresselhaus (RD) effects 20, 21, 22, 23, 24.

In absence of the inversion symmetry, the crystal feels an effective magnetic
field due to SOC. This field coupled with spin moment leads to the momentum
dependent splitting of bands, known as Rashba-Dresselhaus (RD) splitting.
Dresselhaus 25 and Rashba 26 effects were originally reported in zinc-blende
and wurtzite structures, respectively. The primary difference between both
these effects resides in the origin of the noncentrosymmetry. In Rashba
effect, there is site inversion asymmetry, whereas bulk inversion asymmetry is
present in Dresselhaus effect. During the past decade, RD effect has been the
subject of intense research due to its potential applications in the emergent
field of spintronics 27, 28. In view of this, lead halide perovskites have
been studied as promising functional materials for spin based applications 29,
30, 31, 32, 33. Recently, a significant Rashba effect has been reported in
tetragonal phase of MAPbI3 (MA = CH3NH3+) due to the rotations of MA ion 34,
35. The effect has also been predicted in CsPbBr3 and MAPbBr3 perovskites 36,
37. Lately, there are several reports on ferroelectric-coupled Rashba effect
in halide perovskites (viz. $\beta$-MAPbI3, $\beta$-MASnI3, ortho-MASnBr3,
FASnI3 (FA = HC(NH2)2+), opening new possibilities for perovskite based spin
devices 29, 38. However, due to the volatile nature of organic molecules,
these perovskites suffer from poor stability towards heat and moisture 39.

Therefore, in our present work, we intend to explore inorganic FE perovskites
in terms of RD splitting. Note that, the interplay between ferroelectricity
and Rashba effect give rise to electric control of bulk Rashba effect, leading
to an exciting class of Ferroelectric Rashba Semiconductors (FERSCs) 40, 41,
42. This effect can be theoretically quantified by employing k.p perturbation
theory. According to this theory, the lowest-order RD Hamiltonian is given by
43, 44

$H_{{RD}}(\textbf{{k}})=\alpha_{{R}}(\sigma_{x}k_{y}-\sigma_{y}k_{x})+\alpha_{{D}}(\sigma_{x}k_{x}-\sigma_{y}k_{y})$
(1)

Here, $\sigma_{x}$ and $\sigma_{y}$ are Pauli spin matrices, $k_{i}$ is the
crystal momentum ($i=x,y,z$), $\alpha_{{R}}$ and $\alpha_{{D}}$ represent the
Rashba and Dresselhaus coefficients, respectively. On solving the free-
electron Hamiltonian with these terms, we get two spin-split polarized states
having opposite spin polarization. Though Rashba and Dresselhaus exhibit a
similar type of splitting, projection of spin in momentum space i.e., spin
texture gives the idea on the nature of the splitting. Striving towards
sizable Rashba-type splitting, CsPbF3 perovskite may be a desirable candidate.
To the best of our knowledge, the available literature does not encompass the
detailed quantitative study of RD effect in this material. The presence of Pb
and ferroelectricity in noncentrosymmetric phase of CsPbF3 indicates the
possibility of RD effect in this material.

Motivated by this idea, we have undertaken a theoretical study based on the
perturbative k.p formalism and backed by the first-principles calculations. In
the present work, we have studied Pm$\bar{3}$m and R3c phases of CsPbF3.
Firstly, we have examined the electronic band structures in the above
mentioned phases. After that, we have estimated the band gap of these phases
using first-principles based approaches combined with SOC, viz. density
functional theory (DFT) 45, 46 with semi-local exchange-correlation
($\epsilon_{xc}$) functional (PBE 47), hybrid DFT with HSE06 48, 49 and
single-shot GW (G0W0) 50, 51 under the many body perturbation theory (MBPT)
and Wannier model approach. Subsequently, we have analyzed the electronic band
structure of R3c phase in terms of Rashba splitting under the combined
framework of DFT and perturbative k.p formalism. We have also investigated the
effect of strain on electronic band gap and Rashba parameters of R3c phase.
Finally, we have determined the minimum energy pathway of the FE transition
using climbing image nudged elastic band (CINEB) method.

Figure 1: (a) Optimized crystal structure of CsPbF3 in rhombohedral R3c phase.
Cs, Pb and F atoms are indicated by red, grey and yellow colors, respectively.
(b) The first hexagonal Brillouin zone showing the high symmetry path for band
structure calculations in R3c phase of CsPbF3. Electronic band structure of
CsPbF3 for R3c phase, calculated using (c) PBE (d) PBE+SOC. The conduction and
valence bands considered in the discussion are indicated by orange color. (e)
Schematic representation of bands showing Rashba splitting. (f) Splitting of
conduction band minimum (CBm) and valence band maximum (VBM) of the chosen
bands along the M-$\Gamma$-K path. The inset shows the enlarged view of CBm.
(g) Projected density of states (pDOS) in R3c phase of CsPbF3 calculated using
HSE06+SOC. The Fermi energy is set to zero in the energy axis.

CsPbF3 mainly manifests in cubic (Pm$\bar{3}$m) and rhombohedral (R3c) phases
52, 53. The Pm$\bar{3}$m phase is centrosymmetric i.e., contains an inversion
center. On the other hand, the rhombohedral R3c phase (see Fig. 1(a)) is
noncentrosymmetric and exhibits FE behavior due to distortion of cations away
from anionic polyhedra 53. The calculated change in FE polarization of the
rhombohedral (R3c) phase relative to the centrosymmetric structure is 34
$\mu$C/cm2 along the [0001] direction in hexagonal setting (along the [111]
direction in rhombohedral setting). The details of optimized crystal
structures are provided in Sec. I of Supplemental Information (SI). Next, we
have investigated the electronic band structures of Pm$\bar{3}$m and R3c
phases. The band structure of Pm$\bar{3}$m phase in presence of SOC reveals
that there is no momentum-dependent splitting owing to its centrosymmetric
structure (band structure and pDOS of Pm$\bar{3}$m phase are given in Sec. II
of SI). In addition, the cubic Pm$\bar{3}$m phase is not dynamically stable
(see Sec. III for phonon band structures). Therefore, we have explored RD
splitting in the latter phase (i.e. R3c). We have plotted the band structure
of FE R3c phase along the high symmetry path using hexagonal setting (as shown
in Fig. 1(b)). Firstly, we have performed non-spin polarized calculations.
After that we have considered SOC in the calculation of electronic band
structure. Fig. 1(c) and 1(d) show the electronic band structures calculated
using PBE and PBE+SOC, respectively. A direct band gap of 3.26 eV is observed
without SOC, whereas on including SOC, the band gap is reduced to 2.42 eV
(indirect) around the $\Gamma$ point. This change in band gap is attributed to
SOC arising from the presence of Pb-6$p$ orbitals in the conduction band (see
pDOS in Fig. 1(g)). Hence, SOC is indispensable and duly considered in all
further calculations. Note that PBE $\epsilon_{xc}$ functional is well known
to underestimate the band gap due to its inability to capture the electron’s
self-interaction error. Therefore, we have also calculated the band gap with
hybrid $\epsilon_{xc}$ functional HSE06 (with SOC), which comes out to be 3.57
eV. Until date, there is no theoretical or experimental report on the band gap
of R3c phase. In view of this, for the better estimation of the band gap, we
have employed G0W0 approximation on top of HSE06+SOC orbitals, that results in
the band gap of 5.01 eV (see Fig. 5 (a)).

Figure 2: (a) Band structure of $R3c$ phase using G0W0@HSE06+SOC. (b)
Imaginary part (Im ($\varepsilon$)) of the dielectric function for R3c phase
of CsPbF3 calculated using HSE06+SOC and G0W0@HSE06+SOC. Table 1: Band gap (in
eV) of $Pm\bar{3}m$ and $R3c$ phases using different $\epsilon_{xc}$
functionals.

Structure PBE PBE+SOC HSE06 HSE06+SOC G0W0@HSE06+SOC $Pm\bar{3}m$ 2.62 1.69
3.6154, 55 2.71 4.38 $R3c$ 3.26 2.42 4.34 3.57 5.01

Subsequently, we have also plotted the imaginary part (Im ($\varepsilon$)) of
dielectric function with converged k-grid (for k-grid convergence, see Sec. IV
in SI) for R3c phase of CsPbF3 using HSE06+SOC and G0W0@HSE06+SOC (see Fig. 5
(b)). The first peak of Im ($\varepsilon$) corresponds to the electronic band
gap, which is the same as we get from band structure calculations. For
imaginary part (Im ($\varepsilon$)) of dielectric function of Pm$\bar{3}$m
phase, see Sec. V in SI. The values of band gaps are listed in Table 1. It is
worth quoting that HSE06+SOC/G0W0@HSE06+SOC only enhances the band gap without
any notable change in the nature of band structure and strength of Rashba
splitting 27, 34. To confirm this, we have compared the band structures of
Pm$\bar{3}$m phase using PBE+SOC, HSE06+SOC, G0W0@HSE06+SOC and have found
that the band profile remains same (see Sec. VI in SI for details). Therefore,
in view of computational cost PBE+SOC is considered to compute RD parameters.
Inclusion of SOC leads to the splitting of bands in $k_{z}$=$\frac{\pi}{c}$
plane, which is perpendicular to the direction of polarization (see Fig.
1(d)). On the contrary, splitting is completely absent along $\Gamma$-A i.e.,
in direction parallel to polarization axis, indicating that the momentum
dependent splitting around $\Gamma$ point is the Rashba-type splitting. As a
consequence, the CBm and VBM are located slightly off the $\Gamma$ point. The
CBm shifts from $\Gamma$ towards M and K by 0.075 Å-1 and 0.061 Å-1,
respectively. The VBM shifts from $\Gamma$ towards M and K by 0.022 Å-1 and
0.012 Å-1, respectively (see Fig. 1(f)). This shift from $\Gamma$ point in
either direction is known as offset momentum ($\delta$$k$) (see Fig. 1(e)).
The difference of energies at $\Gamma$ and extremum point is known as Rashba
spin-splitting energy ($\delta$$E$).

To grasp the overall nature of splitting, the constant energy 2D contour plots
of spin texture are plotted in $k_{x}$-$k_{y}$ plane centered at $\Gamma$
point (see Fig. 3(a) for schematic representation of spin textures). Fig. 3(b)
and 3(c) show $x$, $y$ and $z$ components of spin at constant energies around
CBm and VBM, respectively. As we can see from spin textures of CBm and VBM,
the in-plane spin components (Sx and Sy) are perpendicular to the crystal
momentum and out-of-plane component (Sz) is completely absent. This results in
the helical shape of spin texture with inner and outer bands having opposite
orientation, which confirms the existence of dominant Rashba splitting.

Figure 3: (a) Schematic representation of spin textures in Rashba splitting.
Spin projected constant energy contour plots of spin texture calculated in
$k_{x}$-$k_{y}$ plane centered at $\Gamma$ point. The upper and lower panels
represent the spin textures calculated at constant energy: (b)
E=E${}_{\textrm{F}}+2.5$ eV and (c) E=E${}_{\textrm{F}}-0.2$ eV, respectively.
Electronic band structures showing spin-splitting of (d) CBm and (e) VBM
around $\Gamma$ point, respectively. Band structure is plotted along
($\frac{2\pi}{a}0.25,0,0$) $-$ (0, 0, 0) $-$
($\frac{2\pi}{a}0.16,\frac{2\pi}{b}0.16,0$) direction of momentum space, which
is M $-$ $\Gamma$ $-$ K direction. DFT and k.p band structures are plotted
with dashed lines and dots, respectively. In the color scale, red depicts
spin-up while blue depicts spin-down states.

In order to have a quantitative study of the RD effect, we have considered the
k.p Hamiltonian. The model Hamiltonian of $R3c$ structure possessing $C_{3v}$
point group symmetry near $\Gamma$ point in the presence of SOC can be written
as 56, 57, 58:

$\displaystyle\begin{split}H_{\Gamma}(\textbf{{k}})&=H_{o}(\textbf{{k}})+\alpha\sigma_{y}k_{x}+\beta\sigma_{x}k_{y}+\gamma\sigma_{z}[(k^{3}_{x}+k^{3}_{y})-3(k_{x}k^{2}_{y}+k_{y}k^{2}_{x})]\end{split}$
(2)

Here $H_{o}(\textbf{{k}})$ is the Hamiltonian of the free electrons with
eigenvalues $E_{o}(\textbf{{k}})$. $\alpha$, $\beta$ are the coefficients of
the linear term and $\gamma$ is the coefficient of the higher order term in
the Hamiltonian. Since we have not noticed out-of-plane spin component in the
spin texture, therefore, we have neglected the higher order terms in the
Hamiltonian (see Sec. VII in SI for more details). Hence, the Hamiltonian for
$\Gamma$ point (considering only linear terms) can now be written as:

$\displaystyle\begin{split}H_{\Gamma}(k)&=H_{o}(k)+\alpha
k_{x}\sigma_{y}+\beta k_{y}\sigma_{x}\\\
&=H_{o}(k)+\alpha_{R}(\sigma_{x}k_{y}-\sigma_{y}k_{x})\\\
&+\>\alpha_{D}(\sigma_{x}k_{x}-\sigma_{y}k_{y})\end{split}$ (3)

The energy eigen values corresponding to this Hamiltonian is given as,
$E_{o}(k)=\frac{\hbar^{2}k_{x}^{2}}{2m^{*}_{x}}+\frac{\hbar^{2}k_{y}^{2}}{2m^{*}_{y}}$
and $m^{*}_{x}$, $m^{*}_{y}$ are the effective masses in $x$ and $y$
directions, respectively. Rashba and Dresselhaus coefficients are defined as
$\alpha_{R}=(\alpha-\beta)/2$ and $\alpha_{D}=(\alpha+\beta)/2$, respectively
59. On diagonalizing Eq. 3, we get energy eigenvalues as:

$E_{\pm}({k})=E_{o}(k)\pm\sqrt{\alpha^{2}k_{x}^{2}+\beta^{2}k_{y}^{2}}$ (4)

The band structures obtained from DFT and the model Hamiltonian around
$\Gamma$ point are shown with dashed lines and dots in Fig. 3(d) and 3(e),
respectively. The good agreement between both the band structures justifies
the reliability of the chosen Hamiltonian. The DFT calculations for CBm give
$\delta E$ = 43.5 meV, ${\delta k}$ = 0.075 Å-1 along $\Gamma$-M direction,
where $k_{y}$ = 0. Hence, the value of $\alpha$ = $\frac{2\delta E}{\delta k}$
= 1.16 eVÅ. Along $\Gamma$-K direction, where $k_{x}$ = $k_{y}$, $\delta E$ =
43.5 meV, ${\delta k}$ = 0.061 Å-1 and the calculated value of
$\sqrt{\alpha^{2}+\beta^{2}}$ = $\frac{2\delta E}{\delta k}$ = 1.43 eVÅ. On
putting the value of $\alpha$, we get $\beta$ = –0.84 eVÅ, which results in
$\alpha_{R}$ = 1.00 eVÅ and $\alpha_{D}$ = 0.16 eVÅ. Similarly, for VBM,
$\delta E$ = 3.4 meV, ${\delta k}$ = 0.022 Å-1 along $\Gamma$-M direction. So,
the value of $\alpha$ = $\frac{2\delta E}{\delta k}$ = 0.31 eVÅ. Along
$\Gamma$-K direction, $\delta E$ = 3.4 meV and ${\delta k}$ = 0.012 Å-1.
Therefore, $\sqrt{\alpha^{2}+\beta^{2}}$ = $\frac{2\delta E}{\delta k}$ = 0.57
eVÅ, which gives $\beta$ = –0.48 eVÅ. Using $\alpha$ and $\beta$ parameters,
the estimated values of $\alpha_{R}$ and $\alpha_{D}$ are 0.40 eVÅ and
$\alpha_{D}$ = 0.09 eVÅ, respectively. The values are summed up in Table 2. As
we can see from the values of RD parameters, the Rashba effect dominates in
the conduction band. The origin of large Rashba effect in CBm can be
attributed to the stronger SOC stemming from higher contribution of Pb-6$p$
orbital in conduction band (see Fig. 1(g)). This large contribution of SOC in
CBm compared to VBM is in well agreement with the previous findings 60. On
fitting the DFT band structure around $\Gamma$ point for CBm, we find $\alpha$
= –0.90 eVÅ and $\beta$ = 1.20 eVÅ. This gives $\alpha_{R}$ = 1.05 eVÅ and
$\alpha_{D}$ = 0.15 eVÅ. Similarly, fitting the band structure for VBM, gives
$\alpha$ = –0.33 eVÅ and $\beta$ = 0.49 eVÅ and hence $\alpha_{R}$ = 0.41 eVÅ
and $\alpha_{D}$ = 0.08 eVÅ. These values are in well agreement with the
predicted values based on DFT calculation. The calculated values of RD
parameters also confirm that the splitting around $\Gamma$ point is mainly
dominated by the Rashba effect.

Figure 4: Band structures of CsPbF3 ($R3c$ phase) under uniaxial strain of
$\pm$5%, calculated using PBE+SOC for path (a) $\Gamma$-M-K-$\Gamma$-A-L and
(b) M-$\Gamma$-K. (c) Rashba spin-splitting energy ($\delta E$) (d) offset
momentum ($\delta k$) (e) Rashba parameters ($\alpha_{R}$ and $\alpha_{D}$) as
a function of strain. The values are calculated for CBm in $R3c$ phase. Note
that $\delta E$ is same along both M-$\Gamma$ and $\Gamma$-K directions. Table
2: Rashba parameters for band-splitting at $\Gamma$ point in $R3c$ phase.

Position $\delta E$ (meV) $\delta$k${}_{\Gamma-\textrm{M}}$ (Å-1)
$\delta$k${}_{\Gamma-\textrm{K}}$ (Å-1) $\alpha_{R}$ (eVÅ) $\alpha_{D}$ (eVÅ)
CBm 43.5 0.075 0.061 1.05 0.15 VBM 3.4 0.022 0.012 0.41 0.08

After thorough analysis of RD spin-splitting, we have investigated the effect
of strain on the band structure of $R3c$ phase. For this, we have applied
uniaxial strain in $z$-direction, which is defined as

$x=\frac{c-c_{0}}{c_{0}}\times 100\%$ (5)

where $c_{0}$ is the equilibrium lattice constant and $c$ is the strained
lattice constant. The lattice vector $c$ is varied from -5% to +5%, where “+”
and “$-$” are used to denote tensile and compressive strain, respectively.
After optimizing the structures at a given strain, we have plotted the band
structures along high symmetry path. Fig. 4(a) shows the band structure of
$R3c$ phase along $\Gamma$-M-K-$\Gamma$-A-L path, under the uniaxial strain of
$\pm$5%. To clearly examine the shift, we have also plotted the band
structures along M-$\Gamma$-K (see Fig. 4 (b)). Here, we have focused on CBm
as there is large Rashba splitting in comparison to VBM. From the band
structures in Fig. 4 (b), we infer that strain causes a shift in momentum
($\delta k$) on either side of $\Gamma$ point. The arrows represent the
direction of shift. We have taken the equilibrium band structure (i.e., with
0% strain) as the reference for all the strained band structures.
Interestingly, under compressive strain, the bands shift off from $\Gamma$ on
either side towards M and K. On the contrary, under tensile strain, the bands
shift towards $\Gamma$ from either side. This in turn changes $\delta k$ and
$\delta E$ as a function of strain, and the overall effect of these parameters
will change $\alpha_{R}$ and $\alpha_{D}$ accordingly. To quantify the effect
of strain on Rashba parameters, we have calculated their values at a given
strain within the framework of DFT. Fig. 4 (c-e) show $\delta E$, $\delta k$,
$\alpha_{R}$ and $\alpha_{D}$ as a function of strain (for details, see Sec.
VIII in SI). From Fig. 4 (c), we have observed that under compressive strain
$\alpha_{R}$ is significantly enhanced from 1.05 to 1.48 eVÅ making the
material tunable for spintronics application. Also, we have seen notable
change in electronic band gap on the application of strain. The band gap
values are mentioned in Fig.4(b) (for more quantitative details, see Sec. IX
in SI).

Figure 5: (a) Climbing image nudged elastic band (CINEB) calculation for the
polarization switching process in CsPbF3 perovskite. Two ferroelectric (FE)
structures in the ground state with opposite direction of electric
polarization are shown. Eb is the activation barrier energy for the
polarization switching process. Reversible in-plane spin textures calculated
at constant energy E=EF+2.5 eV with oppposite spin polarization: (b) -P (c)
+P.

Lastly, we have explored the possibility of polarization switching in FE
CsPbF3 using strain effects. To predict the feasibility of this phenomenon, we
have analyzed the minimum energy pathway of the FE transition using CINEB
method 61, 62. For this, we have chosen the structure deformation path between
two FE states with opposite spontaneous polarization through a centrosymmetric
reference. As shown in Fig.5, the Eb denotes the activation barrier for the
polarization switching, which comes out to be 0.75 eV. This suggests that a
switchable FE polarization is plausible in the material. The strain tunability
and reversible spin textures owing to FE switching in CsPbF3 give electrical
control of spins and provide an all-semiconductor design for spintronic
devices such as in spin-field effect transistor 63, 64.

In summary, using first-principles based methodologies viz. PBE, HSE06 and
many-body perturbation theory (G0W0), we have systematically studied the
electronic structure of Pm$\bar{3}$m and R3c phases of CsPbF3. Note that SOC
is duly considered in all the calculations. The R3c phase being
noncentrosymmetric is investigated in terms of RD splitting. The DFT
calculations are well corroborated by symmetry adapted k.p model Hamiltonian.
The helical nature of in-plane spin components in spin textures indicates the
Rashba-type splitting. The calculated values of RD parameters suggest that the
Rashba effect is predominant in our material while Dresselhaus effect has a
negligible contribution. In addition, a larger Rashba effect is observed in
CBm than VBM of R3c phase owing to the larger contribution of Pb-6$p$ orbital
in conduction band. Further, we have noticed that the Rashba parameters viz.
$\delta k$, $\delta E$, $\alpha_{R}$, and $\alpha_{D}$ almost linearly
increase under compressive strain. The band gap of CsPbF3 is significantly
tuned with the application of strain. Also, we have observed reversible spin
textures via FE switching. These results indicate that FE CsPbF3 is having
enough potential as an excellent amenable material in the field of
spintronics.

##  1 Computational Methods

The DFT based first-principles calculations have been performed using the
Vienna ab initio simulation package (VASP) 65, 66. The ion-electron
interactions in all the elemental constituents are described using projector
augmented wave (PAW) 67, 68 method as implemented in VASP. We have considered
pseudopotentials with the following valence states: Cs, 5s25p66s1; Pb,
6s25d106p2; F, 2s22p5. The structures are optimized using Perdew-Burke-
Ernzerhof generalized gradient approximation (PBE-GGA), relaxing all ions
until Hellmann-Feynman forces are less than 0.001 eV/Å. The cutoff energy of
520 eV is used for the plane-wave basis set such that the total energy
calculations are converged within 10-5 eV. The $\Gamma$-centered
6$\times$6$\times$6 and 9$\times$9$\times$4 k-grid are used to sample the
irreducible Brillouin zones of cubic phase with Pm$\bar{3}$m space group and
rhombohedral phase with R3c space group of bulk CsPbF3, respectively. The
phonon calculations are performed for 3$\times$3$\times$3 and
2$\times$2$\times$2 supercells in Pm$\bar{3}$m and R3c phases using PHONOPY
package 69, 70. In order to predict band gap, single-shot GW (G0W0)
calculations have been performed on top of the orbitals obtained from
HSE06+SOC $\epsilon_{xc}$ functional [G0W0@HSE06+SOC]. For this, we have used
6$\times$6$\times$6 and 4$\times$4$\times$2 k-grids in Pm$\bar{3}$m and
R${3}$c phases, respectively. The number of bands is set to four times the
number of occupied bands. The polarizability calculations are performed on a
grid of 50 frequency points. FE polarization is evaluated in the framework of
Berry-phase theory of polarization 71, 72. The minimum energy pathways of FE
transitions are determined through the climbing image nudged elastic band
(CINEB) method 73. Spin textures are plotted by calculating expectation values
of spin operators $S_{i}$ (i = x, y, z), given by

$\langle S_{i}\rangle=\frac{1}{2}\bra{\Psi_{k}}\sigma_{i}\ket{\Psi_{k}}$ (6)

where $\sigma_{i}$ are the pauli matrices and $\Psi_{k}$ is the spinor
eigenfunction obtained from noncollinear spin calculations. The spin texture
is calculated with closely spaced 12$\times$12 k-grid around high symmetry
points. The PyProcar code is used to calculate the constant energy contour
plots of the spin texture 74.
P.B. acknowledges UGC, India, for the senior research fellowship [grant no.
1392/(CSIR-UGC NET JUNE 2018)]. D.G. acknowledges UGC, India, for the senior
research fellowship [grant no. 1268/(CSIR-UGC NET JUNE 2018)]. S.S.
acknowledges CSIR, India, for the junior research fellowship [grant no.
09/086(1432)/2019-EMR-I]. S.B. acknowledges the financial support from SERB
under core research grant (grant no. CRG/2019/000647). We acknowledge the High
Performance Computing (HPC) facility at IIT Delhi for computational resources.
Optimized structures of Pm$\bar{3}$m and R${3}$c phases; Band structure and
pDOS (projected density of states) of cubic Pm$\bar{3}$m phase; k-grid
convergence in Pm$\bar{3}$m and R${3}$c phases; Imaginary part
(Im($\varepsilon$)) of the dielectric function for cubic Pm$\bar{3}$m phase;
Comparison of PBE+SOC and HSE06+SOC band structures of cubic Pm$\bar{3}$m
phase; RD parameters as a function of strain in R${3}$c phase; Band gap as a
function of strain in R${3}$c phase.

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