# Instructions for EMNLP 2021 Proceedings

Oleg Vasilyev, John Bohannon
Primer Technologies Inc.
San Francisco, California
<EMAIL_ADDRESS>

# Estimation of Summary-to-Text Inconsistency by Mismatched Embeddings

Oleg Vasilyev, John Bohannon
Primer Technologies Inc.
San Francisco, California
<EMAIL_ADDRESS>

###### Abstract

We propose a new reference-free summary quality evaluation measure, with
emphasis on the faithfulness. The measure is designed to find and count all
possible minute inconsistencies of the summary with respect to the source
document. The proposed ESTIME, Estimator of Summary-to-Text Inconsistency by
Mismatched Embeddings, correlates with expert scores in summary-level SummEval
dataset stronger than other common evaluation measures not only in Consistency
but also in Fluency. We also introduce a method of generating subtle factual
errors in human summaries. We show that ESTIME is more sensitive to subtle
errors than other common evaluation measures.

##  1 Introduction

Summarization must preserve the factual consistency of the summary with the
text. Human annotation of factual consistency can be accompanied with detailed
classification of factual errors, thus giving a hope that the annotation
scores are reasonably objective Kryscinski et al. (2020); Huang et al. (2020);
Vasilyev et al. (2020b); Gabriel et al. (2020); Maynez et al. (2020).

Factual consistency, or ’faithfulness’ of a summary is one of several summary
qualities; for the purpose of human annotation these qualities can be
specified in different ways Xenouleas et al. (2019b); Kryscinski et al.
(2020); Fan et al. (2018); Vasilyev et al. (2020b); Fabbri et al. (2020).
Summarization models nowadays create satisfactorily fluent, coherent and
informative summaries, but the factual consistency has a lot of room for
improvement. Some factual errors (swapped named entities and crude
hallucinations) are easily noticeable and make a summary look very bad right
away; other factual errors could be hardly noticeable even for annotators Lux
et al. (2020); Vasilyev et al. (2020b) \- which is arguably even worse.

Existing summary evaluation measures are based on several approaches, which
may be more sensitive to one or another quality. A question-answering based
evaluation estimates how helpful is the summary in answering questions about
the source text Xenouleas et al. (2019a); Eyal et al. (2019); Scialom et al.
(2019a); Deutsch et al. (2020); Durmus et al. (2020); Wang et al. (2020). A
text reconstruction approach estimates how helpful is the summary in guessing
parts of the source text Vasilyev et al. (2020a, b); Egan et al. (2021).
Evaluation measures that use some kind of text similarity can estimate how
similar is the summary to special human-written reference summaries Zhang et
al. (2020); Zhao et al. (2019); Lin (2004), or, more realistically, how
similar is the summary to the source text Gao et al. (2020); Louis and Nenkova
(2009).

In order to assess how well the summary factual consistency is evaluated, it
is necessary either to have a dataset of human-annotated imperfect machine-
generated summaries Bhandari et al. (2020); Fabbri et al. (2020), or to have a
dataset of artificially introduced factual errors in originally factually
correct human-written summaries Kryscinski et al. (2020).

Given: $summary$; $text$; $model$, $tokenizer$; parameters $W=450$, $L=8$,
$M=50$, $H=24$
---
$tokens\\_text=tokenizer.tokenize(text)$
$tokens\\_summary=tokenizer.tokenize(summary)$
# Get all embeddings from the text:
Initialize $embeddings\\_text$ of the same length as $tokens\\_text$
While there are $text$ tokens with embeddings not taken:
Select leftmost token $t$ with embedding not yet taken
Make input widow of size $W$ start to the left by $M$ tokens from the index of
$t$
Starting from token $t$, in the input window mask each $L$th token with
embedding not yet taken
Take the input by $model$, take $H$th hidden layer embeddings for all the
masked tokens
Place the obtained embeddings into their corresponding locations in
$embeddings\\_text$
# Get all embeddings from the summary:
Repeat the above for the $summary$. Result: $embeddings\\_summary$
# Count summary-to-text inconsistencies:
$num\\_inconsistencies=0$
For each embedding $E_{s}$ in $embeddings\\_summary$:
Find in $embeddings\\_text$ an embedding $E_{t}$ with highest similarity (dot-
product) to $E_{s}$.
If the tokens corresponding to $E_{s}$ and $E_{t}$ are not equal, then
$num\\_inconsistencies\mathrel{{+}{=}}1$
Figure 1: ESTIME: Estimation of Summary-to-Text Inconsistency by Mismatched
Embeddings.

In this paper we focus on presenting a new evaluation measure with emphasis on
factual consistency. Our contribution:

  1. 1.

We introduce ESTIME, Estimator of Summary-to-Text Inconsistency by Mismatched
Embeddings. Using human-annotated machine-generated summaries from the datset
SummEval Fabbri et al. (2020), we compare ESTIME with other common or
promising evaluation measures.

  2. 2.

We introduce a natural no-heuristics method of generating subtle factual
errors. We use it here to compare the performance of ESTIME with other
measures on human-written summaries with generated subtle errors.

##  2 Methods

In order to estimate consistency of a summary with the text, we attempt to
find all the summary tokens that could be related to a factual error. Our
motivation is that transformer-made token embeddings are highly contextual
Ethayarajh (2019). We assume that even if phrased somewhat differently in the
summary, the context should suggest approximately the same token embedding as
an embedding that would be suggested by the corresponding context in the text.
Thus, we check embeddings of all tokens of the summary that have one or more
occurrences in the text. For each token embedding we find its match: the most
similar embedding in the text. We assume that if the summary is factually
correct then the matched tokens are the same. If the tokens are not the same,
we count such mismatch as an indicator of inconsistency of the summary to the
text. The algorithm for ESTIME is shown in more detail in Figure 1.

This approach is different from matching similar embeddings for sake of
measuring similarity (e.g. similarity between a summary and a reference
summary in BERTScore Zhang et al. (2020)). It also differs from using a model
trained to replace wrong tokens with correct ones Cao et al. (2020);
Kryściński et al. (2019).

In preliminary evaluations, similar to the ones presented in the next
sections, we have not found any improvement from adding up one or another
flavour of heuristics, such as covering only named entities or only certain
parts of speech. For finding most similar embedding we are using simple
unnormalized dot-product, - replacing it by cosine similarity makes all the
results presented in the next sections slightly worse. The embeddings are
taken using the pretrained BERT model Devlin et al. (2018) bert-large-uncased-
whole-word-masking of transformers library Wolf et al. (2020). While there is
no crucial difference with other varieties of BERT, ALBERT and RoBERTa, this
model showed a better overall performance, and we used it for evaluations in
the next sections. We found that there is no strong dependency on the
parameters $W$, $L$, $M$ of ESTIME, we set the input size $W=450$ close to
maximal BERT input length, and the distances $L=8$ and $M=50$ reasonably
large. We present results for $H=12$ as ESTIME-12, and for $H=24$ as
ESTIME-24, corresponding to the embeddings taken from the middle and from the
top of the large BERT.

Table 1: Summary level correlations $\rho$ (Spearman) and $\tau$ (Kendall Tau-c) of quality estimators with human experts scores. The top rows evaluation measures are reference-free, separated from the lower rows evaluation measures, which need human references. In each column the highest correlation is bold-typed. The only p-values above 0.01 in this table are p=0.03 for BERTScore-P and p=0.01 for ROUGE-2. measure | consistency | relevance | coherence | fluency
---|---|---|---|---
| $\rho$ | $\tau$ | $\rho$ | $\tau$ | $\rho$ | $\tau$ | $\rho$ | $\tau$
BLANC-AXXL | 0.200 | 0.098 | 0.246 | 0.179 | 0.127 | 0.093 | 0.115 | 0.066
BLANC-BLU | 0.207 | 0.102 | 0.217 | 0.156 | 0.116 | 0.085 | 0.112 | 0.065
(-)ESTIME-12 | 0.374 | 0.184 | 0.140 | 0.100 | 0.238 | 0.173 | 0.343 | 0.198
(-)ESTIME-24 | 0.358 | 0.176 | 0.117 | 0.084 | 0.187 | 0.134 | 0.363 | 0.209
(-)J-Shannon | 0.193 | 0.095 | 0.406 | 0.298 | 0.289 | 0.213 | 0.125 | 0.072
SummaQA-F | 0.174 | 0.085 | 0.16 | 0.113 | 0.089 | 0.065 | 0.12 | 0.069
SummaQA-P | 0.197 | 0.097 | 0.179 | 0.127 | 0.112 | 0.082 | 0.133 | 0.076
SUPERT | 0.297 | 0.147 | 0.306 | 0.222 | 0.236 | 0.175 | 0.175 | 0.101
BERTScore-F | 0.109 | 0.053 | 0.371 | 0.273 | 0.377 | 0.277 | 0.142 | 0.082
BERTScore-P | 0.055 | 0.027 | 0.268 | 0.196 | 0.323 | 0.238 | 0.126 | 0.072
BERTScore-R | 0.164 | 0.081 | 0.423 | 0.309 | 0.345 | 0.253 | 0.12 | 0.069
BLEU | 0.095 | 0.047 | 0.213 | 0.153 | 0.176 | 0.128 | 0.140 | 0.080
ROUGE-L | 0.115 | 0.057 | 0.241 | 0.174 | 0.170 | 0.124 | 0.079 | 0.045
ROUGE-1 | 0.137 | 0.067 | 0.302 | 0.220 | 0.184 | 0.134 | 0.080 | 0.046
ROUGE-2 | 0.129 | 0.063 | 0.245 | 0.177 | 0.146 | 0.105 | 0.063 | 0.036
ROUGE-3 | 0.149 | 0.073 | 0.251 | 0.180 | 0.160 | 0.116 | 0.066 | 0.038

##  3 Performance on human-annotated machine-generated summaries

We used SummEval dataset 111https://github.com/Yale-LILY/SummEval Fabbri et
al. (2020) for comparing ESTIME with a few well known or promising evaluation
measures. The part of SummEval dataset that we use consists of 100 texts, each
text is accompanied by 16 summaries generated by 16 different models, making
altogether 1600 text-summary pairs. Each text-summary pair is annotated (on
scale 1 to 5) by 3 experts for 4 qualities: consistency, relevance, coherence
and fluency. We took average of the expert scores for each quality of a text-
summary pair. Each text is also accompanied by 11 human-written reference
summaries, for the measures that need them.

We calculated scores of ESTIME and other measures for all the 1600 summaries,
and presented their correlations with the average expert scores in Table 1.
The measures in Table 1 are split into the group of reference-free measures
(top) and the measures requiring human-written references (bottom). BLANC-help
Vasilyev et al. (2020a) is calculated in two
versions222https://github.com/PrimerAI/blanc#blanc-on-summeval-dataset, which
differ by the underlying models: BLU - bert-large-uncased, and AXXL - albert-
xxlarge-v2. ESTIME and Jensen-Shannon Louis and Nenkova (2009) values are
negated. SummaQA Scialom et al. (2019b) is represented by SummaQA-P (prob) and
SummaQA-F (F1 score)333https://github.com/recitalAI/summa-qa. SUPERT Gao et
al. (2020) is calculated as single-doc with 20 reference sentences
’top20’444https://github.com/yg211/acl20-ref-free-eval. BLEU Papineni et al.
(2002) is calculated with NLTK. BERTScore Zhang et al. (2020) is represented
as F1 (BERTScore-F), precision (BERTScore-P) and recall
(BERTScore-R)555https://github.com/Tiiiger/bert_score. For ROUGE Lin (2004)
the ROUGE-L is calculated as rougeLsum666https://github.com/google-
research/google-research/tree/master/rouge.

By design ESTIME should perform well for consistency, and indeed it beats
other measures in the table. Being a one-sided summary-to-text estimator of
inconsistencies, ESTIME should not and does not perform well for relevance.
Interestingly, ESTIME performs better than other measures for fluency, and
reasonably well for coherence.

Table 2: System level correlation $\rho$ (Spearman) and $\tau$ (Kendall Tau-c) of quality estimators with human experts scores of consistency. Top rows show reference-free evaluation measures. measure | $\rho$ | $\tau$
---|---|---
BLANC-AXXL | 0.812 | 0.617
BLANC-BLU | 0.724 | 0.567
(-)ESTIME-12 | 0.756 | 0.583
(-)ESTIME-24 | 0.815 | 0.633
(-)J-Shannon | 0.753 | 0.533
SummaQA-F | 0.862 | 0.667
SummaQA-P | 0.912 | 0.750
SUPERT | 0.832 | 0.633
BERTscore-F | -0.029 | -0.017
BERTscore-P | -0.329 | -0.217
BERTscore-R | 0.885 | 0.733
BLEU | -0.150 | -0.017
ROUGE-L | 0.376 | 0.283
ROUGE-1 | 0.694 | 0.517
ROUGE-2 | 0.779 | 0.600
ROUGE-3 | 0.888 | 0.717

In Table 2 we show correlations on system level, meaning that the scores (of
automated measures and of human experts) are averaged over the 100 texts, so
that each array of scores has length only 16 rather than 1600 Fabbri et al.
(2020). The purpose of this would be a comparison of the summarization models.
We present results for consistency in the table; for other qualities some
measures have p-value higher than 0.05.

##  4 Performance on human summaries with generated subtle errors

Machine-generated summaries, even by abstractive summarization models,
generally follow the source text by frequently reproducing large spans from
it. It is no surprise that the most similar context should point to the same
token, thus helping ESTIME to be a good factual consistency measure.

Human summaries are more varied in describing the source text, and it is
interesting how useful can be ESTIME for evaluating them. Fundamentally, we
are asking how flexible are the embeddings in understanding the context. In
order to answer this question, we made random selection of 2000 text-summary
pairs from CNN/Daily Mail dataset Hermann et al. (2015). For each human-
written summary we then added the same summary modified by generated factual
errors. We thus made 4000 text-summary pairs. We assigned the ’golden’ scores
as 1 to each clean summary, and 0 to each summary with errors.

Our ’subtle errors’ generation method is simple, heuristic-free and easily
reproducible. In order to generate an error, we randomly select a whole-word
token in the summary, mask and predict it by an LM model (we used bert-base-
cased). We then select the top predicted candidate that is not equal to the
real token, and substitute it for the real token. The resulting ’subtle
errors’ are indeed subtle and similar to real machine-generated mishaps and
hallucinations, with the fluency preserved.

Thus, we made the evaluation task double difficult: the summaries are human-
written, and the errors are subtle. For the evaluation presented in Table 3 we
created 3 subtle errors in each ’score=0’ summary. ESTIME is more sensitive to
such subtly generated errors than other measures, as shown in Table 3. All
p-values in the table are less than $10^{-3}$, except $0.023$ for BLANC-AXXL,
$0.002$ for Jensen-Shannon and $0.001$ for SummaQA-P.

Table 3: Correlation $\rho$ (Spearman) and $\tau$ (Kendall Tau-c) of quality estimators with the presence of generated subtle errors in human summary. The dataset of 4000 text-summary pairs was created by random pick of 2000 test-summary pairs from CNN / Daily Mail dataset, duplicating these 2000 pairs, and by generating subtle errors in the 2000 duplicated summaries. measure | $\rho$ | $\tau$
---|---|---
BLANC-AXXL | 0.036 | 0.042
BLANC-BLU | 0.076 | 0.087
(-)ESTIME-12 | 0.138 | 0.159
(-)ESTIME-24 | 0.169 | 0.195
(-)J-Shannon | 0.048 | 0.055
SummaQA-F | 0.055 | 0.064
SummaQA-P | 0.054 | 0.062
SUPERT | 0.107 | 0.123

##  5 Conclusion

We introduced ESTIME: estimator of summary-to-text inconsistency by mismatched
embeddings, - a new reference-free summary quality measure, with emphasis on
measuring factual inconsistency of the summary with respect to the text. In
view of its good performance, we intend to use ESTIME for improving
faithfulness of summarization.

We also introduced and used for evaluation a method for generating ’subtle
errors’; the method has a potential for creating consistent and realistic
benchmark datasets for factual consistency.

## References

  * Bhandari et al. (2020) Manik Bhandari, Pranav Narayan Gour, Atabak Ashfaq, Pengfei Liu, and Graham Neubig. 2020.  Re-evaluating evaluation in text summarization.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing_ , pages 9347–9359. Association for Computational Linguistics.
  * Cao et al. (2020) Meng Cao, Yue Dong, Jiapeng Wu, and Jackie Chi Kit Cheung. 2020.  Factual error correction for abstractive summarization models.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing_ , pages 6251–6258. Association for Computational Linguistics (2020).
  * Deutsch et al. (2020) Daniel Deutsch, Tania Bedrax-Weiss, and Dan Roth. 2020.  Towards question-answering as an automatic metric for evaluating the content quality of a summary. _arXiv_ , arXiv:2010.00490.
  * Devlin et al. (2018) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018.  Bert: Pre-training of deep bidirectional transformers for language understanding. _arXiv_ , arXiv:1810.04805.
  * Durmus et al. (2020) Esin Durmus, He He, and Mona Diab. 2020.  Feqa: A question answering evaluation framework for faithfulness assessment in abstractive summarization. In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 5055–5070. Association for Computational Linguistics.
  * Egan et al. (2021) Nicholas Egan, Oleg Vasilyev, and John Bohannon. 2021.  Play the shannon game with language models: A human-free approach to summary evaluation. _arXiv_ , arXiv:2103.10918.
  * Ethayarajh (2019) Kawin Ethayarajh. 2019.  How contextual are contextualized word representations? comparing the geometry of bert, elmo, and gpt-2 embeddings.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing_ , pages 55–65. Association for Computational Linguistics (Hong Kong, China, 2019).
  * Eyal et al. (2019) Matan Eyal, Tal Baumel, and Michael Elhadad. 2019.  Question answering as an automatic evaluation metric for news article summarization. In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1_ , pages 3938–3948. Association for Computational Linguistics.
  * Fabbri et al. (2020) Alexander R. Fabbri, Wojciech Kryściński, Bryan McCann, Caiming Xiong, Richard Socher, and Dragomir Radev. 2020.  Summeval: Re-evaluating summarization evaluation. _arXiv_ , arXiv:2007.12626v4.
  * Fan et al. (2018) Lisa Fan, Dong Yu, and Lu Wang. 2018.  Robust neural abstractive summarization systems and evaluation against adversarial information. _arXiv_ , arXiv:1810.06065.
  * Gabriel et al. (2020) Saadia Gabriel, Asli Celikyilmaz, Rahul Jha, Yejin Choi, and Jianfeng Gao. 2020\.  Go figure! a meta evaluation of factuality in summarization. _arXiv_ , arXiv:2010.12834.
  * Gao et al. (2020) Yang Gao, Wei Zhao, and Steffen Eger. 2020.  Supert: Towards new frontiers in unsupervised evaluation metrics for multi-document summarization. In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 1347–1354. Association for Computational Linguistics.
  * Hermann et al. (2015) Karl Moritz Hermann, Tomáš Kočiský, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa Suleyman, and Phil Blunsom. 2015.  Teaching machines to read and comprehend.  In _Advances in Neural Information Processing Systems 28_ , pages 1693–1701. Curran Associates, Inc.
  * Huang et al. (2020) Dandan Huang, Leyang Cui, Sen Yang, Guangsheng Bao, Kun Wang, Jun Xie, and Yue Zhang. 2020.  What have we achieved on text summarization? In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing_ , pages 446–469. Association for Computational Linguistics.
  * Kryscinski et al. (2020) Wojciech Kryscinski, Bryan McCann, Caiming Xiong, and Richard Socher. 2020.  Evaluating the factual consistency of abstractive text summarization. In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing_ , pages 9332–9346. Association for Computational Linguistics.
  * Kryściński et al. (2019) Wojciech Kryściński, Bryan McCann, Caiming Xiong, and Richard Socher. 2019.  Evaluating the factual consistency of abstractive text summarization. _arXiv_ , arXiv:1910.12840.
  * Lin (2004) Chin-Yew Lin. 2004.  Rouge: A package for automatic evaluation of summaries. In _Proceedings of Workshop on Text Summarization Branches Out_ , pages 74–81. Association for Computational Linguistics.
  * Louis and Nenkova (2009) Annie Louis and Ani Nenkova. 2009.  Automatically evaluating content selection in summarization without human models. In _Proceedings of the 2009 Conference on Empirical Methods in Natural Language Processing_ , pages 306–314. Association for Computational Linguistics.
  * Lux et al. (2020) Klaus-Michael Lux, Maya Sappelli, and Martha Larson. 2020.  Truth or error? towards systematic analysis of factual errors in abstractive summaries.  In _Proceedings of the First Workshop on Evaluation and Comparison of NLP Systems (Eval4NLP)_ , pages 1–10. Association for Computational Linguistics (2020).
  * Maynez et al. (2020) Joshua Maynez, Shashi Narayan, Bernd Bohnet, and Ryan McDonald. 2020.  On faithfulness and factuality in abstractive summarization.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 1906–1919. Association for Computational Linguistics (2020).
  * Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. 2002.  Bleu: a method for automatic evaluation of machine translation.  In _Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics (ACL)_ , pages 311–318, Philadelphia. Association for Computational Linguistics.
  * Scialom et al. (2019a) Thomas Scialom, Sylvain Lamprier, Benjamin Piwowarski, and Jacopo Staiano. 2019a.  Answers unite! unsupervised metrics for reinforced summarization models. In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing_ , pages 3246–3256. Association for Computational Linguistics.
  * Scialom et al. (2019b) Thomas Scialom, Sylvain Lamprier, Benjamin Piwowarski, and Jacopo Staiano. 2019b.  Answers unite! unsupervised metrics for reinforced summarization models.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 3246–3256, Hong Kong, China. Association for Computational Linguistics.
  * Vasilyev et al. (2020a) Oleg Vasilyev, Vedant Dharnidharka, and John Bohannon. 2020a.  Fill in the blanc: Human-free quality estimation of document summaries. In _Proceedings of the First Workshop on Evaluation and Comparison of NLP Systems_ , pages 11–20. Association for Computational Linguistics.
  * Vasilyev et al. (2020b) Oleg Vasilyev, Vedant Dharnidharka, Nicholas Egan, Charlene Chambliss, and John Bohannon. 2020b.  Sensitivity of blanc to human-scored qualities of text summaries. _arXiv_ , arXiv:2010.06716.
  * Wang et al. (2020) Alex Wang, Kyunghyun Cho, and Mike Lewis. 2020.  Asking and answering questions to evaluate the factual consistency of summaries.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages s 5008–5020. Association for Computational Linguistics (2020).
  * Wolf et al. (2020) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush. 2020.  Transformers: State-of-the-art natural language processing.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 38–45. Association for Computational Linguistics (2020).
  * Xenouleas et al. (2019a) Stratos Xenouleas, Prodromos Malakasiotis, Marianna Apidianaki, and Ion Androutsopoulos. 2019a.  Sum-qe: a bert-based summary quality estimation model. In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing_ , pages 6005–6011. Association for Computational Linguistics.
  * Xenouleas et al. (2019b) Stratos Xenouleas, Prodromos Malakasiotis, Marianna Apidianaki, and Ion Androutsopoulos. 2019b.  Sumqe: a bert-based summary quality estimation model.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 6005–6011, Hong Kong, China. Association for Computational Linguistics.
  * Zhang et al. (2020) Tianyi Zhang, Varsha Kishore, Felix Wu, Kilian Q. Weinberger, and Yoav Artzi. 2020\.  Bertscore: Evaluating text generation with bert. _arXiv_ , arXiv:1904.09675v3.
  * Zhao et al. (2019) Wei Zhao, Maxime Peyrard, Fei Liu, Yang Gao, Christian M. Meyer, and Steffen Eger. 2019.  Moverscore: Text generation evaluating with contextualized embeddings and earth mover distance. In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing_ , pages 563–578. Association for Computational Linguistics.


# Topological states in multi-orbital HgTe honeycomb lattices

W. Beugeling  Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer
Straße 38, 01187 Dresden, Germany  E. Kalesaki  IEMN- Dept. ISEN, UMR CNRS
8520, 41 boulevard Vauban, 59046 Lille Cedex, France  Physics and Materials
Science Research Unit, University of Luxembourg, 162a avenue de la Faïencerie,
L-1511 Luxembourg  C. Delerue  IEMN- Dept. ISEN, UMR CNRS 8520, 41 boulevard
Vauban, 59046 Lille Cedex, France  Y.-M. Niquet  Univ. Grenoble Alpes, INAC-
SP2M, L_Sim, Grenoble, France and CEA, INAC-SP2M, L_Sim, 17 avenue des
Martyrs, 38054 Grenoble, France  D. Vanmaekelbergh  Debye Institute for
Nanomaterials Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht,
The Netherlands  C. Morais Smith  Institute for Theoretical Physics, Center
for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4,
3584 CE Utrecht, The Netherlands

###### Abstract

Research on graphene has revealed remarkable phenomena arising in the
honeycomb lattice. However, the quantum spin Hall effect predicted at the $K$
point could not be observed in graphene and other honeycomb structures of
light elements due to an insufficiently strong spin-orbit coupling. Here we
show theoretically that 2D honeycomb lattices of HgTe can combine the effects
of the honeycomb geometry and strong spin-orbit coupling. The conduction
bands, experimentally accessible via doping, can be described by a tight-
binding lattice model as in graphene, but including multi-orbital degrees of
freedom and spin-orbit coupling. This results in very large topological gaps
(up to $35\,\mathrm{meV}$) and a flattened band detached from the others.
Owing to this flat band and the sizable Coulomb interaction, honeycomb
structures of HgTe constitute a promising platform for the observation of a
fractional Chern insulator or a fractional quantum spin Hall phase.

Correspondence and requests for materials should be addressed to C.D. (email:
<EMAIL_ADDRESS>or to C.M.S. (email: C.deMoraisSmith@uu.nl).

The discovery of graphene has confronted us with a material that exhibits
fascinating electronic properties CastroNetoEA2009 , such as zero-mass
carriers, persisting conductivity at vanishing density at the Dirac point
Nair08 , Klein tunnelling Katsnelson06 , and an anomalous quantum Hall effect
NovoselovEA2005 ; DuEA2009 ; BolotinEA2009 . Nevertheless, the absence of a
band gap in its spectrum prevents its use as a field-effect transistor, and
its weak spin-orbit coupling (SOC) hampers the possibility to realize the
quantum spin Hall effect (QSHE) KaneMele2005PRL95-22  and use it for quantum
spintronics. The prospect of artificial graphene samples Polini13  that
display the lacking properties has motivated research in various types of
honeycomb lattice, such as arrays of ultracold atoms SoltanPanahiEA2011 ,
molecular graphene GomesEA2012 , organometallic lattices Wang13 , and 2D
electron gases subject to a geometric array of gates Gibertini09 ;
ParkLouie2009 . Recently, an alternative path came from the self-assembly of
semiconductor nanocrystals forming atomically coherent 2D structures with a
long-range honeycomb pattern; the thickness and honeycomb period are defined
by the size of the nanocrystals, which is in the range of 5 nm EversEA2013 ;
BoneschanscherEA2014 . Honeycomb lattices of PbSe and CdSe nanocrystals have
been fabricated, with astonishing atomic coherence due to the oriented
attachment of the nanocrystals. Theoretical investigations have shown that
CdSe superlattices formed in such a way exhibit Dirac cones at two energies
and nearly dispersionless bands. Unfortunately, these flat bands are connected
to the nearby higher energy bands, and the SOC gaps in the conduction band are
very small KalesakiEA2014 .

In this work, we propose a design for robust topological insulators that
combine the properties of the honeycomb lattice and strong SOC. We consider
three different types of HgTe layers with superimposed honeycomb geometry and
present atomistic tight-binding (TB) calculations of their conduction band
structure that we accurately describe by a 16-band effective model. Such
lattices take advantage of the multi-orbital degrees of freedom in the
honeycomb setup, allied to the strong SOCZhangEA2014 . The access to multi-
orbital degrees of freedom allows for further manipulation of the topological
properties. While the Haldane Haldane1988  and Kane-Mele KaneMele2005PRL95-22
models originally concerned honeycomb lattices characterized by a single
orbital per site (e.g., $p_{z}$ in graphene) and isotropic nearest-neighbour
(NN) hopping integrals between them, multi-orbital models have attracted much
attention recently, in particular because they seem to be a paradigm to
generate topologically non-trivial flat bands WuEA2007 ; SunEA2011 ;
Venderbos11 ; Yang12 ; Olschlager13 . The flat band structure opens the way to
the realization of interesting strongly correlated states, such as fractional
QSHE, fractional Chern insulators, or ferromagnetic fractional Chern
insulators NeupertEA2011 ; SunEA2011 ; HuEA2011 ; Goerbig2012 ;
RegnaultBernevig2011 ; TangEA2011 ; Xiao2011 ; Yang12 ; HeEA2012 ; Bergholtz13
. The currently proposed HgTe lattices exhibit conduction bands characterized
by large topological gaps and an isolated flat band. We conclude that,
depending on the position of the Fermi level, not only QSHE could be observed
in these structures, but also fractional QSHE or fractional Chern insulator
phases, as on-site and NN Coulomb-interaction parameters are found in the
energy range required for their realization.

Results

Figure 1: Nanocrystal lattices and their conduction band dispersions. a,
honeycomb nanoribbon formed by the HgTe (CdSe) nanocrystals. Hg (Cd) atoms are
in yellow, Te (Se) atoms are in grey. Each nanocrystal has a truncated
nanocube shape, the vertices of which are given by all permutations of $[\pm
1,\pm(1-q),\pm(1-q)]l$, where $q$ is the truncation factor and $2l$ is the
size of the original nanocube before truncation. The honeycomb lattice spacing
$a$, i.e., the center-to-center distance between neighbor nanocrystals, is
defined as $a=(2N+1)a_{0}/\sqrt{2}$ where $N$ is an integer and $a_{0}$ is the
cubic lattice parameter of HgTe (CdSe). The nanocrystals are attached via
$\langle 110\rangle$ facets ($a=\sqrt{2}(2-q)l$). The arrows along the ribbon
indicate the electron propagation in the helical edge states present in the
quantum spin Hall phase. Red and blue colors correspond to top and bottom edge
for spin up, bottom and top edge for spin down, respectively. b,c, band
dispersions for the bulk resulting from the atomistic tight-binding
calculations with $q=0.5$ ($a=5.0\,\mathrm{nm}$ for HgTe, $4.7\,\mathrm{nm}$
for CdSe). d, same for the HgTe superlattice resulting from the effective
model. Insets in (c,d) show the $s$ bands in the gap region with higher
magnification
$\left[2K/3=\left(\frac{4\pi}{9a},\frac{4\pi}{9\sqrt{3}a}\right)\right]$.

Design of the HgTe honeycomb lattices. The three structures that we consider
have in common that the $[111]$ direction of the zincblende lattice is
perpendicular to the plane. The first system is inspired by recent results
obtained for nanocrystal self-assembly EversEA2013 ; BoneschanscherEA2014 ,
i.e., a graphene-type superlattice of truncated cubic nanocrystals attached
via $\langle 110\rangle$ facets. The second system consists of spheres
connected by short cylinders, which allow us to vary the coupling between
honeycomb lattice sites in a convenient way. The third system corresponds to a
honeycomb array of cylinders. These two last systems may be experimentally
realized by gas-phase deposition and lithography. We will show below that the
multi-orbital topological effects are common to these three different
structures.

The HgTe superlattices proposed here differ fundamentally from HgTe/CdTe
quantum wells, where the QSHE has been predicted BernevigEA2006  and
experimentally observed KonigEA2007 . In the latter system, the appearance of
the QSHE at the $\Gamma$ point is connected to band inversion and vanishes for
quantum wells of thickness below 6 nm. Here, instead, the effect occurs at the
$K$ point, and it is driven by the honeycomb nanogeometry, allied to the
strong SOC of the composing HgTe nanocrystals. This distinction is important
because zero modes (Majoranas) bind to topological lattice defects when the
band gap opens at a non-$\Gamma$ point in the Brillouin zone RanEA2009 ;
JuricicEA2012 , and hence, the underlying topological order can be detected by
measuring the structure of the topological defect RueggEA2013 ; SlagerEA2013 .

Band structure of lattices of HgTe nanocrystals. To unveil the topological
properties of these systems, we have performed atomistic tight-binding (TB)
band-structure calculations. We use a basis of twenty atomic orbitals on each
atom of the nanocrystals, including the spin degree of freedom. The
methodology is described in Ref. Allan12,  and is summarized in the Methods
section. In comparison to bulk HgTe, the electronic structure of HgTe
superlattices is characterized by a large bandgap due to the strong quantum
confinement. Conduction and valence bands close to this gap are composed of
several minibands and minigaps due to the periodic scattering of the
electronic waves in the honeycomb structure KalesakiEA2014 . In the following,
we only discuss the physics of the sixteen lowest conduction bands (Fig. 1).

A typical honeycomb lattice composed of HgTe nanocrystals is shown in Fig. 1a
and the related conduction-band structure is displayed in Fig. 1c. In order to
visualize the effects of SOC, we compare it to the band structure of the same
honeycomb structure composed of CdSe (Fig. 1b). The strong SOC in HgTe gives
rise to effects which are absent in CdSe KalesakiEA2014 . The overall
behaviour of the band structure can be understood as follows. Each individual
nanocrystal is characterized by two states with $s$ envelope wave-function and
six $p$ states at higher energy. In the honeycomb structure, strong coupling
between the wave-functions of neighbour nanocrystals leads to the formation of
sixteen bands grouped into two manifolds of four ($s$) and twelve ($p$) bands
that are well separated. The $s$ bands have the same type of dispersion as the
$\pi$ and $\pi^{\star}$ bands in real graphene CastroNetoEA2009 . In the case
of CdSe (Fig. 1b), these bands are spin-degenerate and are connected at the
$K$ and $K^{\prime}$ points of the Brillouin zone, where their dispersion is
linear (Dirac points). In HgTe, instead, the $s$ bands exhibit a small gap
($5.7\,\mathrm{meV}$) at the $K$ points and have a quadratic dispersion (see
Fig. 1c). In addition, they are characterized by a visible spin splitting at
all points of the Brillouin zone, except at $\Gamma$ and $M$. Among the twelve
$p$ bands of CdSe nanocrystal superlattices, eight have a small dispersion and
the other four basically behave like the (Dirac) $s$ bands. Four flat bands
are built from the $p_{z}$ states perpendicular to the lattice, which are not
very dispersive because $p_{z}$-$p_{z}$ ($\pi$) interactions are weak. Four
other bands ($p_{x,y}$), respectively above and below the $p$ Dirac band, are
flat due to destructive interferences of electron hopping induced by the
honeycomb geometry WuEA2007 ; SunEA2011 . In honeycomb lattices of HgTe
nanocrystals, the SOC induces spin splitting, opens a large gap at $K$ in the
$p$-like Dirac bands, and produces a considerable detachment of the lowest
flat $p$-band from the Dirac $p$-band (Fig. 1c). The effects of the SOC are so
strong that it is hardly possible to recognize the Dirac bands.

These unexpected features allow for the realization of several topological
states of matter, by doping the system using a field-effect transistor or
electrolyte gel gating YeEA2010 . At zero energy, the undoped system is a
semiconductor with a trivial gap of about 0.4 eV. Upon doping the material
with one electron per nanocrystal, the small $s$-like topological gap may be
reached, whereas for fillings between 2 and 3 electrons per nanocrystal, the
fractional quantum (spin) Hall regime may be realized at the flat band. For a
doping level of 4 electrons/nanocrystal one reaches the QSHE gap. At this
point, we should emphasize that doping of nanocrystals with up to 10 electrons
has already been demonstrated experimentally Ger2003 ; therefore, all the
interesting regimes that we discuss are at reach with the existing technology,
at a simple switch of the doping level. Other examples of band structures for
lattices of HgTe nanocrystals with different size or truncation factors are
presented in Supplementary Fig. 1 and are discussed in Supplementary Note 1.
They all show large topological gaps, especially in the $p$ sector.

Figure 2: Topological edge states and non-trivial gaps in honeycomb lattices
of HgTe nanocrystals. a, conduction bands calculated using the atomistic
tight-binding method for a zigzag ribbon composed of sixteen nanocrystals per
unit cell ($q=0.5$, body diagonal of $5.0\,\mathrm{nm}$, cell length
$l=8.7\,\mathrm{nm}$). b, same but computed from the effective Hamiltonian. c,
spin orientation on each site for a selection of states calculated at
$k=31\pi/(30l)$. A vertical arrow indicates that the spin is along the $z$
direction, perpendicular to the lattice. The size of the circles represents
the weight of the wave function on each site. In each figure, the colour
indicates the expectation value $\langle y\sigma_{z}\rangle$, i.e., red and
blue correspond to top and bottom edge for spin up (bottom and top edge for
spin down), respectively. At each energy $E$, there are two states which live
on opposite edges with opposite spin, therefore they are represented by the
same colour. The bulk states are grey.

The topological properties of the bands are most transparently studied through
an edge-state analysis in a 1D nanoribbon BeugelingEA2012PRB86-07 . We
consider a zigzag ribbon composed of sixteen nanocrystals ($34740$ atoms) per
unit cell. Figure 2a shows that edge states are crossing the three gaps
between the $p$ bands as well as the gap between the $s$ bands. These results
also hold for armchair ribbons. In Supplementary Fig. 2a, we present the band
structure for another nanoribbon, which has two inequivalent edges. Still,
helical edge states characteristic of the QSHE are found, as shown by the 2D
plots of the wavefunctions (Supplementary Figs. 2b–e).

Effective model. The band structures resulting from the atomistic TB
calculations are accurately described by a sixteen-band effective model (Fig.
1d), where each nanocrystal is treated as one site on a honeycomb lattice. The
effective TB model is written in the basis of the four aforementioned orbitals
($s,p_{x},p_{y},p_{z}$) per site as
$H_{\mathrm{eff}}=H_{\mu}+H_{\mathrm{NN}}+H_{\mathrm{ISO}}+H_{\mathrm{RSO}}$.
Here, $H_{\mu}$ incorporates the on-site energies $E_{s}$,
$E_{p_{x}}=E_{p_{y}}$, and $E_{p_{z}}$. The edge nanocrystals have a slightly
different value of $E_{s}$ compared to the bulk, in order to account for the
different number of neighbours. The term

$H_{\mathrm{NN}}=\sum_{\left\langle
i,j\right\rangle}\sum_{\begin{subarray}{c}\alpha\\\
b,b^{\prime}\end{subarray}}c^{\dagger}_{i,b,\alpha}V_{i,b;j,b^{\prime}}c_{j,b^{\prime},\alpha},$
(1)

encodes NN hopping, where $\left\langle i,j\right\rangle$ denotes NN sites,
$\alpha=\mathord{\uparrow},\mathord{\downarrow}$ denotes spin, and $b$ and
$b^{\prime}$ the orbitals. The coupling coefficients $V_{i,b;j,b^{\prime}}$
are expressed in terms of the hopping parameters $V_{ss\sigma}$,
$V_{pp\sigma}$, $V_{pp\pi}$, and $V_{sp\sigma}$, following the notations of
Ref. SlaterKoster1954, .

The intrinsic SOC term $H_{\mathrm{ISO}}$ couples the electron orbital angular
momentum $\mathbf{L}$ and spin $\mathbf{S}=\bm{\sigma}/2$. In the $p$ sector,
it is encoded through the on-site term
$\lambda_{\mathrm{ISO}}^{p}\mathbf{L}\cdot\bm{\sigma}$. There is no on-site
term in the $s$ sector because the orbital angular momentum is “frozen”. For
the same reason, in graphene, the on-site intrinsic SOC term is absent because
the $sp^{2}$ hybridization freezes the orbital momentum in the $p_{z}$ state.
However, as shown by Kane and Mele KaneMele2005PRL95-22 , the intrinsic SOC
introduces a next-nearest-neighbour (NNN) hopping term, which is written as

$H_{\mathrm{ISO}}=\mathrm{i}\lambda_{\mathrm{ISO}}^{s}\sum_{\left\langle\langle
i,j\right\rangle\rangle}\sum_{\alpha}c^{\dagger}_{i,s,\alpha}\sigma^{z}_{\alpha\alpha}\nu_{ij}c_{j,s,\alpha}.$
(2)

Here, the summation is over NNNs, and $\nu_{ij}=\pm 1$, with the sign
depending on the outer product of the two NN vectors that connect sites $i$
and $j$. The Rashba SOC term, proportional to the cross product of momentum
and spin, $\mathbf{p}\times\mathbf{S}$, is written as a NN-hopping term

$H_{\mathrm{RSO}}=\mathrm{i}\sum_{\left\langle
i,j\right\rangle}\sum_{\begin{subarray}{c}\alpha,\beta\\\
b,b^{\prime}\end{subarray}}c^{\dagger}_{i,b,\alpha}\gamma_{i,b;j,b^{\prime}}[\hat{z}\cdot(\bm{\sigma}\times\mathbf{r}_{ij})]_{\alpha\beta}c_{j,b^{\prime},\beta}.$
(3)

The coupling coefficients $\gamma_{i,b;j,b^{\prime}}$ have the same structure
as the $V_{i,b;j,b^{\prime}}$ for the ordinary NN hopping and are expressed in
terms of $\gamma_{ss\sigma}$, $\gamma_{pp\sigma}$, and $\gamma_{pp\pi}$. The
$sp$ term may be neglected due to the large energy separation between the $s$
and $p$ bands.

on-site | NN hopping | Rashba SOC | intrinsic SOC
---|---|---|---
$E_{s}^{\mathrm{bulk}}=0.365\,\mathrm{eV}$ | $V_{ss\sigma}=-26.4\,\mathrm{meV}$ | $\gamma_{ss\sigma}=0.56\,\mathrm{meV}$ | $\lambda_{\mathrm{ISO}}^{s}=0.71\,\mathrm{meV}$
$E_{s}^{\mathrm{edge}}=0.370\,\mathrm{eV}$ | $V_{pp\sigma}=45.6\,\mathrm{meV}$ | $\gamma_{pp\sigma}=1.50\,\mathrm{meV}$ | $\lambda_{\mathrm{ISO}}^{p}=15.8\,\mathrm{meV}$
$E_{p_{x}}=0.691\,\mathrm{eV}$ | $V_{pp\pi}=-2.7\,\mathrm{meV}$ | $\gamma_{pp\pi}=0.80\,\mathrm{meV}$ |
$E_{p_{y}}=0.691\,\mathrm{eV}$ | $V_{sp\sigma}=31.1\,\mathrm{meV}$ |  |
$E_{p_{z}}=0.747\,\mathrm{eV}$ |  |  |
Table 1: Parameters of the effective model. Parameters derived for the lattice
of HgTe nanocrystals described in Fig. 1. $E_{s}$, $E_{p_{x}}$, $E_{p_{y}}$,
and $E_{p_{z}}$ are the on-site energies on the $s$, $p_{x}$, $p_{y}$, and
$p_{z}$ orbitals, respectively. In ribbons, the edge nanocrystals have a
slightly different value of $E_{s}$ compared to the bulk
($E_{s}^{\mathrm{bulk}}\neq E_{s}^{\mathrm{edge}}$). $V_{ss\sigma}$,
$V_{pp\sigma}$, $V_{pp\pi}$, and $V_{sp\sigma}$ are the hopping parameters,
following the notations of Ref. SlaterKoster1954, . $\gamma_{ss\sigma}$,
$\gamma_{pp\sigma}$, and $\gamma_{pp\pi}$ are the terms describing the Rashba
SOC, following the same notations. The intrinsic SOC is defined by
$\lambda_{\mathrm{ISO}}^{s}$ and $\lambda_{\mathrm{ISO}}^{p}$ on $s$ and $p$
orbitals, respectively.

In Table 1, we present typical values for the parameters obtained numerically
using least-squares fitting to the band structure of Fig. 1c. The band
structure of the effective model is shown in Fig. 1d. As expected, the on-site
term $\lambda_{\mathrm{ISO}}^{p}$ is much larger than the NNN term
$\lambda_{\mathrm{ISO}}^{s}$, explaining the opening of a very large gap at
the $K$ point and the detachment of the flat bands in the $p$ sector. Using
the effective model, the non-trivial topology of the bands is confirmed by the
calculation of the $Z_{2}$ topological invariant, the spin Chern number
(Methods).

The effective model yields a band structure for the ribbon (Fig. 2b) in
excellent agreement with the atomistic TB calculations (Fig. 2a). The red and
blue colours in Figs. 2a,b indicate the expectation value $\langle
y\sigma_{z}\rangle=\langle\psi_{i}|\hat{y}\hat{\sigma}_{z}|\psi_{i}\rangle$,
where $y$ is the coordinate perpendicular to the ribbon edges. This
expectation value allows us to identify helical edge states that come in pairs
with identical dispersion, opposite spin, and live on opposite edges. Both
atomistic and effective TB models show that the gaps in the $s$ and $p$
sectors exhibit helical edge states, characteristic of the QSHE.

When the Rashba coupling is neglected in the effective model, all states are
spin degenerate. The Rashba term induces a small splitting in energy, and
tilts the spins slightly away from the perpendicular direction. In Figs. 2c,
we plot the spin direction on each site of the zigzag ribbon for four sets of
four edge states (two in each edge). The spin direction is always
perpendicular to the edge, i.e., the spin lies in the $yz$ plane if we choose
the $x$ direction to be parallel to the edge. The localization of the selected
states on the edges is visible from the weight of the wave function, indicated
by the size of the circles in the figure. The colours of the circles are
determined by the local value of $y\sigma_{z}$, and correspond to the colours
in the dispersion (Fig. 2a,b). The site dependence of the spin direction leads
to interesting spin textures. For the edge states, the typical spin texture is
almost smooth: Going from one edge to the other, the spin direction
interpolates between (almost) up and (almost) down in a rotational manner. A
slight tilt of $\sim 3^{\circ}$ is observed for the edge states in the $p$
bands. In the $s$ bands, the tilt is stronger, similarly to graphene
BeugelingEA2012PRB86-07 . Here, the spin vector at the edge site points $\sim
30^{\circ}$ away from the vertical. This difference in tilt can be explained
by the much larger intrinsic SOC in the $p$ sector than in the $s$ one,
whereas the two Rashba couplings $\gamma_{ss\sigma}$ and $\gamma_{pp\sigma}$
are of similar order of magnitude. Strictly speaking, one should denote this
state a $Z_{2}$ topological insulator, but since the Rashba SOC is extremely
small, one can think of an approximate QSHE.

Flat band and Coulomb interactions. The large gaps in the $p$ sector are
mainly due to the intrinsic SOC, which, contrarily to the Kane-Mele model, is
described by an on-site term
($\lambda_{\mathrm{ISO}}^{p}=15.8\,\mathrm{meV}$). In particular, the
intrinsic SOC separates the lowest flat band from the other $p$ bands, with
large gaps, e.g., $35\,\mathrm{meV}$ in the case of Fig. 1c. This gap ranges
from 13 meV to 36 meV when we vary the nanocrystal size and shape
(Supplementary Table 1).

$\varepsilon_{\mathrm{in}}$ | $\varepsilon_{\mathrm{out}}$ | $U$ | $V$ | Band width | Absolute gap | Gap at $\Gamma$
---|---|---|---|---|---|---
$14$ | $14$ | $48\,\mathrm{meV}$ | $23\,\mathrm{meV}$ | $20\,\mathrm{meV}$ | $35\,\mathrm{meV}$ | $56\,\mathrm{meV}$
$14$ | $6$ | $76\,\mathrm{meV}$ | $43\,\mathrm{meV}$
Table 2: Coulomb energies. On-site ($U$) and nearest-neighbor ($V$) Coulomb
interaction energies calculated for a honeycomb lattice of HgTe nanocrystals
(with lattice parameters $q=0.5$, $a=5\,\mathrm{nm}$) compared to the width of
the lowest $p$ band, the absolute gap between the lowest $p$ bands (the gap
between band extrema), and the vertical gap at $\Gamma$ ($\mathbf{k}=0$)
between them. $\varepsilon_{\mathrm{in}}$ and $\varepsilon_{\mathrm{out}}$ are
the dielectric constants of the materials composing the lattice (HgTe) and its
environment, respectively. Figure 3: Honeycomb lattice of HgTe spheres and its
conduction band structure. a,b, top view of an assembly of spheres connected
by cylinders, forming a honeycomb lattice of HgTe. Hg atoms are in yellow, Te
atoms are in grey. The lattice spacing $a$, i.e., the center-to-center
distance between neighbor spheres, is equal to the diameter $D$ of the spheres
and $d$ is the diameter of the cylinders. c,d,e, conduction band dispersions
resulting from the atomistic tight-binding calculation for
$D=5.0\,\mathrm{nm}$ and $d/D=0.3$ (c), $d/D=0.6$ (d), or $d/D=0.8$ (e). Non-
trivial gaps are indicated by pink shaded regions. f, evolution of the gap
between the lowest $p$ bands versus $d/D$ for three values of $D$ (green
triangles: $D=5.9\,\mathrm{nm}$; blue circles: $D=5.0\,\mathrm{nm}$; red
crosses: $D=4.1\,\mathrm{nm}$).

Under partial filling of the lowest-energy flat $p$ band by electrons,
fascinating phenomena such as fractional QSHE are predicted in presence of
strong correlations, when the strength of Coulomb interactions between
electrons is large compared to the band width and smaller or comparable to the
energy gap between the flat band and the next higher one (for a recent review,
see Ref. Bergholtz13, ). In addition, it has been shown that also for large
Coulomb interactions, a fractional Chern insulator phase cannot be ruled out
Kourtis14 . We have thus calculated the on-site ($U$) and NN ($V$) electron-
electron interaction energies for the honeycomb lattice of Fig. 1a, assuming
two different dielectric environments around the nanocrystals (Methods). Table
2 shows that Coulomb energies are larger than the band width and are
comparable to the gap between $p$ bands. Therefore, the gap sizes open the
possibility to the experimental observation of strongly correlated quantum
phases including the long sought fractional QSHENeupertEA2011 ; SunEA2011 ;
HuEA2011 ; RegnaultBernevig2011 ; Goerbig2012 .

Band structure for other types of HgTe lattices. In order to understand the
effects of the electronic coupling between honeycomb lattice sites, we have
studied a second type of superlattices with a simpler geometry, consisting of
tangent spheres connected by horizontal cylinders (Fig. 3a,b). The HgTe
spheres have their $[111]$ axis orthogonal to the lattice plane and the
cylinders are oriented along $\langle 110\rangle$ directions perpendicular to
$[111]$. Figures 3c,d,e depict the evolution of the band structure with the
ratio between the diameters of cylinders ($d$) and spheres ($D$) (see also
Supplementary Fig. 3). An increase of $d/D$ induces larger NN hopping terms,
broader $s$ and $p$ bands, and stronger $sp$ hybridization, as shown by the
deformation of the $s$ band. It also results in a larger NNN term
$\lambda_{\mathrm{ISO}}^{s}$, explaining why the non-trivial gap in the $s$
sector only exists for $d/D>0.3$. On the contrary, topological gaps are always
present in the $p$ sector, even for small values of $d/D$, because they are
mainly determined by the on-site term $\lambda_{\mathrm{ISO}}^{p}$. At high
values of $d/D$ ($>0.7$), the spin splitting of the bands in the entire
Brillouin zone becomes particularly important due to increased Rashba
couplings. In general, the lowest $p$ band is rather flat and has a maximum
separation from the next higher one for $d/D$ close to $0.4$–$0.5$. In that
case, the energy separation can be as large as $35\,\mathrm{meV}$ (Fig. 3f).
HgTe honeycomb structures with moderate coupling between lattice sites should
provide the most suitable gap and band widths to observe the strongly
correlated phases associated to the flatness of the bands.

In Supplementary Fig. 4, we show further results for a third type of honeycomb
structure made of overlapping HgTe cylinders parallel to each other. Once
again, the band structure can be described by the effective model
(Supplementary Note 3 and Supplementary Fig. 5). In that case, the NN
couplings are even stronger, there is no gap in the $s$ sector due to a large
Rashba term, but the non-trivial gap above the lowest $p$ band remains. We can
conclude that the topological effects in the $p$ sector are robust against
changes in the electronic coupling and honeycomb period.

Discussion

In summary, we have performed atomistic TB calculations of the band structure
of 2D honeycomb lattices of HgTe. We demonstrate that the strong SOC of HgTe
combined with the honeycomb structure results in several topological phases.
The calculated band structure can be described by a honeycomb lattice model as
in graphene but including multi-orbital degrees of freedom that generate in
particular a topologically non-trivial flat band. By taking advantage of these
features, we show that in the same structure not only the QSHE, but
potentially also the elusive fractional QSHE could be observed, just by
varying the electron density. Both topological effects turn out to be
protected by a gap as large as $35\,\mathrm{meV}$, and can thus be observed at
high temperatures. Honeycomb superlattices of HgTe are therefore platforms of
high interest to study electrons on a multi-orbital honeycomb lattice under
strong SOC. Such structures could be fabricated by nanocrystal self-assembly
(in a similar way as for PbSe and CdSe EversEA2013 ; BoneschanscherEA2014 ) or
by a combination of gas-phase deposition and lithography. Our results open the
path towards high-temperature quantum spintronics in artificial graphene.

Methods

Atomistic TB methodology. The electronic structure of HgTe superlattices is
calculated within the TB approximation, as described in detail in Ref.
Allan12, . The TB Hamiltonian matrix is written in a basis of atomic orbitals
($sp^{3}d^{5}s^{*}$ for each spin orientation) as function of parameters that
have been obtained by fitting to two reference band structures: Close to the
Fermi level, we use the $\mathbf{k}\cdot\mathbf{p}$ band structure of Ref.
Man91,  whereas elsewhere we use the band structure of Ref. Svane11,  obtained
using a quasi-particle self-consistent GW approximation in a hybrid scheme. In
the present work, we have used the TB parameters that give the band structure
of HgTe at $300\,\mathrm{K}$. The surfaces of the superlattices are saturated
by pseudo-hydrogen atoms that push surface states far from the energy regions
of interest in this study. Therefore surface states do not interfere with edge
states predicted in ribbons. Due to the large size of the systems that we have
studied (up to $\approx 10^{5}$ atoms per unit cell), only near-gap
eigenstates are calculated using the numerical methods described in Ref.
Niquet00, .

Calculation of the $Z_{2}$ topological invariant. The $Z_{2}$ topological
invariant (spin Chern number $C_{s}$) for the bands of interest is calculated
using the model Hamiltonian following the methodology proposed in Ref.
Fukui07,  and derived from Ref. Fu06, . This approach works even for systems
without inversion symmetry, which is the case here. $C_{s}$ is given by a sum
of terms calculated on a regular lattice in the Brillouin zone. We have
checked that the results converge for a mesh denser than $21\times 21$ $k$
vectors. In all cases, the invariants that we have computed for the bands are
consistent with the number of edge states we observe in the bulk gaps.

Coulomb interactions. The Coulomb repulsion between electrons in honeycomb
lattices of HgTe nanocrystals can be characterized as follows. For simplicity,
we consider electrons in the $s$ band, since in the case of individual
nanocrystals, it was shown theoretically Niquet02  and experimentally Banin99
that the Coulomb integrals are almost identical for states with $s$ and $p$
envelope functions. The Coulomb interaction associated to electrons on
nanocrystals $i$ and $j$ is calculated as

$\int|\Psi_{i}(\mathbf{r})|^{2}|\Psi_{j}(\mathbf{r}^{\prime})|^{2}{\cal
V}(\mathbf{r},\mathbf{r}^{\prime})d\mathbf{r}d\mathbf{r}^{\prime}$ (4)

where $\Psi_{i}(\mathbf{r})$ is the $s$ state on the nanocrystal $i$, and
${\cal V}(\mathbf{r},\mathbf{r}^{\prime})$ is the screened Coulomb energy of
two interacting electrons at $\mathbf{r}$ and $\mathbf{r}^{\prime}$. Taking
into account that there is just one (spin-degenerate) $s$ state per
nanocrystal, $\Psi_{i}$ and $\Psi_{j}$ are simply defined as the components on
nanocrystals $i$ and $j$ of the wavefunction calculated for the lowest $s$
band at $\Gamma$ (or another $k$ vector), normalized on their respective
nanocrystals. Coulomb matrix elements are decomposed in the basis of the
atomic orbitals and are calculated following the methodology described in Ref.
Delerue97, , by using usual approximations in the TB description, i.e.,
neglecting overlaps between atomic orbitals and considering atomic charges as
point-like charges. For the potential ${\cal V}$, we consider two
configurations: First, a dielectrically homogeneous system, for which ${\cal
V}(\mathbf{r},\mathbf{r}^{\prime})=e^{2}/\left(\varepsilon_{\mathrm{in}}|\mathbf{r}-\mathbf{r}^{\prime}|\right)$,
where $\varepsilon_{\mathrm{in}}=14$ is the dielectric constant of HgTe;
Second, a dielectrically inhomogeneous system, for which ${\cal V}$ is
calculated by solving the Poisson equation, with the dielectric constant
inside (outside) the lattice equal to $\varepsilon_{\mathrm{in}}$
($\varepsilon_{\mathrm{out}}$). We have chosen $\varepsilon_{\mathrm{out}}=6$,
a typical value taken to simulate the complex dielectric environment around
semiconductor nanocrystals Niquet02 . The on-site ($U$) and NN ($V$) terms are
presented in Table 2. As expected, larger values are obtained for
$\varepsilon_{\mathrm{out}}=6$ than for
$\varepsilon_{\mathrm{out}}=\varepsilon_{\mathrm{in}}$. Longer-range Coulomb
terms are expected to decay as the inverse of the distance between
nanocrystals. However, it is important to note that these values do not take
into account the extra screening induced by the electrons filling the bands.
This could be computed, for example, in the random-phase approximation, but
this is clearly beyond the scope of the present work. Long-range interactions
will be strongly screened, while short-range ones will be only slightly
reduced Nozieres99 . In this context, for a band filling of the order of
$1/3$, correlations will be mainly governed by short-range effects.

## References

  * (1) Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K.  The electronic properties of graphene.  _Rev. Mod. Phys._ 81, 109–162 (2009).
  * (2) Nair, R. R. _et al._ Fine structure constant defines visual transparency of graphene.  _Science_ 320, 1308 (2008).
  * (3) Katsnelson, M. I., Novoselov, K. S. & Geim, A. K.  Chiral tunnelling and the Klein paradox in graphene.  _Nat. Phys._ 2, 620–625 (2006).
  * (4) Novoselov, K. S. _et al._ Two-dimensional gas of massless Dirac fermions in graphene.  _Nature_ 438, 197–200 (2005).
  * (5) Du, X., Skachko, I., Duerr, F., Luican, A. & Andrei, E. Y.  Fractional quantum Hall effect and insulating phase of Dirac electrons in graphene.  _Nature_ 462, 192–195 (2009).
  * (6) Bolotin, K. I., Ghahari, F., Shulman, M. D., Stormer, H. L. & Kim, P.  Observation of the fractional quantum Hall effect in graphene.  _Nature_ 462, 196–199 (2009).
  * (7) Kane, C. L. & Mele, E. J.  Quantum spin Hall effect in graphene.  _Phys. Rev. Lett._ 95, 226801 (2005).
  * (8) Polini, M., Guinea, F., Lewenstein, M., Manoharan, H. C. & Pellegrini, V.  Artificial honeycomb lattices for electrons, atoms and photons.  _Nat. Nano._ 8, 625–633 (2013).
  * (9) Soltan-Panahi, P. _et al._ Multi-component quantum gases in spin-dependent hexagonal lattices.  _Nat. Phys._ 7, 434–440 (2011).
  * (10) Gomes, K. K., Mar, W., Ko, W., Guinea, F. & Manoharan, H. C.  Designer Dirac fermions and topological phases in molecular graphene.  _Nature_ 483, 306–310 (2012).
  * (11) Wang, Z. F., Su, N. & Liu, F.  Prediction of a two-dimensional organic topological insulator.  _Nano Lett._ 13, 2842–2845 (2013).
  * (12) Gibertini, M. _et al._ Engineering artificial graphene in a two-dimensional electron gas.  _Phys. Rev. B_ 79, 241406 (2009).
  * (13) Park, C.-H. & Louie, S. G.  Making massless Dirac fermions from a patterned two-dimensional electron gas.  _Nano Lett._ 9, 1793–1797 (2009).
  * (14) Evers, W. H. _et al._ Low-dimensional semiconductor superlattices formed by geometric control over nanocrystal attachment.  _Nano Lett._ 13, 2317–2323 (2013).
  * (15) Boneschanscher, M. P. _et al._ Long-range orientation and atomic attachment of nanocrystals in 2D honeycomb superlattices.  _Science_ 344, 1377–1380 (2014).
  * (16) Kalesaki, E. _et al._ Dirac cones, topological edge states, and nontrivial flat bands in two-dimensional semiconductors with a honeycomb nanogeometry.  _Phys. Rev. X_ 4, 011010 (2014).
  * (17) Zhang, G.-F., Li, Y. & Wu, C.  Honeycomb lattice with multiorbital structure: Topological and quantum anomalous hall insulators with large gaps.  _Phys. Rev. B_ 90, 075114 (2014).
  * (18) Haldane, F. D. M.  Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”.  _Phys. Rev. Lett._ 61, 2015–2018 (1988).
  * (19) Wu, C., Bergman, D., Balents, L. & Das Sarma, S.  Flat bands and Wigner crystallization in the honeycomb optical lattice.  _Phys. Rev. Lett._ 99, 070401 (2007).
  * (20) Sun, K., Gu, Z., Katsura, H. & Das Sarma, S.  Nearly flatbands with nontrivial topology.  _Phys. Rev. Lett._ 106, 236803 (2011).
  * (21) Venderbos, J. W. F., Daghofer, M. & van den Brink, J.  Narrowing of topological bands due to electronic orbital degrees of freedom.  _Phys. Rev. Lett._ 107, 116401 (2011).
  * (22) Yang, S., Gu, Z.-C., Sun, K. & Das Sarma, S.  Topological flat band models with arbitrary Chern numbers.  _Phys. Rev. B_ 86, 241112 (2012).
  * (23) Ölschläger, M. _et al._ Interaction-induced chiral $p_{x}\pm ip_{y}$ superfluid order of bosons in an optical lattice.  _New Journal of Physics_ 15, 083041 (2013).
  * (24) Neupert, T., Santos, L., Chamon, C. & Mudry, C.  Fractional quantum Hall states at zero magnetic field.  _Phys. Rev. Lett._ 106, 236804 (2011).
  * (25) Hu, X., Kargarian, M. & Fiete, G. A.  Topological insulators and fractional quantum Hall effect on the ruby lattice.  _Phys. Rev. B_ 84, 155116 (2011).
  * (26) Goerbig, M. O.  From fractional Chern insulators to a fractional quantum spin Hall effect.  _Eur. Phys. J. B_ 85, 1–8 (2012).
  * (27) Regnault, N. & Bernevig, B. A.  Fractional Chern insulator.  _Phys. Rev. X_ 1, 021014 (2011).
  * (28) Tang, E., Mei, J.-W. & Wen, X.-G.  High-temperature fractional quantum Hall states.  _Phys. Rev. Lett._ 106, 236802 (2011).
  * (29) Xiao, D., Zhu, W., Ran, Y., Nagaosa, N. & Okamoto, S.  Interface engineering of quantum hall effects in digital transition metal oxide heterostructures.  _Nat. Commun._ 2, 596 (2011).
  * (30) He, J., Wang, B. & Kou, S.-P.  Ferromagnetism and antiferromagnetism of a correlated topological insulator with a flat band.  _Phys. Rev. B_ 86, 235146 (2012).
  * (31) Bergholtz, E. J. & Liu, Z.  Topological flat band models and fractional Chern insulators.  _Int. J. Mod. Phys. B_ 27, 1330017 (2013).
  * (32) Bernevig, B. A., Hughes, T. L. & Zhang, S.-C.  Quantum spin Hall effect and topological phase transition in HgTe quantum wells.  _Science_ 314, 1757–1761 (2006).
  * (33) König, M. _et al._ Quantum spin Hall insulator state in HgTe quantum wells.  _Science_ 318, 766–770 (2007).
  * (34) Ran, Y., Zhang, Y. & Vishwanath, A.  One-dimensional topologically protected modes in topological insulators with lattice dislocations.  _Nat. Phys._ 5, 298–303 (2009).
  * (35) Juričić, V., Mesaros, A., Slager, R.-J. & Zaanen, J.  Universal probes of two-dimensional topological insulators: Dislocation and $\pi$ flux.  _Phys. Rev. Lett._ 108, 106403 (2012).
  * (36) Rüegg, A. & Lin, C.  Bound states of conical singularities in graphene-based topological insulators.  _Phys. Rev. Lett._ 110, 046401 (2013).
  * (37) Slager, R.-J., Mesaros, A., Juričić, V. & Zaanen, J.  The space group classification of topological band-insulators.  _Nat. Phys._ 9, 98–102 (2013).
  * (38) Allan, G. & Delerue, C.  Tight-binding calculations of the optical properties of HgTe nanocrystals.  _Phys. Rev. B_ 86, 165437 (2012).
  * (39) Ye, J. T. _et al._ Liquid-gated interface superconductivity on an atomically flat film.  _Nat. Mater._ 9, 125–128 (2010).
  * (40) Germeau, A. _et al._ Optical transitions in artificial few-electron atoms strongly confined inside ZnO nanocrystals.  _Phys. Rev. Lett._ 90, 097401 (2003).
  * (41) Beugeling, W., Goldman, N. & Morais Smith, C.  Topological phases in a two-dimensional lattice: Magnetic field versus spin-orbit coupling.  _Phys. Rev. B_ 86, 075118 (2012).
  * (42) Slater, J. C. & Koster, G. F.  Simplified LCAO method for the periodic potential problem.  _Phys. Rev._ 94, 1498–1524 (1954).
  * (43) Kourtis, S., Neupert, T., Chamon, C. & Mudry, C.  Fractional Chern insulators with strong interactions that far exceed band gaps.  _Phys. Rev. Lett._ 112, 126806 (2014).
  * (44) Man, P. & Pan, D. S.  Infrared absorption in HgTe.  _Phys. Rev. B_ 44, 8745–8758 (1991).
  * (45) Svane, A. _et al._ Quasiparticle band structures of $\beta$-HgS, HgSe, and HgTe.  _Phys. Rev. B_ 84, 205205 (2011).
  * (46) Niquet, Y. M., Delerue, C., Allan, G. & Lannoo, M.  Method for tight-binding parametrization: Application to silicon nanostructures.  _Phys. Rev. B_ 62, 5109–5116 (2000).
  * (47) Fukui, T. & Hatsugai, Y.  Quantum spin Hall effect in three dimensional materials: Lattice computation of Z2 topological invariants and its application to Bi and Sb.  _J. Phys. Soc. Jpn._ 76, 053702 (2007).  eprint http://journals.jps.jp/doi/pdf/10.1143/JPSJ.76.053702.
  * (48) Fu, L. & Kane, C. L.  Time reversal polarization and a $Z_{2}$ adiabatic spin pump.  _Phys. Rev. B_ 74, 195312 (2006).
  * (49) Niquet, Y. M., Delerue, C., Allan, G. & Lannoo, M.  Interpretation and theory of tunneling experiments on single nanostructures.  _Phys. Rev. B_ 65, 165334 (2002).
  * (50) Banin, U., Cao, Y., Katz, D. & Millo, O.  Identification of atomic-like electronic states in indium arsenide nanocrystal quantum dots.  _Nature_ 400, 542–544 (1999).
  * (51) Delerue, C., Lannoo, M. & Allan, G.  Calculations of the electron-energy-loss spectra of silicon nanostructures and porous silicon.  _Phys. Rev. B_ 56, 15306–15313 (1997).
  * (52) Nozières, P. & Pines, D.  _Theory Of Quantum Liquids_.  Advanced Books Classics Series (Westview Press, 1999).

Acknowledgements

This work was supported by the French National Research Agency (ANR) project
”ETSFG” (ANR-09-BLAN-0421-01). W.B. is funded by the Max-Planck-Gesellschaft
through the visitor’s programme at MPI-PKS. E.K. acknowledges funding by the
University of Luxembourg Research Office. D.V. and C.M.S. wish to acknowledge
the Dutch FOM for financial support via the Program 13DDC01 ”Designing Dirac
Carriers in Honeycomb Superlattices”. The work of C.M.S. is part of the D-ITP
consortium, a program of the Netherlands Organisation for Scientific Research
(NWO) that is funded by the Dutch Ministry of Education, Culture and Science
(OCW).

Author contributions

W.B., E.K., and C.D. performed the calculations. Y.M.N. contributed to the
development of the codes and methodologies. C.D., D.V. and C.M.S. supervised
the project. All authors were involved in writing of the manuscript.

Additional information

Competing financial interests: The authors declare no competing financial
interests.

How to cite this article: Beugeling, W.  _et al._ , Topological states in
multi-orbital HgTe honeycomb lattices. _Nat. Commun._ 6:6316
doi:10.1038/ncomms7316 (2015).

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Supplementary Figure 1: Conduction bands in lattices of HgTe nanocrystals.
Band dispersion resulting from the atomistic tight-binding calculations.
Topologically non-trivial gaps are indicated by pink shaded regions. Each
nanocrystal has a truncated nanocube shape. Truncation factor: $q=0.25$ (a,c)
or $q=0.45$ (b,d). Honeycomb lattice spacing: $a=5.9\,\mathrm{nm}$ (a,b) or
$a=6.8\,\mathrm{nm}$ (c,d).

.
Supplementary Figure 2: Ribbon with broken inversion symmetry. Atomistic
tight-binding calculations for a honeycomb ribbon of HgTe nanocrystals, with
broken inversion symmetry. The nanocrystals have a truncated nanocube shape
(truncation factor $q=0.5$, honeycomb lattice spacing $a=5.0\,\mathrm{nm}$). A
single plane of atoms is removed from the right side of the ribbon in order to
break the inversion symmetry (not shown). f, schematic view of the ribbon. The
unit cell of 16 nanocrystals, shown in magenta, is reproduced periodically
along the direction indicated by the arrow. a, dispersion of the p bands. The
position of the bulk bands is indicated by pink vertical bars along the left
axis. b,c,d,e, 2D plots of the wave functions of four states calculated at
$k=0.3\times 2\pi/l$ where $l$ is the length of the unit cell. The labels 1
(b), 2 (c), 3 (d) and 4 (e) refer to the states indicated in a. The plots are
restricted to the unit cell of the ribbon. The white dots indicate the atoms.

Supplementary Figure 3: Band structure for lattices of spherical nanocrystals.
Conduction band dispersion resulting from the atomistic tight-binding
calculation for honeycomb superlattices of spheres (diameter
$D=5\,\mathrm{nm}$) connected by cylinders (diameter $d=0.4D$). a: bulk (gap
between the lowest $p$ bands $=25\,\mathrm{meV}$). b: armchair nanoribbon
composed of sixteen nanocrystals per unit cell.

.

Supplementary Figure 4: Band structures for lattices of cylinders. a, top view
of an assembly of vertical cylinders forming a honeycomb lattice of HgTe
(cylinder diameter $d=1.02a=7.0\,\mathrm{nm}$, layer thickness
$t=5.3\,\mathrm{nm}$, $a$ is the lattice spacing). Crystallographic axes of
HgTe are indicated. b, corresponding conduction band dispersion resulting from
the atomistic tight-binding calculation. Non-trivial gaps are indicated by
pink shaded regions. c, band structure of a zigzag ribbon formed of $12$
cylinders per unit cell.

Supplementary Figure 5: Effective model for a lattice of cylinders. a,
conduction band dispersion calculated using the model Hamiltonian. b, band
structure of a zigzag ribbon formed by $12$ sites per unit cell.

| $a=5.0\,\mathrm{nm}$ | $a=5.9\,\mathrm{nm}$ | $a=6.8\,\mathrm{nm}$
---|---|---|---
$q=0.25$ | $17.2$ | $13.1$ | $13.7$
$q=0.30$ | $17.2$ | $13.6$ | $13.7$
$q=0.35$ | $15.3$ | $13.6$ | $32.0$
$q=0.40$ | $15.3$ | $36.2$ | $32.0$
$q=0.45$ | $35.4$ | $36.2$ | $15.7$
$q=0.50$ | $35.4$ | $18.6$ | $15.7$

Supplementary Table 1: Energy gap in the $p$ bands. Energy gap
($\mathrm{meV}$) between the lowest $p$ bands in honeycomb lattices of HgTe
nanocrystals for different values of the truncation factor $q$ and the
honeycomb lattice spacing $a$. The same gap predicted for two successive
values of $q$ means that the honeycomb lattices are identical, the number of
atoms per nanocrystal varying discontinuously with $q$.

on-site | NN hopping | Rashba SOC | intrinsic SOC
---|---|---|---
$E_{s}=266$ | $V_{ss\sigma}=-17.2$ | $\gamma_{ss\sigma}=5.11$ | $\lambda^{s}_{\mathrm{ISO}}=0.3$
$E_{p_{x}}=493$ | $V_{pp\sigma}=28.9$ | $\gamma_{pp\sigma}=4.77$ | $\lambda^{p}_{\mathrm{ISO}}=14.1$
$E_{p_{y}}=493$ | $V_{pp\pi}=-0.6$ | $\gamma_{pp\pi}=0.00$ |
$E_{p_{z}}=698$ | $V_{sp\sigma}=24.2$ |  |

Supplementary Table 2: Parameters of the effective model. Parameters
($\mathrm{meV}$) used for the model for the HgTe superlattice described in
Supplementary Fig. 5a. $E_{s}$, $E_{p_{x}}$, $E_{p_{y}}$, and $E_{p_{z}}$ are
the onsite energies on the $s$, $p_{x}$, $p_{y}$, and $p_{z}$ orbitals,
respectively. $V_{ss\sigma}$, $V_{pp\sigma}$, $V_{pp\pi}$, and $V_{sp\sigma}$
are the hopping parameters, following the notations of Ref. [1].
$\gamma_{ss\sigma}$ , $\gamma_{pp\sigma}$, and $\gamma_{pp\pi}$ are the terms
describing the Rashba SOC, following the same notations. The intrinsic SOC is
defined by $\lambda^{s}_{\mathrm{ISO}}$ and $\lambda^{p}_{\mathrm{ISO}}$ on
$s$ and $p$ orbitals, respectively.

Supplementary Note 1
Influence of size on lattices of HgTe nanocrystals.

When we vary the lattice spacing $a$ ($3$–$8\,\mathrm{nm}$) and the nanocube
truncation ($q$ between $0.25$ and $0.5$) in lattices of HgTe nanocrystals,
all band structures have similar behaviour. Examples of band structure are
shown in Supplementary Fig. 1. The presence of well-separated $s$ and $p$
bands with helical gaps is quite general, even if the width of the gaps may
vary substantially depending on the geometry of the superlattices. The overall
shape of the $s$ bands is always the same, whereas for the $p$ bands the
variations are more important because not only the nearest- neighbour hopping,
but also the respective positions of the $p_{x}$, $p_{y}$ and $p_{z}$ states
depend on nanocrystal size and truncation. However, the lowest $p$ band is
always detached from the next higher one. The width of the lowest gap in the
$p$ bands is given in Supplementary Table 1 for different values of $a$ and
$q$.

Supplementary Note 2
Edge localization in ribbons of HgTe nanocrystal superlattices

In this section, we present additional results on ribbons made from lattices
of HgTe nanocrystals. We consider the same nanocrystals as in Figs. 1 and 2
($q=0.5$, $a=5.0\,\mathrm{nm}$) but we investigate a ribbon in which the two
edges are not symmetric by inversion. First, we discuss the effect of this
asymmetry on the band structure. Second, we present plots of the wave
functions. When the two edges of the ribbon are equivalent by inversion
symmetry, the topological edge states on the opposite sides of the ribbon are
quasi-degenerate for each value of $k$, the coupling between opposite edge
states being negligible (Fig. 2). In order to observe the effect of geometry
on the results, we have also considered a ribbon in which the inversion
symmetry has been broken. For that purpose, we have removed all atoms of the
last atomic plane on the right side of the ribbon and we have saturated the
broken bonds with pseudo-hydrogen atoms. Supplementary Fig. 2a shows that the
edge states are preserved thanks to their topological protection but their
degeneracy at a given $k$ is lifted due to the asymmetry between the two sides
of the ribbon. The 2D plots of the wavefunctions of the four states denoted
1–4 in Supplementary Fig. 2a for the asymmetric ribbon are shown in
Supplementary Figs. 2b–e. These states calculated at $k=0.3\times 2\pi/l$ are
strongly localized on the edges of the ribbon. State 3 is more delocalized
than the other three states because it lies very close to the bulk band edge.

Supplementary Note 3
Band structures of honeycomb lattices of vertical cylinders

We have investigated a third type of honeycomb structure composed of HgTe
cylinders. The axes of the cylinders are parallel to each other and are
organized on a honeycomb lattice (Supplementary Fig. 4a). Such structures
could be fabricated from a HgTe layer, grown for example by gas-phase
approaches. The honeycomb nanogeometry is then defined using nanoscale
lithography. Quite similar band structures are obtained for these lattices
(Supplementary Fig. 4b). Once again, $s$ and $p$-like bands can be easily
identified. The Rashba SOC is in general much larger than for lattices of
nanocrystals due to a stronger coupling between neighbouring sites.
Interestingly, a very similar behaviour was found for superlattices of spheres
connected by cylinders when the coupling between neighbouring spheres is
strong, i.e., for large values of $d/D$ (Fig. 3). As a consequence of the
large Rashba SOC, the gap in the $s$ sector is closed. Non- trivial gaps
remain in the $p$ bands, for many configurations that we have investigated, in
spite of larger spin splitting. However, the larger Rashba SOC tends to
increase the dispersion of the lowest $p$ band. Once again, the results of the
atomistic TB calculations are well described by the effective model
(Supplementary Table 2 and Supplementary Fig. 5a). Only the highest bands of
Supplementary Fig. 4b are not reproduced by the effective model because they
involve higher-energy orbitals which are not considered in the model. The
topological properties of the bands are demonstrated by edge-state analyses in
ribbons, using the atomistic TB calculations (Supplementary Fig. 4c) or the
effective TB model (Supplementary Fig. 5b) which give results in excellent
agreement. The non-trivial topology of the bands is confirmed by calculations
of the $Z_{2}$ topological invariants using the effective-model Hamiltonian.

Supplementary Reference

[1] Slater, J. C. & Koster, G. F. Simplified LCAO method for the periodic
potential problem. _Phys. Rev._ 94, 1498–1524 (1954).