# Keyphrase Generation: A Text Summarization Struggle

Erion Çano
Institute of Formal and Applied
Linguistics, Charles University,
Prague, Czech Republic
<EMAIL_ADDRESS>
&Ondřej Bojar
Institute of Formal and Applied
Linguistics, Charles University,
Prague, Czech Republic
<EMAIL_ADDRESS>

###### Abstract

Authors’ keyphrases assigned to scientific articles are essential for
recognizing content and topic aspects. Most of the proposed supervised and
unsupervised methods for keyphrase generation are unable to produce terms that
are valuable but do not appear in the text. In this paper, we explore the
possibility of considering the keyphrase string as an abstractive summary of
the title and the abstract. First, we collect, process and release a large
dataset of scientific paper metadata that contains 2.2 million records. Then
we experiment with popular text summarization neural architectures. Despite
using advanced deep learning models, large quantities of data and many days of
computation, our systematic evaluation on four test datasets reveals that the
explored text summarization methods could not produce better keyphrases than
the simpler unsupervised methods, or the existing supervised ones.

##  1 Introduction

A valuable concept for searching and categorizing scientific papers in digital
libraries is the _keyphrase_ (we use _keyphrase_ and _keyword_
interchangeably), a short set of one or few words that represent concepts.
Scientific articles are commonly annotated with keyphrases based on taxonomies
of concepts and the authors’ judgment. Finding keyphrases that best describe
the contents of a document is thus essential and rewarding.

Most of the proposed keyphrase extraction solutions tend to be unsupervised
Florescu and Caragea (2017); Nguyen and Luong (2010); Rose et al. (2010);
Bougouin et al. (2013); Campos et al. (2018) and generate terms by selecting
the most appropriate candidates, ranking the candidates based on several
features and finally returning the top $N$. Another way is to utilize datasets
of texts and keywords for training supervised models with linguistic or other
features to predict if candidates are keywords or not Witten et al. (1999);
Turney (2000); Medelyan (2009); Hulth (2003).

All above methods propose $N$ keyphrases for each article which are joined
together with “,” (or other separator like “;”) to form the _keyphrase string_
of that article. They suffer from various problems or discrepancies. First,
they are unable to find an optimal value for $N$ and require it as a preset
parameter. Furthermore, semantic and syntactic properties of article phrases
are analyzed separately. The meaning of paragraphs, sections or entire
document is thus missed. Lastly, only phrases that do appear in the article
are returned. Meng et al. (2017) recently proposed a deep supervised keyphrase
generation solution trained on a big dataset. It successfully solves the last
two problems above, but not the first one.

Motivated by recent advances in neural machine translation and abstractive
text summarization Vaswani et al. (2017); Foster et al. (2018); Rush et al.
(2015); See et al. (2017), in this paper, we explore the possibility of
considering keyphrase generation as an abstractive text summarization task.
Instead of generating keywords one by one and linking them to form the
keyphrase string, we consider the later as an abstractive summary of the
concatenated paper title and abstract. Different recently-proposed text
summarization architectures are tried on four test datasets of article
keyphrases Tanti et al. (2017); Rush et al. (2015); See et al. (2017). We
trained them with a newly created dataset of 2.2 million article titles,
abstracts and keyphrase strings that we processed and
released.111http://hdl.handle.net/11234/1-2943

The selected text summarization models are compared with popular unsupervised
and supervised methods using ROUGE Lin (2004) and full-match F1 metrics. The
results show that though trained with large data quantities for many days, the
tried text summarization methods could not produce better keywords than the
existing supervised or deep supervised predictive models. In our opinion, a
possible explanation for this is the fact that the title and the abstract may
not carry sufficient topical information about the article, even when joined
together. In contrast, when assigning keywords annotations of their paper,
authors are highly influenced by the topic aspects of it.

This paper carries several contributions, despite the fact that no progressive
result scores were reached. It is the first work that considers keyphrase
generation as an abstractive text summarization task. We produced a large
dataset of article titles, abstracts, and keywords that can be used for
keyword generation, text summarization or similar purposes. Finally, we
evaluated the performance of different neural network architectures on
summarization of article keyword strings, comparing them with popular
unsupervised methods.

##  2 Scientific Paper Datasets

Because of the open source and open data initiatives, many public datasets
from various domains can be found online Çano and Morisio (2015). Among the
several collections of scientific articles, some of them have gained
considerable popularity in research literature. In Meng et al. (2017), we
found a recent and big collection of 20K paper abstracts and keyphrases. These
metadata belong to articles of computer science from ACM Digital Library,
ScienceDirect, and Web of Science. In Hulth (2003), we found a collection of
2000 (1500 for train/val and 500 for testing) abstracts in English, together
with titles and authors’ keywords. The corresponding articles were published
from 1998 to 2002 and belong to the discipline of _Information Technology_.
Furthermore, Krapivin et al. (2010) released a dataset of 2000 (1600 for
train/val and 400 for testing) full articles published by ACM from 2003 to
2005 in Computer Science domain. More information about similar keyphrase data
collections or other available resources can be found in Hasan and Ng (2014)
and in online repositories.

Regarding text summarization, some of the most popular datasets are: DUC-2004
222https://duc.nist.gov/duc2004/ mainly used for testing, English Gigaword
Napoles et al. (2012), CNN/Daily Mail described in Section 4.3 of Nallapati et
al. (2016) and Newsroom, a heterogeneous bundle of news articles described in
Grusky et al. (2018). These datasets are frequently used for the task of
predicting titles from abstracts or short stories. However, no keyphrases are
provided; they do not serve to our purpose. ArnetMiner is a recent attempt to
crawl scientific paper data from academic networks Tang et al. (2008). The
system extracts profiles of researchers from digital resources and integrates
their data in a common network. A spin-off is the Open Academic Graph (OAG)
data collection Sinha et al. (2015).

Attribute | Train | Val | Test | Fullset
---|---|---|---|---
Records | 2M | 100K | 100K | 2.2M
Keyphrases | 12M | 575K | 870K | 13.4M
Title tokens | 24M | 1.3M | 1.6M | 27M
Abstract tokens | 441M | 21M | 37M | 499M
Av. Keyphrase | 6 | 5.8 | 8.7 | 6.1
Av. Title | 12.1 | 12.8 | 15.9 | 12.3
Av. Abstract | 220 | 211 | 372 | 227
Table 1: Statistics of OAGK dataset

To produce a usable collection for our purpose, we started from OAG. We
extracted _title_ , _abstract_ and _keywords_. The list of keywords was
transformed into a comma-separated string and a language identifier was used
to remove records that were not in English. Abstracts and titles were
lowercased, and Stanford CoreNLP tokenizer was used for tokenizing. Short
records of fewer than 20 tokens in the abstract, 2 tokens in the title and 2
tokens in the keywords were removed. For the test portion, we selected
documents of at least 27, 3 and 2 tokens in each field. Data preprocessing
stopped here for the release version (no symbol filtering), given that many
researchers want to filter text in their own way. This new dataset named OAGK
can be used for both text summarization (predicting title from abstract) and
keyphrase extraction (unsupervised, supervised or deep supervised) tasks. Some
rounded measures about each set of released data are presented in Table 1.

##  3 Keyphrase Extraction Strategies

###  3.1 Unsupervised and Supervised Methods

TopicRank is an extractive method that creates topic clusters using the graph
of terms and phrases Bougouin et al. (2013). Obtained topics are then ranked
according to their importance in the document. Finally, keyphrases are
extracted by picking one candidate from each of the most important topics. A
more recent, unsupervised and feature-based method for keyphrase extraction is
Yake! Campos et al. (2018). It heuristically combines features like _casing_ ,
_word position_ or _word frequency_ to generate an aggregate score for each
phrase and uses it to select the best candidates.

One of the first supervised methods is Kea described by Witten et al. (1999).
It extracts those candidate phrases from the document that have good chances
to be keywords. Several features like _TF-IDF_ are computed for each candidate
phrase during training. In the end, Naïve Bayes algorithm is used to decide if
a candidate is a keyword or not (binary classification). An improvement and
generalization of Kea is Maui Medelyan (2009). Additional features are
computed, and bagged decision trees are used instead of Naïve Bayes. The
author reports significant performance improvements in precision, recall and
F1 scores.

The above keyphrase extraction methods and others like Florescu and Caragea
(2017) or Nguyen and Luong (2010) reveal various problems. First, they are not
able to find an optimal value for $N$ (number of keywords to generate for an
article) based on article contents and require it as a preset parameter.
Second, the semantic and syntactic properties of article phrases (considered
as candidate keywords) are analyzed separately. The meaning of longer text
units like paragraphs or entire abstract/paper is missed. Third, only phrases
that do appear in the paper are returned. In practice, authors do often assign
words that are not part of their article.

Meng et al. (2017) overcome the second and third problem using an encoder-
decoder model (CopyRnn) with a bidirectional Gated Recurrent Unit (GRU) and a
forward GRU with attention. They train it on a datasets of hundred thousands
of samples, consisting of abstract-keyword (one keyword only) pairs. The model
is entirely data-driven and can produce terms that may not appear in the
document. It still produces one keyword at a time, requiring $N$ (first
problem) as parameter to create the full keyphrase string.

|  Hulth (500) Krapivin (400) Meng (20K) OAGK (100K)
---|---
Method | F1@5 | F1@7 | F1@5 | F1@7 | F1@5 | F1@7 | F1@5 | F1@7
Yake! | 19.35 | 21.47 | 17.98 | 17.4 | 17.11 | 15.19 | 15.24 | 14.57
TopicRank | 16.5 | 20.44 | 6.93 | 6.92 | 11.93 | 11.72 | 11.9 | 12.08
Maui | 20.11 | 20.56 | 23.17 | 23.04 | 22.3 | 19.63 | 19.58 | 18.42
CopyRnn | 29.2 | 33.6 | 30.2 | 25.2 | 32.8 | 25.5 | 33.06 | 31.92
Merge | 6.85 | 6.86 | 4.92 | 4.93 | 8.75 | 8.76 | 11.12 | 13.39
Inject | 6.09 | 6.08 | 4.1 | 4.11 | 8.09 | 8.09 | 9.61 | 11.22
Abs | 14.75 | 14.82 | 10.24 | 10.29 | 12.17 | 12.09 | 14.54 | 14.57
PointCov | 22.19 | 21.55 | 19.87 | 20.03 | 20.45 | 20.89 | 22.72 | 21.49
Table 2: Full-match scores of predicted keyphrases by various methods |  Hulth (500) Krapivin (400) Meng (20K) OAGK (100K)
---|---
Method | $\boldsymbol{R_{1}F_{1}}$ | $\boldsymbol{R_{L}F_{1}}$ | $\boldsymbol{R_{1}F_{1}}$ | $\boldsymbol{R_{L}F_{1}}$ | $\boldsymbol{R_{1}F_{1}}$ | $\boldsymbol{R_{L}F_{1}}$ | $\boldsymbol{R_{1}F_{1}}$ | $\boldsymbol{R_{L}F_{1}}$
Yake! | 37.48 | 24.83 | 26.19 | 18.57 | 26.47 | 17.36 | 20.38 | 14.54
TopicRank | 32.0 | 20.36 | 14.08 | 11.47 | 21.68 | 15.94 | 17.46 | 13.28
Maui | 36.88 | 27.16 | 28.29 | 23.74 | 34.33 | 28.12 | 32.16 | 25.09
CopyRnn | 44.58 | 35.24 | 39.73 | 30.29 | 42.93 | 34.62 | 43.54 | 36.09
Merge | 15.19 | 9.45 | 9.66 | 7.14 | 16.53 | 12.31 | 17.3 | 14.43
Inject | 14.15 | 8.81 | 9.58 | 6.79 | 15.6 | 11.21 | 14.3 | 11.08
Abs | 27.54 | 19.48 | 25.59 | 18.2 | 28.31 | 22.16 | 29.05 | 25.77
PointCov | 37.16 | 33.69 | 35.81 | 29.52 | 38.47 | 35.06 | 38.66 | 34.04
Table 3: Rouge scores of predicted keyphrases by various methods

###  3.2 Text Summarization Methods

To overcome the three problems mentioned in Section 3.1, we explore
abstractive text summarization models proposed in the literature, trained with
article abstracts and titles as sources and keyword strings as targets. They
are expected to learn and paraphrase over entire source text and produce a
summary in the form of a keyphrase string with no need for extra parameters.
They should also introduce new words that do not appear in the abstract. Two
simple encoder-decoder variants based on LSTMs are described in Figure 3 of
Tanti et al. (2017). Merge (Figure 3.a) encodes input and the current summary
independently and merges them in a joint representation which is later decoded
to predict the next summary token. Inject model (Figure 3.b) on the other hand
injects the source document context representation to the encoding part of the
current summary before the decoding operation is performed.

Abs is presented in Figure 3.a of Rush et al. (2015). The encoder (Figure 3.b)
takes in the input text and a learned soft alignment between the input and the
summary, producing the context vector. This soft alignment is the attention
mechanism Bahdanau et al. (2014). To generate the summary words, Rush et al.
apply a beam-search decoder with a window of $K$ candidate words in each
position of the summary.

Pointer-Generator network (PointCov) depicted in Figure 3 of See et al. (2017)
is similar to Abs. It is composed of an attention-based encoder that produces
the context vector. The decoder is extended with a pointer-generator model
that computes a probability $p_{gen}$ from the context vector, the decoder
states, and the decoder output. That probability is used as a switch to decide
if the next word is to be generated or copied from the input. This model is
thus a compromise between abstractive and extractive (copying words from
input) models. Another extension is the coverage mechanism for avoiding word
repetitions in the summary, a common problem of encoder-decoder summarizers Tu
et al. (2016).

##  4 Results

We performed experiments with the unsupervised and supervised methods of
Section 3 on the first three datasets of Section 2 and on OAGK. All supervised
methods were trained with the 2M records of OAGK train part. An exception was
Maui which could be trained on 25K records at most (memory limitation). In
addition to the processing steps of Section 2, we further replaced digit
symbols with # and limited source and target text lengths to 270 and 21
tokens, respectively. Vocabulary size was also limited to the 90K most
frequent words.

The few parameters of the unsupervised methods (length and windows of
candidate keyphrases for Yake!, ranking strategy for TopicRank) were tuned
using the validation part of each dataset. For the evaluation, we used F1
score of full matches between predicted and authors’ keywords. Given that the
average number of keywords in the data is about 6, we computed F1 scores on
top 5 and top 7 returned keywords (F1@5, F1@7).

Before each comparison, both sets of terms were stemmed with Porter Stemmer
and duplicates were removed. In the case of summarization models, keyphrases
were extracted from their comma-separated summaries. We also computed ROUGE-1
and ROUGE-L F1 scores ($\boldsymbol{R_{1}F_{1}}$, $\boldsymbol{R_{L}F_{1}}$)
that are suitable for evaluating short summaries Lin (2004). The keywords
obtained from the unsupervised methods were linked together to form the
keyphrase string (assumed summary). This was later compared with the original
keyphrase string of the authors.

Full-match results on each dataset are reported in Table 2. From the
unsupervised models, we see that Yake! is consistently better than TopicRank.
The next two supervised models perform even better, with CopyRnn being
discretely superior than Maui.

Results of the four summarization models seem disappointing. Merge and Inject
are the worst on every dataset, with highest score 13.39 %. Various
predictions of these models are empty or very short, and some others contain
long word repetitions which are discarded during evaluation. As a result,
there are usually fewer than five predicted keyphrases. This explains why F1@5
and F1@7 scores are very close to each other.

Abs works slightly better reaching scores from 10.24 to 14.75 %. PointCov is
the best of the text summarizers producing keyphrase predictions that are
usually clean and concise with few repetitions. This is probably the merit of
the coverage mechanism. There is still a considerable gap between PointCov and
CopyRnn. Rouge-1 and Rouge-L F1 scores are reported in Table 3. CopyRnn is
still the best but PointCov is close. Abs scores are also comparable to those
of Maui and Yake!. TopicRank, Merge and Inject are again the worst.

Regarding the test datasets, the highest result scores are achieved on Hulth
and the lowest on Krapivin. We checked some samples of the later and observed
that each of them contains separation tags (e.g., –T, –A, –B, Figure etc.) for
indicating different parts of text in the original paper. A more intelligent
text cleaning step may be required on those data.

##  5 Discussion

The results show that the tried text summarization models perform poorly on
full-match keyword predictions. Their higher ROUGE scores further indicate
that the problem is not entirely in the summarization process. Observing a few
samples, we found differences between the two evaluation strategies. For
example, suppose we have the predicted keyword _“intelligent system”_ compared
against authors’ keyword _“system design”_. Full-match evaluation adds nothing
to F1@5 and F1@7 scores. However, in the case of ROUGE evaluation, the
prediction is partially right and a certain value is added to
$\boldsymbol{R_{1}F_{1}}$ score. In follow up works, one solution to this
discrepancy could be to try partial-match comparison scores like overlap
coefficients.

Another detail that has some negative effect in full-match scores is keyword
separation. The predicted string:

_“health care,,,,immune system; human -; metabolism, immunity,,,,”_

produces [“health care”, “immune system”, “human”, “metabolism”, “immunity”]
as the list of keywords after removing the extra separators. Instead, we
expected [“health care”, “immune system”, “human metabolism”, “immunity”].
This again penalizes full-match scores but not $\boldsymbol{R_{1}F_{1}}$
score. A more intelligent keyword separation mechanism could thus help for
higher full-match result scores.

A third reason could be the fact that we used the title and abstract of papers
only. This is actually what most researchers do, as it is hard to find high
quantities of article full texts for free. Article body is usually restricted.
Abstractive summarization methods could still benefit from longer source
texts. Using default hyperparameters for the models may have also influenced
the results. Some parameter tuning could be beneficial, though.

The main reason could be even more fundamental. We trained abstractive
summarization models on abstracts and titles with authors’ keyphrases
considered as golden ones. There might be two issues here. First, when setting
their keywords, authors mostly consider the topical aspects of their work
rather than paraphrasing over the contents. Abstracts and titles we used may
not carry enough topical information about the article, even when joined
together. Second, considering authors’ keywords as golden ones may not be
reasonable. One solution is to employ human experts and ask them to annotate
each article based on what they read. This is however prohibitive when hundred
thousands of samples are required. Extensive experiments on this issue may
provide different facts and change the picture. For the moment, a safe way to
go seems developing deep supervised generative models like the one of Meng et
al. (2017) that predict one keyphrase at each step independently.

##  6 Conclusions

In this paper, we experimented with various unsupervised, supervised, deep
supervised and abstractive text summarization models for predicting keyphrases
of scientific articles. To the best of our knowledge, this is the first
attempt that explores the possibility of conceiving article string of keywords
as an abstractive summary of title and abstract. We collected and produced a
large dataset of 2.2 million abstracts, titles and keyphrase strings from
scientific papers available online. It can be used for future text
summarization and keyphrase generation experiments. Systematic evaluation on
four test datasets shows that the used summarization models could not produce
better keywords than the supervised predictive models. Extensive experiments
with more advanced summarizaiton methods and better parameter optimization may
still reveal a different view of the situation.

## Acknowledgments

This research work was supported by the project No.
CZ.02.2.69/0.0/0.0/16_027/0008495 (International Mobility of Researchers at
Charles University) of the Operational Programme Research, Development and
Education, grant 19-26934X (NEUREM3) of the Czech Science Foundation and
H2020-ICT-2018-2-825460 (ELITR) of the EU.

## References

  * Bahdanau et al. (2014) Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. 2014.  Neural machine translation by jointly learning to align and translate.  _CoRR_ , abs/1409.0473.
  * Bougouin et al. (2013) Adrien Bougouin, Florian Boudin, and Béatrice Daille. 2013.  Topicrank: Graph-based topic ranking for keyphrase extraction.  In _Proceedings of the Sixth International Joint Conference on Natural Language Processing_ , pages 543–551. Asian Federation of Natural Language Processing.
  * Campos et al. (2018) Ricardo Campos, Vítor Mangaravite, Arian Pasquali, Alípio Mário Jorge, Célia Nunes, and Adam Jatowt. 2018.  Yake! collection-independent automatic keyword extractor.  In _Advances in Information Retrieval_ , pages 806–810. Springer International Publishing.
  * Çano and Morisio (2015) Erion Çano and Maurizio Morisio. 2015.  Characterization of public datasets for recommender systems.  In _2015 IEEE 1st International Forum on Research and Technologies for Society and Industry Leveraging a better tomorrow (RTSI)_ , pages 249–257.
  * Florescu and Caragea (2017) Corina Florescu and Cornelia Caragea. 2017.  Positionrank: An unsupervised approach to keyphrase extraction from scholarly documents.  In _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics_ , pages 1105–1115. Association for Computational Linguistics.
  * Foster et al. (2018) George Foster, Ashish Vaswani, Jakob Uszkoreit, Wolfgang Macherey, Lukasz Kaiser, Orhan Firat, Llion Jones, Noam Shazeer, Yonghui Wu, Ankur Bapna, Melvin Johnson, Mike Schuster, Zhifeng Chen, Macduff Hughes, Niki Parmar, and Mia Xu Chen. 2018.  The best of both worlds: Combining recent advances in neural machine translation.  In _ACL (1)_ , pages 76–86. Association for Computational Linguistics.
  * Grusky et al. (2018) Max Grusky, Mor Naaman, and Yoav Artzi. 2018.  Newsroom: A dataset of 1.3 million summaries with diverse extractive strategies.  In _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 708–719. Association for Computational Linguistics.
  * Hasan and Ng (2014) Kazi Saidul Hasan and Vincent Ng. 2014.  Automatic keyphrase extraction: A survey of the state of the art.  In _Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics_ , pages 1262–1273.
  * Hulth (2003) Anette Hulth. 2003.  Improved automatic keyword extraction given more linguistic knowledge.  In _Proceedings of the 2003 conference on Empirical methods in natural language processing_ , pages 216–223. Association for Computational Linguistics.
  * Krapivin et al. (2010) Mikalai Krapivin, Aliaksandr Autayeu, Maurizio Marchese, Enrico Blanzieri, and Nicola Segata. 2010.  Keyphrases extraction from scientific documents.  In _The Role of Digital Libraries in a Time of Global Change_.
  * Lin (2004) Chin-Yew Lin. 2004.  Rouge: A package for automatic evaluation of summaries.  In _Proc. ACL workshop on Text Summarization Branches Out_ , page 10.
  * Medelyan (2009) Olena Medelyan. 2009.  _Human-competitive automatic topic indexing_.  The University of Waikato, Phd Thesis.
  * Meng et al. (2017) Rui Meng, Sanqiang Zhao, Shuguang Han, Daqing He, Peter Brusilovsky, and Yu Chi. 2017.  Deep keyphrase generation.  In _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics_ , pages 582–592. Association for Computational Linguistics.
  * Nallapati et al. (2016) Ramesh Nallapati, Bowen Zhou, Cicero dos Santos, Caglar Gulcehre, and Bing Xiang. 2016.  Abstractive text summarization using sequence-to-sequence rnns and beyond.  In _Proceedings of The 20th SIGNLL Conference on Computational Natural Language Learning_ , pages 280–290. Association for Computational Linguistics.
  * Napoles et al. (2012) Courtney Napoles, Matthew Gormley, and Benjamin Van Durme. 2012.  Annotated gigaword.  In _Proceedings of the Joint Workshop on Automatic Knowledge Base Construction and Web-scale Knowledge Extraction_ , AKBC-WEKEX ’12, pages 95–100, Stroudsburg, PA, USA. Association for Computational Linguistics.
  * Nguyen and Luong (2010) Thuy Dung Nguyen and Minh-Thang Luong. 2010.  Wingnus: Keyphrase extraction utilizing document logical structure.  In _Proceedings of the 5th International Workshop on Semantic Evaluation_ , SemEval ’10, pages 166–169, Stroudsburg, PA, USA. Association for Computational Linguistics.
  * Rose et al. (2010) Stuart Rose, Dave Engel, Nick Cramer, and Wendy Cowley. 2010.  Automatic keyword extraction from individual documents.  In Michael W. Berry and Jacob Kogan, editors, _Text Mining. Applications and Theory_ , pages 1–20. John Wiley and Sons, Ltd.
  * Rush et al. (2015) Alexander M. Rush, Sumit Chopra, and Jason Weston. 2015.  A neural attention model for abstractive sentence summarization.  In _Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing_ , pages 379–389. Association for Computational Linguistics.
  * See et al. (2017) Abigail See, Peter J. Liu, and Christopher D. Manning. 2017.  Get to the point: Summarization with pointer-generator networks.  In _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics_ , pages 1073–1083. Association for Computational Linguistics.
  * Sinha et al. (2015) Arnab Sinha, Zhihong Shen, Yang Song, Hao Ma, Darrin Eide, Bo-June (Paul) Hsu, and Kuansan Wang. 2015.  An overview of microsoft academic service (mas) and applications.  In _Proceedings of the 24th International Conference on World Wide Web_ , WWW ’15 Companion, pages 243–246, New York, NY, USA. ACM.
  * Tang et al. (2008) Jie Tang, Jing Zhang, Limin Yao, Juanzi Li, Li Zhang, and Zhong Su. 2008.  Arnetminer: Extraction and mining of academic social networks.  In _Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ , KDD ’08, pages 990–998, New York, NY, USA. ACM.
  * Tanti et al. (2017) Marc Tanti, Albert Gatt, and Kenneth P. Camilleri. 2017.  What is the role of recurrent neural networks (rnns) in an image caption generator? _CoRR_ , abs/1708.02043.
  * Tu et al. (2016) Zhaopeng Tu, Zhengdong Lu, Yang Liu, Xiaohua Liu, and Hang Li. 2016.  Modeling coverage for neural machine translation.  In _Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics_ , pages 76–85. Association for Computational Linguistics.
  * Turney (2000) Peter D. Turney. 2000.  Learning algorithms for keyphrase extraction.  _Inf. Retr._ , 2(4):303–336.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Ł ukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, _Advances in Neural Information Processing Systems 30_ , pages 5998–6008. Curran Associates, Inc.
  * Witten et al. (1999) Ian H. Witten, Gordon W. Paynter, Eibe Frank, Carl Gutwin, and Craig G. Nevill-Manning. 1999.  Kea: Practical automatic keyphrase extraction.  In _Proceedings of the Fourth ACM Conference on Digital Libraries_ , DL ’99, pages 254–255, NY, USA. ACM.


Zeeman_Cs2CoBr4_18-05-2023_SUPP.pdf

# Confinement of fractional excitations in a triangular lattice
antiferromagnet

L. Facheris<EMAIL_ADDRESS>Laboratory for Solid State Physics, ETH
Zürich, 8093 Zürich, Switzerland  S. D. Nabi  Laboratory for Solid State
Physics, ETH Zürich, 8093 Zürich, Switzerland  A. Glezer Moshe  U. Nagel  T.
Rõõm  National Institute of Chemical Physics and Biophysics, Akadeemia tee 23,
12618 Tallinn, Estonia  K. Yu. Povarov  Laboratory for Solid State Physics,
ETH Zürich, 8093 Zürich, Switzerland  Present address: Dresden High Magnetic
Field Laboratory (HLD-EMFL) and Würzburg-Dresden Cluster of Excellence
ct.qmat, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  J. R.
Stewart  ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot,
OX11 0QX, United Kingdom  Z. Yan  Laboratory for Solid State Physics, ETH
Zürich, 8093 Zürich, Switzerland  A. Zheludev<EMAIL_ADDRESS>http://www.neutron.ethz.ch/ Laboratory for Solid State Physics, ETH Zürich,
8093 Zürich, Switzerland

###### Abstract

High-resolution neutron and THz spectroscopies are used to study the magnetic
excitation spectrum of Cs2CoBr4, a distorted-triangular-lattice
antiferromagnet with nearly XY-type anisotropy. What was previously thought of
as a broad excitation continuum [Phys. Rev. Lett. 129, 087201 (2022)] is shown
to be a series of dispersive bound states reminiscent of “Zeeman ladders” in
quasi-one-dimensional Ising systems. At wave vectors where inter-chain
interactions cancel at the Mean Field level, they can indeed be interpreted as
bound finite-width kinks in individual chains. Elsewhere in the Brillouin zone
their true two-dimensional structure and propagation are revealed.

In conventional magnetic insulators the dynamic response is typically
dominated by coherent single-particle $S=1$ excitations, aka magnons or spin
waves. In many low-dimensional and highly frustrated quantum spin systems
elementary excitations carry fractional quantum numbers, be they spinons in
Heisenberg spin chains [1, 2, 3, 4], Majorana fermions in the now-famous
Kitaev model [5, 6, 7], or pseudo-charge defects in spin ice-like Coulomb
phases [8]. The physical excitation spectrum, such as that measured by neutron
spectroscopy, is then dominated by broad multi-particle continua [9, 10, 11,
12, 13]. In addition to the continuum, fractional excitations may also form
bound states due to attractive interactions between them. A spectacular new
phenomenon emerges when interactions are confining, i.e. do not fall off with
distance, much like strong forces that bind quarks in hadrons [14]. This
produces a hierarchical series of bound states inside the resulting potential
well. An example is the sequence of domain wall (kink) bound states in quasi-
one-dimensional (quasi-1D) Ising spin chains [15, 16, 17]. The confining
potential for this model is linear and results from 3D couplings, which
generate an effective field acting on individual chains [16]. The binding
energies are, in supreme mathematical elegance, spaced according to the
negative zeros of the Airy function [15, 17]. The best-known experimental
examples of such “Zeeman ladder” spectra are the quasi-1D Ising ferromagnet
CoNb2O6 [18] and antiferromagnet (AF) BaCo2V2O8 [19], as well as the
isostructural compound SrCo2V2O8 [20], where as many as 8 consecutive bound
states are observed. Shorter sequences have been found in another prototypical
Ising spin chain material, RbCoCl3 [21].

Bound states are always easier to form in 1D, where even an arbitrary-shallow
potential well provides at least one robust localized state for a propagating
particle. Fractional excitations can also form bound states in higher
dimensional spin systems [22, 23]. Their structure and propagation are much
more complex than that of simple paired domain walls in spin chains.
Unfortunately, no Zeeman ladder analogues, i.e. no hierarchical binding
sequences of fractional excitations have been found in higher dimensions to
date. In the present work we report the observation of such a phenomenon in a
quasi-2D distorted-triangular-lattice AF. That the quintessentially 1D physics
of bound kinks survives in 2D is remarkable. We argue that it is “rescued” at
certain special wave vectors by the intrinsic frustration in triangular
lattice geometry. Elsewhere in the Brillouin zone the bound states are no
longer restricted to single chains and are to be viewed as 2D objects
propagating in the entire triangular plane.

Figure 1: (a)-(b) Neutron scattering intensity (solid symbols) measured at
$T=40$ mK versus energy transfer at the 1D AF zone-centers
$\mathbf{q}=(0,0.5,0.5)$ and $\mathbf{q}=(0,1,0.5)$, respectively. The data
are integrated fully along $h$ direction and in $\pm 0.025$ r.l.u. and $\pm
0.25$ r.l.u. along $k$ and $l$, respectively. Solid lines are fits to a series
of Gaussian peaks. Dashed Gaussians represent the calculated experimental
energy resolution. Black dotted lines indicate the fitted flat background. (c)
Measured THz absorption (solid line) versus absorbed photon energy for light
propagating along the $\bm{c}$ axis at $0.2$ K. Dashed areas highlight the
Gaussian components of the fit to extract peak positions. (d) Measured
excitation energy plotted versus the value of negative roots of the Airy
function. The solid line is a linear fit as described in the text. The blue
area highlights the points used for the fit. Inset: cartoons of the magnetic
ground state and a representative $m=3$ 2-kink bound state.

The material in question, Cs2CoBr4 (space group P$nma$, $a=10.19$, $b=7.73$,
$c=13.51$ Å), is a very interesting $J-J^{\prime}$ model distorted-triangular-
lattice AF [24, 25]. Despite a prominent triangular motif in its structure, it
demonstrates certain 1D features such as a field-induced incommensurate spin
density wave with Tomonaga-Luttinger spin liquid type dynamics and a
propagation vector controlled by a one-dimensional nesting in the spinon Fermi
sea. Its true 2D nature is manifest in the presence of a robust $m=1/3$
magnetization plateau, typical of a triangular AF. The model magnetic
Hamiltonian is described in detail in Refs. [24, 25]. The key structural
features are chains of Co2+ ions that run along the crystallographic $\bm{b}$
axis of the orthorhombic lattice (see Fig. 1 in Ref. [24]). The chains are
coupled in the $(bc)$ plane in a zigzag fashion to form a distorted triangular
network (inset of Fig. 1(d)). Easy-plane single-ion anisotropy ensures that
the low-energy physics of the spin-$3/2$ Co2+ ions can be described in terms
of effective $S=1/2$ pseudo-spins. The components of the effective exchange
coupling constants are subject to restrictions imposed by the pseudo-spin
projection. A simplistic spin-wave analysis of previous inelastic neutron data
provided a rough estimate for the nearest-neighbor in-chain AF exchange tensor
components: $J^{XX}\sim J$, $J^{YY}\sim 1.1J$, $J^{ZZ}\sim 0.25J$, $J=0.8$ meV
[25]. Here $Y$ is chosen along the $\bm{b}$ crystallographic direction, and
$X$ and $Z$ alternate between adjacent chains, where anisotropy planes are
almost orthogonal. Note that this is practically a planar exchange anisotropy,
with only a tiny in-plane Ising component to account for the $\Delta\sim 0.4$
meV spectral gap found in this system. The frustrated inter-chain coupling
$J^{\prime}$ is significant, of the order of $0.45J$, and is of predominantly
Ising ($YY$) character. Inter-plane interactions $J^{\prime\prime}$ are not
frustrated. The material orders magnetically in a collinear stripe-type
structure, with an ordering wavevector $(0,1/2,1/2)$ (see inset in Fig. 1(d)).
The Néel temperature $T_{\text{N}}=1.3$ K allows us to estimate
$J^{\prime\prime}$. If this were the only coupling between chains with no
additional frustration due to $J^{\prime}$, we could expect
$k_{\mathrm{B}}T_{\mathrm{N}}\sim 2\Delta/\ln(\Delta/J^{\prime\prime})$ [26].
The actual value of $J^{\prime\prime}$ must be larger than thus obtained, as
the in-plane frustration interferes with the emerging magnetic structure. A
certain upper estimate is given by the mean field picture where
$k_{\mathrm{B}}T_{\mathrm{N}}\sim 2J^{\prime\prime}S(S+1)$. This leads us to
conclude that $3\cdot 10^{-4}~{}\mathrm{meV}\lesssim J^{\prime\prime}\lesssim
0.075$ meV $\ll J$, confirming the quasi-2D character of the material.

Our previous inelastic neutron scattering experiments indicated that the
excitation spectrum in zero applied field is a gapped continuum of states,
with intensity concentrated on its lower bound, and a strong dispersion along
the chain axis [25]. The central finding of the present work is that this
“continuum” is actually a sequence of at least 9 sharp bound states that
previously could not be observed due to poor experimental energy resolution.
New neutron data were collected at the LET time-of-flight spectrometer at ISIS
(UK), using $2.35$ meV incident energy neutrons in repetition-rate-
multiplication mode [27]. We used the same $1.16$ g single crystal as in [25]
mounted on a 3He-4He dilution refrigerator. All measurements were performed at
a base temperature of $40$ mK. In the experiment the sample was rotated
$180^{\circ}$ around the $\bm{a}$ axis in steps of $1^{\circ}$. The spectra
were measured for $\sim 10$-minute counting time at each sample position.

We first focus on the 1D AF zone-centers ($\mathbf{q}\bm{b}=0,\pi$), where
inter-chain interactions within the triangular planes cancel out at the Mean
Field-RPA level, and where spin wave theory predicts no transverse dispersion
or intensity modulation of excitations. Fig. 1(a),(b) show
constant-$\mathbf{q}$ cuts through the data at wave vectors
$\mathbf{q}=(0,0.5,0.5)$ and $\mathbf{q}=(0,1,0.5)$, respectively. A sequence
of sharp peaks is clearly apparent in both cases. A fit to the data using
empirical Gaussian profiles yields an accurate measure of the peak positions
and shows that their widths are essentially resolution-limited. In Fig.
1(a),(b) this is emphasized by the shaded Gaussians representing the computed
experimental resolution [28].

Corroborative evidence is also obtained by THz spectroscopy. The experiment
was performed with a Martin-Puplett-type interferometer and a 3He-4He dilution
refrigerator with base temperature of $150$ mK using a 3He-cooled Si bolometer
at $0.3$ K. The sample was a circular plate approximately $1$ mm thick in
$\bm{c}$ direction and $4$ mm in diameter. THz radiation propagating along the
crystal $\bm{c}$ axis was unpolarized and the apodized instrumental resolution
was $0.025$ meV. The THz absorption spectrum is shown in Fig. 1(c). It is
calculated as a difference of spectra measured at $0.2$ K and $2$ K, i.e. in
the magnetically ordered phase and above $T_{\text{N}}$. The THz spectrum
appears to have some features absent in the neutron spectrum 111The extra
features are superficially similar to field-induced satellites observed in
[38], but appear in zero field in our case. They are more likely related to
different polarization channels, as discussed below., but all peaks found in
the latter are also present here. The positions of these peaks were determined
in Gaussian fits (shaded peaks) in a narrow range $\pm 0.025$ meV near each
peak value. Comparing data measured at low temperature to that collected at
$2$ K [28] reveals that the series of peaks is endemic to the ordered phase
and disappears above $T_{\text{N}}$.

The spacing between the excitation peaks present in both measurements
corresponds to confinement in an approximately linear one-dimensional
potential. To demonstrate this, we plot the excitation energies deduced from
neutron spectra at several wave vectors, as well as the positions of
corresponding THz peaks, versus the negative roots $z_{i}$ of the Airy
function in Fig. 1(d). For a precise linear attractive potential $\lambda|x|$
between the dispersive particles 222See discussion of lattice effects in the
Supplemental Material., near the minimum
$\epsilon(k)=m_{0}+\hbar^{2}k^{2}/2\mu$ we expect the excitation energies to
be [18, 17]

$m_{i}=2m_{0}+(\hbar\lambda)^{2/3}\mu^{-1/3}z_{i}\,\,\,\,\text{with}\,\,\,\,i=1,2,\dots.$
(1)

In the actual data, the linear dependence is apparent for all but the first
few points. As will be addressed in more detail below, this slight deviation
indicates that the confining force increases somewhat at short distances. From
a linear fit to the higher-energy peaks we can immediately extract the slope
$0.072(3)$ meV and the energy of a single particle $m_{0}=0.18(1)$ meV (half-
intercept). Using the single-particle kinetic mass $\hbar^{2}/\mu=0.39$
meV$\times b^{2}$ [28], we estimate the confining force constant
$\lambda=0.031(2)$ meV/$b$ 333This force of $\sim 6$ fN corresponds to the
gravity pull between two average humans at a separation of $8$ km..

The next point that we make is that the observed bound states at the 1D AF
zone-center are essentially 1D objects. This is concluded by analyzing the
neutron spectra shown in Figs. 2(a),(b) and Supp. Fig. 5(a) in [28]. The bound
states do not propagate in either transverse direction and thus have an
essentially flat dispersion. Moreover, their intensity shows a monotonic
falloff and no obvious transverse modulation, as plotted for the first two
modes in Figs. 2(e),(f), and Supp. Fig. 5(b) [28]. This implies that these
excitations do not involve cross-chain correlations and are confined to a
single chain.

For the first mode, the measured transverse wave vector dependence is
precisely accounted for (orange solid lines in Figs. 2(e),(f), and Supp. Fig.
5(b)) by the combined effects of i) the magnetic form factor of Co2+ and ii) a
neutron polarization factor for spin components perpendicular to the chain
axis (to the direction of ordered moments in the ground state). Longitudinal
excitations would become stronger for scattering vectors deviating from the
chain axis (orange dashed line). Already for the second mode the
identification of polarization is ambiguous (green lines). Unlike in Ising
chains, in Cs2CoBr4 the conservation of the chain-axis projection of angular
momentum is not expected, meaning that excitations cannot be unambiguously
classified as longitudinal nor transverse.

Figure 2: (a)-(d) False color plot of neutron scattering intensity measured at
$T=40$ mK plotted versus energy transfer and momentum transfer transverse to
the crystallographic $\bm{b}$-axis. Gray areas mask regions of elastic-
incoherent scattering. Background subtraction has been performed as described
in [28]. The orange regions represent energy-integration windows used to
extract the cuts in panels below. (e)-(h) Intensity-momentum cuts (solid
symbols) for the first two modes. The orange and green lines show the product
of the magnetic form-factor-squared for Co2+ and calculated neutron
polarization factor for excitations polarized perpendicular (solid) and
parallel (dashed) to the ordered moment direction, respectively. The overall
scale factor of the lines is arbitrary, but always consistent between the two
polarizations and consistent between panels (e) and (f).

The 1D nature of zone-center excitations prompts a simple interpretation.
Similarly to the situation in CoNb2O6 and BaCo2V2O8, the observed modes are
bound states of two kinks (domain walls) in individual chains. Naively,
ignoring the kinetic energy of the kinks as in the purely Ising case, such an
excitation can be visualized as in the cartoon in the inset of Fig. 1(d). The
energy $m_{0}$ is to be associated with that of a single domain wall. As a
consistency check, we can compare that to the computed energy of a domain wall
in a classical spin chain. Using $J^{YY}/J^{XX}\sim 1.1$ as estimated for
Cs2CoBr4, with a trivial numerical classical-energy minimization procedure we
get $m_{0}\sim 0.9JS^{2}=0.18$ meV, in excellent agreement with the measured
value.

Geometric frustration ensures that at the magnetic zone-center these strings
of flipped spins within a single chain incur no energy cost due to
interactions with adjacent chains within the triangular lattice. Moreover, any
transverse dispersion is Suppressed. At the same time, the interaction energy
due to unfrustrated inter-layer coupling is proportional to the string length,
resulting in confinement. In this simplistic picture, the confining force is
$\lambda=2J^{\prime\prime}S^{2}/b$. This yields an inter-layer coupling
constant $J^{\prime\prime}=0.062(4)$ meV, inside the possible range deduced
from $T_{\mathrm{N}}$.

The apparent difference between our data and those for the Ising-chain AF
BaCo2V2O8 [19] is the presence of two Zeeman ladder series in the latter
material. The two series correspond to transverse and longitudinal
excitations, the first transverse mode being a spin flip, i. e. a conventional
magnon. They are offset in energy by $\lambda b$ in our notation [28]. For
Cs2CoBr4 this amounts to as little as $0.03$ meV. The two series can not be
resolved within the resolution and counting statistics of our neutron
experiment but may account for the extra shoulder peaks seen in the THz data
in Fig. 1(c). A similar estimate for BaCo2V2O8 yields a larger offset of about
$0.34$ meV that is easily resolved experimentally [19].

The deviation from linear-potential behavior at low energies is also readily
explained. Since the material is almost planar, the domain walls are not
confined to a single bond as in the ideal Ising case, but have a
characteristic size $l$ 444We define the domain wall width in a spin-$S$ chain
as the distance over which the $z$-axis spin component changes from $S/2$ to
$-S/2$ near its center.. We can estimate that quantity in a classical spin
chain using the above-mentioned anisotropy parameters: $l\sim 2b$. The energy
of the first few bound states is thus modified due to a physical overlap of
the two bounding domain walls. Once the kinks are separated by a distance of
more than $\sim l$, this interaction becomes negligible and the confinement
potential becomes linear.

In summary, at the 1D AF zone-center the simplistic kink model from 1D Ising
physics consistently explains the observed excitation energies. The first mode
is transversely polarized and shows an intensity patterns very consistent with
that of a magnon in spin wave theory [25], being strongly suppressed at the 1D
structural zone-centers (integer $k$). The intensities of higher modes are not
that easily interpreted: they may have a mixed polarization and are less
suppressed at integer $k$ values (Fig. 1(a),(b)).

Away from the 1D AF zone-centers, the excitations are considerably more
complex. This is very clear in the longitudinal dispersion of the bound states
shown in Fig. 3(a),(b). Other than at $\mathbf{q}\bm{b}=0,\pi$ ($k=0,1/2$) the
$m_{1}$ mode splits into two branches, each with an asymmetric dispersion
relation. In fact, the $m_{1}$ state at $\mathbf{q}\bm{b}=\pi$ seems to be
continuously connected to the $m_{2}$ excitations at $\mathbf{q}\bm{b}=2\pi$
($k=1$) and vice versa. Fitting the dispersion of the strongest low-energy
mode in the vicinity of $\mathbf{q}\bm{b}=\pi$ to a Lorentz-invariant
relativistic form

${(\hbar\omega_{\mathbf{q}})}^{2}=\hbar^{2}\Delta\,(\mathbf{q}\bm{b})^{2}\,/\mu+\Delta^{2},$
(2)

yields the value of kinetic mass quoted above.

A look at the intensities reveals that other than at the special wave vectors,
the bound states can no longer be seen as strings in a single chain, but are
“dressed” with correlations extending to several neighboring chains in the
triangular plane. This conclusion is reached from Fig. 2(c),(d), that show a
transverse cut of the spectrum at $\mathbf{q}\bm{b}=5\pi/4$ and
$\mathbf{q}\bm{b}=3\pi/2$, respectively. As plotted in Fig. 2(g),(h), the
measured intensity of the first two modes now shows a much steeper transverse
wave vector dependence than computed from just the polarization and form
factors (solid lines). The second mode even seems to show signs of intensity
oscillations, with a possible mixing/hybridization and resulting intensity
transfer with the first mode.

Figure 3: (a)-(b) False color plot of neutron scattering intensity measured at
$T=40$ mK plotted versus energy transfer and momentum transfer along
$\mathbf{q}=(0,k,0.5)$ and $\mathbf{q}=(0,k,1)$ respectively. The data were
fully integrated along $h$, and in the range $\pm 0.25$ r.l.u. along $l$
around the central value. The gray areas mask regions where the incoherent
scattering dominates the signal. Background subtraction has been performed as
described in [28].

Our data reveal that away from the special wave vectors the bound states also
propagate in two dimensions, albeit with a small bandwidth. Indeed, in Fig.
2(d) one can see that at $\mathbf{q}\bm{b}=3\pi/2$ the bound states develop a
non-zero dispersion along the $\bm{c}^{*}$ direction, in contrast to what is
seen at $\mathbf{q}\bm{b}=0,\pi$. Although the bandwidth of transverse
dispersion, $0.08$ meV, is at the limit of our experimental resolution,
qualitatively one can say that $\mathbf{q}\bm{c}=0,4\pi$ are dispersion minima
for the $m_{1}$ mode, while the maximum is at $\mathbf{q}\bm{c}=2\pi$. That
periodicity is consistent with having two chains per unit cell along the
$\bm{c}$-axis direction in the crystal structure.

Overall, the differences between our results and spectra of Ising spin chains
[18, 19] are striking. In the latter, all bound states, including the first
one, are much less dispersive than the lower edge of the entire spectrum,
which approximately corresponds to the lower edge of the two-kink continuum in
the absence of long-range order. As a result, each bound state persists only
in a restricted area in the Brillouin zone. In contrast, in Cs2CoBr4 a few of
the lower-energy bound states are highly dispersive and span across the entire
zone.

In summary, we demonstrate that “Zeeman ladders” of confined fractional
excitations can exist in a bona fide quasi-2D system. These states are
inherently related to those in 1D models, as revealed at special wave vectors
where 2D interactions are canceled by geometric frustration. However,
elsewhere in reciprocal space their true 2D character is manifest. Once again,
the distorted triangular lattice model provides a link between 1D and 2D
quantum magnetism.

###### Acknowledgements.

This work was Supported by a MINT grant of the Swiss National Science
Foundation. We acknowledge Support by the Estonian Research Council grants
PRG736 and MOBJD1103, and by European Regional Development Fund Project No.
TK134. Experiments at the ISIS Neutron and Muon Source were Supported by beam
time allocation RB2210048 from the Science and Technology Facilities Council
[37].

## References

  * Faddeev and Takhtajan [1981] L. D. Faddeev and L. A. Takhtajan, What is the spin of a spin wave?, Phys. Lett. A 85, 375 (1981).
  * Müller  _et al._ [1981] G. Müller, H. Thomas, H. Beck, and J. C. Bonner, Quantum spin dynamics of the antiferromagnetic linear chain in zero and nonzero magnetic field, Phys. Rev. B 24, 1429 (1981).
  * Stone  _et al._ [2003] M. B. Stone, D. H. Reich, C. Broholm, K. Lefmann, C. Rischel, C. P. Landee, and M. M. Turnbull, Extended Quantum Critical Phase in a Magnetized Spin-$\frac{1}{2}$ Antiferromagnetic Chain, Phys. Rev. Lett. 91, 037205 (2003).
  * Giamarchi [2004] T. Giamarchi,  _Quantum Physics in One Dimension_ (Clarendon Press, U.K., 2004).
  * Kitaev [2006] A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
  * Winter  _et al._ [2017] S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. Gegenwart, and R. Valentí, Models and materials for generalized Kitaev magnetism, J. Phys.: Cond. Mat. 29, 493002 (2017).
  * Takagi  _et al._ [2019] H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Concept and realization of Kitaev quantum spin liquids, Nat. Rev. Phys. 1, 264 (2019).
  * Henley [2010] C. L. Henley, The “Coulomb Phase” in Frustrated Systems, Annual Review of Condensed Matter Physics 1, 179 (2010).
  * Mourigal  _et al._ [2013] M. Mourigal, M. Enderle, A. Klöpperpieper, J.-S. Caux, A. Stunault, and H. M. Rønnow, Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain, Nat. Phys. 9, 435 (2013).
  * Schmidiger  _et al._ [2013] D. Schmidiger, P. Bouillot, T. Guidi, R. Bewley, C. Kollath, T. Giamarchi, and A. Zheludev, Spectrum of a Magnetized Strong-Leg Quantum Spin Ladder, Phys. Rev. Lett. 111, 107202 (2013).
  * Dai  _et al._ [2021] P.-L. Dai, G. Zhang, Y. Xie, C. Duan, Y. Gao, Z. Zhu, E. Feng, Z. Tao, C.-L. Huang, H. Cao, A. Podlesnyak, G. E. Granroth, M. S. Everett, J. C. Neuefeind, D. Voneshen, S. Wang, G. Tan, E. Morosan, X. Wang, H.-Q. Lin, L. Shu, G. Chen, Y. Guo, X. Lu, and P. Dai, Spinon Fermi Surface Spin Liquid in a Triangular Lattice Antiferromagnet ${\mathrm{NaYbSe}}_{2}$, Phys. Rev. X 11, 021044 (2021).
  * Tennant [2019] D. A. Tennant, Studies of Spinons, Majoranas, and Monopoles in Spin Liquid and Quantum Critical Magnets with Neutrons, J. Phys. Soc. Jpn. 88, 081009 (2019).
  * Plumb  _et al._ [2019] K. W. Plumb, H. J. Changlani, A. Scheie, J. W. Krizan, J. A. Rodriguez-Rivera, Y. Qiu, B. Winn, J. R. Cava, and C. L. Broholm, Continuum of quantum fluctuations in a three-dimensional $S=1$ Heisenberg magnet, Nat. Phys. 15, 54 (2019).
  * Wilczek [1989] F. Wilczek, Quantum Chromodynamics: The Modern Theory of the Strong Interaction, Ann. Rev. Nucl. Part. Sci. 32, 177 (1989).
  * McCoy and Wu [1978] B. M. McCoy and T. T. Wu, Two-dimensional Ising field theory in a magnetic field: Breakup of the cut in the two-point function, Phys. Rev. D 18, 1259 (1978).
  * Shiba [1980] H. Shiba, Quantization of Magnetic Excitation Continuum Due to Interchain Coupling in Nearly One-Dimensional Ising-Like Antiferromagnets, Prog. Theor. Phys. 64, 466 (1980).
  * Rutkevich [2008] S. B. Rutkevich, Energy Spectrum of Bound-Spinons in the Quantum Ising Spin-Chain Ferromagnet, J. Stat. Phys. 131, 917 (2008).
  * Coldea  _et al._ [2010] R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry, Science 327, 177 (2010).
  * Grenier  _et al._ [2015] B. Grenier, S. Petit, V. Simonet, E. Canévet, L.-P. Regnault, S. Raymond, B. Canals, C. Berthier, and P. Lejay, Longitudinal and Transverse Zeeman Ladders in the Ising-Like Chain Antiferromagnet ${\mathrm{BaCo}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$, Phys. Rev. Lett. 114, 017201 (2015).
  * Bera  _et al._ [2017] A. K. Bera, B. Lake, F. H. L. Essler, L. Vanderstraeten, C. Hubig, U. Schollwöck, A. T. M. N. Islam, A. Schneidewind, and D. L. Quintero-Castro, Spinon confinement in a quasi-one-dimensional anisotropic Heisenberg magnet, Phys. Rev. B 96, 054423 (2017).
  * Mena  _et al._ [2020] M. Mena, N. Hänni, S. Ward, E. Hirtenlechner, R. Bewley, C. Hubig, U. Schollwöck, B. Normand, K. W. Krämer, D. F. McMorrow, and C. Rüegg, Thermal Control of Spin Excitations in the Coupled Ising-Chain Material ${\mathrm{RbCoCl}}_{3}$, Phys. Rev. Lett. 124, 257201 (2020).
  * Matan  _et al._ [2022] K. Matan, T. Ono, S. Ohira-Kawamura, K. Nakajima, Y. Nambu, and T. J. Sato, Breakdown of linear spin-wave theory and existence of spinon bound states in the frustrated kagome-lattice antiferromagnet, Phys. Rev. B 105, 134403 (2022).
  * Ghioldi  _et al._ [2022] E. A. Ghioldi, S.-S. Zhang, Y. Kamiya, L. O. Manuel, A. E. Trumper, and C. D. Batista, Evidence of two-spinon bound states in the magnetic spectrum of ${\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}$, Phys. Rev. B 106, 064418 (2022).
  * Povarov  _et al._ [2020] K. Y. Povarov, L. Facheris, S. Velja, D. Blosser, Z. Yan, S. Gvasaliya, and A. Zheludev, Magnetization plateaux cascade in the frustrated quantum antiferromagnet ${\mathrm{Cs}}_{2}{\mathrm{CoBr}}_{4}$, Phys. Rev. Research 2, 043384 (2020).
  * Facheris  _et al._ [2022] L. Facheris, K. Y. Povarov, S. D. Nabi, D. G. Mazzone, J. Lass, B. Roessli, E. Ressouche, Z. Yan, S. Gvasaliya, and A. Zheludev, Spin Density Wave versus Fractional Magnetization Plateau in a Triangular Antiferromagnet, Phys. Rev. Lett. 129, 087201 (2022).
  * Carr and Tsvelik [2003] S. T. Carr and A. M. Tsvelik, Spectrum and Correlation Functions of a Quasi-One-Dimensional Quantum Ising Model, Phys. Rev. Lett. 90, 177206 (2003).
  * Bewley  _et al._ [2011] R. Bewley, J. Taylor, and S. Bennington., LET, a cold neutron multi-disk chopper spectrometer at ISIS, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 637, 128 (2011).
  * [28] See Supplemental Material at [url] for detailed discussion of the resolution calculations, additional inelastic neutron scattering data, background subtraction procedure, temperature dependence from THz data, estimate of the kinetic mass for a kink, lattice effects, polarization of the excitations, and additional transverse neutron data, which includes Refs. [29, 30, 31, 32].
  * Arnold  _et al._ [2014] O. Arnold, J. C. Bilheux, J. M. Borreguero, A. Buts, S. I. Campbell, L. Chapon, M. Doucet, N. Draper, R. Ferraz Leal, M. A. Gigg, V. E. Lynch, A. Markvardsen, D. J. Mikkelson, R. L. Mikkelson, R. Miller, K. Palmen, P. Parker, G. Passos, T. G. Perring, P. F. Peterson, S. Ren, M. A. Reuter, A. T. Savici, J. W. Taylor, R. J. Taylor, R. Tolchenov, W. Zhou, and J. Zikovsky, Mantid—Data analysis and visualization package for neutron scattering and $\mu$SR experiments, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 764, 156 (2014).
  * Ewings  _et al._ [2016] R. A. Ewings, A. Buts, M. D. Le, J. van Duijn, I. Bustinduy, and T. G. Perring, Horace: Software for the analysis of data from single crystal spectroscopy experiments at time-of-flight neutron instruments, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 834, 132 (2016).
  * Fishman  _et al._ [2018] R. S. Fishman, J. A. Fernandez-Baca, and T. Rõõm,  _Spin-Wave Theory and its Applications to Neutron Scattering and THz Spectroscopy_ (IOP Concise Physics, Morgan and Claypool Publishers, 2018).
  * Furrer  _et al._ [2009] A. Furrer, J. Mesot, and T. Strässle,  _Neutron Scattering in Condensed Matter Physics_ (World Scientific, 2009).
  * Note [1] The extra features are superficially similar to field-induced satellites observed in [38], but appear in zero field in our case. They are more likely related to different polarization channels, as discussed below.
  * Note [2] See discussion of lattice effects in the Supplemental Material.
  * Note [3] This force of $\sim 6$ fN corresponds to the gravity pull between two average humans at a separation of $8$ km.
  * Note [4] We define the domain wall width in a spin-$S$ chain as the distance over which the $z$-axis spin component changes from $S/2$ to $-S/2$ near its center.
  * [37] L. Facheris, et al.; (2022): Spin-density wave dynamics in a 2D distorted triangular lattice antiferromagnet, STFC ISIS Neutron and Muon Source, https://doi.org/10.5286/ISIS.E.RB2210048 .
  * Amelin  _et al._ [2020] K. Amelin, J. Engelmayer, J. Viirok, U. Nagel, T. Rõõm, T. Lorenz, and Z. Wang, Experimental observation of quantum many-body excitations of ${E}_{8}$ symmetry in the Ising chain ferromagnet ${\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}$, Phys. Rev. B 102, 104431 (2020).

in 1,…,0 See pages \x, of Zeeman_Cs2CoBr4_18-05-2023_SUPP.pdf


* [2] C. Bennett and R. Sharpley.  Interpolation of operators, volume 129 of Pure and Applied Mathematics.  Academic Press, Inc., Boston, MA, 1988.
  * [3] S. Biagini, T. Pennanen, and A.-P. Perkkiö.  Duality and optimality conditions in stochastic optimization and mathematical finance.  Journal of Convex Analysis, 25, 2018.
  * [4] P. Carpentier, J.-P. Chancelier, G. Cohen, and M. De Lara.  Stochastic multi-stage optimization, volume 75 of Probability Theory and Stochastic Modelling.  Springer, Cham, 2015.  At the crossroads between discrete time stochastic control and stochastic programming.
  * [5] R. C. Dalang, A. Morton, and W. Willinger.  Equivalent martingale measures and no-arbitrage in stochastic securities market models.  Stochastics Stochastics Rep., 29(2):185–201, 1990.
  * [6] I. Ekeland and R. Temam.  Convex analysis and variational problems.  North-Holland Publishing Co., Amsterdam, 1976.  Translated from the French, Studies in Mathematics and its Applications, Vol. 1.
  * [7] A. Kozek.  Convex integral functionals on Orlicz spaces.  Comment. Math. Prace Mat., 21(1):109–135, 1980.
  * [8] T. Pennanen.  Arbitrage and deflators in illiquid markets.  Finance and Stochastics, 15(1):57–83, 2011.
  * [9] T. Pennanen.  Convex duality in stochastic optimization and mathematical finance.  Mathematics of Operations Research, 36(2):340–362, 2011.
  * [10] T. Pennanen.  Superhedging in illiquid markets.  Mathematical Finance, 21(3):519–540, 2011.
  * [11] T. Pennanen and A.-P. Perkkiö.  Stochastic programs without duality gaps.  Mathematical Programming, 136(1):91–110, 2012.
  * [12] T. Pennanen and A.-P. Perkkiö.  Shadow price of information in discrete time stochastic optimization.  Math. Program., 168(1-2, Ser. B):347–367, 2018.
  * [13] T. Pennanen and A.-P. Perkkiö.  Duality in convex stochastic optimization.  submitted, 2022.
  * [14] T. Pennanen and A.-P. Perkkiö.  Dynamic programming in convex stochastic optimization.  submitted, 2022.
  * [15] T. Pennanen and A.-P. Perkkiö.  Topological duals of locally convex function spaces.  Positivity, 26(2), 2022.
  * [16] S. R. Pliska.  Duality theory for some stochastic control models.  In Stochastic differential systems (Bad Honnef, 1982), volume 43 of Lect. Notes Control Inf. Sci., pages 329–337. Springer, Berlin, 1982.
  * [17] R. T. Rockafellar.  Level sets and continuity of conjugate convex functions.  Trans. Amer. Math. Soc., 123:46–63, 1966.
  * [18] R. T. Rockafellar.  Integrals which are convex functionals. II.  Pacific J. Math., 39:439–469, 1971.
  * [19] R. T. Rockafellar.  Conjugate duality and optimization.  Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974\.
  * [20] R. T. Rockafellar.  Integral functionals, normal integrands and measurable selections.  In Nonlinear operators and the calculus of variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975), pages 157–207. Lecture Notes in Math., Vol. 543. Springer, Berlin, 1976.
  * [21] R. T. Rockafellar and R. J.-B. Wets.  Nonanticipativity and $L^{1}$-martingales in stochastic optimization problems.  Math. Programming Stud., (6):170–187, 1976.  Stochastic systems: modeling, identification and optimization, II (Proc. Sympos., Univ Kentucky, Lexington, Ky., 1975).
  * [22] R. T. Rockafellar and R. J.-B. Wets.  Stochastic convex programming: relatively complete recourse and induced feasibility.  SIAM J. Control Optim., 14(3):574–589, 1976.
  * [23] R. T. Rockafellar and R. J.-B. Wets.  The optimal recourse problem in discrete time: $L^{1}$-multipliers for inequality constraints.  SIAM J. Control Optimization, 16(1):16–36, 1978.
  * [24] R. T. Rockafellar and R. J.-B. Wets.  Deterministic and stochastic optimization problems of Bolza type in discrete time.  Stochastics, 10(3-4):273–312, 1983.
  * [25] R. T. Rockafellar and R. J.-B. Wets.  Variational analysis, volume 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 1998.
  * [26] W. Schachermayer.  A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time.  Insurance Math. Econom., 11(4):249–257, 1992.
  * [27] A. Shapiro, D. Dentcheva, and A. Ruszczyński.  Lectures on Stochastic Programming: Modeling and Theory, volume 9 of MPS/SIAM Series on Optimization.  SIAM, Philadelphia, 2009.
  * [28] A. Truffert.  Conditional expectation of integrands and random sets.  Ann. Oper. Res., 30(1-4):117–156, 1991.  Stochastic programming, Part I (Ann Arbor, MI, 1989).
  * [29] A. C. Zaanen.  Riesz spaces. II, volume 30 of North-Holland Mathematical Library.  North-Holland Publishing Co., Amsterdam, 1983.
  * [30] C. Zălinescu.  Convex analysis in general vector spaces.  World Scientific Publishing Co., Inc., River Edge, NJ, 2002.