# An Investigation of Noise in Morphological Inflection

Adam Wiemerslage♢ Changbing Yang♣ Garrett Nicolai♣
Miikka Silfverberg♣ Katharina Kann♢
♢University of Colorado Boulder
♣University of British Columbia
{adam.wiemerslage<EMAIL_ADDRESS>
###### Abstract

With a growing focus on morphological inflection systems for languages where
high-quality data is scarce, training data noise is a serious but so far
largely ignored concern. We aim at closing this gap by investigating the types
of noise encountered within a pipeline for truly unsupervised morphological
paradigm completion and its impact on morphological inflection systems: First,
we propose an error taxonomy and annotation pipeline for inflection training
data. Then, we compare the effect of different types of noise on multiple
state-of-the-art inflection models. Finally, we propose a novel character-
level masked language modeling (CMLM) pretraining objective and explore its
impact on the models’ resistance to noise. Our experiments show that various
architectures are impacted differently by separate types of noise, but
encoder-decoders tend to be more robust to noise than models trained with a
copy bias. CMLM pretraining helps transformers, but has lower impact on LSTMs.

##  1 Introduction

Neural morphological inflection has shown impressive results for a huge
variety of languages Cotterell et al. (2016); Kodner et al. (2022).
Performance is impressive even for languages with very little supervised
inflection data, and often generalizes to unseen lemmas. However, the language
settings that arguably stand to benefit the most from these tools, those with
extremely sparse normalized texts, are less likely to have clean, gold-
standard data. Despite this, inflection training data noise is rarely
addressed nor evaluated in popular benchmarks. Noise, like incorrect
annotations, or mixed dialects or orthographies, can arise in inflection data
from web-scraping issues Gorman et al. (2019); McCarthy et al. (2020a), human
error or changes in writing standards in field data Moeller et al. (2020), or
system errors when bootstrapping silver-standard data in an unsupervised
fashion Kann et al. (2020); Erdmann et al. (2020); Wiemerslage et al. (2022).
Unsupervised systems are also prone to over-regularization, where the dataset
contains few or no irregular samples. Datasets derived from FSTs or textbook
examples can also display over-regularization Vylomova et al. (2020).

In this work, we investigate the impact of noise on inflection generation
systems. We build an automatic pipeline for annotating inflection noise and
explore the noise distribution that arises in an unsupervised system for
bootstrapping inflection data. We measure the impact of noise on several
state-of-the-art neural inflection generation systems that we benchmark on the
SIGMORPHON 2017 shared task development set Cotterell et al. (2017). Finally,
we explore a novel character-level masked language modeling (CMLM) pretraining
objective to mitigate the impact of noise during training. By this, we aim to
shed light on which architectures and training methods are how robust to
noise, which types of inflection noise should be targeted in filtering
approaches, and how conservatively unsupervised systems should sample
inflection pairs.

We find that noise related to slot alignment issues is more common, but also
less impactful than noise related to paradigm induction issues. Architectures
with an inductive bias towards copying from the lemma are more effective on
datasets that lack sample diversity, but more typical encoder-decoder models
are more robust to noise. Standard encoder-decoders display better performance
on noisy data when pretrained with CMLM, which is especially effective for the
Transformer. Our code and data are publicly available.
111https://github.com/Adamits/morphological-inflection-noise

##  2 Noisy Training Data

Our data comprises four languages: Icelandic, German, Swedish, and Russian. We
describe the source of training data below.

#### tUMPC

We focus on training data based on Wiemerslage et al. (2022), who propose a
system for truly unsupervised morphological paradigm completion (tUMPC).
Starting with the Bible corpus McCarthy et al. (2020b) for a given language,
tUMPC first clusters data into paradigms using the system from McCurdy et al.
(2021). Second, paradigms are clustered into parts-of-speech. Finally, tUMPC
aligns similarly inflected forms across paradigms that belong to the same POS.
The result is a dataset of paradigms for which each type is marked with its
inflectional slot. We take all possible pairs of words from the tUMPC
paradigms to form inflection training data. This dataset is likely to contain
noise due to errors in the learning process. In § 4, we discuss this noise in
detail.

#### UniMorph

tUMPC relies on frequency thresholds to find productive inflection
transformations, which makes it likely that morpho-phonological variations and
irregular inflections may not be well-attested in the data. To control for the
potential lack of sample diversity in our training data, we create a second
training dataset wherein all samples marked as correct are replaced by a pair
sampled from UniMorph Sylak-Glassman et al. (2015); Batsuren et al. (2022), a
database of morphological paradigms covering hundreds of languages. We sample
pairs from the same MSDs as the tUMPC correct samples. Following recent
results Liu and Hulden (2022); Goldman et al. (2022); Kodner et al. (2022), we
also ensure that the lemma overlap with the evaluation set is the same as in
the original tUMPC data. The resulting dataset has higher lemma diversity,
without introducing missing MSDs, and maintains lemma overlap with the
evaluation set. We combine this data with the noisy training data from tUMPC
to investigate the impact of noise when sample diversity is less pervasive.

| Deu | Isl | Rus | Swe
---|---|---|---|---
Original | 40956 | 32225 | 128326 | 59814
Annotated | 6444 | 4735 | 14171 | 6054
Table 1: Size of training set for each language before and after filtering out
pairs that cannot be annotated.

#### Slot Mapping

tUMPC slots are arbitrary identifiers with no grammatical meaning. In order to
compare tUMPC to UniMorph, and because we evaluate on a benchmark sampled from
UniMorph, we need to map each tUMPC slot to a unique UniMorph slot. UniMorph
slots are bundles of morphological tags that express categories like person or
tense. We refer to these bundles as morpho-syntactic descriptions (MSDs).
Since the tUMPC training pairs have low overlap with UniMorph, we require a
morphological analyzer to get ground truth morphological tags of tUMPC types.
Here, we use morphological analyzers from Apertium Forcada et al. (2011),
which we map to UniMorph MSDs. Then, we apply the mapped Apertium to all tUMPC
types, resulting in ground truth MSDs for every training sample in tUMPC. We
also want the slots predicted by tUMPC expressed as UniMorph MSDs during
training. For this, we compare the unsupervised slot and gold MSD of all tUMPC
types and find the alignment that maximizes their overlap following the
evaluation metric from Jin et al. (2020). Every training sample can then be
assigned the (noisy) MSD from this alignment. For more details see Section A.1
in the appendix.

Many inflection pairs created by tUMPC could not be reliably annotated by our
pipeline and are thus filtered out. Table 1 shows the amount of training data
before and after filtering. These are inflection pairs that are not in the
Apertium lexicon, but are also not determined to be noise. We additionally
filter pairs containing a form with an Apertium analysis that we could not
reliably map to UniMorph due to disagreements between the two resources. We
compare the distribution of morphological tags for our dataset according to
Apertium before and after applying this filtering step, and find no systematic
difference. We conclude that removing forms due to mapping errors does not
bias our data towards certain inflections. These filtering steps remove a
majority of data, but still leave reasonably sized training sets for each
language.

##  3 Related Work

#### Morphological Inflection

Morphological inflection is the task of generating a word form given a lemma
and target MSD. For example, given the verb laugh and the MSD expressing past
tense, the goal is to generate laughed. Encoder-decoder neural approaches have
largely dominated morphological inflection in recent years Faruqui et al.
(2016); Kann and Schütze (2016), where the the full target string is decoded
from a neural network. But neural models that bias the task towards
transducing input strings have been shown to be successful in low-resource
data settings Aharoni and Goldberg (2017); Makarov et al. (2017); Makarov and
Clematide (2018); Sharma et al. (2018). Shared tasks on morphological
inflection Cotterell et al. (2016) have spurred large interest in the task,
and also serve as an evaluation benchmark.

#### Learning from Noisy Data

Several approaches have been proposed for mitigating the impact of noise in
machine learning, for example: confidence weighting Rebbapragada and Brodley
(2007), loss correction Patrini et al. (2017), and noise-contrastive
estimation Gutmann and Hyvärinen (2010). For a recent survey on noise robust
neural networks, see Song et al. (2022). Most approaches consider
classification, framing noise as label corruption. However, there is also work
exploring noise in tasks like machine translation Khayrallah and Koehn (2018);
Michel and Neubig (2018), as well as morphological disambiguation Zalmout et
al. (2018). We focus on morphological inflection, a conditional generation
problem.

#### Morphological Inflection with Training Noise

Morphological inflection with noisy data has largely been unexplored. Moeller
et al. (2020) report gains in performance after manually cleaning training
data that were bootstrapped from interlinear glossed texts. Nicolai and
Silfverberg (2020) find that exposing an inflection model to its own mistakes
leads to better generalizability. Our work explores the impact of different
types of noise that occur in the training data in detail.

##  4 Noise Taxonomy

We develop an automated pipeline to annotate each inflection pair from a
taxonomy of noise, primarily relying on rule-based morphological analyzers for
each language from Apertium. Here we describe each type of noise in our
taxonomy, which we organize into three categories: lemma errors, paired
errors, and MSD errors. For a description of how each type of noise is
detected, see Section § A.2 in the appendix.

###  4.1 Lemma Errors

We first describe noise in which a training sample includes lexical items that
should not be in the inflection training data at all. Lemma noise arises due
to issues inherent to the corpus (e.g. misspellings, etc), or issues in which
lexical items that were sampled from the corpus (e.g. punctuation is not
inflection data). We describe two types of noise that fall into this category.

#### Lexicon Noise

Any word type that is not in the standard vocabulary of a given language is
considered lexicon noise. We expect this noise to come from archaisms,
borrowings, and biblical references. This means that lexicon noise could
follow the regular inflections of a language, and in that case may have a low
impact on downstream inflection systems. It could also have a positive impact
by increasing training data size, likely with higher lemma diversity. However,
lexicon noise could also entail archaic or borrowed inflections, which would
introduce non-existent transformations. In general, lexicon noise could also
occur due to language, dialect, or orthography mixing – though we expect this
to be somewhat rare in the Bible corpora that we build upon.

Figure 1: tUMPC noise distribution for all languages.

#### POS Noise

Not all parts of speech inflect, and this varies by language. However, tUMPC
sometimes induces spurious inflection pairs from words that do not inflect,
like conjunctions in most languages. Any word of a POS that does not inflect
in the given language is thus considered POS noise. This is detected with the
Apertium POS, but ultimately these samples will be assigned an MSD from
UniMorph with a POS that does inflect. See Table 5 for the POS that inflect
according to this study.

While inflection pairs with POS noise may sometimes have real transformations,
it is highly likely that these add spurious inflections to the training data.
For this reason, we expect POS noise to consistently cause mistakes in each
system.

###  4.2 Paired Errors

Next we describe noise types in which valid lexical items erroneously form an
inflection pair. These types of noise occur due to issues in the paradigm
induction algorithm that produced the inflection data.

#### POS Pair Noise

Two words from different POS forming an inflection pair constitutes POS pair
noise. This can occur when tUMPC puts two completely unrelated words in the
same paradigm, but this could also arise due to valid derivations. The latter
case could be considered a morphological transformation that our model should
learn. There is debate around whether or not there is a clearly defined
distinction between inflection and derivation Haspelmath (2023), but since our
goal is to evaluate on SIGMORPHON shared task data that does not include
derivation, we consider this noise in our pipeline. POS pair noise should have
a similar impact as POS noise, except that due to derivations we expect
several of the induced transformations to appear much more commonly in
languages where productive derivation is pervasive.

#### Paradigm Noise

Pairs of forms that do not belong to the same paradigm but share a POS
constitute paradigm noise. An example in English is warp $\rightarrow$ wraps,
where wraps comes from a different paradigm than warp. Because paradigm noise
should contain target words expressing a valid inflection, it is possible that
it will not have much negative impact on an inflection system’s decoder.
However, since paradigm noise also contains source forms that the target is
not actually inflected from, the full transformation from source to target has
the potential to be spurious. For instance, in the warp $\rightarrow$ wraps
example above, consider the apparent a, r metathesis. This could cause models
relying on a bias towards transduction to struggle more with paradigm noise
than models relying predominantly on a decoder.

###  4.3 MSD Errors

Finally we describe noise consisting of assigning the wrong MSD to an
inflection. We have only a single noise type here. It occurs due to issues in
the slot alignment algorithm.

#### Slot Noise

When the target word in an inflection pair has an incorrect MSD, we mark this
as slot noise. For example, consider an inflection cry $\rightarrow$ cried,
which is a valid inflection pair, but if, for example, cried is incorrectly
marked as the third person present, this would be slot noise. Our training
setup makes use of only the target slot – if only the source form has an
incorrect slot then a pair is not marked as slot noise. This error results in
a mismatch of MSDs and inflectional transformations. So, for some paradigms,
the output form for an inflection can be thought of as swapped with another
output form in the same paradigm. If these two forms are different, this is
likely to confuse the inflection system by presenting counter-evidence to the
correct inflection. Additionally, it is possible that there are few to no
correctly tagged instances of rare or irregular inflection classes, causing a
system to confidently learn that, e.g., the past tense inflection is the third
person present tense.

###  4.4 Analysis

Figure 1 presents the noise distribution in the training data for each
language according to our annotation pipeline. In German and Russian, there is
more correct data than noise, but in Icelandic and Swedish, there is more
noise than correct data. We can also consider noise by its source of error in
tUMPC: either resulting from an error in slot alignment, i.e. slot noise; an
error in the corpus, i.e. lexicon noise; or a paradigm induction error, i.e.
all other noise. Slot alignment issues are the most common source of noise,
though, for German and Swedish, there are nearly as many paradigm induction
errors. More specifically, POS pair noise is the most common noise type after
slot noise, and lexicon noise is the least common, most of which occurs in
Swedish.

##  5 Experiments

###  5.1 Models

We compare four neural inflection generation systems. We implement a
bidirectional LSTM with attention (Kann and Schütze, 2016, LSTM), a
Transformer (Wu et al., 2021, Trm), a pointer generator LSTM (Sharma et al.,
2018, PtrGen), and we use the Dynet implementation of (Makarov and Clematide,
2018, M&C): a transducer optimized with minimum risk training. For all other
models, our implementation is based on yoyodyne,222https://github.com/CUNY-
CL/yoyodyne which is built on pytorch Paszke et al. (2019). All models follow
the hyperparameters reported in the original papers, with minor increases in
epochs to ensure that they converge. Explicit hyperparameters are listed in
the appendix in Table 6.

This gives us two classes of models: general encoder-decoders (LSTM and Trm),
which may struggle for low-resource scenarios and under a lack of sample
diversity; and transducer-like models with a bias towards copying from the
lemma (PtrGen and M&C), which are known to perform better in low-resource
scenarios by relying on modeling character transduction. For all results that
we report, we train five of each model on the same five random seeds and
report the mean.

###  5.2 Evaluation

We evaluate on development sets from the SIGMORPHON 2017 shared task Cotterell
et al. (2017). During training, we reuse the target MSDs found through the
mapping described in § 2, which match the SIGMORPHON target MSDs. Though our
training setup considers reinflection, we evaluate on inflection from a lemma.
This small mismatch in task has a minor negative impact on accuracy Cotterell
et al. (2016).

Lang | M&C | LSTM | PtrGen | Trm
---|---|---|---|---
Deu | 28.38 | 23.14 | 31.34 | 21.42
Isl | 21.90 | 16.60 | 19.03 | 16.27
Rus | 38.92 | 31.20 | 33.07 | 32.67
Swe | 35.88 | 22.34 | 26.05 | 25.68
Avg | 31.27 | 23.32 | 27.37 | 24.01
Table 2: Accuracy for the tUMPC training data. Lang | M&C | LSTM | PtrGen | Trm
---|---|---|---|---
Deu | 40.12 | 45.98 | 42.06 | 34.75
Isl | 31.28 | 28.64 | 30.00 | 32.00
Rus | 61.80 | 65.58 | 61.96 | 67.23
Swe | 54.20 | 49.32 | 47.57 | 53.43
Avg | 46.85 | 47.38 | 45.40 | 46.85
Table 3: Accuracy for the UniMorph training data.

(a)  tUMPC

(b)  UniMorph

Figure 2:  Change in accuracy as each dataset is augmented with noise. One
tenth of the noisy data for a given dataset is added to the dataset at each
point, until all noise is added at point 10.

(a)  tUMPC

(b)  UniMorph

Figure 3: Impact of noise by type. Each bar shows the % increase in accuracy
when samples with the corresponding annotation are added to a training set
comprising only correct data. The red line represents the training data size
after adding those samples.

###  5.3 Experiment 1: Training on Noisy Data

We first benchmark each model on all four languages when trained on the full
noisy dataset.

#### tUMPC

In Table 2 we present the results when models are trained on all data that we
were able to classify in our annotation pipeline, cf. Table 1. M&C performs
best on average, with PtrGen performing second best. Notably, both of these
models are designed with an inductive bias towards copying from the lemma.
LSTM and Trm, which both rely on an encoder-decoder that generates from the
vocabulary at every time-step, perform similarly to one another. The largest
training dataset, Russian, is the most accurate for every model, and
Icelandic, the smallest dataset, is the least accurate.

#### UniMorph

Table 3 shows a large increase in accuracy for every language and model when
compared to tUMPC. This indicates that the tUMPC data lacks diversity.
Accuracy is similar for all models on average, but LSTM’s accuracy is highest,
and PtrGen’s is lowest. In a second experiment, we sample UniMorph pairs from
the tUMPC word-length distribution in order to control for the fact that many
UniMorph words tend to be uncharacteristically long. This lowers the type
frequency of the dataset compared the the original UniMorph sampling — which
we interpret as reducing the diversity — and results in a very small increase
in performance compared to training on the tUMPC dataset. For results on
UniMorph, we focus on the initial, more diverse dataset.

###  5.4 Experiment 2: Quantity of Noise

We investigate how introducing randomly sampled noise into a training dataset
affects model performance as the amount of noise increases. This characterizes
the behavior of models as each dataset becomes noisier, and it also shows us
at which quantities noise becomes most problematic. We first train models on
data comprising only samples that were annotated as correct in order to
benchmark performance in the absence of noise. We then partition all of the
noisy samples into ten equally sized splits. The x-axis in Figure 2 represents
the number of partitions that have been added to a given dataset, so, at 10,
we get the results in Tables 2 and 3, and, at 0, we have the results when all
noise is removed. Notice that the amount of noisy data in a given partition
depends on the language, and represents one tenth of the total noise — we
focus our analysis on trends more so than performance at any particular point.
Here we analyze the dotted lines, the solid lines represent the results of
Experiment 4 on CMLM.

#### tUMPC

Every architecture is negatively impacted by noise in German, with a huge
negative impact on M&C. PtrGen behaves sporadically, increasing performance
when the last few noise partitions are introduced, and LSTM suffers the least
from noise. In Swedish, Icelandic, and Russian, model performance tends to
increase as noise is introduced. This is consistent in Trm and LSTM and not
always true for M&C and PtrGen.

The negative impact of noise on M&C in German, and the somewhat sporadic
behavior of PtrGen indicate that the classic encoder-decoder may be more
robust to noise. However, the ranking of models by accuracy does not tend to
change over different amounts of noise, demonstrating that they underperform
on tUMPC. We see consistent increases in performance in Swedish for all
models. One explanation for this could be that the large amount of lexicon
noise in Swedish tends to contain real inflections that our models learn from.
We additionally see several other cases where LSTM and Trm increase in
accuracy as more noise is added. This may simply be because they suffer from a
lack of sample diversity and more data helps even though it is noisy.

#### UniMorph

In the UniMorph dataset, M&C performance steadily, and sometimes drastically,
decreases as noise is introduced for every language. In German, PtrGen behaves
sporadically again, and both Trm and LSTM decrease in performance as noise is
added, with LSTM decreasing very slowly. The same is true for Swedish and
Icelandic, where Trm and LSTM are less impacted by noise than the copy models.
All models besides M&C seem relatively robust to noise in Russian.

This suggests that LSTM may be the most robust to noise in our data on
average, and that the models with an inductive bias towards copying from the
lemma are less robust. In all languages but Russian, there is a distinct
downward trend as noise is added for every architecture. Russian is the
largest dataset and has more correct than noisy data. This suggests that, with
sufficient correct training data, noise is less of a problem. Additionally, a
large portion of Russian noise are slot and paradigm noise, which could
feasibly have a less negative impact on learning. This also shows that
filtering out noisy samples is most useful for M&C, and the negative impact of
noise is more severe for models trained on higher sample diversity.

###  5.5 Experiment 3: Type of Noise

We investigate the impact of each noise type in our annotation schema to see
if certain noise types are more important than others and if they affect
architectures differently. We produce $k$ training data sets, where each set
comprises all of the correct data and all of the samples that have been
annotated with one particular annotation so that we can measure that
annotation’s impact in isolation. Thus, the impact of each annotation is both
a function of the errors entailed by it and the frequency of samples with that
annotation. Because noise types can co-occur on a single training sample, we
consider the unique combination of noise types as a single annotation. Figure
3 presents the percent change in accuracy over training on only the correct
data for each dataset, averaged over all languages. To represent the effect of
data size, we add the red line to track the average training set size when a
given annotation is included.

#### tUMPC

M&C, the best performing model, is most negatively impacted by any single
noise type in both the tUMPC and the UniMorph dataset, which supports the
findings of the previous experiments. POS noise, especially in combination
with slot noise has the largest negative impact. One explanation for this is
that M&C is trained to learn a policy over edits on the input, framing
inflection as a true string transduction task. Since POS noise is likely to
add inflections that are not actually meaningful, this may result in an edit
policy that generates made-up words.

Every model besides M&C gets better when the slot annotation is included in
the tUMPC dataset. This may be due to the fact that there are a huge number of
slot annotations to learn from, meaning each model is trained on a larger
training set. Additionally, M&C may be less sensitive to slot errors as it has
a bias towards copying from the lemma – perhaps the best M&C model trained on
tUMPC ignores some parts of the MSD. However, PtrGen, which also has a bias
towards copying from the lemma, increases in accuracy when any noise,
especially lexicon noise, is added to the data.

#### UniMorph

M&C behaves similarly here, except the slot annotations have a negative
impact. PtrGen still learns from some noise types, but is generally less
affected by any single noise type. LSTM is almost completely unaffected by any
single noise type, which corroborates Experiment 2. Trm behaves similarly to
M&C, though the impact on accuracy is smaller. This indicates that with
sufficiently diverse training data, LSTM is more robust to noise than other
models, and that Trm learns an inductive bias that resembles M&C.

###  5.6 Experiment 4: CMLM Pretraining

We experiment with a simple character masked language modeling (CMLM)
pretraining objective as a method for improving noise-robustness of each
model. Auto-encoding without masking has been shown to impart a helpful
inductive bias towards copying from the lemma in low-resource morphological
inflection Kann and Schütze (2017). Masking can be thought of as additionally
adding a denoising objective to auto-encoding, which may contribute to more
robust learning from noisy training samples. We experiment with CMLM for every
model and analyze its effect on noisy training.

Figure 4:  % change in accuracy from pretraining.

#### Implementation

We follow the BERT masked language modeling objective Devlin et al. (2019)
with only minor adjustments. First, we mask characters rather than subwords.
Second, our sequences will be shorter on average than BERT, so we increase the
mask probability from 0.15 to 0.2. In practice, we follow RoBERTa Liu et al.
(2019) and generate a new mask for each sample dynamically at each epoch.
Every unique type in a given training set, without their MSDs, comprise the
pretraining dataset. We do not include additional types in order to test
whether CMLM itself is effective, rather than the addition of data. Each model
is otherwise trained in the exact same way as finetuning with the same
hyperparameters other than number of epochs. The only exception is that we
train PtrGen with a warm-up scheduler in order to avoid over-fitting to
copying the lemma. Exact hyperparameters are in the appendix in Table 7.

#### Results

Figure 4 shows the average percent change in accuracy for each model in both
datasets when adding the pretraining objective. We find that it has little
effect for PtrGen on average, and actually has a negative impact on M&C. We
attribute this to the fact that these models have an inductive copy bias,
which may be a large part of what CMLM adds to the models. M&C might
additionally be negatively impacted if the imitation learning objective
overfits to the edits during pretraining. More experimentation with
pretraining objectives are needed to understand this negative result, however.
Our focus is on the impact of pretraining on noise robustness.

LSTM is barely affected by pretraining on average. Trm, however, increases in
accuracy in both datasets when pretrained. It additionally increases in
accuracy when training on correct data only, suggesting that CMLM may be
generally beneficial to Trm, not just in noisy settings. We note, however,
that the increase in accuracy is greater when noise is included in both
datasets.

#### CMLM by Noise Quantity

We also reproduce Experiment 2 in order to look more closely at the impact of
pretraining. We focus on the solid lines in Figure 2, and compare them to the
dotted lines that represent the models that were not pretrained.

#### tUMPC

For M&C, we see the negative impact of pretraining is especially strong in
German and Swedish, and pretraining has little effect in other languages.
Pretraining benefits all other architectures in Swedish and Icelandic,
especially as noise is added. In German and Russian, pretraining is not
beneficial to PtrGen, but still helps Trm and, at small amounts of noise,
LSTM. Pretraining is also beneficial when there is no noise in the dataset,
but to a lesser extent and not in every language.

#### UniMorph

M&C shows performance increases in German under small amounts of noise only.
But in every other language pretraining still impacts M&C negatively. Similar
to the tUMPC data, pretraining benefits all other models in Icelandic and
Swedish. Even without noise, LSTM and Trm increase in performance in Icelandic
and to some extent in Swedish and Russian. In German, Trm increases massively
in accuracy from pretraining, including when there is no noise. All other
architectures also benefit from pretraining in German, but largely only at
small amounts of noise. Still, pretrained LSTM is the best overall model for
German in the UniMorph dataset. In Russian, Trm is the only pretrained
architecture that performs better as noise is introduced. Pretraining benefits
LSTM in Russian noise when there is no noise, however.

Trm is the only architecture for which CMLM pretraining helps in every
language and dataset. Overall, pretraining is also clearly beneficial to LSTM,
and sometimes to PtrGen. Pretraining is particularly effective for these three
architectures in Swedish and Icelandic, especially as noise is introduced to
the datasets. Combined with the fact that these two languages have more noisy
than correct data, this implies that pretraining is effective for noise-robust
learning. Still, in many cases pretraining leads to a gain in accuracy when
there is no noise in the dataset, implying that this learning strategy is
generally beneficial, especially for Trm.

##  6 Discussion

We find that low sample diversity has a strong impact on performance of all
models. The tUMPC training setup favors architectures with a copy bias, and
demonstrates that models can learn from noisy training samples when the
dataset is not diverse. We find that the low LSTM and Trm performance is
largely explained by low sample diversity. On the other hand, they seem to be
more robust to noisy data, particularly LSTM, which has stable performance as
noise is added to the UniMorph dataset. Pretraining with CMLM leads to further
gains in performance for LSTM, but the largest gains from CMLM are for Trm. On
average, Trm pretrained with CMLM is the best performing model under noise,
when we have sufficient sample diversity. When we look at specific noise in
the data, we find that slot alignment issues in tUMPC tend to have low impact
on every model. This could be for several reasons: slot noise should be
irrelevant under high amounts of syncretism, which is abundant in German,
Icelandic, and Swedish. Additionally, as models become more corrupted by
noise, they may rely on a bias towards copying from the source form – which is
unaffected by the slot.

We find that certain noise that results from errors in paradigm induction are
particularly impactful for M&C. This is especially true for POS noise, which
may motivate better POS induction in unsupervised morphology systems. Under
greater sample diversity, Trm is similarly impacted by paradigm induction
errors. This suggests that Trm begins to learn an inductive bias similar to
M&C on noisy data. The addition of almost any single noise annotation leads to
an increase in accuracy in PtrGen, and removing noise from the training data
often impacts PtrGen negatively. However, Experiment 2 suggests that PtrGen is
not as robust to increasing amounts of noise as other models. A lot of
reduction in accuracy may come from combinations of different noise types,
which is not captured by Experiment 3. Future work could investigate noise
distributions by type with particular focus on PtrGen behavior. We
additionally find that M&C is negatively impacted by CMLM pretraining. We
believe this may be due to overfitting its copy bias and learning spurious
transductions from the masking objective. However, future work should consider
alternate pretraining strategies for M&C.

Overall this implies that, although copy models are preferred for training on
low sample diversity, classic encoder-decoders are a good choice for noisy
datasets with more diversity. Our results indicate that POS and paradigm
induction components are more important for training data quality than slot
alignment in unsupervised systems and that bootstrapping inflection pairs
should prioritize lemma diversity, even if it may induce noise.

##  7 Conclusion

We have investigated the impact of noise on state-of-the-art neural
morphological inflection models. We find that the noise that arises in an
unsupervised system for bootstrapping inflection pairs is frequently related
to slot alignment errors, but that those also have less impact on the models.
We have also compared two inflection architectures with a copy bias to two
typical encoder-decoder models. We find that, though copy bias is helpful
under low sample diversity, the encoder-decoders are more robust to noise.
Finally, we find that a simple masked pretraining objective makes encoder-
decoders, and especially Transformers, more accurate under noise.

##  8 Limitations

The largest limitation of this study is that our annotation pipeline is
automated. This makes it possible that there are errors in the noise
annotations that we base our analysis on. Additionally, since we capture a
naturally occurring noise distribution, our findings are coupled to the
datasets we study here. Our findings may not generalize to distributions of
noise in other datasets.

## Acknowledgments

We thank the anonymous reviewers for their helpful feedback.

## References

  * Aharoni and Goldberg (2017) Roee Aharoni and Yoav Goldberg. 2017.  Morphological inflection generation with hard monotonic attention.  In _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 2004–2015, Vancouver, Canada. Association for Computational Linguistics.
  * Batsuren et al. (2022) Khuyagbaatar Batsuren, Omer Goldman, Salam Khalifa, Nizar Habash, Witold Kieraś, Gábor Bella, Brian Leonard, Garrett Nicolai, Kyle Gorman, Yustinus Ghanggo Ate, et al. 2022.  Unimorph 4.0: Universal morphology.  _arXiv preprint arXiv:2205.03608_.
  * Cotterell et al. (2017) Ryan Cotterell, Christo Kirov, John Sylak-Glassman, Géraldine Walther, Ekaterina Vylomova, Patrick Xia, Manaal Faruqui, Sandra Kübler, David Yarowsky, Jason Eisner, et al. 2017.  Conll-sigmorphon 2017 shared task: Universal morphological reinflection in 52 languages.  In _Proceedings of the CoNLL SIGMORPHON 2017 Shared Task: Universal Morphological Reinflection_ , pages 1–30.
  * Cotterell et al. (2016) Ryan Cotterell, Christo Kirov, John Sylak-Glassman, David Yarowsky, Jason Eisner, and Mans Hulden. 2016.  The sigmorphon 2016 shared task—morphological reinflection.  In _Proceedings of the 14th SIGMORPHON Workshop on Computational Research in Phonetics, Phonology, and Morphology_ , pages 10–22.
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of deep bidirectional transformers for language understanding.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pages 4171–4186, Minneapolis, Minnesota. Association for Computational Linguistics.
  * Erdmann et al. (2020) Alexander Erdmann, Micha Elsner, Shijie Wu, Ryan Cotterell, and Nizar Habash. 2020\.  The paradigm discovery problem.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 7778–7790, Online. Association for Computational Linguistics.
  * Faruqui et al. (2016) Manaal Faruqui, Yulia Tsvetkov, Graham Neubig, and Chris Dyer. 2016.  Morphological inflection generation using character sequence to sequence learning.  In _Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 634–643.
  * Forcada et al. (2011) Mikel L Forcada, Mireia Ginestí-Rosell, Jacob Nordfalk, Jim O’Regan, Sergio Ortiz-Rojas, Juan Antonio Pérez-Ortiz, Felipe Sánchez-Martínez, Gema Ramírez-Sánchez, and Francis M Tyers. 2011.  Apertium: a free/open-source platform for rule-based machine translation.  _Machine translation_ , 25(2):127–144.
  * Goldman et al. (2022) Omer Goldman, David Guriel, and Reut Tsarfaty. 2022.  (un)solving morphological inflection: Lemma overlap artificially inflates models’ performance.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 864–870, Dublin, Ireland. Association for Computational Linguistics.
  * Gorman et al. (2019) Kyle Gorman, Arya D. McCarthy, Ryan Cotterell, Ekaterina Vylomova, Miikka Silfverberg, and Magdalena Markowska. 2019.  Weird inflects but OK: Making sense of morphological generation errors.  In _Proceedings of the 23rd Conference on Computational Natural Language Learning (CoNLL)_ , pages 140–151, Hong Kong, China. Association for Computational Linguistics.
  * Gutmann and Hyvärinen (2010) Michael Gutmann and Aapo Hyvärinen. 2010.  Noise-contrastive estimation: A new estimation principle for unnormalized statistical models.  In _Proceedings of the thirteenth international conference on artificial intelligence and statistics_ , pages 297–304. JMLR Workshop and Conference Proceedings.
  * Haspelmath (2023) Martin Haspelmath. 2023.  Inflection and derivation as traditional comparative concepts.  _Linguistics (to appear)_.
  * Jin et al. (2020) Huiming Jin, Liwei Cai, Yihui Peng, Chen Xia, Arya D McCarthy, and Katharina Kann. 2020.  Unsupervised morphological paradigm completion.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 6696–6707.
  * Kann et al. (2020) Katharina Kann, Arya D. McCarthy, Garrett Nicolai, and Mans Hulden. 2020.  The SIGMORPHON 2020 shared task on unsupervised morphological paradigm completion.  In _Proceedings of the 17th SIGMORPHON Workshop on Computational Research in Phonetics, Phonology, and Morphology_ , pages 51–62, Online. Association for Computational Linguistics.
  * Kann and Schütze (2016) Katharina Kann and Hinrich Schütze. 2016.  Single-model encoder-decoder with explicit morphological representation for reinflection.  In _Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 555–560, Berlin, Germany. Association for Computational Linguistics.
  * Kann and Schütze (2017) Katharina Kann and Hinrich Schütze. 2017.  Unlabeled data for morphological generation with character-based sequence-to-sequence models.  In _Proceedings of the First Workshop on Subword and Character Level Models in NLP_ , pages 76–81, Copenhagen, Denmark. Association for Computational Linguistics.
  * Karp (1980) Richard M Karp. 1980.  An algorithm to solve the m$\times$ n assignment problem in expected time o (mn log n).  _Networks_ , 10(2):143–152.
  * Khayrallah and Koehn (2018) Huda Khayrallah and Philipp Koehn. 2018.  On the impact of various types of noise on neural machine translation.  In _Proceedings of the 2nd Workshop on Neural Machine Translation and Generation_ , pages 74–83, Melbourne, Australia. Association for Computational Linguistics.
  * Kodner et al. (2022) Jordan Kodner, Salam Khalifa, Khuyagbaatar Batsuren, Hossep Dolatian, Ryan Cotterell, Faruk Akkus, Antonios Anastasopoulos, Taras Andrushko, Aryaman Arora, Nona Atanalov, Gábor Bella, Elena Budianskaya, Yustinus Ghanggo Ate, Omer Goldman, David Guriel, Simon Guriel, Silvia Guriel-Agiashvili, Witold Kieraś, Andrew Krizhanovsky, Natalia Krizhanovsky, Igor Marchenko, Magdalena Markowska, Polina Mashkovtseva, Maria Nepomniashchaya, Daria Rodionova, Karina Scheifer, Alexandra Sorova, Anastasia Yemelina, Jeremiah Young, and Ekaterina Vylomova. 2022.  SIGMORPHON–UniMorph 2022 shared task 0: Generalization and typologically diverse morphological inflection.  In _Proceedings of the 19th SIGMORPHON Workshop on Computational Research in Phonetics, Phonology, and Morphology_ , pages 176–203, Seattle, Washington. Association for Computational Linguistics.
  * Liu and Hulden (2022) Ling Liu and Mans Hulden. 2022.  Can a transformer pass the wug test? tuning copying bias in neural morphological inflection models.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 739–749, Dublin, Ireland. Association for Computational Linguistics.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  Roberta: A robustly optimized bert pretraining approach.  _arXiv preprint arXiv:1907.11692_.
  * Makarov and Clematide (2018) Peter Makarov and Simon Clematide. 2018.  Imitation learning for neural morphological string transduction.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 2877–2882, Brussels, Belgium. Association for Computational Linguistics.
  * Makarov et al. (2017) Peter Makarov, Tatyana Ruzsics, and Simon Clematide. 2017.  Align and copy: Uzh at sigmorphon 2017 shared task for morphological reinflection.  In _Proceedings of the CoNLL SIGMORPHON 2017 Shared Task: Universal Morphological Reinflection_ , pages 49–57.
  * McCarthy et al. (2020a) Arya D. McCarthy, Christo Kirov, Matteo Grella, Amrit Nidhi, Patrick Xia, Kyle Gorman, Ekaterina Vylomova, Sabrina J. Mielke, Garrett Nicolai, Miikka Silfverberg, Timofey Arkhangelskiy, Nataly Krizhanovsky, Andrew Krizhanovsky, Elena Klyachko, Alexey Sorokin, John Mansfield, Valts Ernštreits, Yuval Pinter, Cassandra L. Jacobs, Ryan Cotterell, Mans Hulden, and David Yarowsky. 2020a.  UniMorph 3.0: Universal Morphology.  In _Proceedings of the 12th Language Resources and Evaluation Conference_ , pages 3922–3931, Marseille, France. European Language Resources Association.
  * McCarthy et al. (2020b) Arya D. McCarthy, Rachel Wicks, Dylan Lewis, Aaron Mueller, Winston Wu, Oliver Adams, Garrett Nicolai, Matt Post, and David Yarowsky. 2020b.  The Johns Hopkins University Bible corpus: 1600+ tongues for typological exploration.  In _Proceedings of the 12th Language Resources and Evaluation Conference_ , pages 2884–2892, Marseille, France. European Language Resources Association.
  * McCurdy et al. (2021) Kate McCurdy, Sharon Goldwater, and Adam Lopez. 2021.  Adaptor Grammars for unsupervised paradigm clustering.  In _Proceedings of the 18th SIGMORPHON Workshop on Computational Research in Phonetics, Phonology, and Morphology_ , pages 82–89, Online. Association for Computational Linguistics.
  * Michel and Neubig (2018) Paul Michel and Graham Neubig. 2018.  MTNT: A testbed for machine translation of noisy text.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 543–553, Brussels, Belgium. Association for Computational Linguistics.
  * Moeller et al. (2020) Sarah Moeller, Ling Liu, Changbing Yang, Katharina Kann, and Mans Hulden. 2020.  IGT2P: From interlinear glossed texts to paradigms.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 5251–5262, Online. Association for Computational Linguistics.
  * Nicolai and Silfverberg (2020) Garrett Nicolai and Miikka Silfverberg. 2020.  Noise isn’t always negative: Countering exposure bias in sequence-to-sequence inflection models.  In _Proceedings of the 28th International Conference on Computational Linguistics_ , pages 2837–2846, Barcelona, Spain (Online). International Committee on Computational Linguistics.
  * Paszke et al. (2019) Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. 2019.  Pytorch: An imperative style, high-performance deep learning library.  In _Advances in Neural Information Processing Systems 32_ , pages 8024–8035. Curran Associates, Inc.
  * Patrini et al. (2017) Giorgio Patrini, Alessandro Rozza, Aditya Krishna Menon, Richard Nock, and Lizhen Qu. 2017.  Making deep neural networks robust to label noise: A loss correction approach.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 1944–1952.
  * Rebbapragada and Brodley (2007) Umaa Rebbapragada and Carla E Brodley. 2007.  Class noise mitigation through instance weighting.  In _European conference on machine learning_ , pages 708–715. Springer.
  * Sharma et al. (2018) Abhishek Sharma, Ganesh Katrapati, and Dipti Misra Sharma. 2018.  Iit (bhu)–iiith at conll–sigmorphon 2018 shared task on universal morphological reinflection.  In _Proceedings of the CoNLL–SIGMORPHON 2018 Shared Task: Universal Morphological Reinflection_ , pages 105–111.
  * Song et al. (2022) Hwanjun Song, Minseok Kim, Dongmin Park, Yooju Shin, and Jae-Gil Lee. 2022.  Learning from noisy labels with deep neural networks: A survey.  _IEEE Transactions on Neural Networks and Learning Systems_.
  * Sylak-Glassman et al. (2015) John Sylak-Glassman, Christo Kirov, Matt Post, Roger Que, and David Yarowsky. 2015\.  A universal feature schema for rich morphological annotation and fine-grained cross-lingual part-of-speech tagging.  In _International Workshop on Systems and Frameworks for Computational Morphology_ , pages 72–93. Springer.
  * Vylomova et al. (2020) Ekaterina Vylomova, Jennifer White, Elizabeth Salesky, Sabrina J. Mielke, Shijie Wu, Edoardo Maria Ponti, Rowan Hall Maudslay, Ran Zmigrod, Josef Valvoda, Svetlana Toldova, Francis Tyers, Elena Klyachko, Ilya Yegorov, Natalia Krizhanovsky, Paula Czarnowska, Irene Nikkarinen, Andrew Krizhanovsky, Tiago Pimentel, Lucas Torroba Hennigen, Christo Kirov, Garrett Nicolai, Adina Williams, Antonios Anastasopoulos, Hilaria Cruz, Eleanor Chodroff, Ryan Cotterell, Miikka Silfverberg, and Mans Hulden. 2020.  SIGMORPHON 2020 shared task 0: Typologically diverse morphological inflection.  In _Proceedings of the 17th SIGMORPHON Workshop on Computational Research in Phonetics, Phonology, and Morphology_ , pages 1–39, Online. Association for Computational Linguistics.
  * Wiemerslage et al. (2022) Adam Wiemerslage, Miikka Silfverberg, Changbing Yang, Arya D McCarthy, Garrett Nicolai, Eliana Colunga, and Katharina Kann. 2022.  Morphological processing of low-resource languages: Where we are and what’s next.  In _Findings of the Association for Computational Linguistics: ACL 2022_ , pages 988–1007.
  * Wu et al. (2021) Shijie Wu, Ryan Cotterell, and Mans Hulden. 2021.  Applying the transformer to character-level transduction.  In _Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume_ , pages 1901–1907, Online. Association for Computational Linguistics.
  * Zalmout et al. (2018) Nasser Zalmout, Alexander Erdmann, and Nizar Habash. 2018.  Noise-robust morphological disambiguation for dialectal Arabic.  In _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers)_ , pages 953–964, New Orleans, Louisiana. Association for Computational Linguistics.

##  Appendix A Appendix

Language | C | LEX | PDGM | POS | POS Pair | SLOT | Lemma Overlap | MSD Overlap | Tag Overlap
---|---|---|---|---|---|---|---|---|---
deu | 61.37 | 1.52 | 0.68 | 8.07 | 8.60 | 19.75 | 3.84 | 24.32 | 83.33
isl | 37.95 | 3.91 | 2.11 | 15.59 | 8.17 | 32.27 | 3.24 | 60.61 | 95.65
rus | 50.77 | 0.98 | 6.01 | 4.21 | 4.25 | 33.79 | 3.31 | 68.18 | 100.0
swe | 27.90 | 11.36 | 5.48 | 8.39 | 15.66 | 31.20 | 1.94 | 69.33 | 93.1
Table 4: Statistics for each language’s training data. The % of samples with a given annotation (left). The % overlap with the evaluation set for lemmas, MSDs, or individual tags (right). Isl | verb | adjective | noun |  |
---|---|---|---|---|---
Deu | verb | adjective | noun | prn |
Rus | verb | adjective | noun | prn | numeral
Swe | verb | adjective | noun |  |
Table 5: Parts of speech that inflect in our annotation schema. | M&C | PtrGen | LSTM | Trm
---|---|---|---|---
Epochs | 60 | 60 | 60 | 800
Batch size | 1 | 32 | 20 | 400
Optimizer | AD | adam | AD | adam
LR | 1.0 | 0.001 | 1.0 | 0.001
Scheduler | - | - | - | inv. sqr root
Warmup | - | - | - | 4000
Total Params | 260k | 1.1M | 370k | 7.4M
Table 6: Hyperparameters for each architecture. All other hyperparameters follow their respective publications exactly. AD=ADADELTA. | M&C | PtrGen | LSTM | Trm
---|---|---|---|---
Pretrain epochs | 40 | 40 | 40 | 200
Pretrain scheduler | - | inv. sqr root | - | inv. sqr root
Pretrain warmup | - | 100 | - | 1000
Table 7: Hyperparameters for each architecture during pretraining.

###  A.1 Slot Mapping Details

We begin with each type processed by tUMPC, which has a slot: an arbitrary
identifier for its POS and inflection category, and a (not disambiguated)
morphological analysis from Apertium. For example, given the German verb
tragt, we have a tUMPC slot $2$. We additionally have the analysis from
Apertium with two possibilities: an imperative plural verb (<vblex><imp><pl>)
or a second person present indicative plural verb (<vblex><pri><p2><pl>).

Each possibility in the Apertium analysis is then mapped to a UniMorph MSD via
a mapping we create that translates each tag one at a time. For example, the
tag <vblex> becomes V, in order to match the UniMorph schema. After some
language specific post-processing heuristics, we get a set of UniMorph MSDs
from every Apertium analysis.

We can use these mapped analyses to align tUMPC slots with UniMorph MSDs.
Consider our example above, tragt, where we would end up with two possible
MSDs: V;IMP;2;PL and V;IND;PRS;2;PL. This forms a mapping from the slot $2$,
to both of these MSDS. Due to tUMPC errors, we may also erroneously get a
mapping from slot $2$ to N;ACC;PL via some other word. This gives us a mapping
from three differing UniMorph MSDs to one tUMPC slot. Over all such mappings,
this forms a bipartite graph between tUMPC slots and UniMorph MSDs, where the
same UniMorph MSD may correspond to multiple tUMPC slots. However, one tUMPC
slot represents exactly one MSD. We thus follow Jin et al. (2020) and attain a
one-to-one mapping from tUMPC slots to MSDs by finding the matching that
maximizes the overlap of word types with an aligned slot and MSD. Like them,
we use the algorithm from Karp (1980) to optimize this matching. Finally, the
slot for every training sample can be mapped to it’s MSD according to this
matching.

###  A.2 Noise Detection

Here we describe how our annotation pipeline detects each type of noise. We
rely on Apertium for the entire pipeline. Though most noise is found with
original Apertium analysis, the analyses that have been mapped to UniMorph
MSDs are used for detecting slot noise. .

#### Lexicon Noise

We use three resources to form the lexicon of each language: Apertium,
Wikipedia, and a python spellchecker333https://github.com/dvwright/phunspell
based on hunspell444http://hunspell.github.io/. Any word not in any of these
three is lexicon noise.

#### POS Noise

Here we produce lists of POS that inflect for each language. Any word whose
Apertium analysis does not contain any valid POS according to this list is
considered POS noise. The valid POS for each language are listed in Table 5.

#### POS Pair

Here we consider all POS from Apertium analysis of both words in a pair. If
they have no POS in common, then it is POS pair noise.

#### Paradigm

For a given inflection pair, if both words have no overlapping lemmas of the
same POS (but do have a shared POS) according to Apertium, we consider them to
be from separate paradigms, and thus paradigm noise.

#### Slot

Slot noise occurs when the slot assigned to a word by tUMPC is not in the set
of slots from Apertium analysis. We rely on the version of tUMPC and Apertium
slots that have been mapped to UniMorph for this part of the annotation. Slot
noise considers only the predicted and gold MSD for a slot, so it can can co-
occur on a single sample with any other noise in the taxonomy.


# Quantum phase transitions in the triangular coupled-top model

Liwei Duan1<EMAIL_ADDRESS>Yan-Zhi Wang2 Qing-Hu Chen3,4<EMAIL_ADDRESS>1
Department of Physics, Zhejiang Normal University, Jinhua 321004, China
2 School of Physics and Electronic Information, Anhui Normal University, Wuhu
241002, China
3 Department of Physics, Zhejiang University, Hangzhou 310027, China
4 Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing 210093, China

###### Abstract

We study the coupled-top model with three large spins located on a triangle.
Depending on the coupling strength, there exist three phases: disordered
paramagnetic phase, ferromagnetic phase, and frustrated antiferromagnetic
phase, which can be distinguished by the mean-field approach. The
paramagnetic-ferromagnetic phase transition is accompanied by the breaking of
the global $Z_{2}$ symmetry, whereas the paramagnetic-antiferromagnetic phase
transition is accompanied by the breaking of both the global $Z_{2}$ symmetry
and the translational symmetry. Exact analytical results of higher-order
quantum effects beyond the mean-field contribution, such as the excitation
energy, quantum fluctuation and von Neumann entropy, can be achieved by the
Holstein-Primakoff transformation and symplectic transformation in the
thermodynamic limit. Near the quantum critical point, the energy gap closes,
along with the divergence of the quantum fluctuation in certain quadrature and
von Neumann entropy. Particular attention should be paid to the
antiferromagnetic phase, where the geometric frustration takes effect. The
critical behaviors in the antiferromagnetic phase are quite different from
those in the paramagnetic and ferromagnetic phases, which highlights the
importance of the geometric frustration. The triangular coupled-top model
provides a simple and feasible platform to study the quantum phase transition
and the novel critical behaviors induced by the geometric frustration.

##  I Introduction

Quantum phase transitions are usually accompanied by a qualitative change in
the nature of the ground state when the control parameter of a system passes
through the quantum critical point. In contrast to its classical counterpart
driven by thermal fluctuations, the quantum phase transition occurs at zero
temperature due to the quantum fluctuations, where the non-commuting terms in
the system Hamiltonian play a dominant role [1]. The geometric frustration,
which prohibits simultaneously satisfying every pairwise interaction due to
local geometric constraints, also influences the quantum phase transition
significantly [2, 3]. Frustration is closely related with exotic materials
such as quantum spin liquid and spin ice [3, 4, 5], as well as complex
networks [6] and protein folding [7], etc. The unconventional quantum critical
phenomena and qualitatively new states of matter induced by the frustration
are still an active area of research [8, 9, 10, 11, 12, 13, 14].

As a paradigmatic system to study quantum phase transitions, the transverse-
field Ising model has drawn persistent attention for over half a century [15,
16, 17, 18, 1, 19, 20]. Besides, the transverse-field Ising model with
antiferromagnetic interaction on a two-dimensional triangular lattice offers
itself as an ideal platform to study the frustrated magnetic behaviors [21,
22]. A nature generalization, named coupled-top model, is to replace the
widely used spin-$1/2$s in the transverse-field Ising model with large spins
[23, 24, 25, 26, 27]. The coupled-top model shares some similarities with the
Dicke model, a fundamental model describing the interaction between the light
and matter [28, 29]. (1) Both of them exist the quantum phase transition: the
global $Z_{2}$ symmetry is spontaneously broken with increasing coupling
strength, and the von Neumann entropy is diverged near the quantum critical
point [23, 30]. (2) The level statistics changes from Poissonian distribution
to Wigner-Dyson distribution, which indicates a transition from quasi-
integrability to quantum chaotic [24, 28]. (3) The density of states shows a
singular behavior which corresponds to the excited-state quantum phase
transitions [31, 26]. Recently, the light-matter interacting systems in a
triangular structure have intrigued an enormous amount of attention due to the
exotic behaviors, such as chiral superradiant phase [10, 11], frustrated
superradiant phase and novel critical exponent [12]. We expect that the
triangular coupled-top model also exists rich quantum phases and
unconventional quantum critical phenomena.

In this paper, we study the quantum phase transition in the triangular
coupled-top model. Three large spins are located on a triangle. Each pairwise
interaction can be ferromagnetic or antiferromagnetic. First, we introduce the
mean-field approach which has been widely employed to study the large spin
systems especially in the thermodynamic limit [32, 33]. The mean-field
approach is equivalent to using a product state constructed by SU(2) coherent
states as the trial wavefunction. The ground state is achieved by minimizing
the corresponding energy expectation value. With the mean-field approach, we
confirm the existence of the quantum phase transition by the nonzero order
parameter, which indicates the breaking of global $Z_{2}$ symmetry. Two
critical coupling strengths separate the coupled-top model into three phases:
disordered paramagnetic phase, ferromagnetic phase, and frustrated
antiferromagnetic phase. Then, we introduce the Holstein-Primakoff
transformation and symplectic transformation in order to capture the quantum
fluctuation and quantum entanglement which are ignored by the mean-field
approach. The Holstein-Primakoff transformation maps the collective spin
operators into bosonic ones, which leads to an effective quadratic Hamiltonian
in the thermodynamic limit. The covariance matrix of the effective quadratic
Hamiltonian can be obtained by the symplectic transformation, with which we
can analyze the behaviors of the quantum fluctuation and von Neumann entropy.
We find that the geometric frustration comes into play in the
antiferromagnetic phase, which leads to novel critical behaviors.

The paper is structured as follows. In Sec. II, we introduce the triangular
coupled-top model and the symmetry it possesses. In Sec. III, the quantum
phase transition is confirmed by the mean-field approach. Exact analytical
results of higher-order quantum effects beyond the mean-field contribution, as
well as the corresponding critical behaviors, are given in Sec. IV. A brief
summary is given in Sec. V.

##  II Triangular Coupled-top model

Figure 1:  Schematic illustration of different phases in the coupled-top
model: Three large spins are located on a triangle. Their directions are
specified by the black arrows on the Bloch sphere.
$\lambda_{-}<\lambda<\lambda_{+}$ corresponds to the disordered paramagnetic
phase with all spins aligning in parallel to the external field in $z$ axis.
$\lambda<\lambda_{-}$ corresponds to the ferromagnetic phase where all spins
tend to align in parallel along the $x$ axis. $\lambda>\lambda_{+}$
corresponds to the frustrated antiferromagnetic phase. Two of the large spins
tend to align antiparallel along the $x$ axis, whereas the third one are
frozen in $z$ axis. $\lambda_{+}=1$ and $\lambda_{-}=-\frac{1}{2}$.

The coupled-top model can be regarded as the generalization of the transverse-
field Ising model with spin-$1/2$s replaced by large spins [23, 24, 25, 26,
27]. Previous studies mainly focused on two coupled large spins where the
influences of the geometric structure are negligible. In order to introduce
the geometric frustration, we study the triangular coupled-top model with
$N=3$ large spins, whose Hamiltonian can be written as

$\displaystyle\hat{H}_{\text{CT}}=\sum_{i=1}^{N}\left(-\epsilon\hat{J}_{i}^{z}+\frac{\chi}{J}\hat{J}_{i}^{x}\hat{J}_{i+1}^{x}\right).$
(1)

Here $\epsilon$ is the strength of the external field, and $\chi$ is the
coupling strength.
$\hat{\mathbf{J}}_{i}=\left(\hat{J}_{i}^{x},\hat{J}_{i}^{y},\hat{J}_{i}^{z}\right)$
are the collective spin operators which describe the $i$th large spin. The
eigenvalues of $\hat{\mathbf{J}}_{i}^{2}$ are $J(J+1)$. The periodic boundary
condition is performed which leads to $\hat{J}_{N+1}^{r}=\hat{J}_{1}^{r}$ with
$r=x,y,z$. For simplicity, we introduce the dimensionless form as follows,

$\displaystyle\hat{H}=\hat{H}_{\text{CT}}/\epsilon=\sum_{i=1}^{N}\left(-\hat{J}_{i}^{z}+\frac{\lambda}{J}\hat{J}_{i}^{x}\hat{J}_{i+1}^{x}\right),$
(2)

with the dimensionless coupling strength $\lambda=\chi/\epsilon$. Hamiltonian
(2) commutes with the parity operator $\hat{\Pi}$, with

$\displaystyle\hat{\Pi}=\exp\left[\mathrm{i}\pi\sum_{i=1}^{N}\left(\hat{J}_{i}^{z}+J\right)\right],$
(3)

which indicates a global $Z_{2}$ symmetry. The expectation values satisfy
$J_{i}^{x}=0$ if the $Z_{2}$ symmetry is preserved, whereas $J_{i}^{x}\neq 0$
if the $Z_{2}$ symmetry is broken. Therefore, $J_{i}^{x}$ plays a role of the
order parameter. What’s more, Hamiltonian (2) admits a translational symmetry,
which indicates that $J_{i}^{x}$ doesn’t depend on the site index $i$ in the
symmetric phase.

##  III Mean-field approach

Let us first discuss the coupled-top model in the mean-field picture, which
ignores the correlations between different large spins. We can construct a
mean-field ansatz for the ground state by employing a product state [34, 35],

$\displaystyle\left|{}\psi\right>=\left|{}\theta_{1},\phi_{1}\right>\otimes\left|{}\theta_{2},\phi_{2}\right>\otimes\left|{}\theta_{3},\phi_{3}\right>,$
(4)

with $\left|{}\theta_{i},\phi_{i}\right>$ ($0\leq\theta_{i}\leq\pi$,
$0\leq\phi_{i}<2\pi$) the SU(2) coherent state [36, 37] defined as

$\displaystyle\left|{}\theta_{i},\phi_{i}\right>=\exp\left[\frac{\theta_{i}}{2}\left(\mathrm{e}^{\mathrm{i}\phi_{i}}\hat{J}_{i}^{-}-\mathrm{e}^{-\mathrm{i}\phi_{i}}\hat{J}_{i}^{+}\right)\right]\left|{}J,J\right>.$
(5)

The corresponding expectation values of the collective spin operators can be
written as

$\displaystyle\left(J_{i}^{x},J_{i}^{y},J_{i}^{z}\right)$ $\displaystyle=$
$\displaystyle\left<{}\theta_{i},\phi_{i}\right|\left(\hat{J}_{i}^{x},\hat{J}_{i}^{y},\hat{J}_{i}^{z}\right)\left|{}\theta_{i},\phi_{i}\right>$
$\displaystyle=$ $\displaystyle
J\left(\sin\theta_{i}\cos\phi_{i},\sin\theta_{i}\sin\phi_{i},\cos\theta_{i}\right),$

which corresponds to a point on the Bloch sphere. Note that the external field
creates an energetic bias for one $z$-axis spin direction over the other one,
whereas the coupling strength leads to the alignment or anti-alignment of spin
projections along the $x$ axis. Therefore, the large spins tend to be located
on the $x$-$z$ plane which leads to $\phi_{i}=0$ or $\pi$. Without loss of
generality, we can introduce $\tilde{\theta}_{i}$,

$\displaystyle\tilde{\theta}_{i}=\left\\{\begin{array}[]{c}\theta_{i},\text{
for }\phi_{i}=0,\\\ -\theta_{i},\text{ for }\phi_{i}=\pi.\end{array}\right.$
(9)

The expectation values of the collective spin operators can be simplified as

$\displaystyle\left(J_{i}^{x},J_{i}^{y},J_{i}^{z}\right)$ $\displaystyle=$
$\displaystyle J\left(\sin\tilde{\theta}_{i},0,\cos\tilde{\theta}_{i}\right),$

with which we can achieve the energy expectation value given by

$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{1}{J}\left<{}\psi\right|\hat{H}\left|{}\psi\right>$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N}-\cos\tilde{\theta}_{i}+\lambda\sin\tilde{\theta}_{i}\sin\tilde{\theta}_{i+1}.$

The ground state is determined by minimizing the energy expectation value $E$
with respect to $\left\\{\tilde{\theta}_{i}\right\\}$, namely,

$\displaystyle\frac{\partial E}{\partial\tilde{\theta}_{i}}$ $\displaystyle=$
$\displaystyle\sin\tilde{\theta}_{i}+\lambda\cos\tilde{\theta}_{i}\left(\sin\tilde{\theta}_{i+1}+\sin\tilde{\theta}_{i-1}\right)=0.$
(11)

By carefully analyzing the energy expectation value, we find two critical
coupling strengths $\lambda_{-}=-\frac{1}{2}$ and $\lambda_{+}=1$, which
separate the coupled-top model into three phases: disordered paramagnetic
phase, ferromagnetic phase, and frustrated antiferromagnetic frustrated phase,
as shown in Fig. 1.

Figure 2:  The order parameters $J_{i}^{x}$ as a function of the dimensionless
coupling strength $\lambda$. The blue solid line, red dashed line and yellow
dotted line correspond to $J_{1}^{x}$, $J_{2}^{x}$ and $J_{3}^{x}$
respectively.

###  III.1 Disordered paramagnetic phase

When the coupling strength is weak, the external field plays a dominant role.
The large spins tend to align in parallel to the external field which leads to
a disordered paramagnetic phase. For $\lambda_{-}<\lambda<\lambda_{+}$, the
ground state is nondegenerate with $\tilde{\theta}_{i}=0$ and
$E_{\min}=-3\epsilon$, as given in Eq. (III). The coupled-top model preserves
the global $Z_{2}$ symmetry and the translational symmetry, with the order
parameter $J_{i}^{x}=0$ as shown in Fig. 2.

###  III.2 Ordered phase

When the coupling strength is strong, the large spins prefer the alignment or
anti-alignment of spin projections along the $x$ axis depending on the sign of
$\lambda$.

####  III.2.1 Ferromagnetic phase

For $\lambda<\lambda_{-}=-\frac{1}{2}$, three large spins prefer to align in
parallel along the $x$ axis to decrease the energy expectation value $E$ [Eq.
(III)]. The ground state corresponds to

$\displaystyle\tilde{\theta}_{i}=\pm\arccos\left(-\frac{1}{2\lambda}\right),$
(12)

and it is two-fold degenerate with the order parameter satisfying
$J_{i}^{x}=\pm\sqrt{1-\frac{1}{4\lambda^{2}}}$ respectively. We only show the
positive branch of $J_{i}^{x}$ in Fig. 2. The critical exponent associated
with the order parameter is $\beta=\frac{1}{2}$ due to
$J_{i}^{x}\propto\left(\lambda_{-}-\lambda\right)^{1/2}$ near the quantum
critical point ($\lambda\rightarrow\lambda_{-}$). The global $Z_{2}$ symmetry
is broken in the ferromagnetic phase, whereas the translational symmetry is
preserved.

####  III.2.2 Frustrated antiferromagnetic phase

For $\lambda>\lambda_{+}=1$, the energy expectation value is minimized when
each spin is aligned opposite to its neighbors. However, once two of the large
spins align antiparallel, the third one is frustrated and cannot align
antiparallel with both of them simultaneously due to the triangular
arrangement. In the frustrated antiferromagnetic phase, Both the global
$Z_{2}$ symmetry and the translational symmetry are broken, and the ground
state is six-fold degenerate with

$\displaystyle\tilde{\theta}_{i}=0,\quad\tilde{\theta}_{i+1}=-\tilde{\theta}_{i-1}=\pm\arccos\left(\frac{1}{\lambda}\right).$
(13)

The breaking of $Z_{2}$ symmetry leads to nonzero $J_{i}^{x}$, while the
breaking of translational symmetry leads to dependence of $J_{i}^{x}$ on the
site index $i$. For clarity, we only present one of the six degenerate ground-
state solutions, which corresponds to $J_{1}^{x}=0$ and
$J_{2}^{x}=-J_{3}^{x}=\sqrt{1-\frac{1}{\lambda^{2}}}$, as depicted in Fig. 2.
The order parameter satisfies
$J_{2}^{x}=-J_{3}^{x}\propto\left(\lambda-\lambda_{+}\right)^{1/2}$ near the
quantum critical point ($\lambda\rightarrow\lambda_{+}$), which also yields
the critical exponent associated with the order parameter $\beta=\frac{1}{2}$.

##  IV Beyond mean-field approach

In the thermodynamic limit ($J\rightarrow+\infty$), the mean-field approach is
able to describe the ground-state energy and order parameters very well [32,
33], from which we can distinguish different phases. However, one needs to go
beyond the mean-field approach to study the correlations between different
large spins, as well as the quantum fluctuations. To separate the mean-field
contribution and the higher-order terms, we first perform a unitary
transformation [34, 12, 38], which leads to

$\displaystyle\hat{\bar{H}}$ $\displaystyle=$
$\displaystyle\hat{U}^{\dagger}\hat{H}\hat{U}=\sum_{i}^{N}-\left(\cos\tilde{\theta}_{i}\hat{J}_{i}^{z}+\sin\tilde{\theta}_{i}\hat{J}_{i}^{x}\right)+\frac{\lambda}{J}\left(\cos\tilde{\theta}_{i}\hat{J}_{i}^{x}-\sin\tilde{\theta}_{i}\hat{J}_{i}^{z}\right)\left(\cos\tilde{\theta}_{i+1}\hat{J}_{i+1}^{x}-\sin\tilde{\theta}_{i+1}\hat{J}_{i+1}^{z}\right),$
(14)

with

$\displaystyle\hat{U}=\prod_{i=1}^{N}\exp\left(-\mathrm{i}\tilde{\theta}_{i}\hat{J}_{i}^{y}\right).$
(15)

Note that $\hat{U}$ indicates a rotation by angle $\tilde{\theta}_{i}$ along
$y$ axis due to

$\displaystyle\exp\left(\mathrm{i}\tilde{\theta}\hat{J}_{y}\right)\hat{J}_{x}\exp\left(-\mathrm{i}\tilde{\theta}\hat{J}_{y}\right)$
$\displaystyle=$
$\displaystyle\cos\tilde{\theta}\hat{J}_{x}+\sin\tilde{\theta}\hat{J}_{z},$
(16)
$\displaystyle\exp\left(\mathrm{i}\tilde{\theta}\hat{J}_{y}\right)\hat{J}_{z}\exp\left(-\mathrm{i}\tilde{\theta}\hat{J}_{y}\right)$
$\displaystyle=$
$\displaystyle\cos\tilde{\theta}\hat{J}_{z}-\sin\tilde{\theta}\hat{J}_{x}.$
(17)

The unitary transformation $\hat{U}$ brings the rotated $z$ axis along the
direction of the large spin obtained from the mean-field approach [34].

Then we can apply the Holstein-Primakoff transformation [39], which maps the
collective spin operators to bosonic creation and annihilation operators as

$\displaystyle\hat{J}_{i}^{z}$ $\displaystyle=$ $\displaystyle
J-\hat{a}_{i}^{\dagger}\hat{a}_{i},$ (18a) $\displaystyle\hat{J}_{i}^{+}$
$\displaystyle=$
$\displaystyle\sqrt{2J-\hat{a}_{i}^{\dagger}\hat{a}_{i}}\hat{a}_{i},$ (18b)
$\displaystyle\hat{J}_{i}^{-}$ $\displaystyle=$
$\displaystyle\hat{a}_{i}^{\dagger}\sqrt{2J-\hat{a}_{i}^{\dagger}\hat{a}_{i}},$
(18c)

with $\left[\hat{a}_{i},\hat{a}_{j}^{\dagger}\right]=\delta_{i,j}$. The
thermodynamic limit corresponds to $J\rightarrow+\infty$. Due to
$J\gg\left<{}\hat{a}_{i}^{\dagger}\hat{a}_{i}\right>$, the Holstein-Primakoff
transformation can be approximately simplified as

$\displaystyle\hat{J}_{i}^{z}$ $\displaystyle\approx$ $\displaystyle
J-\hat{a}_{i}^{\dagger}\hat{a}_{i},$ (19a) $\displaystyle\hat{J}_{i}^{+}$
$\displaystyle\approx$ $\displaystyle\sqrt{2J}\hat{a}_{i},$ (19b)
$\displaystyle\hat{J}_{i}^{-}$ $\displaystyle\approx$
$\displaystyle\sqrt{2J}\hat{a}_{i}^{\dagger}.$ (19c)

Substituting Eq. (19) into Eq. (14), we obtain a low-energy effective
Hamiltonian as follows,

$\displaystyle\hat{\bar{H}}$ $\displaystyle\approx$ $\displaystyle
JE+J^{1/2}\hat{\bar{H}}_{1}+J^{0}\hat{\bar{H}}_{2},$ (20)
$\displaystyle\hat{\bar{H}}_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\left(\sin\tilde{\theta}_{i}+\lambda\cos\tilde{\theta}_{i}\left(\sin\tilde{\theta}_{i+1}+\sin\tilde{\theta}_{i-1}\right)\right)$
(21) $\displaystyle\left(\hat{a}_{i}^{\dagger}+\hat{a}_{i}\right),$
$\displaystyle\hat{\bar{H}}_{2}$ $\displaystyle=$
$\displaystyle\sum_{i}^{N}\bar{\epsilon}_{i}\hat{a}_{i}^{\dagger}\hat{a}_{i}+\frac{\bar{\chi}_{i,i+1}}{2}\left(\hat{a}_{i}^{\dagger}+\hat{a}_{i}\right)\left(\hat{a}_{i+1}^{\dagger}+\hat{a}_{i+1}\right)$
(22)

with

$\displaystyle\bar{\epsilon}_{i}$ $\displaystyle=$
$\displaystyle\cos\tilde{\theta}_{i}-\lambda\sin\tilde{\theta}_{i}\left(\sin\tilde{\theta}_{i-1}+\sin\tilde{\theta}_{i+1}\right),$
(23a) $\displaystyle\bar{\chi}_{i,j}$ $\displaystyle=$
$\displaystyle\lambda\cos\tilde{\theta}_{i}\cos\tilde{\theta}_{j}=\chi_{j,i},$
(23b)

where we have ignored terms of order $J^{-l}$ with $l>0$. The first term in
Eq. (20) corresponds to the energy expectation value (III) given by the mean-
field approach. Minimizing the energy expectation value gives rise to the
ground-state energy, which also eliminates $\hat{\bar{H}}_{1}$, as indicated
by Eqs. (11) and (21). Finally, we only need to deal with the quadratic
Hamiltonian $\hat{\bar{H}}_{2}$ which can be solved exactly by the symplectic
transformation [40].

Before performing the symplectic transformation, we first introduce the vector
of canonical operators
$\hat{\mathbf{r}}=\left(\hat{x}_{1},\hat{x}_{2},\hat{x}_{3},\hat{p}_{1},\hat{p}_{2},\hat{p}_{3}\right)^{T}$
with $\hat{x}_{i}=\left(\hat{a}_{i}^{\dagger}+\hat{a}_{i}\right)/\sqrt{2}$ and
$\hat{p}_{i}=\mathrm{i}\left(\hat{a}_{i}^{\dagger}-\hat{a}_{i}\right)/\sqrt{2}$.
$\hat{\mathbf{r}}$ satisfies the canonical commutation relation
$\left[\hat{\mathbf{r}},\hat{\mathbf{r}}^{T}\right]=i\Omega$, where $\Omega$
is given by

$\displaystyle\Omega=\left(\begin{array}[]{rr}O_{3}&I_{3}\\\
-I_{3}&O_{3}\end{array}\right),$ (26)

with $O_{3}$ and $I_{3}$ the $3\times 3$ null and identity matrices
respectively. In terms of the vector of canonical operators
$\hat{\mathbf{r}}$, the quadratic Hamiltonian $\hat{\bar{H}}_{2}$ can be
rewritten as

$\displaystyle\hat{\bar{H}}_{2}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\frac{\bar{\epsilon}_{i}}{2}\left(\hat{x}_{i}^{2}+\hat{p}_{i}^{2}-1\right)+\bar{\chi}_{i,i+1}\hat{x}_{i}\hat{x}_{i+1}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\hat{\mathbf{r}}^{T}H\hat{\mathbf{r}}-\sum_{i=1}^{3}\frac{\bar{\epsilon}_{i}}{2},$

with the Hamiltonian matrix

$\displaystyle H=H_{x}\oplus H_{p}$ (28)

and

$H_{x}=\left(\begin{array}[]{ccc}\bar{\epsilon}_{1}&\bar{\chi}_{1,2}&\bar{\chi}_{1,3}\\\
\bar{\chi}_{2,1}&\bar{\epsilon}_{2}&\bar{\chi}_{2,3}\\\
\bar{\chi}_{3,1}&\bar{\chi}_{3,2}&\bar{\epsilon}_{3}\end{array}\right),\quad
H_{p}=\left(\begin{array}[]{ccc}\bar{\epsilon}_{1}&0&0\\\
0&\bar{\epsilon}_{2}&0\\\ 0&0&\bar{\epsilon}_{3}\end{array}\right).$ (29)

According to Williamson’s theorem [40], for the positive defined real matrix
$H$, there exists a symplectic transformation $S$ ($S^{T}\Omega S=\Omega$)
such that

$S^{T}HS=D,\text{ with
}D=\text{diag}\left(\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{1},\Delta_{2},\Delta_{3}\right).$
(30)

One can introduce a new vector of canonical operators
$\hat{\mathbf{r}}^{\prime}=S^{-1}\hat{\mathbf{r}}$ which split the quadratic
Hamiltonian into decoupled degrees of freedom as

$\displaystyle\hat{\bar{H}}_{2}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\frac{\Delta_{i}}{2}\left(\hat{x}_{i}^{\prime
2}+\hat{p}_{i}^{\prime 2}\right)-\sum_{i=1}^{3}\frac{\bar{\epsilon}_{i}}{2}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\hat{\mathbf{r}}^{\prime
T}D\hat{\mathbf{r}}^{\prime}-\sum_{i=1}^{3}\frac{\bar{\epsilon}_{i}}{2}.$

$\Delta_{i}\geq 0$ corresponds to the excitation energy, as shown in Fig. 3.
Particular attention should be paid to the lowest excitation energies, namely,
$\Delta_{\text{min}}=\min\left(\Delta_{1},\Delta_{2},\Delta_{3}\right)$, as it
corresponds to the energy gap between the ground state and the first excited
state. Clearly, the energy gap closes ($\Delta_{\text{min}}\rightarrow 0$) for
$\lambda\rightarrow\lambda_{\pm}$, which is a characteristic signature of the
quantum phase transition.

Figure 3:  The excitation energy $\Delta_{i}$ as a function of the
dimensionless coupling strength $\lambda$. The blue solid line, red dashed
line and yellow dotted line correspond to $\Delta_{1}$, $\Delta_{2}$ and
$\Delta_{3}$ respectively.

It is well-known that the ground state of the quadratic Hamiltonian is a
Gaussian state, which plays a significant role in continuous variable quantum
information [40]. Instead of dealing with infinite dimension of the associated
Hilbert space, one only needs to deal with the $6\times 6$ covariance matrix
$\sigma$ which is fulfilled to give the complete description of an arbitrary
Gaussian state (up to local unitary operations) [40, 41]. The covariance
matrix $\mathbf{\sigma}$ is defined as

$\displaystyle\mathbf{\sigma}=\frac{1}{2}\left<{}\left\\{\left(\hat{\mathbf{r}}-\left<{}\hat{\mathbf{r}}\right>\right),\left(\hat{\mathbf{r}}-\left<{}\hat{\mathbf{r}}\right>\right)^{T}\right\\}\right>,$
(32)

with $\left<{}\hat{A}\right>$ corresponding to the expectation value of
operator $\hat{A}$. Given the symplectic transformation $S$, it’s easy to
confirm that the covariance matrix can be written as
$\mathbf{\sigma}=SS^{T}/2$ [41]. The quantum fluctuations in $\hat{x}_{i}$ and
$\hat{p}_{i}$ are characterized by the standard deviations which are given by
the diagonal elements of the covariance matrix $\sigma$, namely,

$\displaystyle\left(\Delta x_{i}\right)^{2}$ $\displaystyle=$
$\displaystyle\left<{}\hat{x}_{i}^{2}\right>-\left<{}\hat{x}_{i}\right>^{2}=\sigma_{i,i},$
(33a) $\displaystyle\left(\Delta p_{i}\right)^{2}$ $\displaystyle=$
$\displaystyle\left<{}\hat{p}_{i}^{2}\right>-\left<{}\hat{p}_{i}\right>^{2}=\sigma_{3+i,3+i}.$
(33b)

The quantum fluctuations in $x_{i}$ and $p_{i}$ quadrature are shown in Fig. 4
(a) and (b) respectively. Near the quantum critical point $\lambda_{\pm}$, the
quantum fluctuation in $x_{i}$ quadrature tends to exponential diverge,
whereas that in $p_{i}$ quadrature tends to a finite value. Furthermore,
$\left(\Delta p_{i}\right)^{2}<\frac{1}{2}$ indicates a strong squeezing
effect especially near the quantum critical point [42].

Figure 4:  (a) The quantum fluctuation in $x_{i}$ quadrature $\left(\Delta
x_{i}\right)^{2}$, (b) the quantum fluctuation in $p_{i}$ quadrature
$\left(\Delta p_{i}\right)^{2}$ and (c) the von Neumann entropy
$\mathcal{S}_{i}$ as a function of the dimensionless coupling strength
$\lambda$ for the $i$th large spins. The blue solid line, red dashed line and
yellow dotted line correspond to site index $i=1,2,3$ respectively.

The von Neumann entropy, which characterizes the entanglement between
different subsystems, is directly related to the Heisenberg’s uncertainty
relation for the quadratic Hamiltonian of interacting bosonic systems [43,
41]. In terms of $\Delta x_{i}$ and $\Delta p_{i}$, the von Neumann entropy
$\mathcal{S}_{i}$ can be written as

$\displaystyle\mathcal{S}_{i}$ $\displaystyle=$ $\displaystyle\left(\Delta
x_{i}\Delta p_{i}+\frac{1}{2}\right)\log\left(\Delta x_{i}\Delta
p_{i}+\frac{1}{2}\right)$ (34) $\displaystyle-\left(\Delta x_{i}\Delta
p_{i}-\frac{1}{2}\right)\log\left(\Delta x_{i}\Delta
p_{i}-\frac{1}{2}\right),$

which describes the entanglement between the $i$th large spin and the other
two large spins. A growing interest has recently been devoted to the study of
quantum phase transitions from the entanglement point of view. As illustrated
in Fig. 4 (c), the von Neumann entropy diverges near the quantum critical
point $\lambda_{\pm}$, which indicates the strong correlations between
different large spins ignored by the mean-field approach.

Detailed analytical exact results for the excitation energy, quantum
fluctuation and von Neumann entropy, especially near the quantum critical
point in different phases, are given in the follows.

Figure 5:  Critical behaviors near the quantum phase transition point. (a) and
(d) show the lowest excitation energy $\Delta_{\text{min}}$ as a function of
the coupling strength $\lambda$ near $\lambda_{-}$ and $\lambda_{+}$
respectively. (b) and (e) show the quantum fluctuation in the $x_{1}$
quadrature $\left(\Delta x_{1}\right)^{2}$. (c) and (f) show the von Neumann
entropy $\mathcal{S}_{1}$. The circle, square and triangle correspond to the
analytical exact results obtained from the symplectic transformation for the
paramagnetic phase (PP), ferromagnetic phase (FP) and antiferromagnetic phase
(AFP) respectively. The solid lines refer to the linear fitted results whose
slopes are shown near each line.

###  IV.1 Disordered paramagnetic phase

In the disordered paramagnetic phase ($\lambda_{-}<\lambda<\lambda_{+}$),
$\tilde{\theta}_{i}=0$ leads to $\bar{\epsilon}_{i}=1$ and
$\bar{\chi}_{i,j}=\lambda$ [Eq. (23)]. The quadratic Hamiltonian corresponds
to a translationally invariant harmonic chain with nearest neighbor
interactions. According to the Bloch’s theorem, the wavenumber in the first
Brillouin zone corresponds to $k=0$ and $\pm 2\pi/3$. The symplectic
transformation can be obtained analytically [40], with which we achieve the
excitation energies $\Delta_{1}=\sqrt{1+2\lambda}$ for $k=0$ and
$\Delta_{2}=\Delta_{3}=\sqrt{1-\lambda}$ for $k=\pm 2\pi/3$. The lowest
excitation energy corresponds to $\Delta_{\text{min}}=\Delta_{1}$ for
$\lambda_{-}<\lambda<0$ and $\Delta_{\text{min}}=\Delta_{2}=\Delta_{3}$ for
$0<\lambda<\lambda_{+}$. Therefore, the critical behavior associated with the
excitation energy is
$\Delta_{\text{min}}\propto\left(\lambda-\lambda_{-}\right)^{1/2}$ for
$\lambda\rightarrow\lambda_{-}$, whereas it is
$\Delta_{\text{min}}\propto\left(\lambda_{+}-\lambda\right)^{1/2}$ for
$\lambda\rightarrow\lambda_{+}$, as shown in Fig. 5 (a) and (d).

The quantum fluctuations obtained from the symplectic transformation can be
written as

$\displaystyle\left(\Delta x_{i}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[2\left(1-\lambda\right)^{-1/2}+\left(1+2\lambda\right)^{-1/2}\right],$
(35a) $\displaystyle\left(\Delta p_{i}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[2\left(1-\lambda\right)^{1/2}+\left(1+2\lambda\right)^{1/2}\right].$
(35b)

Near the quantum critical point which corresponds to the paramagnetic-
ferromagnetic phase transition ($\lambda\rightarrow\lambda_{-}$), the quantum
fluctuation in $p_{i}$ quadrature tends to a finite value with $\left(\Delta
p_{i}\right)^{2}\rightarrow\sqrt{6}/6$, whereas the exponentially diverged
quantum fluctuation in $x_{i}$ quadrature is given by

$\displaystyle\left(\Delta
x_{i}\right)^{2}\propto\left(\lambda-\lambda_{-}\right)^{-1/2},$ (36)

as shown in Fig. 5 (b). Near the quantum critical point which corresponds to
the paramagnetic-antiferromagnetic phase transition
($\lambda\rightarrow\lambda_{+}$), the quantum fluctuation in $p_{i}$
quadrature tends to a finite value with $\left(\Delta
p_{i}\right)^{2}\rightarrow\sqrt{3}/6$, whereas the exponentially diverged
quantum fluctuation in $x_{i}$ quadrature is given by

$\displaystyle\left(\Delta
x_{i}\right)^{2}\propto\left(\lambda_{+}-\lambda\right)^{-1/2},$ (37)

as shown in Fig. 5 (e).

The quantum fluctuation in $x_{i}$ quadrature given by Eqs. (36) and (37) can
be unified as

$\displaystyle\left(\Delta
x_{i}\right)^{2}\propto\left|\lambda-\lambda_{\pm}\right|^{-1/2}.$ (38)

Since $\Delta x_{i}\Delta p_{i}\gg 1$ near the quantum critical point, we can
simplify the von Neumann entropy in Eq. (34) as

$\displaystyle\mathcal{S}_{i}\approx\log\Delta x_{i}\Delta
p_{i}\approx-\frac{1}{4}\log\left|\lambda-\lambda_{\pm}\right|+C_{i},$ (39)

with $C_{i}$ a constant independent on the coupling strength, which explains
the divergence in $\mathcal{S}_{i}$ as shown in Fig. 5 (c) and (f).

###  IV.2 Ordered phase

####  IV.2.1 Ferromagnetic phase

In the ferromagnetic phase ($\lambda<\lambda_{-}$), the global $Z_{2}$
symmetry is broken. From Eqs. (12) and (23), we obtain

$\displaystyle\bar{\epsilon}_{i}$ $\displaystyle=$
$\displaystyle\bar{\epsilon}=-2\lambda,$ (40a) $\displaystyle\bar{\chi}_{i,j}$
$\displaystyle=$ $\displaystyle\bar{\chi}=\frac{1}{4\lambda}.$ (40b)

Therefore, $\bar{\epsilon}_{i}$ and $\bar{\chi}_{i,j}$ don’t depend on the
site index, similar to those in the paramagnetic phase. The excitation
energies correspond to $\Delta_{1}=\sqrt{4\lambda^{2}-1}$ and
$\Delta_{2}=\Delta_{3}=\sqrt{4\lambda^{2}+\frac{1}{2}}$. Near the quantum
critical point ($\lambda\rightarrow\lambda_{-}$), the lowest excitation energy
satisfies
$\Delta_{\text{min}}=\Delta_{1}\propto\left(\lambda_{-}-\lambda\right)^{1/2}$,
as shown in Fig. 5 (a). The quantum fluctuations are given by

$\displaystyle\left(\Delta x_{i}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[2\left(1-\frac{\bar{\chi}}{\bar{\epsilon}}\right)^{-1/2}+\left(1+2\frac{\bar{\chi}}{\bar{\epsilon}}\right)^{-1/2}\right],$
(41a) $\displaystyle\left(\Delta p_{i}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left[2\left(1-\frac{\bar{\chi}}{\bar{\epsilon}}\right)^{1/2}+\left(1+2\frac{\bar{\chi}}{\bar{\epsilon}}\right)^{1/2}\right].$
(41b)

Similar to those in the paramagnetic phase, near the quantum critical point
($\lambda\rightarrow\lambda_{-}$), the quantum fluctuation in $p_{i}$
quadrature tends to a finite value with $\left(\Delta
p_{i}\right)^{2}\rightarrow\sqrt{6}/6$, whereas the exponentially diverged
quantum fluctuation in $x_{i}$ quadrature is given by

$\displaystyle\left(\Delta x_{i}\right)^{2}$ $\displaystyle\propto$
$\displaystyle\left(\lambda_{-}-\lambda\right)^{-1/2},$ (42)

as shown in Fig. 5 (b). Accordingly, the von Neumann entropy shares the same
form as that in Eq. (39).

###  IV.3 Antiferromagnetic frustrated phase

In the antiferromagnetic frustrated phase ($\lambda>\lambda_{+}$), both the
global $Z_{2}$ symmetry and the translational symmetry are broken.
$\tilde{\theta}_{i}$ depends on the site index $i$, so do other properties
related with $\tilde{\theta}_{i}$. The exact analytical expressions for
$\Delta_{i}$, $\left(\Delta x_{i}\right)^{2}$, $\left(\Delta p_{i}\right)^{2}$
and $\mathcal{S}_{i}$ become much more tedious. Therefore, we just give the
critical behaviors near the quantum critical point
($\lambda\rightarrow\lambda_{+}$). The lowest excitation energy satisfies
$\Delta_{\text{min}}\propto\left(\lambda-\lambda_{+}\right)$, as verified in
Fig. 5 (d). Near the quantum critical point which corresponds to the
paramagnetic-antiferromagnetic phase transition
($\lambda\rightarrow\lambda_{+}$), the quantum fluctuation in $p_{i}$
quadrature tends to a finite value with $\left(\Delta
p_{i}\right)^{2}\rightarrow\sqrt{3}/6$, whereas the exponentially diverged
quantum fluctuation in $x_{i}$ quadrature is given by

$\displaystyle\left(\Delta x_{i}\right)^{2}$ $\displaystyle\propto$
$\displaystyle\left(\lambda-\lambda_{+}\right)^{-1},$ (43)

as shown in Fig. 5 (e). Accordingly, the von Neumann entropy near the quantum
critical point depicted in Fig. 5 (f) can be written as

$\displaystyle\mathcal{S}_{i}\approx\log\Delta x_{i}\Delta
p_{i}\approx-\frac{1}{2}\log\left(\lambda-\lambda_{+}\right)+C^{\prime}_{i}.$
(44)

Obviously, the critical behaviors of the excitation energy, quantum
fluctuation in $x_{i}$ quadrature and von Neumann entropy in the
antiferromagnetic frustrated phase are different from those in both
paramagnetic and ferromagnetic phases, which demonstrate the significance of
the geometric frustration. A mnemonic summary of the critical behaviors in
different quantum phases is provided in Table 1.

Table 1: Excitation energy $\Delta_{\text{min}}$, quantum fluctuation $\left(\Delta x_{i}\right)^{2}$ and von Neumann entropy $\mathcal{S}_{i}$ near the critical point. | Ferromagnetic phase ($\lambda<\lambda_{-}$) | Paramagnetic phase ($\lambda_{-}<\lambda<\lambda_{+}$) | Antiferromagnetic phase ($\lambda>\lambda_{+}$)
---|---|---|---
$\Delta_{\text{min}}$ | $\left(\lambda_{-}-\lambda\right)^{1/2}$ |  $\left|\lambda-\lambda_{\pm}\right|^{1/2}$111$\left(\lambda-\lambda_{-}\right)^{1/2}$ for $\lambda\rightarrow\lambda_{-}$ and $\left(\lambda_{+}-\lambda\right)^{1/2}$ for $\lambda\rightarrow\lambda_{+}$ | $\left(\lambda-\lambda_{+}\right)$
$\left(\Delta x_{i}\right)^{2}$ | $\left(\lambda_{-}-\lambda\right)^{-1/2}$ |  $\left|\lambda-\lambda_{\pm}\right|^{-1/2}$222$\left(\lambda-\lambda_{-}\right)^{-1/2}$ for $\lambda\rightarrow\lambda_{-}$ and $\left(\lambda_{+}-\lambda\right)^{-1/2}$ for $\lambda\rightarrow\lambda_{+}$ | $\left(\lambda-\lambda_{+}\right)^{-1}$
$\mathcal{S}_{i}$ | $-\frac{1}{4}\log\left(\lambda_{-}-\lambda\right)$ |  $-\frac{1}{4}\log\left|\lambda-\lambda_{\pm}\right|$333$-\frac{1}{4}\log\left(\lambda-\lambda_{-}\right)$ for $\lambda\rightarrow\lambda_{-}$ and $-\frac{1}{4}\log\left(\lambda_{+}-\lambda\right)$ for $\lambda\rightarrow\lambda_{+}$ | $-\frac{1}{2}\log\left(\lambda-\lambda_{+}\right)$

##  V Conclusions

As a nature generalization of the transverse-field Ising model, the coupled-
top model describes large spins with ferromagnetic or antiferromagnetic
interaction. It has been widely employed to study the quantum phase
transition, chaotic phenomena and quantum scar, etc. In this paper, we study
three interacting large spins located on a triangle. Compared with the
original coupled-top model with two large spins, the third large spin cannot
simultaneously satisfy antiferromagnetic interaction with the other two, which
introduces the geometric frustration. Here we focus on the quantum phase
transition in the triangular coupled-top model, especially on the novel
critical behaviors induced by the geometric frustration.

The triangular coupled-top model admits three phases: disordered paramagnetic
phase, ferromagnetic phase, and frustrated antiferromagnetic phase, depending
on whether the ground state breaks or preserves the global $Z_{2}$ symmetry
and translational symmetry. The quantum critical points which separate three
phases, as well as the order parameters $J_{i}^{x}$ in different phases, can
be obtained by the mean-field approach. In the paramagnetic phase
($\lambda_{-}<\lambda<\lambda_{+}$), the ground state is non-degenerate. Both
the global $Z_{2}$ symmetry and translational symmetry are preserved with
$J_{i}^{x}=0$. In the ferromagnetic phase ($\lambda<\lambda_{-}$), only the
translational symmetry is preserved and the ground state is two-fold
degenerate. The breaking of the global $Z_{2}$ symmetry leads to nonzero order
parameters $J_{i}^{x}$. In the antiferromagnetic phase ($\lambda>\lambda_{+}$)
where the geometric frustration comes into play, the ground state is six-fold
degenerate. Neither the global $Z_{2}$ symmetry nor the translational symmetry
is preserved. The order parameter $J_{i}^{x}$ depends on the site index $i$.

Further insight into the quantum effects beyond the mean-field approach can be
achieved by employing the Holstein-Primakoff transformation and the symplectic
transformation. After the Holstein-Primakoff transformation, three interacting
large spins are mapped into coupled harmonic oscillators described by the
quadratic Hamiltonian, which can be solved exactly by the symplectic
transformation. The energy gap between the ground state and first excited
state closes near the quantum critical point. The quantum fluctuation in
$x_{i}$ quadrature tends to exponentially diverge, whereas that in $p_{i}$
quadrature tends to a finite value, which is a signature of the squeezing
effect. The von Neumann entropy also diverges near the critical point, which
indicates a strong correlations between different large spins. It is
noteworthy that the critical behaviors in the antiferromagnetic phase are
different from those in the paramagnetic and ferromagnetic phases. The
triangular coupled-top model opens up new opportunities to investigate novel
critical behaviors induced by the geometric frustration.

###### Acknowledgements.

Y.-Z. W. is supported by the National Natural Science Foundation of China
under Grant No. 12105001 and Natural Science Foundation of Anhui Province
under Grant No. 2108085QA24. Q.-H. C. is supported by the National Science
Foundation of China under Grant No. 11834005.

## References

  * Sachdev [2011] S. Sachdev,  _Quantum Phase Transitions_, 2nd ed. (Cambridge University Press, 2011).
  * Toulouse [1977] G. Toulouse, Commun. Phys. 2, 115 (1977).
  * Balents [2010] L. Balents, Nature 464, 199 (2010).
  * Zhou  _et al._ [2017] Y. Zhou, K. Kanoda, and T.-K. Ng, Rev. Mod. Phys. 89, 025003 (2017).
  * Bramwell and Gingras [2001] S. T. Bramwell and M. J. P. Gingras, Science 294, 1495 (2001).
  * Dorogovtsev  _et al._ [2008] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Rev. Mod. Phys. 80, 1275 (2008).
  * Bryngelson and Wolynes [1987] J. D. Bryngelson and P. G. Wolynes, Proceedings of the National Academy of Sciences 84, 7524 (1987).
  * Lee  _et al._ [2002] S.-H. Lee, C. Broholm, W. Ratcliff, G. Gasparovic, Q. Huang, T. H. Kim, and S.-W. Cheong, Nature 418, 856 (2002).
  * Tokiwa  _et al._ [2015] Y. Tokiwa, C. Stingl, M.-S. Kim, T. Takabatake, and P. Gegenwart, Science Advances 1, e1500001 (2015).
  * Zhang  _et al._ [2021] Y.-Y. Zhang, Z.-X. Hu, L. Fu, H.-G. Luo, H. Pu, and X.-F. Zhang, Phys. Rev. Lett. 127, 063602 (2021).
  * Fallas Padilla  _et al._ [2022] D. Fallas Padilla, H. Pu, G.-J. Cheng, and Y.-Y. Zhang, Phys. Rev. Lett. 129, 183602 (2022).
  * Zhao and Hwang [2022] J. Zhao and M.-J. Hwang, Phys. Rev. Lett. 128, 163601 (2022).
  * Ramirez [1994] A. P. Ramirez, Annual Review of Materials Science 24, 453 (1994).
  * Collins and Petrenko [1997] M. F. Collins and O. A. Petrenko, Canadian Journal of Physics 75, 605 (1997).
  * de Gennes [1963] P. de Gennes, Solid State Communications 1, 132 (1963).
  * Brout  _et al._ [1966] R. Brout, K. Müller, and H. Thomas, Solid State Communications 4, 507 (1966).
  * Pfeuty [1970] P. Pfeuty, Annals of Physics 57, 79 (1970).
  * Stinchcombe [1973] R. B. Stinchcombe, Journal of Physics C: Solid State Physics 6, 2459 (1973).
  * Suzuki  _et al._ [2012] S. Suzuki, J.-I. Inoue, and B. K. Chakrabarti,  _Quantum Ising Phases and Transitions in Transverse Ising Models_ , 2nd ed., Lecture notes in physics (Springer, Berlin, Germany, 2012).
  * Dziarmaga [2005] J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).
  * Moessner and Sondhi [2001] R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).
  * Kim  _et al._ [2010] K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G.-D. Lin, L.-M. Duan, and C. Monroe, Nature 465, 590 (2010).
  * Hines  _et al._ [2005] A. P. Hines, R. H. McKenzie, and G. J. Milburn, Phys. Rev. A 71, 042303 (2005).
  * Mondal  _et al._ [2020] D. Mondal, S. Sinha, and S. Sinha, Phys. Rev. E 102, 020101 (2020).
  * Mondal  _et al._ [2021] D. Mondal, S. Sinha, and S. Sinha, Phys. Rev. E 104, 024217 (2021).
  * Wang and Pérez-Bernal [2021] Q. Wang and F. Pérez-Bernal, Phys. Rev. E 104, 034119 (2021).
  * Mondal  _et al._ [2022] D. Mondal, S. Sinha, and S. Sinha, Phys. Rev. E 105, 014130 (2022).
  * Emary and Brandes [2003a] C. Emary and T. Brandes, Phys. Rev. E 67, 066203 (2003a).
  * Emary and Brandes [2003b] C. Emary and T. Brandes, Phys. Rev. Lett. 90, 044101 (2003b).
  * Lambert  _et al._ [2004] N. Lambert, C. Emary, and T. Brandes, Phys. Rev. Lett. 92, 073602 (2004).
  * Brandes [2013] T. Brandes, Phys. Rev. E 88, 032133 (2013).
  * Lieb [1973] E. H. Lieb, Commun. Math. Phys. 31, 327 (1973).
  * Peng  _et al._ [2019] J. Peng, E. Rico, J. Zhong, E. Solano, and I. n. L. Egusquiza, Phys. Rev. A 100, 063820 (2019).
  * Dusuel and Vidal [2005] S. Dusuel and J. Vidal, Phys. Rev. B 71, 224420 (2005).
  * Bakemeier  _et al._ [2012] L. Bakemeier, A. Alvermann, and H. Fehske, Phys. Rev. A 85, 043821 (2012).
  * Perelomov [2012] A. Perelomov,  _Generalized coherent states and their applications_ , Theoretical and Mathematical Physics (Springer, Berlin, Germany, 2012).
  * Duan [2022] L. Duan, Phys. Rev. A 105, 042215 (2022).
  * Soldati  _et al._ [2021] R. R. Soldati, M. T. Mitchison, and G. T. Landi, Phys. Rev. A 104, 052423 (2021).
  * Holstein and Primakoff [1940] T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940).
  * Serafini [2017] A. Serafini,  _Quantum continuous variables_ (CRC Press, London, England, 2017).
  * Adesso and Illuminati [2007] G. Adesso and F. Illuminati, Journal of Physics A: Mathematical and Theoretical 40, 7821 (2007).
  * Scully and Zubairy [1997] M. O. Scully and M. S. Zubairy,  _Quantum Optics_ (Cambridge University Press, 1997).
  * Nataf  _et al._ [2012] P. Nataf, M. Dogan, and K. Le Hur, Phys. Rev. A 86, 043807 (2012).