# Evaluating Text Summaries Generated by Large Language Models Using OpenAI’s
GPT

Hassan Shakil
University of Colorado - Colorado Springs
Colorado Springs, CO
<EMAIL_ADDRESS>
Atqiya Munawara Mahi
University of Massachusetts Lowell

Phuoc Nguyen
University of Kansas
<EMAIL_ADDRESS>
Zeydy Ortiz
DataCrunch Lab, LLC
Cary, NC
<EMAIL_ADDRESS>
Mamoun T. Mardini
Department of Health Outcomes and Biomedical Informatics, University of
Florida
<EMAIL_ADDRESS>

###### Abstract

This research examines the effectiveness of OpenAI’s GPT models as independent
evaluators of text summaries generated by six transformer-based models from
Hugging Face: DistilBART, BERT, ProphetNet, T5, BART, and PEGASUS. We
evaluated these summaries based on essential properties of high-quality
summary—conciseness, relevance, coherence, and readability—using traditional
metrics such as ROUGE and Latent Semantic Analysis (LSA). Uniquely, we also
employed GPT not as a summarizer but as an evaluator, allowing it to
independently assess summary quality without predefined metrics. Our analysis
revealed significant correlations between GPT evaluations and traditional
metrics, particularly in assessing relevance and coherence. The results
demonstrate GPT’s potential as a robust tool for evaluating text summaries,
offering insights that complement established metrics and providing a basis
for comparative analysis of transformer-based models in natural language
processing tasks.

_Keywords_ Text Summarization $\cdot$ Evaluation $\cdot$ GPT $\cdot$ ChatGPT
$\cdot$ Large Language Models $\cdot$ Transformer Models

##  1 Introduction

In the contemporary era characterized by a deluge of data, the intelligence
community faces the challenge of information overload, needing to process vast
amounts of information swiftly and effectively. The ability to generate
succinct, clear, and actionable summaries from diverse data sources is
crucial, as it often determines the success of strategic objectives in this
information-rich environment. As the demand for systems capable of automating
large-scale text summarization without compromising on quality or relevance
intensifies, the role of such technologies becomes increasingly critical Liu
and Lapata (2019).

Text summarization, a pivotal task within Natural Language Processing (NLP),
has found widespread application across various domains, including news
aggregation and the distillation of extensive documents into manageable
summaries. The exponential growth in data underscores the utility of text
summarization in enhancing content accessibility and comprehension, thus
facilitating more efficient navigation through information landscapes Chouikhi
and Alsuhaibani (2022).

Recent advancements in NLP have been dominated by transformer models, renowned
for their ability to discern intricate textual relationships. Introduced by
Vaswani et al., these models utilize self-attention mechanisms to capture
contextual relationships within text, representing a significant evolution
from traditional sequence-to-sequence approaches Vaswani et al. (2017).
Despite the uniform architecture, transformer models vary considerably in
their specific capabilities and performance characteristics.

This study focuses on the comparative analysis of different transformer-based
models for text summarization, utilizing a benchmark dataset. Specifically, we
examine the efficacy of Open AI’s GPT models in evaluating summaries generated
by six prominent transformer models from Hugging
Face111https://huggingface.co/, including DistilBART Lewis et al.
(2019)222https://huggingface.co/sshleifer/distilbart-cnn-12-6, BERT Devlin et
al. (2018), ProphetNet Qi et al. (2020), T5 Raffel et al. (2020), BART Lewis
et al. (2019), and PEGASUS Zhang et al. (2020). The evaluation metrics will
focus on key attributes of effective intelligence reporting: conciseness,
relevance, coherence, and readability. Established evaluation methods such as
ROUGE Barbella and Tortora (2022), Latent Semantic Analysis (LSA) Dumais
(2004), Flesch-Kincaid Flesch (2007), and Compression Ratio Wang et al. (2019)
will be employed to quantitatively assess the generated summaries.

Innovatively, this study also explores the application of GPT not as a
summarizer, but as an evaluator, assessing the summaries independently without
predefined metrics. This novel approach aims to juxtapose the quality of
summaries generated by different models, leveraging the capabilities of GPT to
offer insights into the efficacy of current summarization technologies.
Through this research, we aim to bridge the gap between traditional summary
evaluation techniques and the cutting-edge developments in AI, exploring how
these advancements can further enhance the strategic capabilities of the
intelligence community in a data-driven world.

##  2 Background and Literature Review

Text summarization, the process of distilling the most essential information
from a text into a condensed form, remains a cornerstone of research in
Natural Language Processing (NLP). This technique is widely applicable across
various sectors including news reporting, automated report generation, and
conversational analysis, demonstrating its versatility and utility in managing
extensive information Liu and Lapata (2019); Widyassari et al. (2022).

Historically, text summarization began with rule-based systems that relied on
heuristics, such as selecting sentences containing frequently occurring words,
a method pioneered by Luhn Luhn (1958). Although these initial approaches were
straightforward, they often failed to grasp the complexities and nuanced
meanings of natural language, leading to the exploration of more sophisticated
machine learning strategies Yadav et al. (2022).

Among the earliest machine learning techniques applied to text summarization
were Recurrent Neural Networks (RNNs), including Long Short-Term Memory (LSTM)
units, known for their efficacy in capturing temporal dependencies within text
sequences Hochreiter and Schmidhuber (1997). Despite their capabilities, RNNs
struggled with long sequences, frequently encountering vanishing and exploding
gradient issues that hindered performance. In response, Nallapati et al.
introduced SummaRuNNer, an innovative extractive summarization model utilizing
a hierarchical RNN framework, designed to mitigate these challenges Nallapati
et al. (2017).

The development of transformer models, as conceptualized by Vaswani et al.,
marked a significant breakthrough in NLP Vaswani et al. (2017). Unlike RNNs,
transformers leverage a self-attention mechanism, allowing for a broader and
more effective capture of contextual relationships across text. The
introduction of BERT by Devlin et al. revolutionized text representation
through self-supervised training on extensive text corpora, although it was
primarily optimized for embedding generation rather than direct text
production Devlin et al. (2018).

Subsequent innovations led to the creation of models capable of both
understanding and generating text, tailored specifically for tasks like
summarization. Notably, models such as BART by Lewis et al. and T5 by Raffel
et al. have demonstrated exceptional performance in summarization benchmarks
due to their robust architecture and training methodologies Lewis et al.
(2019); Raffel et al. (2020). Additionally, models like PEGASUS by Zhang et
al. and ProphetNet by Yan et al. introduced novel pretraining objectives that
further enhanced their summarization capabilities Zhang et al. (2020); Qi et
al. (2020).

Moreover, the advent of knowledge distillation techniques, exemplified by
DistilBART, has facilitated the deployment of large transformer models in
environments with limited resources, maintaining high performance while
reducing computational demands Sanh et al. (2019).

This study uses the CNN/Daily Mail dataset to assess how well OpenAI’s GPT
models evaluate text summaries from leading transformer models like
DistilBART, BERT, ProphetNet, T5, BART, and PEGASUS. By applying traditional
metrics such as ROUGE and Latent Semantic Analysis (LSA) alongside innovative
AI-driven evaluations, the research explores the effectiveness of GPT in
enhancing the quality of automated text summarization. This approach provides
insights into the practical application of AI tools in processing extensive
information environments.

##  3 Methodology

This study rigorously evaluated six transformer-based models, depicted in
Figure 1, to assess their efficiency in text summarization tasks. The
evaluation was structured around two principal methods: the GPT-Based
evaluation and the traditional metrics-based evaluation. The inclusion of the
metrics-based method serves as a benchmark to provide a comparative analysis
against the GPT-based assessment.

Figure 1: Generation of summaries.

Each model selected for this study showcases unique architectural and pre-
training innovations that have been demonstrated to enhance performance across
various NLP tasks. These models were examined using the widely recognized
CNN/Daily Mail dataset, a standard benchmark in the text summarization field,
which consists of a substantial compilation of online news articles Hermann et
al. (2015). Following are the transformer-based models that we used to
generate summaries:

  * •

DistilBART (sshleifer/distilbart-cnn-12-6)
333https://huggingface.co/sshleifer/distilbart-cnn-12-6 DistilBART, a
streamlined version of the BART model, employs knowledge distillation, a
technique introduced by Hinton et al. This process involves training the
distilled model (DistilBART) to emulate the output of the original model
(BART), thereby maintaining high performance while enhancing resource
efficiency Hinton et al. (2015); Lewis et al. (2019)

  * •

BERT-small2BERT-small (mrm8488/bert-small2bert-small-finetuned-cnn_daily_mail-
summarization) 444https://huggingface.co/mrm8488/bert-small2bert-small-
finetuned-cnn_daily_mail-summarization/blob/main/README.md?code=true This
model utilizes a smaller version of BERT, a transformer-based model that
introduced the idea of bi-directional training in transformers. BERT has been
a milestone in transformer-based models due to its impressive performance in
various NLP tasks Devlin et al. Devlin et al. (2018)

  * •

ProphetNet (microsoft/prophetnet-large-uncased-cnndm)
555https://huggingface.co/microsoft/prophetnet-large-uncased-cnndm ProphetNet
introduces a novel self-supervised learning objective known as future n-gram
prediction, coupled with an n-stream self-attention mechanism. These features
enable ProphetNet to predict multiple tokens ahead during pre-training,
enhancing its capability for sequence generation tasks Qi et al. (2020).

  * •

T5-small (t5-small) 666https://huggingface.co/t5-small T5, or Text-to-Text
Transfer Transformer, redefines NLP tasks into a text-to-text framework. The
’small’ variant of T5 offers a reduction in model size while retaining
considerable performance, making it suitable for various applications Raffel
et al. (2020).

  * •

BART-large (facebook/bart-large-cnn) 777https://huggingface.co/facebook/bart-
large-cnn BART integrates the use of a denoising autoencoder for pre-training,
distinguishing itself by learning to reconstruct the original text from
corrupted versions. This unique approach has proven effective across a
multitude of downstream NLP tasks Lewis et al. (2019).

  * •

PEGASUS (google/pegasus-cnn_dailymail)
888https://huggingface.co/google/pegasus-cnn_dailymail PEGASUS employs a
unique pre-training objective where specific sentences are omitted, and the
model is tasked with regenerating them. This method has shown remarkable
effectiveness in abstractive text summarization tasks Zhang et al. (2020).

For the implementation of this study, we utilized the Hugging Face’s
Transformers library, which offers a suite of pre-trained models and APIs for
text processing tasks Wolf et al. (2020). The summarization capabilities of
each transformer model were leveraged to process and summarize texts from the
CNN/Daily Mail dataset.

We evaluated the performance of these models by analyzing the quality of
summaries for selected articles, each limited to 512 words or less. This setup
provided a streamlined method for assessing the effectiveness of each model’s
summarization technique.

###  3.1 GPT-Based Evaluation

The evaluation of generated summaries was based on four essential properties
of high-quality summaries: conciseness, relevance, coherence, and readability.
These properties were aligned with specific comparative benchmarks evaluation
metrics, detailed in Table 1. For the evaluation, we utilized the OpenAI API
to interact with the GPT-3.5 model, a process illustrated in Figure 2.

Figure 2: Illustration of the GPT-Based evaluation.

####  3.1.1 Prompts

The evaluation involved distinct types of prompts to assess the summaries,
each designed to measure different aspects of summary quality. Below is an
example of one such prompt used in the study:

  * •

Zero-shot Evaluation: This prompt type is designed to quickly ascertain the
consistency of the summary with the source article. Consistency here refers to
the presence and accuracy of information from the source article within the
summary.

Prompt: Please determine whether the provided summary is consistent with the
corresponding article. Note that “consistency" refers to how much information
included in the summary is present in the source article.
Article: [Article]
Summary: [Summary]
Answer: (yes or no)

  * •

Chain of thoughts Evaluation: This approach requires a step-by-step rationale,
providing a deeper insight into the decision-making process of the model,
enhancing the interpretability of the evaluation.

Prompt: Please determine whether the provided summary is consistent with the
corresponding article. Note that “consistency" refers to how much information
included in the summary is present in the source article.
Article: [Article]
Summary: [Summary]
Answer: Explain your reasoning step by step then answer the question (yes or
no)

  * •

Score Evaluation: This quantitative approach asks for a direct scoring from 0
to 1, allowing for a precise measurement of consistency, facilitating
comparative analysis across different summaries.

Prompt: Score the following summary given the corresponding article with
respect to consistency from 0 to 1 where 1 means most consistent. Note that
“consistency" refers to how much information included in the summary is
present in the source article.
Article: [Article]
Summary: [Summary]
Score:

Table 1: Evaluation of the summaries using traditional metrics. Property |  Description |  Evaluation metric
---|---|---
Conciseness |  A high-quality summary should effectively convey the most important information from the original source while keeping the length brief. |  Compression Ratio - Calculate the ratio of the length of the summary to the length of the original text.
Relevance |  The information presented in the summary should be relevant to the main topic. |  ROUGE (Recall-Oriented Understudy for Gisting Evaluation) - compares n-gram overlap (unigrams, bigrams, etc.) between the summary and the reference summaries or the source text. It assesses how well the summary captures important content.
Coherence |  A good summary should have a clear structure and flow of ideas, making it easy to understand and follow. |  Latent Semantic Analysis (LSA) to assess the logical connections between sentences or concepts.
Readability |  The sentence used in the summary should be clear and easily understandable. |  Flesch-Kincaid to assess the complexity of sentences in the summary.

###  3.2 Metrics-Based Evaluation

For the metrics-based evaluation of the summaries, we employed several
established quantitative metrics to assess their quality. These metrics
included the compression ratio, ROUGE (Recall-Oriented Understudy for Gisting
Evaluation), Latent Semantic Analysis (LSA), and Flesch-Kincaid readability
tests. Each metric provides a unique lens through which the effectiveness of a
summary can be measured. The specific calculations for each metric are
comprehensively detailed in Table 1 and are visually illustrated in Figure 3.
This approach allows for a multifaceted assessment of the summaries,
highlighting their strengths and areas for improvement in terms of brevity,
content fidelity, semantic preservation, and readability.

Figure 3: Illustration of the Metrics-Based evaluation.

###  3.3 Statistical analysis

In our study, we undertook a comparative analysis between conventional metrics
and the scores generated by GPT. To assess the extent of agreement and
association between these two sets of evaluations, we employed Pearson’s
correlation coefficient Sedgwick (2012) a statistical measure widely used to
evaluate the linear relationships between two continuous variables.

The primary objective was to ascertain the correlation between the
conventional metrics (compression ratio, ROUGE, LSA, and Flesch-Kincaid) and
the scores generated by GPT. By executing Pearson’s correlation analysis, we
aimed to uncover insights into how closely these two methods align in their
assessments of summary quality.

The correlation coefficient resulting from Pearson’s analysis provides a
numerical indicator of the relationship’s strength. A higher correlation
coefficient indicates a strong positive relationship, suggesting that the
evaluations from the conventional metrics and GPT are closely aligned. On the
other hand, a lower correlation coefficient would indicate a weaker
relationship, highlighting potential discrepancies in how the two methods
evaluate summaries. This analysis is crucial for understanding the consistency
and validity of the evaluation methods used in our study.

##  4 Results

The results section presents the outcomes of both the metrics-based and GPT-
based evaluations of summaries generated by the various Large Language Models
(LLMs) studied. These results provide insights into the performance of each
model across different evaluative criteria.

###  4.1 GPT-Based Evaluation

Table 2 displays the GPT-based scores, which also represent the averages
across 30 summaries and are normalized to the 0-1 scale. These scores provide
a direct comparison of how well each model performs according to the AI
evaluator in terms of the same summary properties.

Table 2: Scoring the properties with GPT-3.5. Model |  Conciseness |  Relevance |  Coherence |  Readability
---|---|---|---|---
DISTILBART |  0.24 |  0.78 |  0.70 |  0.79
BERT |  0.31 |  0.72 |  0.64 |  0.73
PROPHETNET |  0.35 |  0.62 |  0.38 |  0.69
T5 |  0.31 |  0.73 |  0.59 |  0.71
BART |  0.26 |  0.81 |  0.72 |  0.82
PEGASUS |  0.24 |  0.79 |  0.68 |  0.78

###  4.2 Metrics-Based Evaluation

The metrics-based scores for each LLM are shown in Table 3. These scores are
averages computed from 30 summaries and have been normalized to a range from 0
to 1, where 0 indicates the lowest performance and 1 the highest. Notably, for
the compression ratio, we adjusted the scores by subtracting them from 1 to
better represent the conciseness as a desirable attribute (lower compression
ratio indicating higher conciseness).

Table 3: Metrics results. Model |  Conciseness (Compression Ratio) |  Relevance (ROUGE) |  Coherence (LSA) |  Readability (Flesch-Kincaid)
---|---|---|---|---
DISTILBART |  0.19 |  0.36 |  0.57 |  0.45
BERT |  0.17 |  0.25 |  0.56 |  0.42
PROPHETNET |  0.05 |  0.08 |  0.29 |  0.38
T5 |  0.15 |  0.29 |  0.59 |  0.43
BART |  0.16 |  0.33 |  0.57 |  0.40
PEGASUS |  0.13 |  0.28 |  0.49 |  0.38

###  4.3 Statistical Analysis of Correlation

The Pearson’s correlation coefficients and P-values, displayed in Table 4,
provide statistical insight into the relationships between the metrics-based
scores and the GPT-3.5-based scores for each summary property.

Table 4: Correlation between Metrics-Based and GPT-3.5-Based Evaluations. Property |  Correlation coefficient |  P Value
---|---|---
Conciseness |  -0.65 |  0.17
Relevance |  0.92 |  0.01
Coherence |  0.85 |  0.03
Readability |  0.11 |  0.83

Figures 4 and 5 visualize the comparative evaluations using traditional
metrics and GPT, respectively, highlighting the performance variations across
the models.

Figure 4: Assessing using traditional metrics. Figure 5: Assessing using GPT.

This comprehensive presentation of results facilitates a nuanced understanding
of how each model’s performance is perceived differently by conventional
evaluation metrics versus AI-driven evaluation, reflecting on the potential
alignments and discrepancies between these two assessment methods.

##  5 Discussion

In this study, we employed traditional metrics such as ROUGE and Latent
Semantic Analysis (LSA) and introduced GPT as an independent evaluator to
assess text summaries. The correlation observed between GPT’s assessments and
traditional metrics, especially in relevance and coherence, highlights GPT’s
effectiveness in evaluating these aspects of summaries. Notably, GPT often
awarded higher scores than traditional metrics, suggesting it evaluates using
a broader set of factors, potentially capturing nuances that traditional
metrics overlook.

The performance of the large language models (LLMs) was relatively uniform,
except for the PROPHETNET model, which received lower scores from both
evaluation types. This can be attributed to PROPHETNET’s highly concise
outputs, which often failed to provide a comprehensive representation of the
original text, underscoring a limitation in capturing detailed content.

These findings demonstrate the utility of integrating AI tools like GPT in the
evaluation process, offering a more nuanced perspective compared to
traditional metrics alone. This approach not only enriches the evaluation
landscape but also helps refine our understanding of model capabilities and
limitations in natural language processing tasks.

##  6 Conclusion

In this study, we employed OpenAI’s GPT model alongside standard metrics such
as ROUGE and LSA to evaluate text summaries. The findings revealed a strong
correlation, particularly in the evaluation of relevance and coherence,
suggesting that GPT can effectively complement traditional assessment tools.
GPT tended to assign higher scores, which likely reflects its ability to
consider a broader range of factors in its evaluations. Notably, summaries
generated by the PROPHETNET model consistently received lower scores,
primarily due to their concise nature and limited content representation.
Overall, the results demonstrate GPT’s potential as an evaluator and its
capacity to enhance traditional metrics, offering valuable insights for future
research in natural language processing. This integration of AI-driven tools
with established metrics could lead to more comprehensive and nuanced
evaluation methods in the field.

##  7 Future Work

Building on the findings of this study, several avenues for future research
present themselves. First, expanding the evaluation framework to include more
diverse NLP tasks, such as sentiment analysis or entity recognition, could
provide a broader understanding of GPT’s capabilities as an evaluator across
different contexts. Additionally, exploring other transformer-based models not
covered in this study could offer insights into how various architectures
influence the efficacy of AI-driven evaluation tools.

Another promising direction would be to refine the methodology for integrating
AI-driven evaluations with traditional metrics, potentially developing a
hybrid model that leverages the strengths of both to produce a more robust
evaluation system.

Moreover, there is an opportunity to explore the psychological and perceptual
aspects of how humans interpret GPT’s evaluations compared to traditional
metrics. Understanding these dynamics could lead to improvements in how AI-
generated evaluations are presented and utilized, making them more intuitive
and actionable for users.

Finally, addressing the limitations observed in models like PROPHETNET, future
work could involve developing strategies to enhance the conciseness without
sacrificing content comprehensiveness. This could include experimenting with
different pre-training and fine-tuning approaches that specifically target the
balance between brevity and detail in summary generation.

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11institutetext:  Université de Franche-Comté, CNRS, Institut FEMTO-ST,
Besançon, France
Department of Physics, Yeshiva University, New York, USA

# Pumping with symmetry

J. A. Iglesias Martínez 11 M. Kadic 11 V. Laude 11 E. Prodan 221122

###### Abstract

Re-configurable materials and meta-materials can jump between space symmetry
classes during their deformations. Here, we introduce the concept of singular
symmetry enhancement, which refers to an abrupt jump to a higher symmetry
class accompanied by an un-avoidable reduction in the number of dispersion
bands of the excitations of the material. Such phenomenon prompts closings of
some of the spectral resonant gaps along singular manifolds in a parameter
space. In this work, we demonstrate that these singular manifolds can carry
topological charges. As a concrete example, we show that a deformation of an
acoustic crystal that encircles a $p11g$-symmetric configuration of an array
of cavity resonators results in an adiabatic cycle that carries a Chern number
in the bulk and displays Thouless pumping at the edges. This points to a very
general guiding principle for recognizing cyclic adiabatic processes with high
potential for topological pumping in complex materials and meta-materials,
which rests entirely on symmetry arguments.

It has been recently recognized that the space symmetry of materials can be a
rich source of topological effects. For example, in topological quantum
chemistry, the materials are divided in homotopy classes such that two systems
from two different classes cannot be continuously deformed into each other
without closing a bulk spectral gap or breaking the space symmetries
associated with the classes [1, 2, 3, 4]. It has also been recognized that
robust topological bulk-boundary correspondences can be induced by space
symmetries under adiabatic pumping conditions [5]. Space symmetries also play
a central role in the topological effects observed in higher order topological
insulators [6] and in other manifestations of bulk-boundary correspondences,
such as in topological corner modes [7] and topological screw dislocations
[8].

Our present work opens another perspective on space symmetries, specifically,
on cyclic deformations of materials in the presence of symmetries. As it is
well known [9], an adiabatic cycle can generate non-trivial topology in the
bulk of a material and a topological spectral flow at its boundaries. The
prototypical source of such phenomena is the Rice-Mele model [10] of
polyacetylene $[{\rm C}_{2}{\rm H}_{2}]_{n}$. It has two parameters that
quantify the fluctuations of the hopping coefficients and an ensuing staggered
potential under possible dimerizations of the ideal chain. The model displays
a gapless energy spectrum at a singular point, where the two parameters are
zero, and a gapped spectrum otherwise. A closed loop encircling this singular
point of the parameter space supports a non-trivial Chern number [9]. This is
often invoked as an example where the symmetry is irrelevant and where the
principles at work are entirely topological. We argue here that this is a very
narrow point of view, which misses a wider picture that can be very revealing
when it comes to identifying materials that support similar topological
effects.

Figure 1: Ball-and-stick model of un-dimerized polyacetylene.

To open the discussion, we point out the glide-reflection symmetry of the un-
dimerized polyacetylene chain, which is quite evident from its structure
reproduced in fig. 1. The singular point in the parameter space of the Rice-
Mele model, which carries the topological charge, exists precisely because of
this symmetry. Away from this singular point, the glide-reflection symmetry of
the polyacetylene chain is removed, so that only discrete translations remain
($p1$ symmetry) and the energy spectrum is gapped. Thus, we are dealing with a
parameter space with predominantly $p1$ symmetry and with one singular point
where the symmetry is enhanced. The main point we want to communicate is that,
even without a tight-binding model for a material or metamaterial, we can
still identify cyclic adiabatic processes with potential for topological
pumping, based entirely on symmetry principles. Indeed, we will show in this
work that the topological adiabatic cycle in polyacetylene is not an isolated
occurrence and that, in fact, topological cycles are prone to occur and very
easy to identify around the sub-manifolds of the parameter space that carry
enhanced symmetries.

Figure 2: (a) Symmetric pattern generated by acting with $p1$-group on the two
seeding shapes shown in color, i.e. using only discrete lattice translations.
$a$ is the lattice constant. (b) Same as (a), but with the seeding shapes
being glide-reflection images of each other. (c) Pattern (b) is reproduced by
acting with $p11g$-group on a single seeding shape, i.e. by repeatedly
applying a half-shift followed by a glide-reflection. Figure 3: Adiabatic
deformations of the materials described in this work can be generated by
modifications of position (hence $x,y$ coordinates) and size (hence scaling by
a parameter $s$) of a seed resonator. When going around the reference
$p11g$-symmetric configuration, shown in gray and corresponding to the origin
of the parameter space $\Delta x=\Delta y=\Delta s=0$: (a) An adiabatic cycle
generated only with displacements inherently intersects the singular manifold
shown in green; (b) whereas an adiabatic cycle generated with displacements
plus scalings can encircle the critical manifold. The shapes in these diagrams
represent self-coupling resonators. In blue, we show a continuous sequence of
configurations indexed by the adiabatic parameter $\Psi$. The diagrams on the
right depict the corresponding loops in parameter space. Configurations drawn
with starred yellow centers are $p11g$-symmetric. Note that, in (b), the
adiabatic loop goes behind at the top and in front at the bottom of the
singular manifold, hence avoiding these $p11g$-symmetric configurations.

To formulate the latter more precisely, let us place the discussion in the
context of the seven frieze groups, which are the discrete space-groups of
planar strips [11], hence appropriate for the investigation of periodic quasi
1-dimensional physical systems. We also switch from molecular systems to meta-
materials, for which we generate symmetric patterns where the positions,
orientations and shapes of the resonators all influence the dynamics of the
collective resonant modes [12]. Specifically, starting from one or more
seeding shapes, later regarded as seeding resonators, we apply all plane
transformations contained in a particular frieze group and generate patterns
displaying a desired symmetry. In regards to the complexity of their
symmetries, at one end stands the $p1$ group, which includes only the discrete
translations of the primitive cell. At the opposite end stands the $p2mm$
group, which incorporates the maximal set of allowed discrete symmetries, that
is, horizontal translations and horizontal/vertical reflections. Regardless of
their complexity, all symmetric patterns can be generated with the $p1$ group,
when the latter acts on an appropriate set of seeding resonators forming the
primitive cell, as illustrated in fig. 2(a). However, if the seeding
resonators have particular shapes, locations and orientations as in fig. 2(b),
the symmetry of the pattern can be enhanced to other frieze groups and a
smaller set of seeding resonators is needed, as illustrated in fig. 2(c) for
the case of the glide-reflection symmetry (frieze group $p11g$). Under the
slightest generic perturbation of the position, shape or orientation of
resonators, the pattern falls back to the $p1$ symmetry. The point we want to
make is that the patterns with symmetries other than $p1$ form isolated
manifolds in the space of symmetric patterns, and these manifolds are
surrounded and connected by the space of $p1$ symmetric patterns. We
demonstrate here that some of these isolated manifolds carry topological
charges.

A decisive factor that must be taken into account is the number of distinct
energy bands that can be produced with a given number of seeding resonators.
If each resonator carries one resonant mode, then the $p1$-symmetric pattern
seen in fig. 2(a) produces two gapped resonant energy bands, generically. In
contrast, the $p11g$-symmetric pattern seen in fig. 2(c) can produce only a
single resonant energy band, regardless of the couplings [13]. Thus, the
spectrum will be ungapped along a manifold of parameters carrying the
$p11g$-symmetry. However, not every enhancement/reduction of symmetry leads to
the phenomena advertised here. If we start with four arbitrary seeding
resonators, we can generate a $p1$-symmetric pattern with four resonators in
the primitive cell. By continuously changing the shape, orientation and
locations of the seeding resonators, we can achieve the $p2mm$ symmetry, in
which case the pattern can be generated from a single seeding resonator. Yet,
both cases can display four separated energy bands, a counting that is based
on the K-theories of these space groups (see e.g. fig. 7.3 in [12]). In this
case, the symmetry enhancement does not display a singular character. In
contradistinction, the enhancement from frieze group $p1m1$ (vertical
reflection) to $p2mg$ (vertical reflection and glide-reflection) does, because
the analysis here is very similar to the one for $p1\to p11g$ enhancement:
$p1m1$ and $p2mg$ belong to the same isomorphism class, hence they have
identical $K$-theories, but the number of needed seeding resonators drops by
one for $p2mg$. Hence, the number of energy bands that can be produced by a
$p2mg$-symmetric pattern is necessarily lower than that produced by a
$p1m1$-symmetric one. While a more thorough analysis will be reported
elsewhere, we can already state the general guiding principle at work here,
namely, the reduction in the number of energy bands that can be produced with
a given pattern when the symmetry enhancement occurs.

Figure 4: Same as fig. 3 but with scalings replaced by rotations. (a) An
adiabatic cycle generated only with displacements inherently intersects the
singular manifold shown in green; (b) whereas an adiabatic cycle generated
with displacements plus rotations (by parameter $\Delta\varphi$) can encircle
the critical manifold. The shapes in these diagrams represent self-coupling
resonators. The configuration shown in gray is the $p11g$-symmetric
configuration at the origin of parameter space, or $\Delta x=\Delta
y=\Delta\varphi=0$. In blue, we show a continuous sequence of configurations
indexed by the adiabatic parameter $\Psi$. The diagrams on the right depict
the corresponding loops in parameter space. Configurations drawn with starred
yellow centers are $p11g$-symmetric.

Once we identified a symmetry enhancement that leads to a singular manifold in
the space of parameters, the next step is to construct adiabatic cycles that
encircle this manifold. They can all be obtained by deforming the seeding
resonators. In general, a seeding resonator has an infinite dimensional
configuration space, so there are many opportunities to engineer deformation
spaces of different topologies and dimensions. In the present study, however,
we restrict ourselves to lower dimensional deformation spaces, by only
allowing specific actions on the resonators. In fig. 3, for example, we
consider a pair of spherical seeding resonators, the first one with constant
radius $r_{0}$ and the second one initially of same radius $r_{0}$ and fixed
to the glide-reflection symmetric position (see the gray configuration in fig.
3). From this $p11g$-symmetric reference configuration, we allow displacements
and scalings (by parameter $s$) of the second seeding resonator. Thus, the
configuration space $({\Delta}x,{\Delta}y,{\Delta}s)$ is 3-dimensional. We
then see that the space of $p11g$-symmetric patterns is 1-dimensional and is
represented by $(0,{\Delta}y,{0})$, or the vertical dashed line in fig. 3(a).
Any closed loop $({\Delta}x(\Psi),{\Delta}y(\Psi),{0})$ around the reference
point $(0,0,{0})$, composed of displacements only, will intersect the singular
manifold at least twice. As shown in fig. 3(b), however, we can encircle the
$p11g$-symmetric phase by using displacements and scalings with a closed loop
$({\Delta}x(\Psi),{\Delta}y(\Psi),{\Delta}s(\Psi))$ that never crosses the
singular manifold and actually encircles it. Note that parameter space is
represented in fig. 3 as $(\Delta x,\Delta y,{\Delta}s)$ so that the reference
configuration $(\Delta x=0,\Delta y=0,{\Delta}s=0$ sits at the origin.

Another example is shown in fig. 4, where the shapes and sizes of the seeding
resonators are matched and fixed, but we allow displacements and rotations
$\Delta\varphi$ of the second seeding resonator. This produces again a closed
loop $({\Delta}x(\Psi),{\Delta}y(\Psi),\Delta\varphi(\Psi))$ in the
3-dimensional configuration space that encircles the critical manifold without
crossing it. Again, the origin of parameter space represents the
$p11g$-symmetric reference phase.

Figure 5: (a) The vectors of the Hilbert space
${\mathbb{C}}^{2}\otimes\ell^{2}({\mathbb{Z}})$ corresponding to the resonant
modes carried by the two resonators; (b-e) The dominant coupling matrices for
a generic $p1$-symmetric pattern with two resonators per primitive cell. $a$
and $a^{\prime}$ are intra- and inter-cell cross-couplings, respectively.
$E_{0}\pm b$ are the on-site self energies. $c\pm d$ are inter-cell couplings
for each site.

The big claim of our work is that, at least in the case when the seed
resonators carry a single mode each, the topologically non-trivial cycles
described above automatically translate into Thouless pumpings provided the
system displays a gap in its resonant spectrum, and no tight-binding models
are needed to understand this phenomenon. Indeed, in the proposed scenario,
the coupling matrices between the resonators are functions of only the three
specified parameters, but their functional dependence can be arbitrary. Now,
during an adiabatic cycle, the collective resonant states can be resolved over
the adiabatic variable $\Psi$ and quasi-momentum $k$, which both live on
circles. Thus, the collective resonant states with frequencies below the
spectral gap supply a vector bundle over the 2-torus. Such bundles are
generated by acting with a $2\times 2$ projection matrix $P(\Psi,k)$ on the
fixed space ${\mathbb{C}}^{2}$ (see e.g. Eq. (2) for an explicit expression).
Every projection obeys the constraints $P=P^{\dagger}=P^{2}$ and, as a result,
they take the form

$P=\begin{pmatrix}\alpha&\sqrt{\alpha(1-\alpha)}e^{i\phi}\\\
\sqrt{\alpha(1-\alpha)}e^{-i\phi}&1-\alpha\end{pmatrix}$ (1)

where $\alpha$ is a real parameter from the interval $[0,1]$ and $\phi$ is
also a real parameter from the interval $[0,2\pi)$. Key here is that, with our
3-dimensional deformation space and with the freedom to choose the functional
dependence of the coupling matrices, we can sample any desired projection,
which requires only two parameters, as seen in Eq. (1). Now, consider one
configuration with gapped spectrum, which can be always expanded into a small
adiabatic loop of configurations without closing the gap. The resulting bundle
over the 2-torus is obviously trivial, hence it carries a Chern number zero.
Consider now an adiabatic loop that encircles the singular manifold carrying
the $p11g$ symmetry. We claim that the resulted bundle over the 2-torus is
topologically distinct from first bundle we previously constructed. Indeed, if
we can modify the functional dependence of the couplings between the
resonators such that the second bundle is deformed into the first one, then we
should be able to contract the second adiabatic loop to a point, without
closing the gap. But this is impossible because, in the process, we will
necessarily touch the singular manifold where there is only one spectral band
in the spectrum. Thus, the second bundle must carry a nontrivial Chern number.

Once we established that the vector bundle supported by the loop encircling
the $p11g$-singular manifold carries a non-trivial Chern number, given the
robustness of the latter, we can consider additional deformations of the
resonators as well as turning on additional degrees of freedom. As long as the
spectral gap remains open for the entire adiabatic cycle, none of the above
actions can destroy the topological character of the loop. This conclusion is
definitely aided by the particular setting we started with. If the resonators
would have carried more than one degree of freedom, then it is very likely
that some of the bands or composite bands will not carry a Chern number under
the proposed scheme. Therefore, we want to be clear that we are not announcing
a theorem here, but rather a guiding principle for detecting cyclic adiabatic
processes with high potential for implementing topological pumps. In general,
a check will still be necessary to confirm the sought properties.

Figure 6: Resonant spectrum (a) and Berry curvature (b) for the loop
$a=1.0+0.4\cos(\Psi)$, $a^{\prime}=1.0+0.4\sin(\Psi)$, $b=0$,
$c=0.4+0.1\cos(\Psi-\pi/4)$ and $d=-0.1\cos(\Psi-\pi/4)$. The expressions of
parameters are consistent with the loop in fig. 4, where $b=0$ because the
resonators are identical and $c+d$ is a constant because the top row is not
modified. The $\pi/4$ phase is an arbitrary choice that makes no difference.

We will confirm the topological nature of the cycle from fig. 4 using both
tight-binding and continuum media simulations. The pattern of resonators
discussed so far have two modes per repeating cell, hence the Hilbert space of
the resonating modes is spanned by the vectors $\xi\otimes|n\rangle$, with
$\xi$ a column vector with two complex entries and $n\in{\mathbb{Z}}$. This
particular Hilbert space is usually denoted as
${\mathbb{C}}^{2}\otimes\ell^{2}({\mathbb{Z}})$. The precise distribution of
these vectors on the resonators is shown in fig. 5(a). In figs. 5(b-e), we
list the nearest-neighbor couplings and their contributions to the dynamical
matrix. According to those couplings, the dynamical matrix $H$ which
determines the resonant pulsations $H|\psi\rangle=\omega^{2}|\psi\rangle$ of
the resonator pattern can be approximated as

$\displaystyle H=$ $\displaystyle E_{0}+a\sigma_{1}\otimes
I+a^{\prime}\,(\sigma_{-}\otimes S+\sigma_{+}\otimes
S^{\dagger})+b\,\sigma_{3}\otimes I$
$\displaystyle+\tfrac{c}{2}\sigma_{0}\otimes(S+S^{\dagger})+\tfrac{d}{2}\sigma_{3}\otimes(S+S^{\dagger}),$

where $S|n\rangle=|n+1\rangle$ is the shift operator, $E_{0}$ is the average
resonant energy of the two seeding resonators, $\sigma_{0}$ is the $2\times 2$
identity matrix and the other $\sigma$’s are Pauli’s matrices. The terms
included encompass intra-cell couplings as well as inter-cell couplings up to
the nearest neighbor. The terms left out involve second and higher nearest
neighbor couplings, which are relatively small. We pass to the momentum space
using the substitution $S^{\pm 1}\to e^{\pm\imath k}$, to find two energy
bands

$E_{\pm}(k)=E_{0}+c\cos(k)\pm\sqrt{(b+d\cos(k))^{2}+|a+a^{\prime}e^{\imath
k}|^{2}}$

separated by a gap. When the symmetry of the pattern is enhanced to $p11g$,
then necessarily $a=a^{\prime}$ and $b=d=0$; $E_{\pm}(k)=E_{0}+(c\pm
2a)\cos(k)$ and the band spectrum is gapless. We recall that this closing of
the energy gap is un-avoidable and that it does not depend on the simplified
Hamiltonian we used. Under an adiabatic cycle parameterized by
$\Psi\in{\mathbb{S}}^{1}$, the Bloch Hamiltonian depends on two parameters
$(\Psi,k)$ that live on torus ${\mathbb{T}}^{2}$, hence we can evaluate a
Chern number as

${\rm Ch}(P)=\int_{{\mathbb{T}}^{2}}d\Psi\,dk\,F(\Psi,k),$

where $P$ is the spectral projector onto the lower dispersion band, which can
be conveniently computed as

$P(\Psi,k)=\frac{H(\Psi,k)-E_{+}(\Psi,k)\,I_{2\times
2}}{E_{-}(\Psi,k)-E_{+}(\Psi,k)},$ (2)

and $F(\Psi,k)$ is the Berry curvature,

$F(\Psi,k)=\tfrac{\imath}{2\pi}\ {\rm
Tr}\Big{(}P(\Psi,k)\big{[}\partial_{\Psi}P(\Psi,k),\partial_{k}P(\Psi,k)\big{]}\Big{)}.$

Note that parameter $c$ does not enter in any of the last three equations
because its corresponding contribution to $H(\Psi,k)$ just shifts
$E_{\pm}(\Psi,k)$ by the same amount. Thus, we are effectively dealing with a
4-dimensional parameter space and, as such, the critical manifold
$a-a^{\prime}=b=d=0$ has dimension 1. Now, with these tools, we can verify
that any loop that encircles the 1-dimensional critical manifold carries a
non-trivial Chern number. Due to the topological character of the statement,
it is enough to check this statement for one loop and this is confirmed in
fig. 6, where a parametrization consistent with the process illustrated in
fig. 4 is considered.

We now demonstrate how our guiding principle unfolds for an actual acoustic
crystal. Its building elements are the C-shaped Helmholtz resonators with the
unit cell shown in fig. 7(a) and full crystal shown in fig. 8(b-e). The out of
page dimension of the resonators is small such that the low frequency resonant
modes are all uniform in the direction perpendicular to the page. We hope the
reader will agree with us that, a priori, it is not clear what cyclic actions
on such acoustic crystal will result in Thouless pumping. According to our
guidelines, the first task is to optimize its $p11g$-symmetric configuration
such that a breaking of the $p11g$-symmetry produces a complete gap in the
dispersion spectrum. In fig. 7(b), we show the evolution of the dispersion
bands as a function of the orientation of resonators in $p11g$-symmetric
configurations and, from that data, we chose the angle $\varphi=54^{\circ}$
giving the dispersion shown in fig. 7(c). The touching of the dispersion bands
marked there is protected by the $p11g$-symmetry and the loss of this
protection will open a local spectral gap. The particular geometry of the
dispersion bands then assures us that this local band splitting develops into
a complete spectral gap.

Figure 7: Tuning the resonating structure: (a) Unit cell of the
$p11g$-symmetric configuration of a chain of C-shaped acoustic resonators. The
gray region indicate the domain of the acoustic wave propagation. Hard-wall
boundary conditions are applied at the top and bottom of the cell, as well as
on the walls of the C-shape. In units of $a$, the values of the marked
parameters are $b=0.85a$, $d=0.15a$, $r=0.3a$, and $R=0.4a$. The glide
parameter $g$ is fixed at $a/2$. (b) Bulk dispersion as function of the
resonator orientations. (c) Bulk dispersion of the tuned structure, with a
mark showing the targeted band splitting. The simulations were generated with
COMSOL MultiPhysics software.

Next we define the deformation path as

$g=a/2-a/5\,\sin(\Psi),\quad\varphi=54^{\circ}+20^{\circ}\cos(\Psi).$

This encloses the singular $p11g$-manifold and it leads to the gapped bulk
dispersion spectrum shown in the inset of fig. 8(a). According to our
prediction, this loop supports a Chern number $+1$ and, as such, the bulk-
boundary correspondence principle [14] assures us of the emergence of
topological edge states in a finite geometry, which display a topological
spectral flow with the pumping parameter $\Psi$. This is confirmed in the
COMSOL simulations reported in fig. 8(a), where two chiral edge bands located
at the opposite ends of finite crystal are clearly visible. Furthermore,
samples of the modes as well as a computation of their center of mass confirm
their expected localization. In contrast, if the pumping cycle does not
encircle the singular $p11g$-manifold, as fig. 9 exemplifies, then the
topological edge state inside the band gap always remains localized on one of
the sides of the finite chain of resonators and never crosses the band gap.

Figure 8: (a) Spectral flow of the resonant spectrum against the pumping
parameter $\Psi$ for a finite acoustic crystal with 31 unit cells. The cycle
parametrized by pumping parameter $\Psi$ encircles the glide symmetric point
in parameter space $(g,\phi)$, as the left inset depicts. Topological edge
modes are seen crossing the bulk gap of the dispersion diagram. Their
localization, as measured by the center of mass $\langle X\rangle$ of the
modes, is encoded in color. (Right inset) Resonant spectrum of the infinite
acoustic crystal (compare with fig. 6(a)). (b-e) Samples of spatial profiles
of modes, reported as the pressure field $\hat{p}$, collected at the arrows
indicated in panel (a). Figure 9: (a) Spectral flow of the resonant spectrum
against the pumping parameter $\Psi$ for a finite acoustic crystal with 31
unit cells. The cycle parametrized by pumping parameter $\Psi$ does not
encircle the glide symmetric point in parameter space $(g,\phi)$, as the inset
depicts. The localization of topological edge modes, as measured by their
center of mass $\langle X\rangle$, is encoded in color. (b-c) Samples of
spatial profiles of modes, reported as the pressure field $\hat{p}$, collected
at the arrows indicated in panel (a).

In conclusion, we announced a guiding principle that enables one to identify
adiabatic cycles with high potential for Thouless pumping, without making
appeal to any analytic tight-binding model. Indeed, the topological cycles
described in Figs. 3 and 4 were produced using only geometric considerations.
Shunting the need of analytic calculation can have great practical
implications. For example, our extremely simple topological adiabatic cycle
for the C-shaped resonators (see [15] for an animation) would have been hard
to guess from an analytic model, yet we discover it without much effort using
the new geometric principles. We anticipate that our guiding principle will be
fruitful as well for wallpaper groups and crystallographic groups in
3-dimensions, and even for finite highly symmetric molecules. The principle
may be also relevant to the electron-phonon coupling in quantum materials. For
example, deformations of the crystalline or molecular structures that encircle
manifolds of enhanced symmetries may supply mechanisms for quantized charge
transfers across extended systems.

###### Acknowledgements.

EP acknowledges financial support from US National Science Foundation through
the grants CMMI-2131760 and DMR-1823800, and US Army Research Office through
contract W911NF2310127. Support from the EIPHI Graduate School [Contract No.
ANR-17-EURE-0002] and from ANR PNanoBot [Contract No. ANR-21-CE33-0015] is
also acknowledged.
COMSOL simulations are available from the authors upon reasonable request.

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