In this paper, we present our approach to tackle QA 2023 shared tasks A and  B.
To address the challenge of low-resourced training data, we rely on transfer learning together with
a voting ensemble to improve prediction stability across multiple runs.
Additionally, we employ different architectures and learning mechanisms for a range of Arabic pre-trained
transformer-based models for both tasks.

To identify unanswerable questions, we propose using a
thresholding mechanism.
Our top-performing systems greatly surpass the baseline performance on the
hidden split, achieving a MAP score of 25.05% for task
A and a *pAP of 57.11% for task B.

§ INTRODUCTION

Ad hoc search is a fundamental task in *IR and serves as the foundation for numerous *QA systems and search engines.
*MRC is a long-standing endeavor in *NLP and plays a significant role in the framework of text-based *QA systems.
The emergence of *BERT and its family of *LM have revolutionized the landscape of transfer learning systems
for *NLP and *IR as a whole <cit.>.

Arabic is widely spoken in the Middle East and North Africa, and among Muslims worldwide.
Arabic is known for its extensive inflectional and derivational features.
It has three main variants: *CA, *MSA, and *DA.

QA 2023 shared task A is a passage retrieval task organized to engage the community in conducting ad hoc search over the Holy  <cit.>.
While QA 2023 shared task B is a ranking-based *MRC over the Holy , which is the second version of QA 2022 shared task <cit.>.

This paper presents our approaches to solve the two tasks A and B.
For task A,
we explore both dual-encoders and cross-encoders for ad hoc search <cit.>.
For task B, we investigate *LMs for extractive *QA using two learning methods <cit.>.
For both tasks, we utilize various pre-trained Arabic *LM variants.
Moreover, we adopt external Arabic resources in our fine-tuning setups <cit.>.
Finally, we employ an ensemble-based approach to account for inconsistencies among multiple runs.

We contribute to the *NLP community by releasing our experiment codes and trained *LMs to GitHub [<https://github.com/mohammed-elkomy/quran-qa>].

In this work, we address the following research questions [A superscript at the end of a RQ refers to one of the tasks. No superscript means the RQ applies for both tasks.]:

RQ1: What is the impact of using external resources to perform pipelined fine-tuning?

RQ2: How does ensemble learning improve the performance obtained?

RQ3: What is the effect of thresholding on zero-answer questions?

RQ4A: What is the impact of hard negatives on the dual-encoders approach?

RQ5B: What is the impact of multi answer loss method on multi-answer cases?

RQ6B: How is post-processing essential for ranking-based extractive question answering?

The structure of our paper is as follows: Sections <ref> and
<ref> provide an overview of the datasets used in our study. In
Section <ref>, we present the system design and implementation
details for both tasks. The main results for both tasks are presented in
Section <ref>. Section <ref> focuses on the analysis
and discussion of our research questions RQs. Finally, Section <ref>
concludes our work.

§ TASK A DATASET DETAILS

2c|Split     Training     Development
# Question-passage
relevance pairs
972     160
1c|4*# Questions     Multi-answer     105 (60%)     15 (60%)

1c|                                           Single-answer     043 (25%)     06 (24%)

1c|                                           Zero-answer       026 (15%)     04 (16%)
1c|                                           Total             174                      25

Task A dataset relevance pairs distribution across training and development splits.
We also include the distribution of answer types per split.

QA 2023 shared task A serves as a test collection for the ad hoc
retrieval task. The divine text is divided into segments known as the *QPC,
where logical segments are formed based on common themes found among consecutive ic verses <cit.>.
In this task, systems are required to provide responses to user
questions in  *MSA by retrieving relevant passages from
the *QPC when possible. This suggests there is a language gap between the questions and the passages, as the passages are in *CA.

Table  <ref> presents the distribution of the dataset across the training and development splits.

The majority of questions in
the dataset are multi-answer questions, meaning that systems can only receive
full credit if they are able to identify all relevant passages for these
Additionally, Table  <ref> provides information on zero-answer questions,
which are unanswerable questions from the entire .
(More information about the dataset distribution of topics in Appendix <ref>)

Task A is evaluated as a ranking task using the standard *MAP metric.
(Additional information about the evaluation process including zero-answers cases
can be found in Appendix <ref>)

§ TASK B DATASET DETAILS

QA 2023 shared task B is a ranking-based SQuADv2.0-like *MRC over the Holy , which extends to the QA 2022 <cit.>.
The dataset is also referred to as *QRCDv1.2.
The same questions from task A are organized as answer span extraction task from relevant passages <cit.>.
(See the dataset distribution of topics in Appendix <ref>)

2c|Split     Training     Development
# Question-passage-answer
1179     220
# Question-
Multi-answer     9134 (14%)0                           1129 (18%)0

1c|     Single-answer       806 (81%)       124 (76%)

1c|     Zero-answer             952 (5%)0                                  110 (6%)0
1c|     Total             992                                  163

Task B dataset pairs and triplets distribution across training and development splits.
For questions-passage pairs, we show the distribution of answer types.

Table <ref> depicts the distribution of dataset pairs and triplets across the training and development splits.
In addition, the table presents the distribution of answer types for the dataset pairs.

Although zero-answer questions account for 15% of the questions in task A test collection,
they only contribute to 5% of the question-passage pairs in task B.
Furthermore, task B has a limited number of unique questions in comparison to their corresponding question-passage pairs as seen from Tables <ref> and <ref>, respectively.

As a consequence, task B can have repeated questions and passages among different samples
and can be even leaked among training and development splits <cit.>.
<cit.> analyzed this phenomenon and identified sources of leakage in *QRCDv1.1.
In *QRCDv1.1, leakage is defined as the presence of passages, questions,
or answers that are shared among multiple samples <cit.>. This can lead to *LM
memorizing or overfitting leaked samples <cit.>.
<cit.> categorized
*QRCDv1.1 into four distinct and mutually exclusive categories based on
the type of leakage: pairs of passage-question, passage-answer, or just questions.
(For more information about leakage in task B, please refer to Appendix <ref>)

We extend the analysis made by <cit.> for *QRCDv1.2. Our main observation is that 90% of the
samples with no answer belong to the trivial leakage group called $D_{(1)}$.
This group refers to samples with duplicate passage-answer or question-answer pairs.
This indicates that zero-answer questions are not just less prevalent in task B but also present a greater challenge in terms of generalization.
Given the four groups defined by <cit.>, they proposed a data re-splitting mechanism for *QRCDv1.1 called faithful splits.
In this work, we extend their resplitting approach and create faithful splits for *QRCDv1.2.
(Please refer to Appendix <ref> for more details about faithful splitting)

Task B is evaluated as a ranking task as well, using a recently proposed measure called
*pAP <cit.>.
(More details about this measure
and zero-answer sample evaluation can be found in Appendix  <ref>)

§ SYSTEM DESIGN

1c|3*Systems     8c|Single Model     2cSelf Ensemble
1c|     1c|     2*MAP     2*MRR                          2*R@10                          2*R@100                          1c|2*
MAP$^\bigstar_{\scaleto{\zeta}{10pt}}$          3c|MAP (Question Type)                                     2*MAP     2*
1c|                                            1c|                                                                                1c|          Zero            Single          1c|Multi
1lLexical Baseline                                                           1l     1l     1l               1l                1l                                                                                1l     1l     1l
1c|102BM$_\sim$     1l|BM25                               18.43     26.40     19.98                               19.98                                1c|26.40                                                                               25.00                     16.67                     1c|17.39               N/A                              N/A
1lDual-encoder                                                               1l     1l     1l     1l     1l     1l     1l     1l
1c|102ARB$^\circledcirc_\sim$     1l|

[-8.5pt] xxx$_{\textit{TASK A+ Random Neg}}$

20.02     42.87     29.72     48.23     1c|20.02     0.00     35.42     1c|19.20     N/A     N/A

1c|102ARB$^\circledcirc_\approx$     1l|

[-8.5pt] xxx$_{\textit{TASK A+ Hard Neg}}$

24.44     35.17     36.09     43.96     1c|24.44     0.00     45.00     1c|22.73               N/A                              N/A
1lCross Encoder                                                              1l     1l     1l     1l     1l     1l     1l     1l
1c|102ELC$^\otimes_\sim$     1l|

[-8.5pt] xxxxxxxxxx$_{\textit{TASK A}}$

8.96     16.51     19.13     42.49     1c|16.48     3.00     10.32     1c|10.01     12.18     16.18

1c|102ELC$^\otimes_\approx$     1l|

[-8.5pt] xx$_{\textit{TyDi}\text{ }\textit{QA}_{\text{AR}}\rightarrow \textit{Tafseer} \rightarrow \textit{TASK A}}$

26.60     41.61     38.52     59.19     1c|31.91     19.00     38.31     1c|23.94     29.13     36.56

1c|102CAM$^\otimes_\sim$     1l|

[-8.5pt] xxxxxxxxxxx$_{\textit{TASK A}}$

23.16     33.52     37.06     55.12     1c|27.45     13.00     36.92     1c|20.36     27.57     32.02

1c|102CAM$^\otimes_\approx$     1l|

[-8.5pt] xx$_{\textit{TyDi}\text{ }\textit{QA}_{\text{AR}}\rightarrow \textit{Tafseer} \rightarrow \textit{TASK A}}$

29.34     42.17     39.93     57.23     1c|33.81     18.00     51.40     1c|23.54     32.77     36.77

1c|102ARB$^\otimes_\sim$     1l|

[-8.5pt] xxxxxxxaxxxx$_{\textit{TASK A}}$

31.76     41.93     46.55     62.71     1c|34.27     46.00     28.16     1c|29.41     36.09     36.87

1c|102ARB$^\otimes_\approx$     1l|

[-8.5pt] xx$_{\textit{TyDi}\text{ }\textit{QA}_{\text{AR}}\rightarrow \textit{Tafseer} \rightarrow \textit{TASK A}}$

34.83     47.09     39.99     60.82     1c|37.55     43.00     46.22     1c|28.10     36.70     40.70

Dev split evaluation results for task A. MAP means $\zeta$ is set to mark 15% of questions as unanswerable. $\bigstar$ accompanied by $\zeta$ refers to applying the best $\zeta$ (see Appendix  <ref>).

Average performance is reported for multiple runs of single models.
Superscripts $\circledcirc$ and $\otimes$ in short form refer to dual-encoder and cross encoder, respectively.
Subscripts $\sim$ and $\approx$ denote direct fine-tuning and pipelined fine-tuning, respectively.

In this work, we fine-tune a variety of pre-trained Arabic *LMs, namely
<cit.>,  <cit.>, and  <cit.>.
We utilize transfer learning and ensemble learning for both tasks.
To determine zero-answer cases, we apply a thresholding mechanism.
(Additional information
on transfer learning and ensemble learning can be found in Appendices <ref> and  <ref>, respectively)

§.§ Task A Architecture

We examine two distinct approaches for neural ranking in ad-hoc search:
dual-encoders and cross-encoders approaches <cit.>.

In dual-encoders, documents and queries are encoded separately into
dense vectors, which are then compared using a metric learning function, such as
cosine distance.
We utilize *STAR with a batch size of 16 queries to
train our dense retrievers <cit.>.

In contrast cross-encoders involve encoding positive and negative
pairs of documents and questions, assigning a relevance score.
This method packs a document and a question into a single input for a sentence similarity *LM <cit.>.
Both methods require negative relevance signals during training.
(Please refer to Figures <ref> and <ref> in Appendix for both approaches.
Additionally, see Appendix <ref> for more details about negative selection criteria and zero-answer prediction)

Although cross-encoders have a higher computational overhead compared to dual-encoders
when used for ranking, the former has a quadratic complexity while the
latter has a linear complexity. However, both methods are still feasible for
low-resource datasets <cit.>.

In both approaches, we utilize the cumulative predicted scores of the top K
documents to calculate the likelihood of each question having an answer. We then
apply a threshold $\zeta$ to identify zero-answer questions.

§.§ Task B Architecture

We fine-tune pre-trained *LMs for span prediction as in SQuADv2.0

We use two different fine-tuning methods: *FAL and *MAL. The *FAL
method focuses on optimizing for the first answer in the ground truth answers,
which is the default approach in standard span prediction implementations for SQuAD <cit.>.
In contrast, *MAL optimizes for multiple answers simultaneously for the multi-answer samples in *QRCDv1.2.
This helps prevent the trained
systems from being overly confident in a single span and distributes the
predicted probability among different spans.
(Refer to Appendix <ref> for more information about these learning methods)

It is worth noting that raw
predictions from span prediction *LMs are suboptimal for ranking *MRC,
as many of them have overlapping content.
To address this, we follow a post-processing mechanism proposed by <cit.>.
(See Appendix <ref> for implementation details)

Similar to task A, we perform thresholding by a hyperparameter $\zeta$ to determine
zero-answer samples using *LM null answer [CLS] token probability <cit.>.
(See Appendix <ref> for more details on zero-answer cases)

§ RESULTS

The results tables for both tasks use the following notational format:
We use short forms to refer to combinations of *LM and
their fine-tuning approaches using superscripts and subscripts.

The subscripts $\sim$ and $\approx$ denote direct fine-tuning and
pipelined fine-tuning, respectively. Additionally, the arrows in model names
subscripts indicate the stages of pipelined fine-tuning, with the learning resources names
listed. Superscripts are used to denote the architectures employed for task A and the learning methods for task B.

Tables <ref> and <ref> present our detailed results on
the development split for both tasks for single and self-ensemble models.

Table <ref> shows the results for cross encoder and dual-encoders for task A.
Our best single model, , achieved a *MAP of 34.83% and an MRR of 47.09%.
self-ensemble achieved the best *MAP of 36.70%.
Table <ref> also presents the R@10 and R@100 metrics.
This represents the upper bound on the reranking stage performance that we can obtain <cit.>.

Table <ref> summarizes the results for task B.
Our best performing model over the standard split,
, attained a *pAP of 53.36% and 55.21% for single model and self-ensemble models, respectively.
Table <ref> also presents results for the faithful validation split we defined previously.
is our best performing single model for the faithful split, achieving a *pAP score of 54.19%.

2c|Systems     8c|Single Model                                                                                3cSelf Ensemble Model
1c|3*Model     3*Method     2c|Faithful     9cStandard Development Split
1c|         2*pAP     1c|2*pAP$_{\textit{Post}}$                               2*pAP                                2*pAP$_{\textit{Post}}$                                1c|2*
pAP$^\bigstar_{\scaleto{\zeta}{10pt}}$                   3c|pAP (Sample Type)                           2*pAP     2*pAP$_{\textit{Post}}$     2*

1c|                                            1c|                                                                                1c|                   Zero     Single     1c|Multi
102ELC$^\textit{F}_\sim$     2*

[-8.5pt] xxxxxxxxxx$_{\textit{TASK B}}$

FAL     34.97     1c|41.23     38.27     44.40     1c|39.26     18.67            41.51              1c|31.18             41.16                            46.50                                              41.72

102ELC$^\textit{M}_\sim$                                  MAL     37.44              1c|42.63     40.55     45.56     1c|41.48     14.67     43.69     1c|36.04     42.01     47.21     43.90
102ELC$^\textit{F}_\approx$     2*

[-8.5pt] xx$_{\textit{TyDi}\text{ }\textit{QA}_{\text{AR}} \rightarrow \textit{TASK B}}$

FAL     52.76     1c|55.45     49.76     53.70     1c|51.99                                                                            10.33           54.36             1c|43.69            50.66                           55.35                                             52.75

102ELC$^\textit{M}_\approx$                               MAL     53.15              1c|55.43     53.36     56.42     1c|55.10     18.33     56.61     1c|51.55     55.21     58.38     57.05
102CAM$^\textit{F}_\sim$     2*

[-8.5pt] xxxxxxxxxxx$_{\textit{TASK B}}$

FAL     41.45     1c|45.76     37.63     42.04     1c|38.36     11.00            40.83              1c|33.13             42.51                            45.50                                              43.18

102CAM$^\textit{M}_\sim$                                  MAL     43.54              1c|47.36     38.57     43.38     1c|39.38     12.67     40.52     1c|39.20     41.66     45.39     43.80
102CAM$^\textit{F}_\approx$     2*

[-8.5pt] xxx$_{\textit{TyDi}\text{ }\textit{QA}_{\text{AR}} \rightarrow \textit{TASK B}}$

FAL     50.64     1c|53.12     41.59     46.50     1c|42.39                                                                            13.67           44.36             1c|39.39            47.03                           49.37                                             47.12

102CAM$^\textit{M}_\approx$                               MAL     52.14              1c|54.01     40.08     44.80     1c|41.30     15.00     41.61     1c|42.18     42.75     46.87     44.23
102ARB$^\textit{F}_\sim$     2*

[-8.5pt] xxxxxxxaxxxx$_{\textit{TASK B}}$

FAL     44.81     1c|48.93     45.66     49.34     1c|46.60                                                                             23.67            49.29              1c|37.74             49.38                            53.05                                              50.01

102ARB$^\textit{M}_\sim$                                  MAL     47.41              1c|50.62     45.71     47.69     1c|46.85     25.67     48.43     1c|41.03     49.69     52.03     51.28
102ARB$^\textit{F}_\approx$     2*

[-8.5pt] xxx$_{\textit{TyDi}\text{ }\textit{QA}_{\text{AR}} \rightarrow \textit{TASK B}}$

FAL     52.97     1c|55.86     50.62     54.43     1c|51.28                                                                            35.33           53.78             1c|42.39            52.20                           55.77                                             53.45

102ARB$^\textit{M}_\approx$                               MAL     54.19     1c|56.55     50.51                          53.32                          1c|51.35              31.33             53.22               1c|45.54              52.13              54.94              52.94

Dev split evaluation results for task B. pAP means fixing $\zeta$ to 0.8. Post subscript identifies post-processing. $\bigstar$ accompanied by $\zeta$ refers to applying the best $\zeta$ (see Appendix  <ref>).

Average performance is reported for multiple runs of single models.
Superscripts F and M in short form indicate *FAL and *MAL methods, respectively.
Subscripts $\sim$ and $\approx$ denote direct fine-tuning and pipelined fine-tuning, respectively.
Underlined values refer to the higher performance when comparing the two learning methods.

Self Ensemble Model                      MAP       MRR
2l|TF-IDF Baseline     9.03     22.60
1c|CAM$^\otimes_\approx$     c]@l@

[-8.5pt] x$_{\textit{TyDi}\text{ }\textit{QA}_{\text{AR}} \rightarrow  \textit{Tafseer}\rightarrow\textit{TASK A}}$
23.02     47.06

1c|ARB$^\otimes_\approx$     c]@l@

[-8.5pt] x$_{\textit{TyDi}\text{ }\textit{QA}_{\text{AR}} \rightarrow  \textit{Tafseer}\rightarrow\textit{TASK A}}$
24.64     49.39

1c|MIX$^\otimes_\approx$     CAM$^\otimes_\approx$ + ARB$^\otimes_\approx$     25.05     46.10

Results on the hidden split for task A.
$\zeta$ is set to mark 15% of questions as unanswerable.

Both tables present comprehensive results for different question types, as well as
the outcomes for a manually set threshold $\zeta$ and $\zeta^\bigstar$, i.e.,
the threshold that yields the best performance.
(See Appendix <ref> for more details about $\zeta$ and optimal $\zeta$ selection)

Considering the question types , experiments of and obtains the best *MAP performance for zero-answer and multi-answer questions for task A.

With regard to the hidden split, Tables <ref> and <ref>
provide a summary of our official submissions.

In task A, as
shown in Table <ref>, we made 3 cross-encoder submissions:
MIX$^\otimes_\approx$, which is an ensemble combining runs from
CAM$^\otimes_\approx$ and ARB$^\otimes_\approx$ cross encoders. MIX$^\otimes_\approx$ achieved a *MAP of
25.05%. In comparison, the TF-IDF baseline only achieved a *MAP of 9.03%.

On the other hand, in task B, we experimented with our two best performing models in Table <ref>.
As shown in Table <ref>,
outperformed with a *pAP of 57.11%. This result
is consistent with the findings from the faithful validation split <cit.> in Table <ref>
for and . Specifically, the
*MAL method outperformed *FAL for all of our models in the faithful
validation split (underlined in Table <ref>).

§ ANALYSIS AND DISCUSSION

Regarding RQ1, external resources always bring significant improvements to the same *LM for
both tasks.
For task A, we have three stages of fine-tuning as indicated by arrows in Table <ref>.
For example, when is fine-tuned with external resources into the *MAP performance
improves from 8.96% to 26.60% for single models as in Table <ref>.
In similar fashion for task B,
outperforms by almost 13% for the standard split in Table <ref>.

To answer our RQ2, ensemble learning consistently outperforms single models for both tasks.
For instance, ensemble surpasses its single model by 3.5% for the *MAP metric for task A.
Similarly, ensemble outperforms its corresponding single model by almost a *pAP of 2% for task B.

With regard to RQ3, the hyperparameter $\zeta$ affects the zero answer type evaluation scores for both tasks.
We make best use of the available data by employing a quantile method to determine the threshold $\zeta$ for both tasks.
However, model *MAP performance improves by 3% when the optimal $\zeta^\bigstar$ is employed for task A.
This suggests that there is a room for improvement for the $\zeta$ parameter.
(Please refer to Appendix <ref> for more details about $\zeta$ selection and RQ3).

In Table <ref>, we experimented with dual-encoders using both random and hard negatives <cit.> to address RQ4.
outperforms by almost 4.5% when we perform hard negatives mining using a fine-tuned checkpoint .

In Table <ref>,
*MAL learning method consistently brings significant improvements to the final performance for all models over the faithful split.
Moreover, it consistently outperforms *FAL learning method for the multi-answer type of samples.
For instance, performs better than , achieving a  *pAP score of 51.55%  compared to 43.69% achieved by
for the subset of multi-answer samples.
However, due to the fact that multi-answer samples make up only 18%
of the development samples in the standard split (Table <ref>),
*MAL does not always outperform *FAL for the standard split overall performance.
This finding addresses RQ5.

With regard to RQ6, the post-processing approach proposed by <cit.> always surpasses the raw prediction score for both single and ensemble models.
This is represented by Post subscript in Table <ref>.
For example,  post-processing improves both single model and self-ensemble *pAP performance by almost 3%.

1c|Method     Self Ensemble Model     pAP@10
3l|Full-passage Baseline     32.68
1c|ELC$^\textit{M}_\approx$     1c|3*MAL     c]@l@

[-8.5pt] xx$_{\textit{TyDi}\text{ }\textit{QA}_{\text{AR}} \rightarrow \textit{TASK B}}$

1c|ARB$^\textit{M}_\approx$     1c|     c]@l@

[-8.5pt] xx$_{\textit{TyDi}\text{ }\textit{QA}_{\text{AR}} \rightarrow \textit{TASK B}}$

1c|MIX$^{\textit{M}}_\approx$     1c|                    ELC$^\textit{M}_\approx$ + ARB$^\textit{M}_\approx$     56.43

Results on the hidden split for task B.
$\zeta$ is set to mark 5% of pairs as unanswerable.

§ CONCLUSION

In this paper, we have presented our solution for both task A and task B  of  QA 2023 shared tasks.
We explored various Arabic *LMs
using different training approaches and architectures.
Our best performing
systems are ensemble-based, enhanced with transfer learning using external learning
Lastly, we addressed a set of RQs that highlight
the main strengths of our work.

§ LIMITATIONS

In this paper, we have adapted conventional learning-based architectures
for Arabic *QA tasks, specifically for *MRC and ad hoc search.
However, we faced several challenges
throughout our study. One significant challenge was the scarcity of training
resources, along with the imbalanced distribution of topics and question types. This
was particularly evident in the zero-answer cases.
As a consequence, our zero-answer thresholding mechanism demonstrated high sensitivity to each
individual model.

Additionally, we noticed significant performance variations
due to the small size of the datasets.
In order to tackle the problem of variations and noisy predictions, we
investigated an ensemble approach. However, we still suggest that
the results we obtained during the development phase may not accurately reflect
the actual performance of learning systems.
Despite the effectiveness of faithful splits for task B, we still suggest exploring n-fold cross-validation for both tasks.
However, our computation resources were significantly limited during the competition phase.

For task B, our models
trained for *MRC were found to be suboptimal for
ranking tasks. Although our post-processing technique improved the raw
predictions, this indicates the necessity for other ranking-based *MRC approaches.
Furthermore, we would like to explore the performance of large *LMs on this particular task.

§ ETHICS STATEMENT

The paper contains facts and beliefs that do not necessarily reflect
the views or opinions of the authors. The information presented is based on
objective analysis and does not aim to promote or endorse any particular
religious interpretation.

§ ACKNOWLEDGEMENTS

We would like to extend our heartfelt appreciation to Dr. Moustafa El Zantout
for his invaluable support and insights during the course of this work.
We would also like to express our deep gratitude to the organizers of
QA for their efforts in promoting research
in Arabic in general, and the most significant Arabic text, the Holy .

§ APPENDIX

§ DATASET ADDITIONAL DETAILS

AyaTEC is a dataset designed to evaluate the performance of retrieval-based
Arabic QA systems over the Holy .
It contains 207 questions and 1,762 corresponding answers, which are categorized into 11 topics covering different aspects of the .
The dataset caters to the information needs of two types
of users: skeptical and curious <cit.>.

The dataset includes single-answer and
multi-answer questions, as well as questions that have no answer.
Both QA 2023 shared tasks are primarily based on an adapted version of AyaTEC <cit.>.
Figure <ref> illustrates an example from task A.
The question asks whether there is a reference in the to the body part used for reasoning.
Four relevant ic segments are annotated to have an answer for this question.
Figure <ref> depicts a question-passage-answer triplet from task B.
The question in this case is about creatures capable of praising God, within the context of the given passage.

A sample from shared task A. We highlight the most relevant part in each ic segment.

A sample from shared task B. We highlight the ground truth answers in the ic passage.

§.§ Topic Distribution for tasks

AyaTEC covers 11 diverse topics referenced in the Holy .
Figure <ref> illustrates the imbalanced nature of those different topics.
Furthermore, the representation of unique questions is significantly limited in comparison to question-passage-answer triplets.
Additionally, it is evident that the ratio of triplets to unique questions varies for each respective topic.
In task B, these factors give rise to common questions across various passages.
Consequently, they result in data leakage between the training and development splits <cit.>.
(Further information regarding this can be found in Appendix <ref>)

§.§ Task A Evaluation Measures

For this ranking task, systems are expected to return up to 10 ic passages for each question when possible.
If the system determines that the question is unanswerable from the entire , a null document is only returned, indicated by -1.
The primary measure for the task is *MAP, which gives full credit only if all relevant documents are retrieved at the top of the ranked answer list.
For the zero-answer questions, full credit is given to successful systems
only when they are unable to find any relevant ic passage to answer the question, and return the null document.
In addition to *MAP, *MRR is also reported, which gives credit just for the first relevant document from the ranked list <cit.>.

In formal notation, we begin by defining the function  $\operatorname{\alpha}(q, p)$, which is a
binary relevance function that indicates whether a passage $p$ is annotated as
relevant to a question $q$ in the test collection. Equ.(<ref>) represents the function that calculates
the total number of relevant  ic passages from the *QPC to $q$.

\begin{equation}
\operatorname{\psi}(q) = {\sum_{p \in \mathcal{QPC}} \operatorname{\alpha}(q, p)}
\label{mypsi}
\end{equation}

Zero-answer questions have a zero value for the function $\psi$, and their
*MAP score is calculated in a different way. Equ.(<ref>) shows the
evaluation measure for *MAP for answerable questions. For a
ranked list $R$, we calculate the precision at each possible cutoff $@i$ at which a
relevant document is present <cit.>.

\begin{equation}
\operatorname{MAP}(R, q)=\frac{\sum_{(i, p) \in R} \operatorname{Prec} @ i(R, q) \cdot \operatorname{\alpha}(q, p)}{ \operatorname{\psi}(q)},
\label{MAP}
\end{equation}

Equ.(<ref>) illustrates the combined *MAP
evaluation measure for task A. In this measure, zero-answer questions are given
full credit only when $R$ is the null document,
represented by $-1$ in the official evaluation script [ The symbol $\equiv$ signifies the equivalence operator between two lists.] <cit.>.

\begin{equation}
\operatorname{MAP_\text{A}}(R, q)=
\begin{cases}
\mathds{1}_{R \equiv [-1]} & \text{if } \psi(q) = 0 \\\\
\operatorname{MAP}(R, q) &  \text{Otherwise}
\end{cases}
\label{MAP_total}
\end{equation}

$\mathds{1}_{C}$ is an indicator function, which returns $1$ if the binary condition $C$ holds and $0$ otherwise.

Distribution of *QRCDv1.2 over the 11 topics for task A questions and task B triplets.

§.§ Task B Evaluation Measures

Standard *MRC tasks, like SQuADv2.0, are evaluated based only on the first prediction.
In contrast, task B is evaluated as a ranking task against a ranked list, rather than relying solely on the top prediction.
As in task A, systems are expected to return up to 10 answer spans from a given ic passage to answer a question when possible.
The primary evaluation metric for this task is *pAP <cit.>.
This metric incorporates partial matching with the traditional rank-based Average Precision measure, i.e., *MAP.
In the case of unanswerable samples, the system receives a full score if it only returns and empty ranked list.

Formally, partial matching is performed over token indexes of two substrings extracted from a given supporting passage.
Based on <cit.>, $F_{1}$ is used to calculate the similarity between the two substrings $R_{k}$ and $g$.
$R_{k}$ represents the $k^{th}$ answer from a ranked list $R$, and $g$ refers to any ground truth answer from the set of ground truth answers $G$.

\begin{equation}
\mathcal{F_{\hspace{-2 pt}{\scaleto{1}{3pt}}}}^{R}_{k}  = \max_{g \in G}\left\{ F_{1}\left(R_{k},g\right)\right\}
\label{eq:f1}
\end{equation}

In terms of Equ.(<ref>), we can define a partial matching version of precision at cutoff $K$, i.e., $\operatorname{pPrec}$ <cit.>.

\begin{equation}
\operatorname{pPrec}@K(R)=\frac{1}{K} \sum_{i=1}^K  \mathcal{F_{\hspace{-2 pt}{\scaleto{1}{3pt}}}}^{R}_{i}
\label{eq:pPrec}
\end{equation}

In their study, <cit.> introduced a method for handling multi-answer
samples. They proposed a string splitting mechanism that ensures only one
correct answer is matched in each entry of $R$.
Equ.(<ref>) presents the *pAP evaluation metric for multi-answer ranking *MRC in terms of $\operatorname{pPrec}$ <cit.>,
which stands as a token-level partial matching version of Equ(<ref>).

\begin{equation}
\operatorname{pAP}(R)=\frac{\sum_{i \in R} \operatorname{pPrec} @ i(R) \cdot \operatorname{\beta}(R,i)}{ |G|},
\label{pAP}
\end{equation}

$\operatorname{\beta}(R,i)$ is a binary function that returns one if $R_i$ is a partially relevant answer.
More specifically,

\begin{equation}
\operatorname{\beta}(R,k)=   \mathds{1}_{\mathcal{F_{\hspace{-2 pt}{\scaleto{1}{3pt}}}}^{R}_{k} > 0}
\end{equation}

In similar fashion, Equ.(<ref>) presents the complete *pAP
evaluation measure for task B. In this measure, zero-answer samples are given
full credit only when $R$ is an empty list <cit.>.

\begin{equation}
\operatorname{pAP_\text{B}}(R)=
\begin{cases}
\mathds{1}_{R \equiv [\text{ }]} & \text{if } |G| = 0 \\\\
\operatorname{pAP}(R) &  \text{Otherwise}
\end{cases}
\label{pAP_total}
\end{equation}

§.§ Leakage in *QRCDv1.2

<cit.> analyzed *QRCDv1.1 and identified instances where passages and questions were repeated.
They classified *QRCDv1.1 into four logical mutually-exclusive categories according to their complexity.
Table <ref> provides a summary of the criteria used and the expected behavior of
trained *LMs for each category.
Additionally, symbols are employed to indicate the levels of complexity within each category, as determined by performance scores obtained by <cit.>.
Based on their analysis, <cit.> solely utilized $D_{(3) \text { ood + hard }}$ for their final development split for *QRCDv1.1.

Category     Criteria     Expected *LM behavior
$D_{(1) \text { in+leakage }} $     c]@c@
Samples with repeated passage-answer
or question-answer pairs
Memorize answers and overfit to
training data $^{\AC}$

$D_{(2) \text { in}+\text {no leakage }}$     c]@c@
Samples with repeated passages but
having unique answers which are
different from  $D_{(1)}$ answers
Reasoning is required to find the
right answer
$D_{(3) \text { ood + hard }}$     c]@c@
Samples with unique passages but
having rarely repeated questions
(appearing 3 times or less)
Some reasoning is required to find
the right answer for rare questions $^{\VHF}$

$D_{(4) \text { ood + easy}}$     c]@c@
Samples with unique passages but
having commonly repeated questions
(more than 3 times)
Lexical matching guides trained
*LMs to find similar answers  $^{\HF}$

Description of the four categories introduced by <cit.> over *QRCDv1.1 dataset.
We show the criteria for identifying each category and the expected behavior for a fine-tuned *LM.
We denote the complexity of each category using symbols.
For instance, -1pt represents the most challenging set for learning systems, while refers the least challenging set.

Category     Splitting Strategy by <cit.>     Our Modified Splitting Strategy
$D_{(1) \text { in+leakage }}$     c]@c@
For duplicate question-answer or passage-answer pairs,
choose only one sample for training and leave the rest
for the development set.
Use it entirely for training, this is due to the fact that
$D_{(1)}$ is trivial for development.
To balance the zero-answer questions ratio, we take entire
zero-answer leakage groups into the development set.
We employ disjoint-set algorithm for this purpose.

$D_{(2) \text { in}+\text {no leakage }}$     c]@c@
Split randomly with a splitting ratio of 86.7%
for training and 13.3% for development,
which corresponds to the original ratio of the data.
Split them into two overlapping sets, as such, confusing examples
with the same passages are distributed among training and
development with different answers.

$D_{(3) \text { ood + hard }}$     c]@c@
Only use it for the development set
(removed from training).
Same as <cit.>
$D_{(4) \text { ood + easy}}$     c]@c@
Split randomly with a splitting ratio of 86.7%
for training and 13.3% for development,
which corresponds to the original ratio of the data.
Use it entirely for training, this is due to the fact that
$D_{(4)}$ is trivial for development.

Description of our modified faithful splitting for *QRCDv1.2 dataset over the four categories introduced by <cit.>.
We also show their proposed splitting approach <cit.>.
Check Table <ref> for more details and reasons behind such splitting strategies.

Dual-encoder generic architecture with metric learning for neural ranking.

Cross-encoder generic architecture for an input pair of a question and a passage with a predicted similarity score.

Diagrams for model architectures for task A.

In this work, we extend their approach for *QRCDv1.2.
We slightly modify this by considering both $D_{(2)}$ and $D_{(3)}$ for the development split.
In addition, we employ disjoint set algorithm to find all leakage groups in $D_{(1)}$.
We use those groups to balance the zero-answer questions ratio in the development split.
This is because 90% of zero-answer questions belong to the trivial leakage group $D_{(1)}$.

In their work, <cit.> also proposed a resplitting approach for *QRCDv1.1.
They reorganized training and development splits using the four logical groups to create what they called faithful splits for *QRCDv1.1.
Faithful splits aim to create more representative evaluations for *QRCDv1.1 dataset.
Table <ref> summarizes the modifications we made for performing evaluation using faithful splits.
Table <ref> presents the distribution of
our faithful split for *QRCDv1.2 based on our modified splitting strategy outlined in
Table <ref>. It also includes the distribution of zero-answer
samples within each group.
As in Table <ref>, we preserve the original ratio of training to development splits.
Additionally, the percentage of zero-answer samples within each
split is preserved compared to the original distribution in Table <ref>.

Category                             Train                   Development             Total
$D_{(1) \text { in+leakage }}$                405 (49)                         007 (7)1     1412 (56)

$D_{(2) \text { in}+\text {no leakage }}$     290 (2)1               095 (1)1     2385 (3)0

$D_{(3) \text { ood + hard }}$                000 (0)5     062 (3)1     1262 (3)0

$D_{(4) \text { ood + easy}}$                 296 (0)1               000 (0)1     1296 (0)0
Total                                         991 (51)                         164 (11)                         1155 (62)
Zero-answer %                                5.15 %                          6.71 %                          5.37 %

*QRCDv1.2 dataset distribution of pairs for our faithful splitting over the four categories introduced by <cit.>.
Parenthesized values refer to the number of zero-answer samples within each category for each split.

§.§ External Learning Resources

We leverage external resources to perform pipelined fine-tuning for both tasks
A and B.

For task A, we utilized interpretation resources (tafseer) from
both Muyassar and Jalalayn, obtained from <cit.>.
We created pairs of *QPC
ic passages and their corresponding interpretations, resulting in
approximately 2.5K relevant pairs.
Additionally, we used the Arabic TyDI-QA Gold$\textbf{P}$ dataset <cit.> to generate pairs of relevant questions
and their supporting evidence passages, resulting in 15K relevant pairs.
task B, we solely relied on the Arabic subset of the TyDI-QA Gold$\textbf{P}$
*MRC dataset <cit.>. This dataset consists of
approximately 15K question-passage-answer triplets.

§ TRANSFER LEARNING

In order to overcome the limited training resources for both tasks, we
incorporate external *QA and interpretation resources
(tafseer) <cit.>.
External resources enhance our learning systems in general by leveraging transfer learning across multiple fine-tuning stages <cit.>.
We use arrows in subscripts in Tables <ref>, <ref>, <ref>, and <ref> to refer to stages of fine-tuning.
(More details about external learning resources and
their construction in Appendix <ref>)

§ ENSEMBLE LEARNING

We utilize a voting self-ensemble technique for a group of fine-tuned models
trained with different seeds <cit.>. We use the raw
predictions without applying a zero-answer threshold.

In task A, for an ensemble $\mathcal{E}$ we aggregate the relevance
scores for a ic passage $p$ and a question $q$ assigned by a model $\varphi$.
The ensemble relevance score $\mathcal{S}$ between $p$ and $q$ is as follows:

\begin{equation}
\operatorname{ \mathcal{S}}(q,p) = {\sum_{\varphi \in \mathcal{E}} \operatorname{\varphi}(q, p)}
\label{ens-a}
\end{equation}

In similar fashion for task B, we leverage a span voting ensemble <cit.>.
For each sample, we aggregate span scores for each span $s$
made by each predictor $\varphi$.

\begin{equation}
\operatorname{ \mathcal{S}}(s) = {\sum_{\varphi \in \mathcal{E}} \operatorname{\varphi}(s)}
\label{ens-b}
\end{equation}

After that, we apply zero-answer thresholding to the aggregated result.

§ ADDITIONAL SYSTEM DETAILS FOR TASK A

We summarize both architectures for task A in Figures <ref> and <ref> for dual-encoders and cross-encoders, respectively.

§.§ Implementation Details

In our  *STAR training process, we incorporate both random in-batch negatives and
hard negatives. Random negatives involve randomly selecting irrelevant documents
for each query, providing positive and negative signals for learning systems <cit.>.
On the other hand, hard negatives refer to the most offending
irrelevant examples predicted by an encoder similarity score <cit.>. In a batch of size 16, we
encode 16 different queries with their corresponding positive documents; in addition,
in-batch negatives are used for all other queries. These negatives can be
chosen randomly or through *STAR hard negative mining.
We use a learning rate of $5\times10^{-5}$ for all of our dual-encoder experiments.
In the case of cross-encoders, we generate question-document pairs.
These pairs have a ratio of one positive pair and three randomly selected negative pairs.
For all of our cross-encoders, we use a learning rate of $1\times10^{-6}$ with a batch size of 16.

§.§ Zero-answer Prediction

We assign a likelihood for each question $q$ to be answerable using
the total relevance scores for the top returned passages $R$.
$\varphi$ refers to a general relevance predictor between $q$ and a passage $p$.

\begin{equation}
\operatorname{\gamma}(q) = -{\sum_{p \in R} \operatorname{\varphi}(q, p)}
\label{eq:threshold-a}
\end{equation}

The negative sign corresponds to the inverse proportional relationship between
high relevance scores and the likelihood of unanswerability.
We then normalize those scores for all questions into ${\operatorname{\bar{\gamma}}(q)}$ and apply
a no answer threshold $\zeta$.
We define a binary threshold function, $\operatorname{\sigma}$,
which applies the threshold to identify unanswerable questions.

\begin{equation}
\operatorname{\sigma}(q)= \mathds{1}_{{\operatorname{\bar{\gamma}}(q)} > \zeta}
\label{eq:threshold-a2}
\end{equation}

§ ADDITIONAL SYSTEM DETAILS FOR TASK B

In this work, we fine-tune *LMs for extractive *MRC as span predictors <cit.>.
The fine-tuning process involves packing each
question-passage pair $x$ together and feeding it to a *LM to predict the start and end token indices from the passage, as shown in Figure <ref>.
To achieve this, a trainable randomly
initialized start vector $S$ and end vector $E$ are stacked on top of the  *LM,
having the $i^{th}$ token hidden-representation $T_{i}$.
The final model with
the newly stacked layers has learnable parameters $\theta$.

Generic architecture illustration of a *LM for ranking *MRC.

Standard (*FAL)

Illustration of Learning Methods.

Thresholding effect against *MAP performance for one of our fine-tuned models.

The dot product between $S$ and $T_{i}$ is chosen to determine the score that the $i^{th}$ token is the start of the answer span.
These scores for all passage tokens are
followed by a softmax layer that produces the probabilities for individual
tokens being the start of the answer span <cit.>.
depicts the probability that the $i^{th}$ token is the start of the answer span.

\begin{equation}
\mathbb{P}\left(i \mid x ; \theta\right)=\frac{e^{S \cdot T_{i}}}{\sum_{j}^{|T|} e^{S \cdot T_{j}}}
\label{eqn:start_softmax}
\end{equation}

Under full-supervision, the training objective is to optimize the log-likelihoods for both the ground truth start and end positions.
For a model with learnable $\theta$, an input $x$, and a single ground truth answer span $y$, the log likelihood for the start token position is as follows:

\begin{equation}
\mathcal{L}_{\text {start}}\left(\theta \mid x, y\right)=-\log \mathbb{P} \left({y^{}_{_{\scaleto{s}{3pt}}}} {\mid x ; \theta}\right)
\label{eqn:loss_single}
\end{equation}

where the subscript $s$ in ${y^{}_{_{\scaleto{s}{3pt}}}}$ refers to the start position of the answer span $y$.

If there are multiple answers for a sample $x$, we rather have a set of plausible answer spans $\mathcal{Y}$.
<cit.> in QA 2022 tackled this by considering any answer span from $\mathcal{Y}$ by taking one at random or the first answer span, namely, $y^1$.
We denote the $i^{th}$ answer from $\mathcal{Y}$ as $y^i$.
We call this learning method First answer loss (FAL). This can be formulated in terms of $\mathcal{Y}$ as denoted below:

\begin{equation}
\mathcal{L}\strut^{\text {FAL}}_{\text {start}}\left(\theta \mid x, \mathcal{Y}\right)=-\log \mathbb{P}\left({y^1_{_{\scaleto{s}{3pt}}}} \mid x ; \theta\right)
\label{eqn:loss_fal}
\end{equation}

Figure <ref> illustrates this learning method.
However, *QRCDv1.2 task B considers a multi-answer *MRC scenario, this leads to discrepancy between
training and testing when *FAL learning method is employed for fine-tuning.
Towards this end, we define *MAL learning method. This learning method takes the multi-answer cases in consideration by
optimizing for all answers altogether.
Mathematically, this generalizes to any $y^i$ from the set $\mathcal{Y}$ and takes the sum of the log likelihood losses for multiple answers
as shown in Equ.(<ref>):

\begin{equation}
\mathcal{L}\strut^{\text {MAL}}_{\text {start}}\left(\theta \mid x, \mathcal{Y}\right)=-\sum_{y^i \in \mathcal{Y}}{\log \mathbb{P}\left({y^i_{_{\scaleto{s}{3pt}}}} \mid x ; \theta\right)}
\label{eqn:loss_mal}
\end{equation}

We show the *MAL learning method in Figure <ref>.

§.§ Implementation Details

To enhance *LM predictions, we employ a post-processing approach.
<cit.> proposed an effective non-maximum suppression post-processing approach at QA 2022 <cit.>.
They also proposed some operations for rejecting uninformative short answers.
For all of our models, we used a learning rate of $2\times10^{-5}$ and a batch size of 16.

§.§ Zero-answer Prediction

*MRC for SQuADv2.0-like datasets uses null answer [CLS] token probability
to give a likelihood for a question to have an answer within the supporting passage
This works by finding the difference between the null answer score of [CLS] token
and the non-empty answer span with the highest score.
$\varphi$ is a general span extractor that operates on a question $q$ and a passage $p$.

\begin{equation}
\operatorname{\gamma}(q,p) =  \operatorname{\varphi}(q, p)_{_{\textit{[CLS]}}} - \operatorname{\varphi}(q, p)_{_{\textit{MAX}}}
\label{eq:threshold-b}
\end{equation}

Upon calculating scores for all samples, we proceed to normalize them into
${\operatorname{\bar{\gamma}}(q)}$ and then apply a threshold value $\zeta$ to
determine if there is no answer. To identify unanswerable questions, we use a
binary threshold function $\operatorname{\sigma}$,

\begin{equation}
\operatorname{\sigma}(q)= \mathds{1}_{{\operatorname{\bar{\gamma}}(q)} > \zeta}
\label{eq:threshold-b2}
\end{equation}

§ $\ZETA$ SELECTION AND $\ZETA^\BIGSTAR$

In this work, we defined $\zeta$ hyperparameter for zero-answer
thresholding. This hyperparameter controls the proportion of samples that are
considered to be zero-answer. Due to the small size of the dataset, we used a
quantile method to set $\zeta$. This method marks a proportion of the samples
according to the statistics of the dataset.
Task B is less sensitive to this parameter because almost 5% of the samples are zero-answer.
In contrast, task A is highly sensitive to this parameter because of the larger proportion of zero-answer cases compared to task A.
Additionally, We are interested in finding the theoretical upperbound
performance for  $\zeta$; this is addressed by RQ3.

In Tables <ref> and <ref> we use $\bigstar$ accompanied by $\zeta$
to refer to the optimal performance of the binary classification problem of has-answer vs. has-no-answer, as explained
in Appendices <ref> and <ref>.
Figure <ref> illustrates the thresholding effect against fine-tuned model performance for task A; this answers RQ3.
As we can see, the $\zeta$ hyperparameter can not be set arbitrarily.
Instead, we can adjust it by considering the outcomes obtained from trained models on the training data.
To find the optimal threshold $\zeta^\bigstar$ for both tasks, we implemented a greedy optimization algorithm for
all possible levels of thresholds made by a given model; check the code for more details [
In both code bases, this is performed by function find_best_thresh.
You may find this function under metrics directory in compute_score_qrcd.py and
helpers.py scripts for tasks A and B, respectively.


# Non-Hermitian non-Abelian topological insulators with $PT$ symmetry

Motohiko Ezawa  Department of Applied Physics, University of Tokyo, Hongo
7-3-1, 113-8656, Japan

###### Abstract

We study a non-Hermitian non-Abelian topological insulator preserving $PT$
symmetry, where the non-Hermitian term represents nonreciprocal hoppings. As
it increases, a spontaneous $PT$ symmetry breaking transition occurs in the
perfect-flat band model from a real-line-gap topological insulator into an
imaginary-line-gap topological insulator. By introducing a band bending term,
we realize two phase transitions, where a metallic phase emerges between the
above two topological insulator phases. We discuss an electric-circuit
realization of non-Hermitian non-Abelian topological insulators. We find that
the spontaneous $PT$ symmetry breaking as well as the edge states are well
observed by the impedance resonance.

##  I Introduction

Topological insulators are one of the most fascinating ideas in contemporary
physicsHasan ; Qi . They are characterized by topological numbers such as the
winding number, the Chern number and the $\mathbb{Z}_{2}$ index. However, all
of these topological numbers are Abelian.

Non-Abelian topological charges are discussed in three-band models protected
by $PT$ symmetryWuScience ; Tiwari ; Guo ; YangPRL ; Leng  or $C_{2}T$
symmetryBouhon ; DWang . They are realized in nodal line semimetalsWuScience ;
Tiwari ; YangPRL ; Leng ; MWang ; BJiang ; HPark  in three dimensions and Weyl
pointsBouhon  in two dimensions. Non-Abelian topological insulators in one
dimension are studied for three-band modelsGuo  and four-band modelsJiang4 .
They are experimentally observed in photonic systemsYangPRL ; DWang , phononic
systemsBJiang ; BPeng  and transmission linesGuo ; Jiang4 . In addition, a
generalization to multi-band theories is proposed in nodal line
semimetalsWuScience . As far as we aware of, there is no study on non-
Hermitian non-Abelian topological phases so far.

Non-Hermitian topological physics have attracted much attentionBender ;
Bender2 ; Malzard ; Konotop ; Rako ; Gana ; Zhu ; Yao ; Jin ; Liang ; Nori ;
Lieu ; UedaPRX ; Coba ; Jiang ; JPhys ; Ashida . In non-Hermitian systems
eigenvalues and eigenfunctions are complex in general. However, they are
restricted to be real if $PT$ symmetry is imposedBender ; Mosta ; Ruter ; Yuce
; LFeng ; Gana ; Weimann . There is a $PT$ symmetry breaking transition, where
the eigenvalues and eigenfunctions become complex. Nonreciprocal hopping is
such a hopping that the right-going and left-going hopping amplitude are
differentHatano . It makes a system non-Hermitian.

In this paper, we study a non-Hermitian non-Abelian topological insulator in
an $N$ band model with $PT$ symmetry. We show that a spontaneous $PT$ symmetry
breaking is induced by increasing the nonreciprocal hoppings from a phase
transition from a real-line-gap topological insulator to an imaginary-line-gap
topological insulator in the case of a perfect-flat band model. Furthermore,
by introducing a band bending term, we may generalize the model to have a
metal with two critical points, where a metallic phase emerges between the
above two topological insulator phases. Finally, we show how to implement the
present model in electric circuits. The edge states and the spontaneous $PT$
symmetry breaking are well signaled by the impedance resonance.

Figure 1: Illustration of the tight-binding Hamiltonian. (a) Hermitian and (b)
non-Hermitian models. Interactions between the $\alpha$ and $\beta$ chains
yield a non-Abelian topological number. All other chains shown in green act as
spectators. Red arrows represent nonreciprocal hoppings.

##  II Non-Hermitian Non-Abelian topological insulators

###  II.1 Hermitian Hamiltonian

Figure 2: Energy spectrum of the non-Hermitian Hamiltonian in nanoribbon
geometry. Eigenvalues of (a1) the perfectly flat $\alpha$ and $\beta$ bands
with $\xi=0$ and (a2) the bended bands with $\xi=0.5$ shown in blue. The red
dots represent the topological edge states. The band structure as a function
of $\xi$ with (b1) $\gamma=0$ and (b2) $\gamma=0.25$. The red lines represent
the topological edge states. (c1) and (c2) Real part of the energy. (d1) and
(d2) Imaginary part of the energy. We have set $\xi=0$ for (c1) and (d1),
while we have set $\xi=0.5$ for (c2) and (d2). The bulk bands are colored in
blue, while the edge states are colored in red. The spectator band is colored
in green. When $\xi=0$, there are two phases, a real-line-gap topological
insulator (real-TI) phase and an imaginary-line-gap topological insulator (im-
TI) phase. When $\xi\not=0$, a metallic phase emerges between these two
topological insulator phases. We have set $\varepsilon_{\alpha}=1$ and
$\varepsilon_{\beta}=2$.

We start with a Hermitian system capable to describe a non-Abelian topological
insulator based of the one-dimensional lattice in Fig.1(a). We consider
generators of $\mathfrak{so}\left(N\right)$ rotation $L_{\alpha\beta}$ indexed
by $\alpha$ and $\beta$, whose $ab$ components are defined by

$\left(L_{\alpha\beta}\right)_{ab}=\delta_{\alpha b}\delta_{\beta
a}-\delta_{\alpha a}\delta_{\beta b}.$ (1)

We consider a $PT$-invariant Hamiltonian in the momentum space given
byWuScience

$H_{\alpha\beta}\left(k\right)=R_{\alpha\beta}\left(\frac{k}{2}\right)\text{diag.}\left(\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{N}\right)R_{\alpha\beta}\left(\frac{k}{2}\right)^{\text{t}},$
(2)

where $0\leq k<2\pi$, $1\leq\alpha,\beta\leq N$,
$\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{N}$ are real, and

$R_{\alpha\beta}\left(\frac{k}{2}\right)=e^{\frac{k}{2}L_{\alpha\beta}}$ (3)

is a rotation matrix given by

$\displaystyle\left(R_{\alpha\beta}\left(\frac{k}{2}\right)\right)_{ab}$
$\displaystyle=\delta_{ab}+\left(\delta_{a\alpha}\delta_{b\alpha}+\delta_{a\beta}\delta_{b\beta}\right)\cos\frac{k}{2}$
$\displaystyle+\left(\delta_{a\beta}\delta_{b\alpha}-\delta_{a\alpha}\delta_{b\beta}\right)\sin\frac{k}{2}.$
(4)

The Hamiltonian (2) is explicitly written as

$\displaystyle H_{\alpha\beta}\left(k\right)$
$\displaystyle=\frac{\varepsilon_{\alpha}+\varepsilon_{\beta}}{2}$
$\displaystyle+\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{2}\left(\delta_{a\alpha}\delta_{b\alpha}-\delta_{a\beta}\delta_{b\beta}\right)\cos
k$
$\displaystyle+\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{2}\left(\delta_{a\beta}\delta_{b\alpha}+\delta_{a\alpha}\delta_{b\beta}\right)\sin
k.$ (5)

It is decomposed into two parts,

$H_{\alpha\beta}\left(k\right)=\bigoplus_{j\neq\alpha,\beta}H_{j}\oplus
H_{\alpha\beta}^{\prime}\left(k\right),$ (6)

where

$\displaystyle H_{j}$ $\displaystyle=\varepsilon_{j}\mathbb{I}_{1},$ (7)
$\displaystyle H_{\alpha\beta}^{\prime}\left(\theta\right)$
$\displaystyle=\left[\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{2}\left(\begin{array}[]{cc}\cos
k&\sin k\\\ \sin k&-\cos
k\end{array}\right)+\frac{\varepsilon_{\alpha}+\varepsilon_{\beta}}{2}\mathbb{I}_{2}\right].$
(10)

The Hamiltonian is nontrivial only for the $\alpha$ and $\beta$ bands, with
eigenvalues $\varepsilon_{\alpha}$ and $\varepsilon_{\beta}$. All other bands
are spectators with respect to the $\alpha$ and $\beta$ bands. See Fig.1.

The energy spectrum of the bulk Hamiltonian does not change by the rotation
(3) and is given by

$E\left(k\right)=\varepsilon_{1},\quad\varepsilon_{2},\quad\cdots,\quad\varepsilon_{N}.$
(11)

The eigenfunctions for the $2\times 2$ matrix
$H_{\alpha\beta}^{\prime}\left(k\right)$ are

$\displaystyle\psi_{a}^{+}$
$\displaystyle=\delta_{a\alpha}\sin\frac{k}{2}+\delta_{a\beta}\cos\frac{k}{2},$
(12) $\displaystyle\psi_{a}^{-}$
$\displaystyle=-\delta_{a\alpha}\cos\frac{k}{2}+\delta_{a\beta}\sin\frac{k}{2},$
(13)

while those for $H_{j}$ are $\psi_{a}=\delta_{aj}$.

The $\alpha$ and $\beta$ bands are perfectly flat. They are ($\ell$–$1$) fold
degenerate in a finite chain, where $\ell$ is the number of sites in the
chain. See Fig.2(a1).

Figure 3: (a1)$\sim$(e1) Real part of the energy and (a2)$\sim$(e2) imaginary
part of the energy. (a1) and (a2) for a real-line-gap topological insulator
(real-TI) with $\gamma=0$; (b1) and (b2) for a phase transition point with
$\gamma=\gamma_{1}=0.3$; (c1) and (c2) for a metal with $\gamma=0.4$; (d1) and
(d2) for a phase transition point with $\gamma=\gamma_{0}=0.5$; (e1) and (e2)
for an imaginary-line-gap topological insulator (im-TI) with $\gamma=0.6$. We
have set $\xi=0.2$ for all figures. See also the caption of Fig.2.

###  II.2 Non-Hermitian Hamiltonian

We generalize the Hermitian non-Abelian system (2) to a non-Hermitian non-
Abelian system, keeping $PT$ symmetry. We consider the Hamiltonian

$H_{\alpha\beta}^{\prime}\left(k;\gamma,\xi\right)=H_{\alpha\beta}^{\prime}\left(k\right)+i\gamma\sigma_{y}+\xi\sigma_{x}\sin
k,$ (14)

whose eigenenergies are

$E_{\alpha\beta}^{\prime}\left(k;\gamma,\xi\right)=\frac{\varepsilon_{\alpha}+\varepsilon_{\beta}\pm\sqrt{g(k;\gamma,\xi)}}{2},$
(15)

with

$g(k;\gamma,\xi)=\left(\varepsilon_{\alpha}-\varepsilon_{\beta}\right)^{2}-\gamma^{2}+4\xi\left(\varepsilon_{\alpha}-\varepsilon_{\beta}+\xi\right)\sin^{2}k.$
(16)

We explain the meanings of the $\gamma$ term and the $\xi$ term. The
Hamiltonian (14) is Hermitian when $\gamma=0$. When $\xi=0$ in addition, the
band structure is highly degenerate as in Fig.2(a1). This degeneracy is
resolved by introducing the $\xi$ term as shown in Fig.2(a2). We show the band
structure with $\gamma=0$ as a function of $\xi$ in Fig.2(b1). The perfect
flat bands at $\varepsilon_{\alpha}$ and $\varepsilon_{\beta}$ become bended
and have dispersions. We also show the band structure with $\gamma=0.25$ as a
function of $\xi$ in Fig.2(b2).

We show Re[$E_{\alpha\beta}^{\prime}\left(k;\gamma,\xi=0\right)$] in Fig.2(c1)
and Im[$E_{\alpha\beta}^{\prime}\left(k;\gamma,\xi=0\right)$] in Fig.2(d1) as
a function of $\gamma$. They are real for $|\gamma|\leq\gamma_{0}$ with

$\gamma_{0}=\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{2},$ (17)

where $PT$ symmetry is preserved. On the other hand, they are complex for
$|\gamma|>\gamma_{0}$, and hence $PT$ symmetry is spontaneously broken there.
Namely, although the Hamiltonian is $PT$-symmetric, eigenvalues and
eigenfunctions are no longer real in the spontaneous symmetry broken phase. We
show the real and imaginary parts of the energy as a function of $\gamma$ in
Fig.2(c2) and (c2), where the bulk band has a finite width. We also show the
real and imaginary parts of the energy as a function of the momentum $k$ in
Fig.3, where the bands have dispersions.

Especially, we have

$E_{\alpha\beta}^{\prime}\left(\frac{\pi}{2};\gamma,\xi\right)=\frac{\varepsilon_{\alpha}+\varepsilon_{\beta}\pm\sqrt{h(\gamma,\xi)}}{2},$
(18)

with

$h(\gamma,\xi)=\left(\varepsilon_{\alpha}-\varepsilon_{\beta}+2\xi-2\gamma\right)\left(\varepsilon_{\alpha}-\varepsilon_{\beta}+2\xi+2\gamma\right).$
(19)

By solving the condition that
$E_{\alpha\beta}^{\prime}\left(\frac{\pi}{2};\gamma,\xi\right)$ is complex, or
$h(\gamma,\xi)<0$, we find a phase transition point $\gamma_{1}$ in addition
to the phase transition point $\gamma_{0}$ as

$\gamma_{1}=\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{2}-\xi=\gamma_{0}-\xi.$
(20)

When $\xi>0$, the bulk energy is real for $\left|\gamma\right|\leq\gamma_{1}$,
complex for $\left|\gamma\right|>\gamma_{1}$. On the other hand, when $\xi<0$,
the bulk energy is real for $\left|\gamma\right|\leq\gamma_{0}$, complex for
$\left|\gamma\right|>\gamma_{0}$.

###  II.3 Tight-binding Hamiltonian

The tight-binding Hamiltonian (10) is written in the coordinate space as

$H_{\alpha\beta}^{\prime}=H_{0}+H_{\gamma}+H_{\xi},$ (21)

with

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{\alpha}-\varepsilon_{\beta}}{2}\sum_{j=1}^{\ell-1}(\left|\alpha_{j}\right\rangle\left\langle\alpha_{j+1}\right|+\left|\beta_{j}\right\rangle\left\langle\beta_{j+1}\right|,$
(22)
$\displaystyle+i\left|\alpha_{j}\right\rangle\left\langle\beta_{j+1}\right|-i\left|\beta_{j}\right\rangle\left\langle\alpha_{j+1}\right|)+\text{h.c.},$
$\displaystyle H_{\gamma}$ $\displaystyle=$
$\displaystyle\gamma\sum_{j=1}^{\ell}(\left|\alpha_{j}\right\rangle\left\langle\beta_{j}\right|-\left|\beta_{j}\right\rangle\left\langle\alpha_{j}\right|),$
(23) $\displaystyle H_{\xi}$ $\displaystyle=$ $\displaystyle
i\xi\sum_{j=1}^{\ell-1}(\left|\alpha_{j}\right\rangle\left\langle\beta_{j+1}\right|-\left|\beta_{j}\right\rangle\left\langle\alpha_{j+1}\right|)+\text{h.c.},$
(24)

where the first two terms in $H_{0}$ represent normal hoppings, while the last
two terms represent spin-orbit-like imaginary hoppings. The $\xi$ term
modifies the spin-orbit-like imaginary hoppings. The $\gamma$ term represents
nonreciprocal hoppings, which make the system non-Hermitian.

The tight-binding Hamiltonians for the spectator bands are simply given by

$H_{j\neq\alpha,\beta}=\sum_{j=1}^{\ell}\varepsilon_{j}\left|j\right\rangle\left\langle
j\right|+\sum_{j=1}^{\ell-1}t_{j}\left|j\right\rangle\left\langle
j+1\right|+\text{h.c.},$ (25)

where $\varepsilon_{j}$ is the on-site energy and $t_{j}$ is the hopping
parameter.

In this sense, it is enough to consider only the $\alpha$ and $\beta$ bands
for an arbitrary $N$ band system. We illustrate the tight-binding model in
Fig.1.

Figure 4: Non-Abelian topological charge marked in red as a function of
$\gamma$ for various $\xi$. (a) $\xi=0$, (b) $\xi=-0.2$, (c) $\xi=-0.4$, (d)
$\xi=0.2$ and (e) $\xi=0.4$. It is quantized at $1/2$ except for the metallic
phase. In the figures, real-TI (im-TI) stands for real(imaginary)-line-gap
topological insulator phase.

###  II.4 Edge states for non-Hermitian model

We illustrate the tight-binding model (21) in Fig.1(b). In a finite chain, two
localized states emerge at the edges with the energy

$E(\xi,\gamma)=\frac{\varepsilon_{\alpha}+\varepsilon_{\beta}}{2}\pm i\gamma$
(26)

in the presence of the $\gamma$ term and the $\xi$ term. They are degenerate
only in the Hermitian limit ($\gamma=0$). In contrasted to the bulk band, the
eigenenergy (26) is complex once $\gamma$ is introduced even for the $PT$
symmetric phase. We show Eq.(26) as a function of $\gamma$ in Fig.2. In
contrast to the bulk band, the eigenenergy (26) has no $\xi$ dependence: See
Figs.2(d1) and (d2).

When $\xi=0$, the eigenfunctions for the edge states
$\psi_{\alpha}\left(j\right)$ and $\psi_{\beta}\left(j\right)$ at the $j$ site
are perfectly localized at the edges and given by

$\psi_{\alpha}\left(j\right)=\frac{1}{\sqrt{2}}\delta_{1,j},\qquad\psi_{\beta}\left(j\right)=\frac{-i}{\sqrt{2}}\delta_{1,j}$
(27)

for the left edge, and

$\psi_{\alpha}\left(j\right)=\frac{1}{\sqrt{2}}\delta_{\ell,j},\qquad\psi_{\beta}\left(j\right)=\frac{i}{\sqrt{2}}\delta_{\ell,j}$
(28)

for the right edge. Here, $1$ in $\delta_{1,j}$ represent the left edge, while
$\ell$ in $\delta_{\ell,j}$ represents the right edge. The perfectly localized
edge states for $\xi=0$ are transformed to edge states with finite penetration
depth for $\xi\neq 0$.

###  II.5 Non-Hermitian Non-Abelian topological charges

We define a non-Hermitian non-Abelian Berry connection or a non-Hermitian
Berry-Wilczek-Zee (BWZ) connection byWZ

$A_{\alpha\beta}^{\text{RL}}\left(\theta\right)=\left\langle\psi_{\alpha}^{\text{R}}\right|\partial_{\theta}\left|\psi_{\beta}^{\text{L}}\right\rangle,$
(29)

where

$H\left|\psi_{\alpha}^{\text{L}}\right\rangle=\varepsilon_{\alpha}\left|\psi_{\alpha}^{\text{L}}\right\rangle$
(30)

is the left eigenfunction, and

$H^{\dagger}\left|\psi_{\alpha}^{\text{R}}\right\rangle=\varepsilon_{\alpha}\left|\psi_{\alpha}^{\text{R}}\right\rangle$
(31)

is the right eigenfunction.

Figure 5: (a) Real and imaginary parts of the bulk-band energy in the
$(\gamma,\xi)$ plane. (b) Topological phase diagram in the $(\gamma,\xi)$
plane. Metallic phase appears except for $\xi=0$. In the figure, real-TI (im-
TI) stands for real(imaginary)-line-gap topological insulator phase.

We define a non-Hermitian BWZ phase by

$\Gamma_{\alpha\beta}^{\text{RL}}=\frac{1}{2\pi}\int_{0}^{2\pi}\text{Re}\left[A_{\alpha\beta}^{\text{RL}}\left(\theta\right)\right]d\theta,$
(32)

which we use as a non-Hermitian non-Abelian topological charge. The
eigenfunctions are analytically solved as

$\displaystyle\left|\psi_{\pm}^{\text{L}}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{\left(\psi_{1}^{\text{L}}\right)^{2}+\left(\psi_{2}^{\text{L}}\right)^{2}}}\left\\{\psi_{1}^{\text{L}},\psi_{2}^{\text{L}}\right\\},$
(33) $\displaystyle\psi_{1}^{\text{L}}$ $\displaystyle=-\gamma_{0}\cos
k\pm\sqrt{\gamma_{0}^{2}-\gamma^{2}+\xi\left(2\gamma+\xi\right)\sin^{2}k},$
(34) $\displaystyle\psi_{2}^{\text{L}}$
$\displaystyle=\gamma-\left(\gamma_{0}+\xi\right)\sin k,$ (35)

and

$\displaystyle\left|\psi_{\pm}^{\text{R}}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{\left(\psi_{1}^{\text{R}}\right)^{2}+\left(\psi_{2}^{\text{R}}\right)^{2}}}\left\\{\psi_{1}^{\text{R}},\psi_{2}^{\text{R}}\right\\},$
(36) $\displaystyle\psi_{1}^{\text{R}}$ $\displaystyle=\gamma_{0}\cos
k\pm\sqrt{\gamma_{0}^{2}-\gamma^{2}+\xi\left(2\gamma+\xi\right)\sin^{2}k},$
(37) $\displaystyle\psi_{2}^{\text{R}}$
$\displaystyle=\gamma+\left(\gamma_{0}+\xi\right)\sin k.$ (38)

When $\gamma=0$ and $\xi=0$, using (33) and (36), we calculate the non-
Hermitian BWZ connection (29) numerically and find that

$A_{\alpha\beta}^{\text{RL}}=\frac{1}{2}\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right),$ (39)

which leads to the non-Hermitian BWZ phase (32) as

$\Gamma_{\alpha\beta}^{\text{RL}}=\frac{1}{2}\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right),$ (40)

where we have explicitly written only the nontrivial $2\times 2$ submatrix
within the $N\times N$ matrix. Hence, the topological charges are given by

$\Gamma_{\alpha\beta}^{\text{RL}}=-\frac{1}{2}L_{\alpha\beta}$ (41)

with Eq.(1). The topological charges $\Gamma_{\alpha\beta}^{\text{RL}}$ obey
essentially the same non-Abelian algebra as $L_{\alpha\beta}$.

When $\gamma\neq 0$ and $\xi=0$, we calculate the non-Hermitian BWZ connection
(29) to find that it is no longer a constant. However, the non-Hermitian BWZ
phase (32) is calculated as in Eq.(40), and hence the topological charge is
quantized as in Eq.(41) for any $\gamma$. Nevertheless, the eigenfunctions as
well as the eigenvalues are real (i.e., real-line-gap topological insulator
phase) only for $\gamma^{2}\leq\gamma_{0}^{2}$, while the eigenvalues and the
eigenfunctions are complex (i.e. imaginary-line-gap topological insulator
phase) for $\gamma^{2}>\gamma_{0}^{2}$. Hence, PT symmetry is preserved only
for $\gamma^{2}\leq\gamma_{0}^{2}$, and it is spontaneously broken for
$\gamma^{2}>\gamma_{0}^{2}$. The system undergoes a phase transition at
$\gamma=\pm\gamma_{0}$.

When $\gamma\neq 0$ and $\xi\neq 0$, we have numerically calculated the
topological charge (32) with the use of (33) and (36). We have shown the
$\left(2,1\right)$ component of the $2\times 2$ matrix
$\Gamma_{\alpha\beta}^{\text{RL}}$ for various values of $\xi$ in Fig.4. It is
quantized to be 1/2 for $\gamma^{2}\leq\gamma_{1}^{2}$ and
$\gamma^{2}>\gamma_{0}^{2}$ when $\xi>0$, while $\gamma^{2}\leq\gamma_{0}^{2}$
and $\gamma^{2}>\gamma_{1}^{2}$ when $\xi<0$, where
$\gamma_{1}=\gamma_{0}-\xi$ as in Eq.(20). On the other hand, it is not
quantized for the metallic phase that emerges between $\gamma_{0}$ and
$\gamma_{1}$, as in Fig.4. It is concluded that the topological charges are
quantized and given by Eq.(41) in the insulator phases.

###  II.6 Topological phase diagram

In non-Hermitian systems, there are line-gap insulators and point-gap
insulatorsUedaPRX ; Kawabata  in general. In the point-gap insulator, there is
a gap in $|E|$. On the other hand, there are two types of line-gap insulators.
A real-line gap topological insulator has a gap in Re$\left[E\right]$, while
an imaginary-line-gap topological insulator has a gap in Im$\left[E\right]$.
We first consider the case $\xi>0$. For $\left|\gamma\right|<\gamma_{1}$, the
system is a non-Hermitian line-gap topological insulator along the
Re$\left[E\right]$. The systems is metallic for
$\gamma_{1}\leq\left|\gamma\right|\leq\gamma_{0}$. For
$\left|\gamma\right|>\gamma_{0}$, the system is a non-Hermitian line-gap
topological insulator along the Im$\left[E\right]$. If $\xi<0$, the system is
a real-line-gap topological insulator for $\left|\gamma\right|<\gamma_{0}$, it
is a metal for $\gamma_{0}\leq\left|\gamma\right|\leq\gamma_{1}$ and it is an
imaginary-line-gap topological insulator for $\left|\gamma\right|>\gamma_{1}$.
We show the topological phase diagram in Fig.5(b). It is consistent with the
real and imaginary parts of the energy in the $\gamma$-$\xi$ plane as shown in
Fig.5(a).

Figure 6: (a) Illustration of the electric circuit corresponding to the
lattice in Fig.1(b). The hopping along the $\alpha$-chain ($\beta$-chain) is
represented by the inductance $L$ (the capacitance $C$). (b) Negative
impedance converter $R_{X}$ represents an imaginary hoppingHofmann . (c)
Operational amplifier circuit $C_{\gamma}$ represents a nonreciprocal
hoppingNonR .

Figure 7: Real and imaginary parts of impedance $Z_{aa}$ at the edge as a
function of frequency $\omega$. (a1) $\sim$(a4) Hermitian model with
$\gamma=0$. (b1)$\sim$(b4) Non-Hermitian model with $\gamma=0.025$,
(c1)$\sim$(c4) with $\gamma=0.04$ (phase transition point $\gamma_{1}$),
(d1)$\sim$(d5) with $\gamma=0.045$, (e1)$\sim$(e4) with $\gamma=0.05$ (phase
transition point $\gamma_{0}$), and (f1)$\sim$(f4) with $\gamma=0.075$. We
have used a finite chain with open boundary condition (red) and periodic
boundary condition (cyan). (a1)$\sim$(f2) $\xi=0$. (a3)$\sim$(f4) $\xi=0.2$.
We have set $\varepsilon_{\alpha}=1$ and $\varepsilon_{\beta}=1.1$. The length
of the chain is 20.

##  III Electric circuit simulation

An electric circuit is described by the Kirchhoff current law. By making the
Fourier transformation with respect to time, the Kirchhoff current law is
expressed as

$I_{a}\left(\omega\right)=\sum_{b}J_{ab}\left(\omega\right)V_{b}\left(\omega\right),$
(42)

where $I_{a}$ is the current between node $a$ and the ground, while $V_{b}$ is
the voltage at node $b$. The matrix $J_{ab}\left(\omega\right)$ is called the
circuit Laplacian. Once the circuit Laplacian is given, we can uniquely setup
the corresponding electric circuit. By equating it with the Hamiltonian $H$
asTECNature ; ComPhys

$J_{ab}\left(\omega\right)=i\omega H_{ab}\left(\omega\right),$ (43)

it is possible to simulate various topological phases of the Hamiltonian by
electric circuitsTECNature ; ComPhys ; Hel ; Lu ; EzawaTEC ; EzawaLCR ;
Research ; Zhao ; EzawaSkin ; Garcia ; Hofmann ; EzawaMajo ; Tjunc ; NonR .
The relations between the parameters in the Hamiltonian and in the electric
circuit are determined by this formula.

In the present problem, only the $\alpha$-chain and the $\beta$-chain are
active in the tight-binding Hamiltonian as in Fig.1. Thus, we need only a
$2\times 2$ matrix. The circuit Laplacian follows from the Hamiltonian (14) as

$J_{\alpha\beta}^{\prime}\left(k\right)=i\omega\left[\left(\begin{array}[]{cc}-\frac{L}{\omega^{2}}\cos
k&f_{+}\\\ f_{-}&C\cos
k\end{array}\right)+\frac{\varepsilon_{\alpha}+\varepsilon_{\beta}}{2}\mathbb{I}_{2}\right],$
(44)

with

$f_{\pm}=\frac{1}{\omega R_{X}}\left(1+\xi\right)\sin k\pm\gamma.$ (45)

We may design the electric circuit to realize this circuit Laplacian as in
Fig.6. The main part consists of the $\alpha$-channel and the $\beta$-channel
corresponding to the $\alpha$-chain and the $\beta$-chain in the lattice in
Fig.1. Additionally, each node in the $i$-channel is connected to the ground
via a set of inductor $L_{i}$ and capacitor $C_{i}$, where $i=\alpha$ or
$\beta$, in order to realize the diagonal term
$\varpropto(\varepsilon_{\alpha}+\varepsilon_{\beta})$ in Eq.(44).

Hopping terms along the $\alpha$-chain and the $\beta$-chain are described by
the diagonal terms in Eq.(44), where $\pm\cos k=\pm(e^{ik}+e^{-ik})/2$
represents the plus (minus) hopping in the tight-bind model. To simulate the
positive and negative hoppings in the Hamiltonian, we replace them with the
capacitance $i\omega C$ and the inductance $1/i\omega L$, respectively.

Hopping terms across the $\alpha$-chain and the $\beta$-chain are described by
the off-diagonal terms $f_{\pm}$ in Eq.(44), which consist of two terms
proportional to $\sin k$ and $\gamma$.

(i) The term proportional to $\sin k$ produces the cross hopping, where $\sin
k=(e^{ik}-e^{-ik})/2i$ represents an imaginary hopping in the tight-bind
model. The imaginary hopping is implemented by a negative impedance converter
$R_{X}$ with current inversionHofmann , as is constructed based on an
operational amplifier with resistors: See Fig.6(b). The voltage-current
relation is given by

$\left(\begin{array}[]{c}I_{1}\\\
I_{2}\end{array}\right)=\frac{1}{R_{X}}\left(\begin{array}[]{cc}-\nu&\nu\\\
-1&1\end{array}\right)\left(\begin{array}[]{c}V_{1}\\\
V_{2}\end{array}\right),$ (46)

with $\nu=R_{b}/R_{a}$, where $R_{X}$, $R_{a}$ and $R_{b}$ are the resistances
in an operational amplifier. We note that the resistors in the operational
amplifier circuit are tuned to be $\nu=1$ in the literatureHofmann  so that
the system becomes Hermitian, where the corresponding Hamiltonian represents a
spin-orbit interaction. It produces the Hamiltonian

$H=\frac{1}{\omega R_{X}}\left(\begin{array}[]{cc}i&-i\\\
i&-i\end{array}\right)$ (47)

for the Hermitian limit.

(ii) The term $\varpropto\gamma$ produces the nonreciprocal hopping terms,
which are vertical hoppings represented by red arrows in Fig.1(b). The
nonreciprocal hopping is constructed by a combination of an operational
amplifier and capacitorsNonR ,

$\left(\begin{array}[]{c}I_{ij}\\\ I_{ji}\end{array}\right)=i\omega
C_{\gamma}\left(\begin{array}[]{cc}-1&1\\\
-1&1\end{array}\right)\left(\begin{array}[]{c}V_{i}\\\
V_{j}\end{array}\right),$ (48)

as in Fig.6(c). It corresponds to the Hamiltonian

$H=C_{\gamma}\left(\begin{array}[]{cc}-1&1\\\ -1&1\end{array}\right).$ (49)

In this way, the tight-binding Hamiltonian for the present non-Hermitian non-
Abelian topological system is implemented in the electric circuit given in
Fig.6.

###  III.1 Impedance resonance

The band structure as well as edge states are well observed by impedance
resonance, which is definedTECNature ; ComPhys ; Hel  by

$Z_{ab}=V_{a}/I_{b}=G_{ab},$ (50)

where $G=J^{-1}$ is the Green function. Taking the nodes $a=b$ at an edge, we
show the real and imaginary parts of the impedance for a finite chain as a
function of $\omega$ in Fig.7, which are marked in red. For comparison, we
also show the impedance for a periodic boundary condition in cyan, where the
edge states are absent.

We first study the Hermitian case ($\gamma=0$) with $\xi=0$, where the
impedance is shown in Fig.7(a1) and (a2). The edge impedance resonance is
clear by comparing the periodic boundary condition and the open boundary
condition. There are only two bulk peaks in cyan at
Re[$E_{\alpha\beta}^{\prime}\left(k;\gamma,\xi\right)$]. On the other hand,
there is an additional peak in red due to the edge states between two bulk
peaks, as corresponds to Fig.2(a1).

Next, we show the impedance for various nonreciprocity $\gamma$ with $\xi=0$
in Fig.7(a1)$\sim$(f1) and (a2)$\sim$(f2). The edge impedance resonance
rapidly decreases as the increase of $\gamma$, as shown in Fig.7(b1). This is
due to the imaginary contribution in Eq.(26). Then, the distance between two
bulk peaks becomes narrower, which is consistent with
Re[$E_{\alpha\beta}^{\prime}\left(k;\gamma,\xi=0\right)$] as shown in
Fig.2(c1). The two bulk peaks merge into one peak at the spontaneous $PT$
symmetry breaking point $\gamma_{0}$, as shown in Fig.7(e1). The bulk
impedance resonance is very strong due to the gap closing of the bulk band. We
also observe the edge impedance resonance in the imaginary-line gap
topological insulating phase, where the impedance resonance is weak comparing
to Fig.7(a1) as shown in Fig.7(f1). This is also the imaginary contribution in
Eq.(26).

We also show the impedance for finite $\xi$ in Fig.7(a3)$\sim$(f3) and
(a4)$\sim$(f4), as corresponds to Fig.2(c2). The bulk impedance peaks become
broad, which reflects the broadening of the bulk bands. As a result, the edge
impedance peak becomes clearer as in Fig.7(a3) in comparison to Fig.7(a1).
There are strong cyan resonances at the phase transition point $\gamma_{1}$
point as shown in Fig.7(c3) and (c4). It is due to the gap closing of the bulk
band. In Fig.7(d3) and (d4), the impedance structure is complicated, which
reflects the metallic band structure. The effect of the $\xi$ term is
negligible for the imaginary-line-gap topological phase as shown in Fig.7(f3)
and (f4) since the peak of the impedance is broad even for $\xi=0$ in
Fig.7(f1) and (f2). Here, note that $\xi$ appears only in the form of
$(1+\xi)$ in Eq.(45).

##  IV Conclusion

We have proposed a non-Hermitian non-Abelian topological insulator model by
imposing $PT$ symmetry in one dimension. It describes a real-line-gap
topological insulator with real eigenvalues in the Hermitian limit. The system
undergoes a spontaneous breakdown of $PT$ symmetry as the non-Hermitian term
increases, and turns out to describe an imaginary-line-gap topological
insulator, when the bulk bands are perfectly flat. When we introduce a bulk
bending term, there are two phase transitions with the emergence of a metal
with complex eigenvalues between the above two topological insulators.
Finally, we have presented how to construct these models in electric circuits.
We have shown that the spontaneous $PT$ symmetry breaking as well as
topological edge states are well signaled by measuring the frequency
dependence of the impedance.

## Acknowledgement

The author is very much grateful to N. Nagaosa for helpful discussions on the
subject. This work is supported by the Grants-in-Aid for Scientific Research
from MEXT KAKENHI (Grants No. JP17K05490 and No. JP18H03676). This work is
also supported by CREST, JST (JPMJCR16F1 and JPMJCR20T2).

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