# Bhasha-Abhijnaanam: Native-script and Romanized Language Identification for
22 Indic Languages

Yash Madhani1 Mitesh M. Khapra2 Anoop Kunchukuttan3
AI4Bharat1,2,3 IIT Madras1,2,3 Microsoft3
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We create publicly available language identification (LID) datasets and models
in all 22 Indian languages listed in the Indian constitution in both native-
script and romanized text. First, we create Bhasha-Abhijnaanam, a language
identification test set for native-script as well as romanized text which
spans all 22 Indic languages. We also train IndicLID, a language identifier
for all the above-mentioned languages in both native and romanized script. For
native-script text, it has better language coverage than existing LIDs and is
competitive or better than other LIDs. IndicLID is the first LID for romanized
text in Indian languages. Two major challenges for romanized text LID are the
lack of training data and low-LID performance when languages are similar. We
provide simple and effective solutions to these problems. In general, there
has been limited work on romanized text in any language, and our findings are
relevant to other languages that need romanized language identification. Our
models are publicly available at https://ai4bharat.iitm.ac.in/indiclid under
open-source licenses. Our training and test sets are also publicly available
at https://ai4bharat.iitm.ac.in/bhasha-abhijnaanam under open-source licenses.

##  1 Introduction

In this work, we focus on building a language identifier for the 22 languages
listed in the Indian constitution. With increasing digitization, there is a
push to make NLP technologies like translation, ASR, conversational
technologies, etc. Bose (2022) available as a public good at population scale
Chandorkar (2022). A good language identifier is required to help build
corpora in low-resource languages. For such languages, language identification
is far from a solved problem due to noisy web crawls, small existing datasets,
and similarity to high-resource languages Caswell et al. (2020).

Existing publicly available LID tools like
CLD3111https://github.com/google/cld3,
LangID222https://github.com/saffsd/langid.py Lui and Baldwin (2011),
FastText333https://fasttext.cc/docs/en/language-identification.html Joulin et
al. (2016) and
NLLB444https://github.com/facebookresearch/fairseq/tree/nllb#lid-model NLLB
Team et al. (2022) have some shortcomings with respect to Indian languages.
They do not cover all the above-mentioned 22 languages. In social media and
chats, it is also common to use the roman script for most Indian languages
leading to substantial user-generated content in roman script. However, none
of the LIDs have any support for the detection of romanized Indian language
text (except cld3 support for Latin Hindi). The widespread use of romanization
implies that accurate romanized Language Identification models are a critical
component in the NLP stack for Indian languages, given that this affects over
735 million internet users KPMG and Google (2017). Therefore, our work on
developing accurate and effective romanized Language Identification models has
the potential to make a significant impact in the NLP space for Indian
languages, particularly in the social media and chat application domains.
Hence, we undertake the task of creating a LID for these 22 Indian languages.
The main contributions of our work are as follows:

$\bullet$ We create Bhasha-Abhijnaanam555The word means language-
identification in Sanskrit., a language identification test set for native-
script as well as romanized text which spans 22 Indic languages. Previous
benchmarks for native script do not cover all these languages NLLB Team et al.
(2022); Roark et al. (2020). The Dakshina test set for romanized text covers
only 11 languages and there are ambiguous instances in the test set like named
entities that cannot be assigned to a particular language Roark et al. (2020).

$\bullet$ We also train, IndicLID, an LID for all the above-mentioned
languages in both native and romanized script. For native-script training
data, we sample sentences from diverse sources and oversample low-resource
languages. IndicLID native-script model has better language coverage than
existing LIDs and is competitive or better than other LIDs with 98% accuracy
and at least 6 times better throughput.

$\bullet$ To the best of our knowledge, ours is one of the first large-scale
efforts for romanized LID in any language, a task that has not received much
attention. A major challenge for romanized text LID is the lack of romanized
training data. We show that synthetic romanized training data created via
transliteration can help train a reasonably good LID for romanized text. A
simple linear classifier does not perform well for romanized text. Hence, we
combine a simple but fast text classifier with a slower but more accurate
classifier based on a pre-trained language model to achieve a good trade-off
between accuracy and speed.

Our findings are relevant to other languages that need LID for romanized text.
We require native script data and a transliteration model to create the
synthetic romanized data for the target language. This romanized data serves
as training data for the romanized LID.

Language | Script | Native | Roman
---|---|---|---
Assamese | Bengali | 1012 | 512
Bangla | Bengali | 5606 | 4595
Bodo | Devanagari | 1500 | 433
Dogri | Devanagari | 1498 | 512
Gujarati | Gujarati | 5797 | 4785
Hindi | Devanagari | 5617 | 4606
Kannada | Kannada | 5859 | 4848
Kashmiri | Perso-Arabic | 2511 | 450
Devanagari | 1012
Konkani | Devanagari | 1500 | 444
Maithili | Devanagari | 2512 | 439
Malayalam | Malayalam | 5628 | 4617
Manipuri | Bengali | 1012 | 442
Meetei Mayek | 1500
Marathi | Devanagari | 5611 | 4603
Nepali | Devanagari | 2512 | 423
Oriya | Oriya | 1012 | 512
Punjabi | Gurmukhi | 5776 | 4765
Sanskrit | Devanagari | 2510 | 448
Santali | Ol Chiki | 2512 | 0
Sindhi | Perso-Arabic | 5893 | 4881
Tamil | Tamil | 5779 | 4767
Telugu | Telugu | 5751 | 4741
Urdu | Perso-Arabic | 6883 | 4371

Table 1: Summary of the Bhasha-Abhijnaanam benchmark. Number of romanized and
native-script sentences are reported. The cells in bold indicate the datasets
newly contributed by this work. Romanized Santali testset has not been created
since Santhali annotators we contacted did not use roman script and spoke
Bengali as a second language. NLLB Team et al. (2022) also cite a similar
experience.

##  2 Bhasha-Abhijnaanam benchmark

We describe the creation of the Bhasha-Abhijnaanam LID benchmark for 22 Indian
languages in native and roman script. Table 1 describes the statistics of the
Bhasha-Abhijnaanam benchmark. We build upon existing benchmarks to fill in the
coverage and quality gaps and cost-efficiently cover all languages.

###  2.1 Native script test set.

We compile a native script test set comprising 19 Indian languages and 11
scripts from the FLORES-200 devtest NLLB Team et al. (2022) and Dakshina
sentence test set Roark et al. (2020). We create native text test sets for the
remaining three languages (Bodo, Konkani, Dogri) and one script (Manipuri in
Meetei Mayek script) not covered in these datasets. For these new languages we
first sample the English sentences from Wikipedia and ask in-house,
professional translators to translate the sentences to respective languages.
This method ensured the quality and accuracy of our test samples, as well as
minimizing any potential noise in the data.

###  2.2 Roman script test set.

We propose a new benchmark test set to evaluate roman-script language
identification for 21 Indian languages. Out of these, 11 languages are
represented in the Dakshina romanized sentence test set Roark et al. (2020),
which comprises native script sentences from Wikipedia along with their
romanization. However, this test set includes short sentences which are just
named entities and English loan words which are not useful for romanized text
LID evaluation. To address this issue, we manually validate the Dakshina test
sets for the languages we are interested in and filter out about 7% of the
sentences. Section 2.3 describes the details of the filtering process. To
create a benchmark test set for the remaining 10 Indian languages, we sampled
sentences from IndicCorp Doddapaneni et al. (2022) and asked annotators to
write the same in roman script. We did not specify any transliteration
guidelines and annotators were free to transliterate in the most natural way
they deemed fit. We additionally asked annotators to skip the sentence if they
find it invalid (wrong language, offensive, truncated, etc.).

###  2.3 Romanized Dakshina testset filtering

The Dakshina romanized sentence test set includes short sentences which are
just named entities and English loan words which are not useful for romanized
text LID evaluation. To address this issue, we manually validate the Dakshina
test sets for the languages we are interested in. We first identified
potentially problematic sentences from the romanized Dakshina test set by
applying two constraints: (i) sentences shorter than 5 words, and (ii) native
LID model is less confident about the native language sentence (prediction
score less than 0.8). These sentences were then validated by native language
annotators. The annotators were asked to read the roman sentences and
determine whether they were named entities or sentences where they could not
determine the language. Such entries were filtered out. About 7% of the
sentences were filtered. Table 2 describes the filtering statistics.

Language | Total samples | Valid samples | %filtered
---|---|---|---
Bengali | 5001 | 4600 | 8.0183
Gujarati | 5001 | 4789 | 4.2391
Hindi | 5001 | 4616 | 7.6984
Kannada | 5001 | 4849 | 3.0393
Malayalam | 5001 | 4627 | 7.4785
Marathi | 5001 | 4617 | 7.6784
Punjabi | 5001 | 4782 | 4.3791
Sindhi | 5001 | 4889 | 2.2395
Tamil | 5001 | 4802 | 3.9792
Telugu | 5001 | 4754 | 4.9390
Urdu | 4881 | 4395 | 9.9569

Table 2: Statistics of Dakshina roman filtered test set

##  3 IndicLID Model

Figure 1: IndicLID Classifier Workflow

IndicLID is a classifier specifically for Indic languages that can predict 47
classes (24 native-script classes and 21 roman-script classes plus English and
Others). We create three classifier variants: a fast linear classifier, a
slower classifier finetuned from a pre-trained LM, and an ensemble of the two
models which trades off speed v/s accuracy.

###  3.1 Training dataset creation

Native-script training data. We compiled the training data sentences from
various sources viz. IndicCorp Doddapaneni et al. (2022), NLLB NLLB Team et
al. (2022), Wikipedia, Vikaspedia 666https://vikaspedia.in and internal
sources. To ensure a diverse and representative training dataset, we sampled
100k sentences per language-script combination in a balanced way across all
these sources. We used oversampling for languages with less than 100k
sentences. We tokenized and normalized the sentences using IndicNLP library
777https://github.com/anoopkunchukuttan/indic_nlp_library Kunchukuttan (2020)
with default settings.

Romanized training data. There is hardly any romanized corpora for Indian
languages in the public domain888CC-100 has romanized versions for 4 Indian
languages, but a manual analysis suggested that it contains a lot of profane
content.. Hence, we explored the use of transliteration for creating synthetic
romanized data. We create romanized training data by transliterating the
native script training data into roman script using the multilingual
IndicXlit999https://github.com/AI4Bharat/IndicXlit transliteration model
(Indic-to-En version) Madhani et al. (2022), The authors have provided results
on the transliteration quality of the IndicXlit model. We rely on this
analysis to ensure the quality of generated training data.

###  3.2 Linear classifier

Linear classifiers using character n-gram features are widely used for LIDs
Jauhiainen et al. (2021). We use FastText Joulin et al. (2016) to train our
fast, linear classifier. It is a lightweight and efficient linear classifier
that is well-suited for handling large-scale text data. It utilizes character
n-gram features which enables it to utilize sub-word information. This makes
it particularly useful for dealing with rare words and allows it to
discriminate between similar languages having similar spellings. We trained
separate classifiers for native script (IndicLID-FTN) and roman script
(IndicLID-FTR). We chose 8-dimension word-vector models after experimentation
as they maintain small model sizes without losing model accuracy (refer
Appendix A for results).

###  3.3 Pretrained LM-based classifier

For romanized text, we observed that linear classifiers do not perform very
well. Hence, we also experimented with models having larger capacity.
Particularly, we finetuned a pretrained LM on the romanized training dataset.
We evaluated the following LMs: XLM-R Conneau et al. (2020), IndicBERT-v2
Doddapaneni et al. (2022) and MuRIL Khanuja et al. (2021). The last two LMs
are specifically trained for Indian languages and MuRIL also incorporates
synthetic romanized data in pre-training. Hyperparameters for finetuning are
described in Appendix B. We used IndicBERT-based classifier as the LM-based
classifier (henceforth referred to as IndicLID-BERT) since it was amongst the
best-performing romanized text classifiers and had maximum language coverage.

###  3.4 Final Ensemble classifier

Our final IndicLID classifier is an pipeline of multiple classifiers. Figure 1
shows the overall workflow of the IndicLID classifier. The pipeline works as
described here: (1) Depending on the amount of roman script in the input text,
we invoke either the native-text or romanized linear classifier. IndicLID-FTR
is invoked for text containing $>$50% roman characters. (2) For roman text, if
IndicLID-FTR is not confident about its prediction, we redirect the request to
the IndicLID-BERT. We resort to this two-stage approach for romanized input to
achieve a good trade-off between classifier accuracy and inference speed. The
fast IndicLID-FTR’s prediction is used if the model is confident about its
prediction (probability of predicted class $>0.6$ ), else the slower but more
accurate IndicLID-BERT is invoked. This threshold provides a good trade-off
(See Appendix C for more details).

##  4 Results and Discussion

We discuss the performance of various models on the benchmark and analyze the
results. To prevent any overlap between the test/valid and train sets, we
excluded the Flores-200 test set NLLB Team et al. (2022), Dakshina test set
Roark et al. (2020) while sampling native train samples from various sources.
Additionally, we removed the training samples from the benchmark samples when
collecting sentences for the benchmark test set. We also made sure that there
was no overlap between the test and valid sets. To create the romanized
training set, we simply transliterated the native training set. As the
Dakshina test set Roark et al. (2020) provided parallel sentences for the
native and roman test sets, there was no overlap between the roman train and
test sets.

Model | P | R | F1 | Acc | Throughput | Size
---|---|---|---|---|---|---
IndicLID-FTN-8-dim (24) | 98.11 | 98.56 | 98.31 | 98.55 | 30,303 | 318M
Comparing our IndicLID-FTN model with CLD3 model (12)
IndicLID-FTN-4-dim | 99.43 | 98.40 | 98.89 | 98.33 | 47,619 | 208M
IndicLID-FTN-8-dim | 99.73 | 98.67 | 99.18 | 98.62 | 33,333 | 318M
CLD3 | 98.52 | 98.14 | 98.31 | 98.03 | 4,861 | -
Comparing our IndicLID-FTN model with NLLB model (20)
IndicLID-FTN-4-dim | 97.78 | 98.10 | 97.92 | 98.19 | 41,666 | 208M
IndicLID-FTN-8-dim | 98.13 | 98.59 | 98.34 | 98.56 | 29,411 | 318M
NLLB | 99.28 | 98.65 | 98.95 | 98.78 | 4,970 | 1.1G

Table 3: Benchmarking on the Bhasha-Abhijnaanam native-script testset. For
fair comparison with NLLB and CLD3, we restrict the comparison to languages
that are common with IndicLID-FTN (count of common languages is indicated in
brackets). Throughput is number of sentence/second.

###  4.1 Native script LID

We compare IndicLID-FTN with the NLLB model NLLB Team et al. (2022) and the
CLD3 model. As we can see in Table 3, the LID performance of IndicLID-FTN is
comparable or better than other models. Our model is 10 times faster and 4
times smaller than the NLLB model. The model’s footprint can be further
reduced by model quantization Joulin et al. (2016) which we leave for future
work.

###  4.2 Roman script LID

Table 4 presents the results of different model variants on the romanized test
set (see Appendix D for language-wise results). IndicLID-BERT is significantly
better than IndicLID-FTR, but the throughput decreases significantly. The
ensemble model (IndicLID) maintains the same LID performance as IndicLID-BERT
with a 3x increase in the throughput over IndicLID-BERT. Further speedups in
the model throughput can be achieved by creating distilled versions, which we
leave for future work.

Figure 2: Confusion matrix (IndicLID, roman testset)

Model | P | R | F1 | Acc | Throughput | Size
---|---|---|---|---|---|---
IndicLID-FTR (dim-8) | 63.12 | 78.01 | 63.28 | 71.49 | 37,037 | 357 M
IndicLID-BERT (unfeeze-layer-1) | 72.70 | 84.01 | 74.52 | 80.04 | 3 | 1.1 GB
IndicLID (threshold-0.6) | 72.74 | 84.50 | 74.72 | 80.40 | 10 | 1.4 GB

Table 4: Performance of IndicLID-FTR on Bhasha-Abhijnaanam roman script test
set. Throughput is number of sentence/second.

LID confusion analysis The confusion matrix for IndicLID is shown in Figure 2.
We see that major confusions are between similar languages. Some examples of
such language clusters that can be observed are (1) Hindi and very close
languages like Maithili, Urdu and Punjabi, (2) Konkani and Marathi, (3) Sindi
and Kashmiri. Improving romanized LID between very similar languages is thus
an important direction of improvement.

Testset | P | R | F1 | Acc
---|---|---|---|---
Original | 72.74 | 84.50 | 74.72 | 80.40
Synthetic | 90.79 | 97.24 | 93.43 | 95.96

Table 5: Comparison of results on Synthetic vs. original Romanized test sets
for IndicLID model

Impact of synthetic training data To understand the impact of synthetic
training data, we generate a machine-transliterated version of the romanized
test set using IndicXlit. We compare the LID accuracy on the original and
synthetically generated test sets. Table 5 shows that the results on the
synthetic test set are significantly better than the original test set
(approaching accuracy levels in the 90s). The data characteristics of the
synthetic test set are much closer to the training data than the original test
set. Closing the training-test distribution gap (by representing original
romanized data in the training data and/or improved generation of synthetic
romanized data to reflect true data distribution) is critical to improving
model performance.

The confusion matrix gives further insights into the impact of synthetic
training data. Hindi is confused with languages like Nepali, Sanskrit, Marathi
and Konkani using the same native script as Hindi (Devanagari). Since a
multilingual transliteration model with significant Hindi data was used to
create the synthetic romanized training data, it may result in the synthetic
romanized forms of these languages being more similar to Hindi than would be
the case with original romanized data.

Figure 3: Effect of input length on romanized testset

Impact of input length Figure 3 plots the LID accuracy for various input
length buckets. The LID is most confused for short inputs (<10 words) after
which the performance is relatively stable.

##  5 Conclusion

We introduce an LID benchmark and models for native-script and romanized text
in 22 Indian languages. These tools will serve as a basis for building NLP
resources for Indian languages, particularly extremely low-resource ones that
are "left-behind" in the NLP world today Joshi et al. (2020). Our work takes
first steps towards LID of romanized text, and our analysis reveals directions
for future work.

## Acknowledgements

We would like to thank the Ministry of Electronics and Information Technology
of the Government of India for their generous grant through the Digital India
Bhashini project. We also thank the Centre for Development of Advanced
Computing for providing compute time on the Param Siddhi Supercomputer. We
also thank Nilekani Philanthropies for their generous grant towards building
datasets, models, tools and resources for Indic languages. We also thank
Microsoft for their grant to support research on Indic languages. We would
like to thank Jay Gala and Ishvinder Sethi for their help in coordinating the
annotation work. Most importantly we would like to thank all the annotators
who helped create the Bhasha-Abhijnaanam benchmark.

## Limitations

The benchmark for language identification for the most part contains clean
sentences (grammatically correct, single script, etc.). Data from the real
world might be noisy (ungrammatical, mixed scripts, code-mixed, invalid
characters, etc.). A better representative benchmark might be useful for such
use cases. However, the use cases captured by this benchmark should suffice
for the collection of clean monolingual corpora. This also represents a first
step for many languages where no LID benchmark exists.

The use of synthetic training data seems to create a gap in performance due to
divergence in train/test data distributions. Acquisition of original native
romanized text and methods to generate better romanized text are needed.

Note that the romanized LID model does not support Dogri since the IndicXlit
transliteration model does not support Dogri. However, since Dogri is written
in the Devanagari script using the transliterator for Hindi which uses the
same script might be a good approximation to generate synthetic training data.
We will explore this in the future.

This work is limited to the 22 languages listed in the 8th schedule of the
Indian constitution. Further work is needed to extend the benchmark to many
more widely used languages in India (which has about 30 languages with more
than a million speakers).

## Ethics Statement

For the human annotations on the dataset, the language experts are native
speakers of the languages and from the Indian subcontinent. They were paid a
competitive monthly salary to help with the task. The salary was determined
based on the skill set and experience of the expert and adhered to the norms
of the government of our country. The dataset has no harmful content. The
annotators were made aware of the fact that the annotations would be released
publicly and the annotations contain no private information. The proposed
benchmark builds upon existing datasets. These datasets and related works have
been cited.

The annotations are collected on a publicly available dataset and will be
released publicly for future use. The IndicCorp dataset which we annotated has
already been checked for offensive content.

All the datasets created as part of this work will be released under a CC-0
license101010https://creativecommons.org/publicdomain/zero/1.0 and all the
code and models will be released under an MIT
license.111111https://opensource.org/licenses/MIT

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##  Appendix A Hyperparameter tuning for Roman script linear classifier

We train the IndicLID-FTR model using 100k samples. While deciding the
configuration IndicLID-FTR model, we experimented with fixing the dimension of
IndicLID-FTR model and tuning on the rest of the hyperparameters. As we can
see from table 6 model size increases with the increase of IndicLID-FTR
dimension. However, beyond 8 dimensions, there is not much improvement
observed. Therefore, we chose the model with 8 dimensions, taking into account
the model size.

Dimension | Precision | Recall | F1-Score | Accuracy | Throughput | Model Size
---|---|---|---|---|---|---
4 | 60.01 | 74.56 | 61.09 | 67.52 | 50000 | 171M
8 | 63.13 | 78.02 | 63.29 | 71.49 | 37037 | 357M
16 | 63.67 | 78.33 | 64.32 | 71.58 | 30303 | 578M
32 | 64.62 | 78.67 | 65.16 | 71.95 | 15625 | 1.6G
64 | 64.54 | 78.58 | 65.10 | 71.93 | 14085 | 1.9G
128 | 64.55 | 78.45 | 65.03 | 71.77 | 9901 | 3.3G
256 | 64.60 | 78.54 | 65.13 | 71.89 | 7463 | 7.3G
512 | 63.89 | 78.29 | 64.58 | 71.49 | 4608 | 11G
768 | 64.37 | 78.63 | 65.07 | 72.04 | 3876 | 22G
1024 | 64.30 | 78.53 | 65.07 | 71.94 | 3322 | 29G

Table 6: IndicLID-FTR performance on Bhasha-Abhijnaanam roman script test set.
IndicLID-FTR are hyper-tuned by fixing different dimensions. Throughput is
number of sentence/second.

Model | Precision | Recall | F1-Score | Accuracy
---|---|---|---|---
XLMR Conneau et al. (2020) | 63.19 | 70.92 | 59.49 | 65.15
MuRIL Khanuja et al. (2021) | 66.70 | 79.08 | 67.77 | 73.70
IndicBERT Doddapaneni et al. (2022) | 68.07 | 80.52 | 68.91 | 75.81

Table 7: Bhasha-Abhijnaanam roman script test set results on roman script
Language models finetuned by freezing all the layers

Model | Precision | Recall | F1-Score | Accuracy
---|---|---|---|---
unfreezed-layer-1 | 72.70 | 84.01 | 74.53 | 80.04
unfreezed-layer-2 | 69.84 | 83.84 | 72.44 | 79.55
unfreezed-layer-4 | 69.53 | 83.44 | 72.12 | 79.47
unfreezed-layer-6 | 68.41 | 81.89 | 70.02 | 77.08
unfreezed-layer-8 | 67.46 | 81.88 | 68.42 | 76.04
unfreezed-layer-11 | 70.55 | 83.73 | 72.63 | 79.88

Table 8: Bhasha-Abhijnaanam roman script test set results on IndicLID-BERT
finetuned with unfreezing different numbers of layers

##  Appendix B Model selection for Roman script LM-based classifier

We experimented with three different pre-trained language models: IndicBERT
Doddapaneni et al. (2022), XLM-R Conneau et al. (2020), and MuRIL Khanuja et
al. (2021). In the initial experiment, we froze all the layers except for the
last softmax layer and finetuned the model with our training data. To fine-
tune the language model, we added one softmax layer to the end of the model
and used our roman script training data to finetune the model. The results for
these experiments are shown in Table 7. We found that IndicBERT and MuRIL
performed similarly among these three models for our roman LID task. MuRIL
leverages the advantage of roman text training data, while IndicBERT was
trained on the only native script but performed similarly. However, IndicBERT
supports 24 Indian languages, while MuRIL only supports 17 Indian languages.
Therefore, we selected IndicBERT due to its superior coverage and performance.
We then further experimented with IndicBERT by unfreezing 1, 2, 4, 6, 8, and
11 layers. The results and comparison of all the experiments are described in
Table 8. We found that unfreezing 1 layer was enough for our task and that
unfreezing more layers did not provide any additional benefit.

Thresholds | P | R | F1 | Acc | Throughput
---|---|---|---|---|---
threshold 0.1 | 63.13 | 78.02 | 63.29 | 71.49 | 50000
threshold 0.2 | 63.43 | 78.18 | 63.63 | 71.77 | 379
threshold 0.3 | 65.50 | 79.64 | 66.15 | 73.84 | 54
threshold 0.4 | 68.39 | 81.84 | 69.77 | 76.84 | 22
threshold 0.5 | 70.99 | 83.60 | 72.87 | 79.15 | 14
threshold 0.6 | 72.74 | 84.51 | 74.72 | 80.4 | 10
threshold 0.7 | 73.60 | 84.80 | 75.54 | 80.93 | 9
threshold 0.8 | 73.88 | 84.81 | 75.77 | 80.96 | 8
threshold 0.9 | 73.51 | 84.50 | 75.35 | 80.62 | 6

Table 9: Trade-off between inference time and accuracy with different
thresholds. Throughput is number of sentence/second.

##  Appendix C Analysis of speed/accuracy tradeoff

We experimented IndicLID with different thresholds. If the probability score
is below a certain threshold we invoke a more powerful model IndicLID-BERT,
otherwise, we go with IndicLID-FTR model prediction. IndicLID-FTR model is
quite fast as compared to IndicLID-BERT model. We can see a good trade-off
between throughput and accuracy in table 9 as we increase the threshold. As
the threshold increases, the input is more likely to go towards the IndicLID-
BERT model, as we are making the model less reliant on the IndicLID-FTR model.

##  Appendix D Language-wise analysis for Roman script classifiers

Table 10 illustrates the language-specific performance of IndicLID-FTR,
IndicLID-BERT and IndicLID models in detail. As we can see IndicLID-BERT has
better representation than IndicLID-FTR for almost all the languages which
leads better F1 score for IndicLID. However, for the languages of Sanskrit and
Manipuri, the IndicLID-FTR model has a better representation than the
IndicLID-BERT model, which is an interesting finding that warrants further
investigation in future studies.

| IndicLID-FTR (8 dim) | IndicLID-BERT (unfreeze 1) | IndicLID (threshold 0.6)
---|---|---|---
| Precision | Recall | F1 | Precision | Recall | F1 | Precision | Recall | F1
Assamese | 37.72 | 93.55 | 53.76 | 66.81 | 91.21 | 77.13 | 72.41 | 92.77 | 81.34
Bangla | 76.63 | 94.10 | 84.47 | 97.12 | 88.14 | 92.41 | 94.94 | 93.95 | 94.44
Bodo | 70.88 | 98.38 | 82.40 | 84.78 | 99.08 | 91.37 | 85.66 | 99.31 | 91.98
Konkani | 24.62 | 95.72 | 39.17 | 38.35 | 99.32 | 55.33 | 40.90 | 97.75 | 57.67
Gujarati | 89.52 | 78.70 | 83.76 | 95.88 | 85.20 | 90.23 | 95.16 | 86.69 | 90.73
Hindi | 65.46 | 15.68 | 25.29 | 76.32 | 60.40 | 67.43 | 77.16 | 53.32 | 63.06
Kannada | 89.66 | 96.41 | 92.91 | 95.79 | 95.71 | 95.75 | 95.29 | 96.78 | 96.03
Kashmiri | 18.74 | 91.56 | 31.12 | 39.45 | 93.11 | 55.42 | 34.80 | 94.67 | 50.90
Maithili | 07.81 | 38.95 | 13.01 | 29.00 | 41.69 | 34.21 | 21.97 | 43.74 | 29.25
Malayalam | 89.75 | 94.46 | 92.04 | 92.19 | 95.32 | 93.73 | 91.33 | 95.36 | 93.30
Manipuri | 64.84 | 98.87 | 78.32 | 50.06 | 98.42 | 66.36 | 58.85 | 99.32 | 73.91
Marathi | 87.21 | 79.58 | 83.22 | 96.35 | 80.80 | 87.89 | 95.86 | 82.92 | 88.92
Nepali | 19.55 | 82.51 | 31.61 | 43.25 | 93.85 | 59.21 | 36.94 | 93.62 | 52.98
Oriya | 41.88 | 95.70 | 58.26 | 64.09 | 95.51 | 76.71 | 62.96 | 97.27 | 76.44
Punjabi | 78.52 | 37.21 | 50.49 | 84.71 | 64.64 | 73.32 | 85.62 | 62.62 | 72.34
Sanskrit | 49.32 | 96.43 | 65.26 | 32.55 | 99.33 | 49.04 | 36.88 | 99.11 | 53.75
Sindhi | 80.00 | 61.05 | 69.25 | 86.39 | 71.91 | 78.49 | 87.88 | 72.51 | 79.46
Tamil | 97.32 | 90.56 | 93.82 | 97.15 | 93.06 | 95.06 | 97.50 | 92.64 | 95.01
Telugu | 94.24 | 87.68 | 90.84 | 95.25 | 88.76 | 91.89 | 95.89 | 89.50 | 92.58
Urdu | 78.88 | 33.24 | 46.77 | 88.53 | 44.84 | 59.53 | 86.87 | 46.31 | 60.41
Avg | 63.13 | 78.02 | 63.29 | 72.70 | 84.01 | 74.53 | 72.74 | 84.51 | 74.72

Table 10: Precision, recall and F1-score of IndicLID-FTR, IndicLID-BERT and
IndicLID roman script model. All scores are calculated on Bhasha-Abhijnaanam
roman script test set. Bold indicates the best language representation among
IndicLID-FTR, IndicLID-BERT and IndicLID roman script model for individual
languages.


# Hugenholtz-Pines theorem for Bose–Einstein condensates
with internal degrees of freedom

Shohei Watabe  Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku,
Tokyo, 162-9601, Japan

###### Abstract

The Hugenholtz-Pines (HP) theorem for Bose–Einstein condensates (BECs) with
internal degrees of freedom has been an open problem, in particular, in the
field of quantum gases for spinor BECs. We here exactly clarified the HP
theorem where an original Hamiltonian has a U$(1)$$\times$SO$(N)$ or
U$(1)$$\times$SU$(N)$ symmetry, but applied to it are SO$(N)$ or SU$(N)$
symmetry breaking external fields. This identity serves to organize the HP
theorem of the spin-$f$ spinor BEC in the presence of a magnetic field with
broken U$(1)$$\times$SO$(3)$ symmetry.

A Ward-Takahashi identity — an exact relation between correlation functions —
is very important for physics, which reflects the symmetry that a system has.
The Hugenholtz-Pines (HP) theorem — a relation between self-energies of a one-
particle Green’s function — is one of those Ward-Takahashi identities for a
Bose–Einstein condensate (BEC) Hugenholtz and Pines (1959), which ensures a
gapless Nambu-Goldstone mode owing to the spontaneously broken U$(1)$
symmetry. This theorem plays a criterial role in developing theories for
scalar BECs Griffin (1993), such as superfluid liquid 4He as well as ultracold
gaseous atomic BECs without internal degrees of freedom. In recent ultracold
atomic gases, on the other hand, spin-$1,2$, and $3$ BECs have been realized
Chang  _et al._ (2004); Stenger  _et al._ (1998); Black  _et al._ (2007);
Kuwamoto  _et al._ (2004); Schmaljohann  _et al._ (2004); Pasquiou  _et al._
(2011), where U$(1)$$\times$SO$(3)$ symmetry is broken. Nevertheless, the HP
theorem in the spinor BECs has been an unsolved problem except the polar phase
in the spin-$1$ Bose gas Kis-Szabó  _et al._ (2007). We have had no criteria
for developing theories for spinor BECs, and here encounter the first
question; (i) What is the HP theorem for spinor BECs with spontaneously broken
U$(1)$$\times$SO$(3)$ symmetry?

In ultracold atomic spinor BECs, on the other hand, a magnetic field breaks
SO$(3)$ symmetry. In ferromagnetic spinor BECs, for example, a transverse spin
excitation is gapped in the presence of the magnetic field Ohmi and Machida
(1998); Kawaguchi and Ueda (2012), although the nature of this ferromagnetic
spin mode may be still related to the SO$(3)$ symmetry. However, we have had
no exact words that explain the connection between this gapped excitation and
the SO$(3)$ symmetry in spinor BECs. Inspired by this problem, we here
encounter a general question; (ii) What theorem is behind those gapped
excitations and the symmetry in the case where symmetry breaking external
fields are remained?

The HP theorem is thus now highly desired for the spin-$f$ spinor BEC with
broken U$(1)$$\times$SO$(3)$ symmetry for the needs of recent experiments as
well as theories. In the absence of a magnetic field, the problem (i) for
spontaneously broken continuous symmetry case may be solved by directly
applying the Goldstone–Salam–Weinberg equality Goldstone  _et al._ (1962).
However, since this equality is derived by using the invariance of the
effective action under the symmetry transformation, this proof is not directly
applied to the spinor BECs in the presence of a magnetic field that breaks
SO$(3)$ symmetry. We need other routes to derive a generalized identity in the
case where symmetry breaking external fields are remained.

We here exactly clarified the HP theorem with broken U$(1)$$\times$SO$(N)$ or
U$(1)$$\times$SU$(N)$ symmetry in the presence of SO$(N)$ or SU$(N)$ symmetry
breaking external fields. This generalized HP theorem, the main result in this
Letter, is summarized in the following way:

$\displaystyle
G^{-1}(0)\mathfrak{G}_{\alpha}\Phi=[\mathfrak{G}_{\alpha},H^{\prime}]\Phi+G^{-1}(0)\Gamma_{[\mathfrak{G}_{\alpha},H^{\prime}]}(0),$
(1)

where $G^{-1}(0)$ is the inverse Green’s function in the static and low-
momentum limits, $\mathfrak{G}_{\alpha}$ a generator of U$(1)$$\times$SO$(N)$
or U$(1)$$\times$SU$(N)$ symmetry, $\Phi$ the order parameter, $H^{\prime}$
the SO$(N)$ or SU$(N)$ symmetry breaking external fields,
$\Gamma_{[\mathfrak{G}_{\alpha},H^{\prime}]}(0)$ the three-point correlation
function in the static and low-momentum limits including the commutation
relation $[\mathfrak{G}_{\alpha},H^{\prime}]$. These quantities are given in
the Nambu-space, and we have omitted $\hbar$ for simplicity. The effect of the
symmetry breaking external fields is included through the non-commutativity
between the symmetry generator and the symmetry breaking external fields. The
identity (1) serves to deductively organize the HP theorem for not only
spin-$f$ spinor BECs with the broken U$(1)$$\times$SO$(3)$ symmetry, but also
various kinds of superfluids, widely ranging from condensed matter physics to
particle physics, with broken more complex continuous symmetry, which answers
two questions (i) and (ii) raised above.

We prove the identity (1) by making use of the linear response theory.
Consider that the action $S_{0}[\psi_{i}^{*},\psi_{i}]$ has a
U$(1)$$\times$SO$(N)$ or U$(1)$$\times$SU$(N)$ symmetry, which may include
two-body and higher-body interactions of bosonic fields with internal degrees
of freedom $\psi_{i=1,\cdots,r}$. Let $\mathfrak{g}_{\alpha=1,\cdots,n}$ be
generators of the symmetry group of the action $S_{0}$, where $n$ is the
dimension of this symmetry group. The number of the internal degrees of
freedom $r$ corresponds to the dimension of the representation of the
generator $\mathfrak{g}_{\alpha}$, and it is convenient to introduce the Nambu
representation
$\Psi(x)\equiv(\psi_{1}(x),\cdots,\psi_{r}(x),\psi_{1}^{*}(x),\cdots,\psi_{r}^{*}(x))^{\rm
T}$ in real space and imaginary time with $x\equiv({\bf r},\tau)$. Since the
spontaneously broken continuous symmetry is described by source currents
$J(x)\equiv(J_{1}(x),\cdots,J_{r}(x),J_{1}^{*}(x),\cdots,J_{r}^{*}(x))^{\rm
T}$, we consider the action $S_{J}[\psi_{i}^{*},\psi_{i}]\equiv\frac{1}{2}\int
dx[J^{{\dagger}}(x)\Psi(x)+\Psi^{\dagger}(x)J(x)]$, where $\int
dx\equiv\int_{0}^{\hbar\beta}d\tau\int d{\bf r}$. We also introduce the action
$S^{\prime}$ that includes the SO$(N)$ or SU$(N)$ symmetry breaking external
fields $h_{ij}^{\prime}$, given in the form
$S^{\prime}[\psi_{i}^{*},\psi_{i}]\equiv\sum\limits_{ij=1}^{r}\int dx\int
dx^{\prime}\psi_{i}^{*}(x)h_{ij}^{\prime}(x-x^{\prime})\psi_{j}(x^{\prime})$,
where $h_{ij}^{\prime}(x-x^{\prime})\equiv h_{ij}^{\prime}({\bf r}-{\bf
r}^{\prime})\delta(\tau-\tau^{\prime})=h_{ij}^{\prime}({\bf r}^{\prime}-{\bf
r})\delta(\tau-\tau^{\prime})$. The action $S^{\prime}$ may be rewritten as
$S^{\prime}=\int dx\int
dx^{\prime}\Psi^{\dagger}(x)H^{\prime}(x-x^{\prime})\Psi(x^{\prime})/2$, where

$\displaystyle H^{\prime}(x-x^{\prime})=\begin{pmatrix}h^{\prime}({\bf r}-{\bf
r}^{\prime})&0\\\ 0&h^{\prime*}({\bf r}-{\bf
r}^{\prime})\end{pmatrix}\delta(\tau-\tau^{\prime}).$ (2)

The total action $S\equiv S_{0}+S^{\prime}+S_{J}$ has the following symmetry.
Let $\mathfrak{G}_{\alpha}$ be a generator of the symmetry group of $S_{0}$ in
the Nambu-space, given by

$\displaystyle\mathfrak{G}_{\alpha}\equiv\mathcal{F}(\mathfrak{g}_{\alpha})\equiv\begin{pmatrix}{\mathfrak{g}}_{\alpha}&0\\\
0&-{\mathfrak{g}}_{\alpha}^{*}\end{pmatrix}.$ (3)

The action $S$ is invariant under a unitary transformation at once
$\Psi(x)\to{\mathcal{U}}_{\alpha}\Psi(x)$,
$J(x)\to{\mathcal{U}}_{\alpha}J(x)$, as well as
$H^{\prime}(x-x^{\prime})\to{\mathcal{U}}_{\alpha}H^{\prime}(x-x^{\prime}){\mathcal{U}}_{\alpha}^{{\dagger}}$,
where
${\mathcal{U}}_{\alpha}\equiv\exp(i\mathfrak{G}_{\alpha}\epsilon_{\alpha})$
with a parameter $\epsilon_{\alpha}$. The action $S$ is, on the other hand,
not invariant under the transformation
$\Psi(x)\to{\mathcal{U}}_{\alpha}\Psi(x)$ alone, owing to the presence of the
symmetry breaking external fields $J$ and $H^{\prime}$.

We study the linear response of the bosonic field $\Psi$ in the case where the
gauge of the symmetry breaking external fields is slightly but globally
turned, such that $J(x)\to{\mathcal{U}}_{\alpha}J(x)$, and
$H^{\prime}(x-x^{\prime})\to{\mathcal{U}}_{\alpha}H^{\prime}(x-x^{\prime}){\mathcal{U}}_{\alpha}^{{\dagger}}$,
where ${\mathcal{U}}_{\alpha}\simeq 1+i\epsilon_{\alpha}\mathfrak{G}_{\alpha}$
with an infinitesimally small parameter $\epsilon_{\alpha}$. The total action
then reads $S+\epsilon_{\alpha}S_{\epsilon_{\alpha}}$, where
$S_{\epsilon_{\alpha}}\equiv
S_{\epsilon_{\alpha},J}+S_{\epsilon_{\alpha},H^{\prime}}$ with

$\displaystyle S_{\epsilon_{\alpha},J}\equiv$ $\displaystyle\frac{i}{2}\int
dx[\Psi^{\dagger}(x)\mathfrak{G}_{\alpha}J(x)-J^{\dagger}(x)\mathfrak{G}_{\alpha}\Psi(x)],$
(4) $\displaystyle S_{\epsilon_{\alpha},H^{\prime}}\equiv$
$\displaystyle\frac{i}{2}\int dx\int
dx^{\prime}\Psi^{{\dagger}}(x)[\mathfrak{G}_{\alpha},H^{\prime}(x-x^{\prime})]\Psi(x^{\prime}).$
(5)

The linear response of the bosonic field with respect to the input action
$\epsilon_{\alpha}S_{\epsilon_{\alpha}}$ may be given by
$\delta\langle\Psi(x)\rangle_{J^{{\dagger}},J,\epsilon_{\alpha}}\equiv\langle\Psi(x)\rangle_{J^{{\dagger}},J,\epsilon_{\alpha}}-\langle\Psi(x)\rangle_{J^{{\dagger}},J}$.
Here, the expectation value of $O(x)$ is defined by $\langle
O(x)\rangle_{J^{{\dagger}},J,\epsilon_{\alpha}}\equiv
Z_{J^{\dagger},J,\epsilon_{\alpha}}^{-1}\int{\mathcal{D}}[\psi_{j}^{*},\psi_{j}]O(x)\exp(-S-\epsilon_{\alpha}S_{\epsilon_{\alpha}})$,
where
$Z_{J^{\dagger},J,\epsilon_{\alpha}}\equiv\int{\mathcal{D}}[\psi_{j}^{*},\psi_{j}]\exp(-S-\epsilon_{\alpha}S_{\epsilon_{\alpha}})$
is the partition function. We have also defined the notation $\langle
O(x)\rangle_{J^{{\dagger}},J}\equiv\langle
O(x)\rangle_{J^{{\dagger}},J,\epsilon_{\alpha}=0}$.

Within the first order of $\epsilon_{\alpha}$, we obtain the relation

$\displaystyle\delta\langle\Psi(x)\rangle_{J^{{\dagger}},J}=i\epsilon_{\alpha}\int
dx^{\prime}G(x-x^{\prime})\mathfrak{G}_{\alpha}J(x)$ $\displaystyle+$
$\displaystyle i\epsilon_{\alpha}\int dx^{\prime}\int
dx^{\prime\prime}G(x-x^{\prime})[\mathfrak{G}_{\alpha},H^{\prime}(x^{\prime}-x^{\prime\prime})]\langle\Psi(x^{\prime\prime})\rangle_{J^{{\dagger}},J}$
$\displaystyle+$ $\displaystyle i\epsilon_{\alpha}\int dx^{\prime}\int
dx^{\prime\prime}\Gamma_{[\mathfrak{G}_{\alpha},H^{\prime}]}(x,x^{\prime},x^{\prime\prime}).$
(6)

Here, the Green’s function $G(x-x^{\prime})$ is given by
$G(x-x^{\prime})=-\langle\phi(x)\phi^{\dagger}(x^{\prime})\rangle_{J^{{\dagger}},J}$,
where $\phi(x)=\Psi(x)-\langle\Psi(x)\rangle_{J^{{\dagger}},J}$ with
$\langle\phi(x)\rangle_{J^{{\dagger}},J}=0$. The three-point correlation
function $\Gamma_{[\mathfrak{G}_{\alpha},H^{\prime}]}$, including the
commutation relation $[\mathfrak{G}_{\alpha},H^{\prime}]$, is defined by

$\displaystyle\Gamma_{[\mathfrak{G}_{\alpha},H^{\prime}]}(x,x^{\prime},x^{\prime\prime})\equiv-\frac{1}{2}\left\langle\phi(x)\phi^{\dagger}(x^{\prime})[\mathfrak{G}_{\alpha},H^{\prime}(x^{\prime}-x^{\prime\prime})]\phi(x^{\prime\prime})\right\rangle_{J^{{\dagger}},J}.$
(7)

In the momentum and bosonic Matsubara frequency space with $k=({\bf
p},i\omega_{n})$, the identity reads

$\displaystyle\delta\langle\Psi(k)\rangle_{J^{{\dagger}},J}=i\epsilon_{\alpha}G(k)\mathfrak{G}_{\alpha}J(k)$
$\displaystyle+$ $\displaystyle
i\epsilon_{\alpha}G(k)[\mathfrak{G}_{\alpha},H^{\prime}]\langle\Psi(k)\rangle_{J^{{\dagger}},J}+i\epsilon_{\alpha}\Gamma_{[\mathfrak{G}_{\alpha},H^{\prime}]}(k),$
(8)

where we have assumed a contact type external field
$H^{\prime}(x-x^{\prime})\equiv H^{\prime}\delta({\bf r}-{\bf
r}^{\prime})\delta(\tau-\tau^{\prime})$. Multiply (8) by $G^{-1}(k)$ with
taking the static limit $i\omega_{n}=0$, and then take the zero momentum limit
${\bf p}\to 0$. If the gauge of the symmetry breaking fields is statically
turned, then the gauge of the output bosonic field is statically dragged with
the same amount as that in the input action, which provides
$\delta\langle\Psi\rangle_{J^{{\dagger}},J,\epsilon_{\alpha}}=i\epsilon_{\alpha}\mathfrak{G}_{\alpha}\langle\Psi\rangle_{J^{{\dagger}},J}$.
Finally make the source currents infinitesimally small $J,J^{{\dagger}}\to 0$.
We have thus proved the HP theorem (1), where
$\Phi=\lim\limits_{J,J^{{\dagger}}\to 0}\lim\limits_{{\bf p}\to
0}\langle\Psi({\bf p},i\omega_{n}=0)\rangle_{J^{{\dagger}},J}$.

The three-point correlation function
$\Gamma_{[\mathfrak{G}_{\alpha},H^{\prime}]}$ in the identity (1) originates
from the fact that the symmetry breaking potential $H^{\prime}$ is of one-
body. If the SO$(N)$ or SU$(N)$ symmetry breaking action $S^{\prime}$ is
extended to an $n$-body potential, then the HP theorem may include
$(2n+1)$-point correlation functions.

It is inevitable to discuss cases in the absence of the symmetry breaking
fields $H^{\prime}$. The HP theorem is given by
$G^{-1}(0)\mathfrak{G}_{\alpha}\Phi=0$, which is identical to the
Goldstone–Salam–Weinberg identity Goldstone  _et al._ (1962), where
$\mathfrak{G}_{\alpha}\Phi$ may be an eigenvector of the zero eigenvalue. In
the spontaneously broken U$(1)$ symmetry case in a scalar BEC, the HP theorem
is given by $G^{-1}(0)\mathfrak{I}\Phi=0$, with
$\mathfrak{I}\equiv\mathcal{F}(1)=\sigma_{3}$ and
$\Phi=(\Phi_{0},\Phi_{0}^{*})^{\rm T}$. Here, $\sigma_{i(=1,2,3)}$ are Pauli
matrices. The inverse Green’s function is given by
$G^{-1}(k)=G_{0}^{-1}(k)-\Sigma(k)$, where $\Sigma(k)$ is the ($2\times 2$)
matrix self-energy, and $G_{0}^{-1}(k)\equiv
i\omega_{n}\sigma_{3}-(\epsilon_{\bf p}-\mu)$ with the chemical potential
$\mu$, and $\epsilon_{\bf p}\equiv{\bf p}^{2}/(2m)$. The HP theorem is thus
given by the well known form $\Sigma^{11(22)}(0)-\Sigma^{12(21)}(0)=\mu$,
where $\Sigma^{11(22)}(0)$ and $\Sigma^{12(21)}(0)$ are the diagonal and off-
diagonal self-energies, respectively. Here, we have assumed the order
parameter to be real $\Phi_{0}=\Phi_{0}^{*}$. This is consistent with the
proof of the HP theorem for the scalar BEC, which was done by Hohenberg and
Martin Hohenberg and Martin (1965); Rickayzen (1980).

We also briefly comment on another HP theorem in the spontaneously broken
U$(1)$ symmetry. In superfluid Fermi gases, it has been known that within the
Gaussian pair fluctuation approximation, a collective mode of the Cooper-pair
fluctuation becomes gapless when the theory satisfies the gap equation He
(2016). The physics behind this is also the HP theorem for the BCS-BEC
crossover. This Hamiltonian with a contact interaction $U(>0)$ is given by
$H=\int d{\bf
r}\sum_{\sigma=\uparrow,\downarrow}\Psi_{\sigma}^{{\dagger}}({\bf
r})\left(-\frac{\hbar^{2}\nabla^{2}}{2m}-\mu\right)\Psi_{\sigma}({\bf
r})-U\int d{\bf r}\Psi_{\uparrow}^{{\dagger}}({\bf
r})\Psi_{\downarrow}^{{\dagger}}({\bf r})\Psi_{\downarrow}({\bf
r})\Psi_{\uparrow}({\bf r})$, where $\Psi_{\sigma}$ is the Grassmann variable
with the pseudo-spin $\sigma=\uparrow,\downarrow$, and $m$ the atomic mass.
Integrating out these fermionic fields after introducing the Hubbard-
Stratonovich transformation with an auxiliary field $\Delta(x)$, we obtain the
effective action $S_{\rm eff}=S_{0}+S_{\rm GF}(\phi,\phi^{*})$ after employing
the stationary phase approximation He (2016), where $S_{0}$ is the stationary
value of the action with the stationary solution
$\Phi\equiv(\Delta_{0},\Delta_{0}^{*})^{\rm T}$. The action including the
Gaussian pair fluctuation is given by $S_{\rm
GF}(\phi,\phi^{*})=-\sum_{k}\phi^{{\dagger}}(k)\Gamma^{-1}(k)\phi(k)/2$, where
$\phi(k)\equiv(\Delta(k),\Delta^{*}(k))^{\rm T}-\Phi$ is fluctuations of the
order parameter, and $\Gamma^{-1}(k)$ the vertex function given by

$\displaystyle\Gamma^{-1}(k)=$
$\displaystyle\begin{pmatrix}\chi^{-+}(k)&\chi^{--}(k)\\\
\chi^{++}(k)&\chi^{+-}(k)\end{pmatrix},$ (9)

with the correlation function
$\chi^{s,s^{\prime}}(k)\equiv-U^{-1}\delta_{s,-s^{\prime}}-\beta^{-1}\sum_{k^{\prime}}{\rm
Tr}\left[\sigma_{s}{\mathcal{G}}(k^{\prime}+k)\sigma_{s^{\prime}}{\mathcal{G}}(k^{\prime})\right]$
for $s,s^{\prime}=\pm$ He (2016). Here, we have taken the system volume to be
unity, and ${\mathcal{G}}(k)\equiv i\bar{\omega}_{n}-(\epsilon_{\bf
p}-\mu)\sigma_{3}+\Delta_{0}\sigma_{1}$ is the mean-field Green’s function,
where $i\bar{\omega}_{n}$ is the fermionic Matsubara frequency, and
$\sigma_{\pm}\equiv(\sigma_{1}\pm\sigma_{2})/2$. The HP theorem is given by
$\Gamma^{-1}(0)\mathfrak{I}\Phi=0$ Haussmann  _et al._ (2007), which provides

$\displaystyle\chi^{ss}(0)-\chi^{s,-s}(0)=$ $\displaystyle 0,$ (10)

where we have assumed the order parameter to be real
$\Delta_{0}=\Delta_{0}^{*}$. This identity (10) may be reduced into the gap
equation $0=U^{-1}-\sum_{{\bf p}}\tanh(\beta E_{\bf p}/2)/(2E_{\bf p})$ in the
Gaussian pair fluctuation approximation, where $E_{\bf
p}\equiv\sqrt{(\epsilon_{\bf p}-\mu)^{2}+\Delta_{0}^{2}}$. Since relations
$\chi^{s,s^{\prime}}(0)=\chi^{-s,-s^{\prime}}(0)$ holds, we obtain ${\rm
det}\Gamma^{-1}(0)=0$, which provides the gapless excitation in the Cooper-
pair fluctuations.

We next discuss the relation between the identity (1) and the Bogoliubov
approximation in the spin-$f$ spinor BEC in the presence of a magnetic field.
In the Bogoliubov approximation, the action is reduced into the quadratic form
given by $S_{\rm B}=S_{0}-\sum_{k}\phi^{{\dagger}}(k)G_{\rm
B}^{-1}(k)\phi(k)/2$, where $S_{0}$ is the stationary value of the action.
Here, the fluctuation in the Nambu-space $\phi(k)$ is given by
$\phi(k)\equiv(\phi_{f}(k),\cdots,\phi_{-f}(k),\phi_{f}^{*}(-k),\cdots,\phi_{-f}^{*}(-k))^{\rm
T}$, the dimension of which is $4f+2$, and $G_{\rm
B}^{-1}(k)=i\omega_{n}\sigma_{3}\otimes\mathbb{I}_{2f+1}-H_{\bf p}^{\rm B}$ is
the Green’s function in the Bogoliubov approximation, where
$\mathbb{I}_{2f+1}$ is the identity matrix of dimension $2f+1$, and the
Hamiltonian has the following form Kawaguchi and Ueda (2012)

$\displaystyle H_{\bf p}^{\rm B}=$ $\displaystyle\begin{pmatrix}H^{(0)}_{\bf
p}+H^{(1)}&H^{(2)}\\\ [H^{(2)}]^{*}&[H^{(0)}_{-\bf
p}+H^{(1)}]^{*}\end{pmatrix}.$ (11)

The dimension of $H^{(0)}_{\bf p}$ as well as $H^{(1,2)}$ is $2f+1$. Since the
expectation value in the Bogoliubov approximation is organized by the Gaussian
integrals, the three-point correlation function
$\Gamma_{[\mathfrak{G}_{\alpha},H^{\prime}]}(0)$ is exactly absent in this
level, and the identity (1) in the Bogoliubov approximation is reduced into
the following form

$\displaystyle G_{\rm
B}^{-1}(0)\mathfrak{G}_{\alpha}\Phi=[\mathfrak{G}_{\alpha},H^{\prime}]\Phi.$
(12)

In the spin-$f$ spinor BEC, a generator $\mathfrak{G}_{\alpha}$ corresponds to
$\mathfrak{I}=\mathcal{F}(\mathbb{I}_{2f+1})$ as well as
$\mathfrak{F}_{i}=\mathcal{F}(\mathfrak{f}_{i})$ for $i=x,y,z$, where
$\mathfrak{f}_{x,y,z}$ are spin matrices, whose dimension of the
representation is $2f+1$.

We apply the identity (12) to spin-$1$ and $2$ spinor BECs in the Bogoliubov
approximation. The Hamiltonian $H^{(0)}_{\bm{p}}$ is given by $H^{(0)}_{\bf
p}\equiv(\epsilon_{\bf p}-\mu)\mathbb{I}_{2f+1}+h^{\prime}$, where
$h^{\prime}\equiv-p\mathfrak{f}_{z}+q\mathfrak{f}_{z}^{2}$ represents the
linear and quadratic Zeeman effects that break the SO$(3)$ symmetry. Here, $p$
is the linear Zeeman energy, and $q$ the quadratic Zeeman energy. In the
spin-1 BEC, we have $H^{(1)}\equiv
n[c_{0}(\rho+\mathbb{I}_{2f+1})+c_{1}\sum_{i=x,y,z}(\mathfrak{f}_{i}\rho\mathfrak{f}_{i}+f_{i}\mathfrak{f}_{i})]$,
and $H^{(2)}\equiv
n[c_{0}\tilde{\rho}+c_{1}\sum_{i=x,y,z}\mathfrak{f}_{i}\tilde{\rho}\mathfrak{f}_{i}^{\rm
T}]$, where $n$ is the total density,
$f_{i}\equiv\zeta^{{\dagger}}\mathfrak{f}_{i}\zeta$,
$\rho\equiv\zeta\zeta^{{\dagger}}$, $\tilde{\rho}\equiv\zeta\zeta^{\rm T}$
with $\zeta=(\Phi_{f},\cdots,\Phi_{-f})^{\rm T}/\sqrt{n}$, and $c_{0,1}$ are
coupling constants of the spin-independent and spin-dependent interaction,
respectively Kawaguchi and Ueda (2012). In the spin-2 BEC, we have
$H^{(1)}\equiv
n[c_{0}(\rho+\mathbb{I}_{2f+1})+c_{1}\sum_{i=x,y,z}(\mathfrak{f}_{i}\rho\mathfrak{f}_{i}+f_{i}\mathfrak{f}_{i})+2c_{2}P_{0}\rho
P_{0}]$, and $H^{(2)}\equiv
n[c_{0}\tilde{\rho}+c_{1}\sum_{i=x,y,z}\mathfrak{f}_{i}\tilde{\rho}\mathfrak{f}_{i}^{\rm
T}+c_{2}a_{00}P_{0}]$, where $c_{2}$ is the coupling constant of the spin-
singlet pair interaction, $a_{00}\equiv\zeta^{\rm T}P_{0}\zeta$ the spin-
singlet pair amplitude per particle, and $P_{0}$ the Clebsch-Gordan
coefficient matrix, given by $(P_{0})_{m_{1},m_{2}}=\langle
0,0|f,m_{1};f,m_{2}\rangle$ for $f=2$ Kawaguchi and Ueda (2012). Taking each
corresponding order parameter Kawaguchi and Ueda (2012), we can analytically
confirm the consistency between the HP theorem (12) and the Bogoliubov theory
in the static and low-momentum limits in the ferromagnetic, antiferromagnetic,
polar, as well as broken-axisymmetry phases in the spin-1 BEC. In the spin-2
BEC with each corresponding order parameter Kawaguchi and Ueda (2012), we can
also analytically confirm that the identity (12) exactly holds in the
ferromagnetic F2 and F1 phases, the uniaxial and biaxial nematic (UN and BN)
phases, the cyclic and C2,3,4 phases, as well as the D${}_{2}^{\prime}$ phase.

In the UN and BN phases in the absence of a magnetic field, quasi-Nambu-
Goldstone modes emerge in the Bogoliubov approximation, for the mean-field
ground state energy has the hidden U$(1)$$\times$SO$(5)$ symmetry Uchino  _et
al._ (2010). We can find that the identity (12) also holds for those quasi-
Nambu-Goldstone modes, given in the form $G_{\rm
B}^{-1}(0)\mathfrak{F}_{13,35}\Phi=[\mathfrak{F}_{13,35},H^{\prime}]\Phi$ with
$\mathfrak{F}_{13,35}\equiv\mathcal{F}(F_{13,35})$, where $F_{13,35}$ is
generators of the SO$(5)$ symmetry Uchino  _et al._ (2010).

We finally study the exact HP theorem for spin-$1$ and $2$ spinor BECs in the
presence of a magnetic field. The three point correlation function is given by
the vector form
$\Gamma_{[\mathfrak{G}_{\alpha},H^{\prime}]}(p)\equiv\Gamma_{\mathfrak{G}_{\alpha}}(p)=(\Gamma_{\mathfrak{G}_{\alpha},f}(p),\cdots,\Gamma_{\mathfrak{G}_{\alpha},-f}(p),\Gamma_{\mathfrak{G}_{\alpha},f}^{*}(-p),\cdots,\Gamma_{\mathfrak{G}_{\alpha},-f}^{*}(-p))$,
where we may give
$\Gamma_{\mathfrak{G}_{\alpha},i}(p)\equiv\sum_{jk}\gamma_{ijk}(p)[\mathfrak{g}_{\alpha},h^{\prime}]_{jk}$
and
$\Gamma_{\mathfrak{G}_{\alpha},i}^{*}(-p)\equiv\sum_{jk}\gamma_{ijk}^{*}(-p)[\mathfrak{g}_{\alpha},h^{\prime}]_{jk}$.
Here,
$\gamma_{ijk}(p)\equiv-\sum_{q}\langle\phi_{i}(p)\phi_{j}^{*}(q)\phi_{k}(q)\rangle_{J^{{\dagger}},J}$
and
$\gamma_{ijk}^{*}(-p)\equiv-\sum_{q}\langle\phi_{i}^{*}(-p)\phi_{j}^{*}(q)\phi_{k}(q)\rangle_{J^{{\dagger}},J}$
are three-point correlation functions, where we take the limit
$J^{{\dagger}},J\to 0$. Since the commutativity
$[\mathbb{I}_{2f+1},h^{\prime}]=[\mathfrak{f}_{z},h^{\prime}]=0$ holds, the
correlation functions satisfy
$\Gamma_{\mathfrak{I}}(q)=\Gamma_{\mathfrak{F}_{z}}(q)=0$. On the other hand,
$\Gamma_{\mathfrak{F}_{\pm}}(q)$ for
$\mathfrak{F}_{\pm}\equiv\mathfrak{F}_{x}\pm i\mathfrak{F}_{y}$ may not
vanish, and the relation
$\Gamma_{\mathfrak{F}_{\pm},i}(q)=-\Gamma_{\mathfrak{F}_{\mp},i}^{*}(-q)$ is
found to hold for the spin-1 and 2 spinor BECs, where we have used
$\gamma_{ijk}(q)=\gamma_{ikj}^{*}(-q)$. In general, the number of the
correlation function $\gamma_{ijk}$ reaches $(2f+1)^{3}$, for example, $27$
for the spin-1 BEC, and $125$ for the spin-2 BEC. However, since the two-body
interaction in the spinor BEC conserves the total spin of two interacting
atoms, the number of $\gamma_{ijk}$ is drastically suppressed. Since matrices
$[\mathfrak{f}_{\pm},h^{\prime}]$ are sparse, furthermore, the number of
$\gamma_{ijk}$ that directly contributes to $\Gamma_{\mathfrak{F}_{\pm}}(p)$
may be further suppressed.

For example, in the ferromagnetic phase of the spin-$1$ BEC with
$\zeta=(1,0,0)^{\rm T}$, the number of $\gamma_{ijk}$ that conserves the total
spin of two interacting atoms is $6$, among which
$\Gamma_{\mathfrak{F}_{+},}(p)$ is given by the linear combination of
$\gamma_{0,+1,0}^{*}$ as well as $\gamma_{0,0,-1}^{*}$, and
$\Gamma_{\mathfrak{F}_{-}}(p)$ is $\gamma_{0,-1,0}$ as well as
$\gamma_{0,0,+1}$. We thus reduce the functions $\Gamma_{\mathfrak{F}_{\pm}}$
into
$\Gamma_{\mathfrak{F}_{+}}=(0,0,0,0,\Gamma_{\mathfrak{F}_{+},0}^{*},0)^{\rm
T}$ and
$\Gamma_{\mathfrak{F}_{-}}=(0,\Gamma_{\mathfrak{F}_{-},0},0,0,0,0)^{\rm T}$.
Because of the total spin conservation of two interacting atoms, the self-
energy matrix is also sparse Phuc  _et al._ (2013a). The identity (1) then
provides the following relations

$\displaystyle\Sigma_{+1,+1}^{11(22)}(0)-\Sigma_{+1,+1}^{12(21)}(0)-\mu-p+q=0,$
(13)
$\displaystyle\Sigma_{0,0}^{11(22)}(0)-\mu-p+q=\frac{\Gamma_{\mathfrak{S}_{-},0}(0)}{\sqrt{2n}}[\Sigma_{0,0}^{11(22)}(0)-\mu].$
(14)

For the representation of the self energy $\Sigma_{i,j}^{ab}$, two subscripts
represent the spin index, and superscripts distinguishes normal and anomalous
self-energies. By solving the Dyson equation $G^{-1}=G_{0}^{-1}-\Sigma$, where
$G_{0}^{-1}\equiv
i\omega_{n}\sigma_{3}\otimes\mathbb{I}_{2f+1}-\mathbb{I}_{2}\otimes H_{\bf
p}^{(0)}$, with (13) and (14), we find that [$G_{+1,+1}^{11(12)}(0)]^{-1}=0$,
where we have used the compact notation $G_{i,j}^{ab}(0)\equiv\lim_{{\bf p}\to
0}G_{i,j}^{ab}({\bf p},i\omega_{n}=0)$. This indicates the presence of a
gapless excitation in $G_{+1,+1}^{11(12)}$, regardless of the presence of a
magnetic field. We also find identities

$\displaystyle G_{0,0}^{11}(0)=$
$\displaystyle-\frac{1-\Gamma_{\mathfrak{S}_{-},0}(0)/\sqrt{2n}}{p-q},$ (15)
$\displaystyle G_{-1,-1}^{11}(0)=$
$\displaystyle-\frac{1}{\Sigma_{-1,-1}^{11}(0)-\mu+p+q}.$ (16)

The ferromagnetic transverse spin mode given by $G_{0,0}^{11}$ has an energy
gap, exactly given by $p-q$, which may exactly turn gapless in the absence of
a magnetic field. On the other hand, the quadrupolar spin mode given by
$G_{-1,-1}^{11}$ is gapful, whose energy gap is
$\Sigma_{-1,-1}^{11}(0)-\mu+p+q$.

In the uniaxial nematic phase of the spin-$2$ BEC with $\zeta=(0,0,1,0,0)^{\rm
T}$, for another example, the number of $\gamma_{ijk}$ that conserve the total
spin of two interacting atoms is $19$, among which
$\Gamma_{\mathfrak{F}_{+},}(p)$ is given by the linear combinations of
$\gamma_{+1,i+1,i}$ as well as $\gamma_{-1,i+1,i}^{*}$, and
$\Gamma_{\mathfrak{F}_{-}}(p)$ is $\gamma_{+1,i-1,i}^{*}$ as well as
$\gamma_{-1,i-1,i}$. We thus reduce the functions
$\Gamma_{\mathfrak{F}_{\pm}}$ into
$\Gamma_{\mathfrak{F}_{+}}=(0,\Gamma_{\mathfrak{F}_{+},+1},0,0,0,0,0,0,\Gamma_{\mathfrak{F}_{+},-1}^{*},0)$
and
$\Gamma_{\mathfrak{F}_{-}}=(0,0,0,\Gamma_{\mathfrak{F}_{-},-1},0,0,\Gamma_{\mathfrak{F}_{-},+1}^{*},0,0,0)$.
With the use of the space form of the self-energy Phuc  _et al._ (2013b), the
generalized HP theorem (1) provides the following identities

$\displaystyle\Sigma_{0,0}^{11(22)}(0)-\Sigma_{0,0}^{12(21)}(0)-\mu=0,$ (17)
$\displaystyle\Sigma_{\pm 1,\pm 1}^{11(22)}(0)-\Sigma_{\pm 1,\mp
1}^{12(21)}(0)-\mu=\frac{A_{\pm}^{11(22)}-B_{\pm}^{12(21)}}{\sqrt{6n}},$ (18)

where $A_{\pm}^{11(22)}\equiv\Gamma_{\mathfrak{S}_{\pm},\pm 1}(0)[\Sigma_{\pm
1,\pm 1}^{11(22)}(0)-\mu\mp p+q]$ and
$B_{\pm}^{12(21)}\equiv\Gamma_{\mathfrak{S}_{\mp},\mp 1}(0)\Sigma_{\pm 1,\mp
1}^{12(21)}(0)$. By solving the Dyson equation with (17) and (18), we find
that $[G_{0,0}^{11(12)}(0)]^{-1}=0$, which indicates the presence of a gapless
excitation. We also find that excitations given by $G_{\pm 2,\pm 2}^{11}(0)$
and $G_{\pm 2,\mp 2}^{12}(0)$ are always gapped. On the other hand,
excitations given by $G_{\pm 1,\pm 1}^{11}$ as well as $G_{\pm 1,\mp 1}^{12}$
are found to be gapped in the presence of a magnetic field, but those exactly
turn gapless in the absence of a magnetic field, when
$\Gamma_{\mathfrak{S}_{\pm},\pm 1}(0)=p=q=0$. The HP theorem has been also
simply determined in the antiferromagnetic and polar phases in the spin-1 BEC,
as well as the ferromagnetic F+2 and F+1, biaxial nematic, as well as cyclic
phases in the spin-2 BEC Watabe (tion).

In summary, we have exactly clarified the Hugenholtz-Pines (HP) theorem with
the broken U$(1)$$\times$SO$(N)$ or U$(1)$$\times$SU$(N)$ symmetry in the
presence of the SO$(N)$ or SU$(N)$ symmetry breaking external fields. By
applying this generalized theorem to the spinor BEC with broken
U$(1)$$\times$SO$(3)$ symmetry, we have deductively organized the HP theorem
for those systems even in the presence of a magnetic field. This theorem may
play a criterial role in developing theories for various kinds of superfluids,
widely ranging from condensed matter physics to particle physics, in the case
where symmetry breaking external fields are remained. Since the derivation is
general, this identity may be extended to ordered phases other than BECs in
the presence of symmetry breaking external fields.

###### Acknowledgements.

The author has been supported by JSPS KAKENHI Grant No. JP16K17774.

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