# Augmenting Customer Support with an NLP-based Receptionist

André Barbosa2, Alan Godoy1

###### Abstract

In this paper, we show how a Portuguese BERT model can be combined with
structured data in order to deploy a chatbot based on a finite state machine
to create a conversational AI system that helps a real-estate company to
predict its client’s contact motivation. The model achieves human level
results in a dataset that contains 235 unbalanced labels. Then, we also show
its benefits considering the business impact comparing it against classical
NLP methods.

##  1 Introduction

Effective customer support is an essential piece for any service company. On
the one hand, as a business grows, it can hire more agents and create
specialized departments focusing on each possible problem a customer may face,
increasing the availability of support for clients. On the other hand, growth
brings new challenges: knowledge about client issues gets scattered across the
company, information regarding processes may not diffuse properly to all
agents, clients may have trouble finding the specific department able to help
them, to name few. As a real estate company, QuintoAndar111A quick
presentation about QuintoAndar and its business model can be found here. has
to deal with a wide range of issues involving individuals as distinct as
tenants (and prospective tenants), landlords, agents (realtors, inspectors,
photographers, etc) and building administrators. The channel of choice for
most contacts is WhatsApp, a messaging app widely adopted in Brazil and other
countries. This introduces even more complexity in terms of user
identification and routing, as any WhatsApp user can start a conversation with
our customer support team simply by sending an open text message.

To improve the routing quality, speed, and automate the collection of all
necessary information before a human analyst is allocated to the ticket, our
team developed a chatbot system that acts as a receptionist for customer
support. This chatbot (presented in section 4) uses a state-machine dialog
manager to integrate multiple machine learning (ML) based classifiers and
business rules to define the messages to be sent and actions to be taken. The
implementation of such service brought a lot of positive results to
QuintoAndar, allowing the full automation of chat triage in customer support,
a task performed previously by a dedicated team, which was reallocated to
other departments.

Beyond presenting a chatbot architecture that combines multiple ML models and
discussing how we merged textual information with structured data in order to
better understand an user’s context, the main contribution of this work is to
bring data about the results achieved by state-of-the-art models in a real-
world application. Much has been discussed about the fantastic results
achieved by recent Natural Language Processing (NLP) models for the English
language both in academic literature an in industry. How these outcomes
transfer to Brazilian Portuguese is not as clear though. We hope that, by
sharing our experience, we may shed some light on how Portuguese-speaking
companies and research groups may use these technologies outside common
benchmark problems.

In summary, the main goal of this paper is to show how we evolve until we
achieve human-level results in a real-world application through combining
Portuguese BERT with structured metadata from our clients with a complete end-
to-end system.

##  2 Related Work

Customer support is a natural candidate to be one of the areas most benefited
by the recent boost in NLP research and application [Wan and Chen 2018, Liu et
al. 2020, Molino et al. 2018]. By having a proper system for customer triage,
a company can match its user to the agent most likely to solve their problem
and also provide the agent with contextual information — e.g.: selected user
information, procedure suggestion and a list of similar issues — to make their
work more effective. Alexandra DeLucia and Elisabeth Moore [DeLucia and Moore
2020] used IT support tickets in a high-performance computing laboratory to
study automatic categorization and similar ticket recommendation. They
combined classical NLP pre-processing, including steps as stop-word removal,
stemming and topic modeling, with a random-forest classifier to select the
most appropriate label among 93 possible categories and retrieve other similar
tickets. A similar work was done by Fotso _et al._ [Fotso et al. 2018], that
created a system to classify client emails in 12 possible categories, using it
to select relevant articles to be automatically sent as response. The team
combined textual information with data regarding the user’s prior interactions
with their products, using word-embeddings, a BiLSTM network and an attention-
based mechanism to predict the most adequate class.

Rather than applying a single machine learning model to classify the user
demand, it’s also possible to create a dialog system that interacts with the
customer to try solving their problem or to ensure that all relevant
information was provided before assigning the ticket to an agent. Such system
demands not only a model that extracts information from user messages but also
a dialog management module to track the current dialog state and define which
action to perform in each moment and also a module to generate the messages
sent to the user [Dai et al. 2020, Jurafsky and Martin 2021, Zhang et al.
2020]. Multiple approaches can be used for each of these modules. For dialog
management they can be as simple as form-filling [Bobrow et al. 1977], in
which the system has a pre-defined list of information that should be provided
for a given task and it will iteratively ask the user to answer each missing
item, or get as complex as using a recurrent neural network to infer a latent
representation of the current dialog state which is fed to a second network
that converts states into policies [Wang et al. 2018].

When considering Brazilian Portuguese, however, there are few works reporting
applications of NLP in industrial applications. Finardi _et al._ [Finardi et
al. 2021] built a BERT language model [Devlin et al. 2019] to be used for
customer support in a large Brazilian bank, evaluating its performance for
sentiment analysis, question answering and named entity recognition (NER)
using datasets extracted from user interactions with the bank’s chatbot.
Azevedo _et al._ [Azevedo et al. 2020] created a system to automatically route
customer emails to four different boxes using a Support Vector Machine (SVM).
Works have also been published regarding applications in law, as [Bonifacio et
al. 2020] — which extensively studies the impacts of fine-tuning of
transformer-based language models using legal texts in NER tasks — and [Dal
Pont et al. 2020] — that provides an empirical study of word embeddings in
legal domain.

On the academic front, however, many works were produced recently related to
dialog systems that can be applied for customer support in Portuguese. Santos
_et al._ [Santos et al. 2020] and Melo and Coheur [Melo and Coheur 2020], for
instance, built retrieval-based conversational agents trained to answer
specific data. Carvalho _et al._ [Carvalho et al. 2020], on the other hand,
created an LSTM-based common-sense module to augment interactions in a dialog
system. A recent significant contribution was also made by Souza _et al._
[Souza et al. 2020] that made open-source a Brazilian Portuguese pre-trained
BERT language model, showing how such mono-lingual model was able to surpass
the performance achieved by a multi-lingual BERT for named entity recognition
(NER).

##  3 Problem Definition and Challenges

A typical customer support flow at QuintoAndar (Fig. 1) starts with a customer
contacting our team through the channel of their choice — namely, phone,
WhatsApp or email/support form. Whichever channel is chosen, the user is
required to state what kind of support is needed, an information that is used
to define to which department the contact should be directed. After the
department is defined, a task allocation system selects among all online
agents who should be responsible for handling the customer’s issue. The role
of such agent is to comprehend the user’s demand and get all relevant
information to solve it — if it is simple enough, it can be resolved
immediately; if it requires more complex actions, a task is created for the
appropriate back-office team.

Figure 1: A typical flow of customer support at QuintoAndar.

In a digital real-state company, however, there are hundreds of different
types of contacts, going from simple questions about using our site to search
for a house to requests for intermediation of complex tenant-owner issues like
negotiating a temporary rent reduction. This means that matching a user to the
right specialist is easier said than done. On the one hand, presenting a menu
with a multitude of customer support departments may be time-consuming and
ineffective, as users may not be familiar with each department’s
responsibilities and are likely to make mistakes when choosing an option. On
the other hand, letting the users freely declare what they need has its own
drawbacks, such as the possibility that the user’s explanation does not have
sufficient context regarding their demand and the issue of how to convert that
explanation into a decision about the destination department of the contact.

For WhatsApp — our main support channel and the focus of the present work —
this issue gets trickier, though: the app is ubiquitously used in Brazil as a
means of communication with family and friends. Added to this is the fact that
Brazilian clients are used to small real estate companies in which commercial
interactions are based on a person-to-person relationship. Contacts, thus, may
start with a simple _“Hi!”_ , with a complete account of the user’s issue or
with a partial description that refers to past contacts.

Prior to this work, triage and department selection was carried out by a team
dedicated solely to verifying that sufficient context was provided, asking
users to give more information if necessary, and directing each chat to the
appropriate department. Of course, this was a far from ideal situation. A
large number of skilled agents were diverted from customer troubleshooting to
perform a highly repetitive task that added latency to proper chat routing and
stiffened any processes changes that incur in changes in departmental
responsibilities, as any such changes would require retraining the triage
team.

To sustain company’s scalability, our team developed a “receptionist” chatbot
that should be able to:

  1. 1.

Determine whether the user has provided enough information in their initial
message, requesting a better description if deemed necessary.

  2. 2.

Combine this textual information with data about the user’s relationship with
QuintoAndar to predict the reason for the current contact.

  3. 3.

Automate the collection of basic data, such as information that helps to
identify users who could not be identified based on their phone number and
data to help the agent verify the authenticity of the user.

  4. 4.

Use business rules together with the predicted contact reason to determine to
which department the chat should be directed.

  5. 5.

Provide the selected agent with the predicted reason for the contact in order
facilitate access to the appropriate guidelines for that service and speed up
the mandatory annotation after the conversation is finished.

This was not without its hurdles. First, there were more than 300 standardized
possible contact reasons with a substantial gray area between them, ensuring a
great deal of noise in labels. Classes were highly unbalanced, with some
containing only a handful of chats in a given year, while others classes
totaling thousands in a single week. Finally, as a Portuguese-speaking
company, there were no open resources available related to customer service,
so we could rely only on pre-trained models and our own private datasets to
develop the chatbot.

##  4 System Architecture

Following the dialog-state architecture [Jurafsky and Martin 2021], the bot
engine we created has the following basic components:

  * •

Dialog memory: a flexible memory to store all relevant information produced by
handlers during the conversation, including special field $\mathcal{S}$ that
indicates the automaton’s current state.

  * •

Handlers: processors used by the chatbot to extract relevant information from
input messages or make decisions about what to do next.

  * •

Finite-state automaton: a recipe that indicates which handler should be called
at each state $\mathcal{S}$ and how to use a policy handler results to select
the next state.

Handlers, in turn, can be of two different types:

  * •

Message processing handlers: responsible for extracting information from input
messages and storing it on dialog memory. The present work focuses on
proposing two machine learning models that act as message processing handlers.

  * •

Policy decision handlers: responsible for deciding what the system should do
given the information stored in dialog memory. Examples of possible actions
are performing some calculations and storing their results in the memory,
sending a response to the user or transferring the chat to a certain
department. When a policy handler decides to send the user a response, it
forwards asks the language generation unit to select a template to use to
create the message.

Each handler has its own implementation, which may range from simple
application of business rules to complex ML models and third-party services.
This work makes use of two different NLP-based handlers that will be explained
on the next subsections.

###  4.1 Context evaluation model

A common behavior by our users was to start the conversation by saying a
greeting, only stating what she needed after being asked by the support agent.
This posed a significant challenge for our receptionist bot, as it would
predict contact reasons and select destination departments based on messages
that were not at all explanatory. A simple rule of thumb to avoid this problem
would be to always discard the first message in a conversation. However, this
would hurt the client’s experience, which is only worsened by the fact that
many customer support contacts are made by already stressed users.

To alleviate this issue, we created a model to predict whether or not a
particular message was sufficiently explanatory. Based on such prediction our
chatbot may decide to ask the client for more input. A dataset with 5348 chats
randomly sampled from QuintoAndar’s ticketing system was manually annotated by
two customer support agents. They could classify a message as one of the
following: _has context_ , _no context_ , _returning client_ and _low value_
(a message that is not intended to create a new conversation, as _“Thanks!”_).
The final dataset contained 2926 chats labeled as its content has enough
context and 2422 labeled as not enough context (_no context_ and _low value_ ;
_returning client_ messages were discarded), endorsing the relevance of this
first model in an efficient reception.

Our goal was to have a simple model that would introduce only a small delay in
response time, so we decided to user bag-of-words (BoW) representation with a
logistic regression (LR) as classifier. Text pre-processing included accent
stripping, uncasing and stop-word removal; no stemming or lemmatization was
used. Bag-of-words representation was then computed for 3-grams with a
vocabulary of 5000 items. LR hyperparameters ($C$, penalty type and class
weight) were selected through Bayesian optimization — the solver used was
‘lbfgs‘ [Zhu et al. 2011]. The resulting model achieved accuracy of 85% on the
testing split, a value considered good enough for production as most errors in
this step could be fixed by human agents after chat allocation.

###  4.2 Contact reason prediction model

The main model for the chatbot is the contact reason prediction model. As
explained above, its main purpose is to combine the user’s message with
tabular information indicating their relationship with QuintoAndar to predict
how likely the contact is to be related to each of 306 possible standardized
contact reasons (see Table 1 for some examples).

`cr_pg` |  Payments of real-estate brokers.
---|---
`ft_ag_alteracao` |  Schedule changes for photographers.
`iq_pr_reserva` |  Issues related to house reservation by a tenant after a proposal is accepted by an owner.
`pp_cm_venda_imovel` |  An owner informs that she intends to sell the house she currently rents.
Table 1: Examples of contact reasons defined by QuintoAndar’s process team.

The association of text and tabular data is important as a way to avoid
requiring users to explain their whole history to the chatbot. Consider the
phrase _“I need to cancel the visit tomorrow.”_. It is a complete yet very
ambiguous message: is it related to a photographer that wants to reschedule a
photo shoot, to a potential tenant who is not interested on visiting an
apartment anymore, or to a real estate agent that will not be able to present
some house and wants to leave it to another colleague? Only by having access
to the user’s relationship with QuintoAndar the model would be able to
accurately predict the contact reason without requesting further context.
Therefore, we used 66 handcrafted features available from our feature store
[Marques 2020, Hermann and Balso 2017]. Some examples are the type of the last
automatic message sent to the user (and time since it was sent), the contact
reason for the last ticket (and time since it was created), whether the user
is a registered agent, number of rented houses (as owner) and number of ended
contracts (as tenant). These features are pre-processed, performing one-hot
encoding on all categorical columns and scaling the numeric ones to have zero
mean and unit variance.

We treat both feature groups (textual and tabular) as different modules that
are fed into a separate classifier. In a first version (V1 - bag-of-words), we
used a simple unigram bag-of-words to extract features from messages. After
analyzing this model errors, we noticed that, despite showing good results, it
had trouble with synonyms and complex sentences. In a second version of the
model (V2 - BERT) we addressed this issue by using for textual feature
extraction a representation extracted by a Portuguese version of a BERT model
[Devlin et al. 2019, Souza et al. 2020].

The overall architecture can be summarized in Figure 2.

Figure 2: An overall picture of system architeture

##  5 Experiments

We briefly discuss the results achieved for version 1 and version 2 of contact
reason prediction model, taking into consideration different setups as well as
analyzing overall model’s performance. For more information regarding
libraries used and hyperparameters, please refer to supplementary information
for more details.

###  5.1 Dataset

The full dataset used to train the contact reason model contains data for
639159 chats between May 2019 and August 2020 manually annotated by support
agents, selecting only user messages sent before an agent entered the
conversation. We used out-of-time split to partition the between training
(511327 chats), validation (63916 chats) and testing (63916 chats) sets. With
respect to classes, originally the data contained 306 distinct ones — after
filtering all tickets belonging to classes with less than 50 samples on
training set the final dataset contained 235 different labels. To fine-tune
BERT on our data, since it is a considerably heavy model, we have used a
smaller portion of dataset, with 178578 samples for training, 19843 for
validation and 66141 for testing.

###  5.2 Metrics

To analyze the model performance, we have calculated top-1 and top-3 test
accuracy both for contact reason and for department222Department selection is
performed by summing predicted probabilities for all contact reasons
associated to a given department. The department with highest score is
selected..

###  5.3 V1 - Bag-of-words

For this first solution approach, we have evaluated three possible
classifiers: logistic regression (LR), random forest (RF) and multilayer
perceptron classifier (MLP). Hyperparameters were selected through Bayesian
optimization with 100 evaluated configurations [Biewald 2020]. Classifiers
comparison are presented in Table 2.

Table 2: Test accuracy for contact reason and department. Higher is better. Model | Top-1 accuracy | Top-3 accuracy | Top-1 dept. accuracy | Top-3 dept. accuracy
---|---|---|---|---
LR | 42.8% | 63.6% | 77.8% | 84.6%
RF | 40.7% | 61.2% | 75.2% | 82.7%
MLP | 44.1% | 65.1% | 78.2% | 85.0%

####  5.3.1 V2 - BERT

As observed in Table 2, the MLP model obtained the best results according to
all metrics. Given the high cost of using a large Transformer-based network
and the fact this new version is very similar to V1, we decided to keep the
same architecture.

The first step we took was to fine-tune the BERT-Large model made available by
Souza _et al._ [Souza et al. 2020] to predict contact reason. Sentences with
more than 64 tokens were truncated. Once this fine-tuned procedure was
completed, we tested different methods in order to replace the bag-of-words of
V1 with a BERT module:

  * •

Use the logits of BERT classification head.

  * •

Use the last layer of BERT language model head as suggested by Devlin _et al._
[Devlin et al. 2019] .

  * •

Concatenate last four layers of BERT language model head as suggested by
Devlin _et al._ [Devlin et al. 2019]

The results are shown in Table 3. First of all, it is relevant to notice the
large gains achieved by combining both textual and tabular features. Also, we
can see that using BERT as textual feature extractor provided better results
than using bag-of-words. Considering engineering constraints we have decided
to use the logits of BERT classification head + tabular data as version 2 of
the contact reason model.

Table 3: Comparison of contact reason accuracy for multiple models. Model | Top-1 accuracy
---|---
BoW alone | 38.14%
BoW + tabular data (V1) | 44.11%
BERT classifier | 45.57%
BERT classifier logit heads + tabular data | 53.10%
Last layer LM BERT + tabular data |  53.20%
Last 4 layers LM BERT concat + tabular data | 53.05%

##  6 Business impacts

To assess the results in production, we have collected data from a triage
human team and for a set of heuristics rules that routed simply based on the
last automatic message sent to the user (e.g., if the message was related to a
visit, then the user was directed to the visits department). The business
metric that we decided to follow to compare was the transference rate — _i.e._
, the rate in which a chat already routed to a given support department should
be transferred to another department.

We have run tests in production comparing human triage and both chatbot
versions. As a security measure to avoid deterioration of user experience, we
only routed automatically the 80% of tickets with highest department score
leaving all low-confidence tickets to human triage. As the results were good
enough (i.e. similar to human performance), we expanded it to 100% to out
clients base. Results are presented in Table 4.

Table 4: Production results. Data for human triage and heuristic rules refer were collected prior the tests. Lower the better for both columns. Model | Transf. rate | Avg. msg. per ticket
---|---|---
Human triage | 12.8% | 18.2
Heuristics rules | 18.3% | 11.2
V1 - Bag-of-words (80%) | 13.9% | 13.2
V2 - BERT (80%) | 10.3% | 13.7
V2 - BERT (100%) | 13.2% | 14.2

Considering these results, we can easily say that the BERT embedding combined
with tabular data achieved human-level performance. The number of message
exchanges until the conversation was ended (avg. message per ticket) was also
substantially smaller for clients routed by the chatbot than for those routed
by humans. An hypothesis regarding the reduction in message numbers is the
fact that by using tabular features the model has access to a large amount of
information not easily consumed by humans.

Such results in a real-world application using business metrics endorses the
potential of modern NLP techniques for Brazilian Portuguese. We hope that
results like these help companies and governments to see that it’s now
feasible to go much beyond the widespread rigid conversational interfaces
based on buttons or simple keywords. By creating applications based on recent
breakthroughs, we can make services both more efficient and accessible.

## Acknowledgments

We would like to thanks Muriel Dias for providing beautiful images that help
in our paper storytelling, as well as Marco Antonio Rocha Vinha for reviewing
some components of system architectures that were showed in section 4.

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Figure 14: Phase-folded TESS SAP light curve from sector 4, exhibiting the
stellar rotation period at 20.5 d for TOI-1201. The TESS SAP light curve from
sector 31 shows a different shape, probably attributed to instrumental effects
or contamination by the companion.  Figure 15: Posteriors of the minimum mass
of the 2.5 d transiting planet around TOI-1201 depending on the model choice
for the RV-only fits. The vertical orange line is the median value
corresponding to the 1 Kep (2.5 d) + 1 Sin (102 d) + dSHO-
GP${}_{\textnormal{19d}}$ model (top). The 25 % and 75 % quartiles are
represented as the blue box, the extending black lines show the rest of the
distribution, and the dots are considered “outliers.” The model names
correspond to those in Table 4. Figure 16: GLS periodograms of the RVs and
various stellar activity indicators from the CARMENES spectroscopic data for
PM J02489–1432E. The vertical solid blue line corresponds to the main signal
found in the RVs (28.5 d). The vertical dotted orange line corresponds to the
41 d alias ($\sim$16.8 d). The horizontal dotted, dot-dashed, and dashed lines
represent the 10 %, 1 %, and 0.1 % FAP levels, respectively.

Figure 17: CARMENES RV data for PM J02489–1432E (TOI-393) with the best-fit
model from the RV-only fit overplotted (dark gray), described in Sect. 6. Top:
RVs time series. The light gray band represents the 68 % credibility interval.
Bottom: RVs phase-folded to the period of the significant signal at 28.5 d.
The bottom panel of each plot shows the residuals after the model is
subtracted. Figure 18: Detection limits on $m\sin i$ following Bonfils et al.
(2013) for the RV residuals of PM J02489–1432E after subtracting out the most
prominent 28.5 d signal (Sect. 6). Any planet above the blue line is excluded
with 99.9% confidence. The orange line is the mass corresponding to a semi-
amplitude of 3 m s-1. TOI-1201 b is indicated by a dashed gray line. Planets
with long-period values greater than the time baseline are marked with a gray
shaded region.

##  Appendix E Data tables

Table 11: CARMENES VIS RV data of TOI-1201.

BJD (TDB**$*$Barycentric dynamical time.) | RV (m s-1) |  $\sigma_{\textnormal{RV}}$ (m s-1)
---|---|---
2458796.50268 | $-11.12$ | $2.86$
2458801.48909 | $-10.09$ | $3.92$
2458804.45542 | $0.33$ | $3.20$
2458808.52929 | $-1.10$ | $4.24$
2458814.47394 | $-4.51$ | $3.76$
2458815.44971 | $-13.25$ | $3.33$
2458816.45605 | $-11.81$ | $2.17$
2458817.43864 | $-6.04$ | $1.91$
2458818.43170 | $-10.98$ | $3.23$
2458829.47849 | $-3.24$ | $3.02$
2458832.40663 | $5.73$ | $2.22$
2458834.41799 | $6.48$ | $3.89$
2458841.37576 | $3.47$ | $1.97$
2458842.38984 | $9.88$ | $2.48$
2458843.41633 | $4.96$ | $2.18$
2458844.39932 | $14.76$ | $4.22$
2458845.34605 | $3.36$ | $2.05$
2458846.37295 | $2.57$ | $1.59$
2458847.37900 | $7.87$ | $1.72$
2458848.34964 | $-5.31$ | $1.77$
2458849.35491 | $4.27$ | $1.52$
2458850.35018 | $-1.90$ | $1.89$
2458855.42501 | $-3.53$ | $1.74$
2458856.35167 | $-0.20$ | $2.26$
2458857.35123 | $6.76$ | $1.73$
2458860.37498 | $2.18$ | $3.63$
2458881.32010 | $-5.56$ | $2.18$
2458882.31533 | $0.88$ | $1.81$
2458890.30180 | $-11.40$ | $3.28$
2458891.29664 | $-8.28$ | $1.95$
2458894.29514 | $-10.75$ | $1.80$
2458895.31304 | $-9.93$ | $2.99$
2458897.30005 | $-6.50$ | $2.31$
292929

Table 12: CARMENES VIS RV data of PM J02489–1432E (TOI-393).

BJD (TDB**$*$Barycentric dynamical time.) | RV (m s-1) |  $\sigma_{\textnormal{RV}}$ (m s-1)
---|---|---
2458801.53021 | $3.20$ | $3.36$
2458804.47688 | $4.59$ | $2.61$
2458808.50608 | $-1.90$ | $4.33$
2458815.47257 | $2.88$ | $2.78$
2458816.48146 | $4.28$ | $1.66$
2458817.47079 | $1.10$ | $3.57$
2458818.45424 | $-6.95$ | $3.32$
2458832.42705 | $2.67$ | $2.22$
2458834.44005 | $7.91$ | $3.78$
2458841.39830 | $2.61$ | $2.11$
2458842.36846 | $2.56$ | $2.87$
2458843.43878 | $-1.09$ | $2.31$
2458844.42653 | $9.05$ | $7.01$
2458845.36897 | $3.67$ | $2.18$
2458846.39647 | $-0.16$ | $1.54$
2458847.40170 | $-1.57$ | $1.63$
2458848.37188 | $-1.86$ | $1.56$
2458849.37805 | $-2.09$ | $1.73$
2458850.37392 | $-1.79$ | $1.59$
2458852.36953 | $-2.85$ | $1.31$
2458855.40269 | $-1.19$ | $2.12$
2458856.37286 | $0.55$ | $2.07$
2458857.37366 | $-1.46$ | $1.99$
303030

Table 13: Astrometric data of the binary WDS J02490–1432 (KPP 2871).

Epoch | $\rho$ | $\theta$ | Origin
---|---|---|---
| (arcsec) | (deg) |
1953.927 | 7.17 $\pm$ 0.30 | 100.7 | POSS-I Red
1986.935 | 7.76 $\pm$ 0.20 | 97.7 | UKST Infrared
1988.838 | 7.90 $\pm$ 0.20 | 96.4 | UKST Blue
1989.742 | 7.81 $\pm$ 0.20 | 100.0 | UKST Red
1998.582 | 8.19 $\pm$ 0.13 | 98.3 | 2MASSa
1999.782 | 8.184 | 99.0 $\pm$ 0.2 | UCAC4b
2000.775 | 8.23 $\pm$ 0.10 | 98.7 | DENIS
2012.110 | 8.39542 | 98.788 | KPPc
2015.000 | 8.4 $\pm$ 1.0 | 98.96 |  Gaia DR1d
2015.500 | 8.391 $\pm$ 0.061 | 98.96 |  Gaia DR2e
2016.000 | 8.408 $\pm$ 0.034 | 98.94 |  Gaia EDR3
313131 $a$$a$footnotetext: WDS tabulates $\rho$ = 8.15 arcsec and $\theta$ =
98.3 deg (Thurgood Marshall High School, priv. comm.). $b$$b$footnotetext:
From Zacharias et al. (2012). $c$$c$footnotetext: From Knapp & Nanson (2019).
$d$$d$footnotetext: WDS tabulates $\rho$ = 8.34363 arcsec and $\theta$ =
99.001 deg (Knapp & Nanson 2019). $e$$e$footnotetext: WDS tabulates $\rho$ =
8.34896$\pm$0.00004 arcsec and $\theta$ = 98.999 deg (El-Badry & Rix 2018).


# Rigorous analysis of the topologically protected edge states in the quantum
spin Hall phase of the armchair ribbon geometry

Mozhgan Sadeghizadeh  Department of Physics, Faculty of Physics, University of
Isfahan, Isfahan 81746-73441, Iran  Morteza Soltani  Department of Physics,
Faculty of Physics, University of Isfahan, Isfahan 81746-73441, Iran
<EMAIL_ADDRESS>Mohsen Amini  Department of Physics, Faculty of
Physics, University of Isfahan, Isfahan 81746-73441, Iran
<EMAIL_ADDRESS>
###### Abstract

Studying the edge states of a topological system and extracting their
topological properties is of great importance in understanding and
characterizing these systems. In this paper, we present a novel analytical
approach for obtaining explicit expressions for the edge states in the Kane-
Mele model within a ribbon geometry featuring armchair boundaries. Our
approach involves a mapping procedure that transforms the system into an
extended Su-Schrieffer-Heeger model, specifically a two-leg ladder, in
momentum space. Through rigorous derivation, we determine various analytical
properties of the edge states, including their wave functions and energy
dispersion. Additionally, we investigate the condition for topological
transition by solely analyzing the edge states, and we elucidate the
underlying reasons for the violation of the bulk-edge correspondence in
relatively narrow ribbons. Our findings shed light on the unique
characteristics of the edge states in the quantum spin Hall phase of the Kane-
Mele model and provide valuable insights into the topological properties of
such systems.

## Introduction

The quantum spin Hall (QSH) effect [1, 2] is a fascinating phenomenon in
condensed matter physics, which manifests as a gapless edge state in a two-
dimensional topological insulator that is protected by time-reversal symmetry.
The Kane-Mele model [3, 4] is one of the simplest model Hamiltonians that
exhibits the QSH effect, and has attracted a lot of attention in recent years
due to its potential applications in spintronics and quantum computing [6, 7,
8, 9]. The emergence of such edge states within the energy band gap in the QSH
systems is intimately connected to the existence of nontrivial topological
properties in their bulk bands which are referred to as topological
invariants. These topological invariants can be associated with a nonzero
topological quantum number and due to the bulk-edge correspondence [10] one
naturally expects the existence of protected edge states. These edge states
are typically localized at the boundaries or interfaces of the material and
are protected from backscattering due to the underlying topological
properties.

Typically, the bulk-edge correspondence serves as a valuable tool to
comprehend various characteristics of topologically protected edge states by
examining the bulk band structure. For instance, the existence and stability
of edge states for various systems like the odd-parity superconductors,
periodically driven quantum systems, non-Hermitian systems, two-dimensional
fermions of high spin, and SU(3) fermions, is investigated by studying
topological properties of the bulk in Refs. [11, 12, 13, 14, 15]. Nonetheless,
it remains intriguing to explore the extent to which information regarding the
system’s topological properties and transitions can be directly inferred
through the study of these edge states. Indeed, understanding the properties
of edge states allows researchers to identify and classify different
topological phases and also provides a way to manipulate and control their
properties [16, 17, 18, 19, 20, 21, 22, 23, 24]. In this regard, from
theoretical point of view, the availability of explicit expressions allows for
a detailed examination of the edge states’ characteristics, such as their
energy dispersion, spatial distribution, and symmetry properties.

Therefore, it is worth analyzing the topological phase transition by rigorous
derivation of the edge states in a QSH system. For this purpose, here, we
investigate the topological properties of an armchair honeycomb ribbon by
considering the Kane-Mele model [3, 4]. This model is known to exhibit edge
states within the energy gap, making it an ideal system for exploring the
characteristics and behaviors of these edge states. The explicit form of the
edge states in the considered model remained elusive until very recently that
the successful determination of the edge state wave functions for the zigzag
ribbon configuration is reported [25]. In this study, by focusing on the
armchair geometry, we specifically target the unique features associated with
the edge states in this type of termination. Through our analysis, we aim to
shed light on the fundamental aspects of the armchair edge states and their
differences with the zigzag edges states. In contrast to the zigzag
termination ribbons in which the edge states obtained perturbatively, the edge
states of the armchair-terminated ribbons can be obtained exactly. This
breakthrough provides us with a valuable opportunity to delve into the
detailed analysis of these edge states and their intriguing properties. By
leveraging this newfound understanding, we aim to identify the topological
transition point as well as the origin of the bulk-edge correspondence
violation for relatively narrow ribbons [26].

## Extended SSH model for the Kane-Mele Hamiltonian of an armchair ribbon

In this section, we delve into the analysis of a Kane–Mele model for the
honeycomb lattice with armchair edges (as shown in Fig. 1(a)), specifically
focusing on the introduction of a mapping technique that enables us to derive
an extended Su-Schrieffer-Heeger (SSH) model for a two-leg ladder structure in
the momentum space. This mapping is of significant importance as it provides a
framework for investigating the properties of the system, including the
emergence of edge states. Prior to delving into the intricacies of the mapping
technique, we provide a concise overview of the fundamental aspects of the
Kane-Mele model. The Hamiltonian that characterizes the dynamics of the Kane-
Mele model is expressed as follows [3, 4]:

$\displaystyle H=t\sum_{\langle
m,l\rangle}{c}_{m}^{\dagger}{c}_{l}+i\lambda_{so}\sum_{\langle\langle
m,l\rangle\rangle}\nu_{m,l}{c}_{m}^{\dagger}\hat{s}_{z}{c}_{l}+i\lambda_{R}\sum_{\langle
m,l\rangle}{c}_{m}^{\dagger}(s\times
d_{m,l})_{z}{c}_{l}+M\sum_{m}\epsilon_{m}c_{m}^{\dagger}c_{m},$ (1)

where ${c}_{m}^{\dagger}=\left(\begin{matrix}{c}_{m\uparrow}^{\dagger}\\\
{c}_{m\downarrow}^{\dagger}\\\ \end{matrix}\right)$ denotes the creation
operator for an electron with spin up/down at the $m$-th site on the honeycomb
lattice and $\left\langle m,l\right\rangle$ and $\left\langle\left\langle
m,l\right\rangle\right\rangle$ stands for summation over the nearest-neighbor
and the next-nearest-neighbor pairs of sites, respectively. The first term
accounts for the nearest-neighbor hopping with an amplitude of $t$ and the
second term captures the spin-orbit couplings with a hopping amplitude denoted
as $\lambda_{SO}$ in which $\widehat{{s}_{z}}$ is the $z$-pauli matrix and
${\nu}_{m,l}$ takes $\pm 1$ depending on $m$ and $l$. The third term is the
Rashba term with coupling ${\lambda}_{R}$ that arises from the interaction
between the electron’s spin and its momentum in the presence of an electric
field gradient. Here, ${d}_{m,l}$ is the unit vector along the nearest
neighbor bond pointing from $m$ to $l$. The last term represents a staggered
on-site potential of strength $M$ in which the parameter $\epsilon_{m}=\pm 1$
depending on the sub-lattice index $m$.

Figure 1: (a) Sketch of the lattice structure of an armchair honeycomb ribbon
with width $N$. The sign of the hopping parameter $\nu$ is indicated by dashed
lines. (b) Schematic representation of the corresponding two-leg ladder model
obtained after performing the Fourier transformation along the horizontal
direction. The new momentum-dependent hopping parameters are represented and
defined as
$\tau_{1}=t,\tau_{2}=i\lambda_{so},\tau_{3}=te^{ik_{x}/2},\tau_{4}=2\lambda_{so}\cos(k_{x}/2)$.
(c) Schematic representation of the transformed two-leg ladder model after
applying a proper unitary transformation described in Eq. (8). The transformed
hopping parameters are denoted with prime notation. The on-site energies
$\pm\Delta=\pm t\sin(k_{x}/2)$ are indicated. This figure illustrates the
step-by-step transformation of the lattice structure and hopping parameters,
leading to an equivalent two-leg ladder model in the momentum space.

In the following analysis, we consider a honeycomb lattice with a finite width
of $N$ (the number of armchair chains) in the $y$ direction, while maintaining
periodic boundary conditions in the $x$ direction, as depicted in Fig. 1. For
the sake of simplicity and obtaining analytical results, we omit the third
term in this Hamiltonian due to its relatively small influence. In the absence
of the Rashba term and at a finite $M$, the system is in the gapped phase
without zero energy states when $\lambda_{SO}=0$. The introduction of a small
$\lambda_{SO}$ initially has no significant impact on the situation. However,
the gap undergoes a closure and the system exhibits the QSH phase when the
magnitude of the intrinsic spin-orbit coupling parameter crosses a critical
value of $M/3\sqrt{3}$ [3, 4]. This critical condition,
$M<3\sqrt{3}\lambda_{SO}$, signifies a threshold for the strength of the
intrinsic spin-orbit coupling, beyond which the QSH phase emerges. Hence, we
initially set $M=0$; however, we will subsequently examine the case where
$M\neq 0$ to investigate the occurrence of a phase transition.

Figure 2:  Numerical energy bands for the Kane-Mele model in a ribbon with
armchair boundary plotted as a function of $k_{x}$, the wave vector parallel
to the armchair direction. The spin-orbit coupling strength is set to
$\lambda_{SO}=0.3t$ and $M=0$. Additionally, we include the analytical
expression obtained in Eq. (25) for the energy spectrum of the edge states in
the same system, demonstrating an excellent agreement between the numerical
and analytical results. The plot showcases the presence of well-defined energy
bands and validates the accuracy of the analytical model in capturing the
behavior of the edge states.

To provide a preliminary understanding, it is informative to numerically
examine the band structure of ribbons with armchair termination prior to
delving into further analysis. Fig. 2 represents the energy spectrum
$E(k_{x})$ where $k_{x}$ is the momentum along the ribbon edges for an
armchair ribbon with a width of $N=50$ and a spin-orbit coupling strength of
$\lambda_{SO}=0.3t$ in the absence of staggered potential ($M=0$). Let us
consider the conventional unit cell as shown in Fig. 1 and choose the unit-
cell length to be $a=1$ to ensure simplicity. As depicted in Fig. 2, the bulk
band-structure displays a distinct gap, while concurrently exhibiting the
presence of gapless edge states. This observation aligns precisely with our
expectations for the QSH phase.

To initiate our mapping procedure, let us rewrite the Hamiltonian of Eq. (1)
as the following:

$\displaystyle H=\sum_{j}H_{j,j}+H_{j,j+1}+H_{j-1,j},$ (2)

where $H_{j,j}$ represents the coupling matrix within the $j$-th unit cell,
while $H_{j,j+1}$ refers to the coupling between two adjacent unit cells $j$
and $j+1$ and can be defined as

$\displaystyle H_{j,j}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}t(a_{i,j}^{\dagger}b_{i,j}+b_{i,j}^{\dagger}c_{i,j}+c_{i,j}^{\dagger}d_{i,j}+b_{i,j}^{\dagger}a_{i+1,j}+c_{i,j}^{\dagger}d_{i+1,j})$
$\displaystyle+\sum_{i=1}^{N}\lambda_{so}(a_{i,j}^{\dagger}c_{i,j}+a_{i,j}^{\dagger}a_{i+1,j}+b_{i,j}^{\dagger}d_{i,j}+b_{i,j}^{\dagger}b_{i+1,j}+b_{i,j}^{\dagger}d_{i+1,j}+c_{i,j}^{\dagger}a_{i+1,j}+c_{i,j}^{\dagger}c_{i+1,j}+d_{i,j}^{\dagger}d_{i+1,j})+H.C,$

and

$H_{j,j+1}=\sum_{i=1}^{N}t(d_{i,j}^{\dagger}a_{i,j+1})+\lambda_{so}(d_{i,j}^{\dagger}b_{i,j+1}+c_{i,j}^{\dagger}a_{i,j+1}+c_{i,j}^{\dagger}a_{i+1,j+1}+d_{i+1,j}^{\dagger}b_{i,j+1}),$
(4)

where H.C stands for the hermitian conjugate and it is obvious that
$H_{j,j-1}=H_{j,j+1}^{\dagger}$. In this notation, we label the four distinct
sites in the unit cell for convenience as $a,b,c,$ and $d$ (as illustrated in
Fig. 1 (a)) and index $i$ runs over all the armchair chains in the unit cell.
Due to the translational invariant in the $x$-direction, we can introduce the
following Fourier transformation of the fermion annihilation operators of the
four basis of the unit cell as the following,

$\displaystyle a(b)_{i,k_{x}}$ $\displaystyle=$
$\displaystyle\sum_{j}e^{i{k_{x}}j}a(b)_{i,j},$ $\displaystyle c(d)_{i,k_{x}}$
$\displaystyle=$ $\displaystyle\sum_{j}e^{i{k_{x}}(j+\frac{1}{2})}c(d)_{i,j},$
(5)

in which to get rid of the extra phase factor in the definition of $c(d)$
operators we have used the proper gauge transformation [27]. Inserting Eq.
(Extended SSH model for the Kane-Mele Hamiltonian of an armchair ribbon) into
Eqs. (Extended SSH model for the Kane-Mele Hamiltonian of an armchair ribbon)
and (4) one can easily obtain the $k$-space Hamiltonian as

$\displaystyle H_{k_{x}}$ $\displaystyle=$ $\displaystyle
t\sum_{i=1}^{N}[a_{i}^{\dagger}b_{i}+d_{i}^{\dagger}c_{i}+d_{i}^{\dagger}a_{i}e^{-ik_{x}/2}+c_{i}^{\dagger}b_{i}e^{ik_{x}/2}+b_{i}^{\dagger}a_{i+1}+c_{i}^{\dagger}d_{i+1}]$
$\displaystyle+2i\lambda_{so}cos(k_{x}/2)\sum_{i=1}^{N}[d_{i}^{\dagger}b_{i}-a_{i}^{\dagger}c_{i}-c_{i}^{\dagger}a_{i+1}+b_{i}^{\dagger}d_{i+1}]+i\lambda_{so}\sum_{i=1}^{N}[a_{i}^{\dagger}a_{i+1}-d_{i}^{\dagger}d_{i+1}-b_{i}^{\dagger}b_{i+1}+c_{i}^{\dagger}c_{i+1}]+H.C.$

It becomes evident that for every value of momentum $k_{x}$, the Hamiltonian
corresponds to an extended SSH model for a two-leg ladder system, as depicted
in Fig. 1(b). To enhance convenience, we can now utilize the Pauli matrices
$\hat{\sigma}_{\lambda}$ (where $\lambda=x,y,z$) along with the identity
operator $\hat{I}_{\sigma}$ to rewrite the aforementioned Hamiltonian in the
following manner:

$\displaystyle H_{k_{x}}$ $\displaystyle=$
$\displaystyle\tau_{1}\begin{bmatrix}a_{i}^{\dagger}&d_{i}^{\dagger}\end{bmatrix}\hat{I}_{\sigma}\begin{bmatrix}b_{i}\\\
c_{i}\end{bmatrix}+\tau_{2}\begin{bmatrix}a_{i}^{\dagger}&d_{i}^{\dagger}\end{bmatrix}\hat{\sigma}_{z}\begin{bmatrix}a_{i+1}\\\
d_{i+1}\end{bmatrix}+\tau_{2}^{*}\begin{bmatrix}b_{i}^{\dagger}&c_{i}^{\dagger}\end{bmatrix}\hat{\sigma}_{z}\begin{bmatrix}b_{i+1}\\\
c_{i+1}\end{bmatrix}+\tau_{4}\begin{bmatrix}a_{i}^{\dagger}&d_{i}^{\dagger}\end{bmatrix}\hat{\sigma}_{y}\begin{bmatrix}b_{i}\\\
c_{i}\end{bmatrix}-\tau_{4}\begin{bmatrix}b_{i}^{\dagger}&c_{i}^{\dagger}\end{bmatrix}\hat{\sigma}_{y}\begin{bmatrix}a_{i+1}\\\
d_{i+1}\end{bmatrix}+H.C$ (7)
$\displaystyle+\tau_{1}\begin{bmatrix}a_{i}^{\dagger}&d_{i}^{\dagger}\end{bmatrix}(\hat{\sigma}_{x}\cos(k_{x}/2)-\hat{\sigma}_{y}\sin(k_{x}/2))\begin{bmatrix}a_{i}\\\
d_{i}\end{bmatrix}+\tau_{1}\begin{bmatrix}b_{i}^{\dagger}&c_{i}^{\dagger}\end{bmatrix}(\hat{\sigma}_{x}\cos(k_{x}/2)+\hat{\sigma}_{y}\sin(k_{x}/2))\begin{bmatrix}b_{i}\\\
c_{i}\end{bmatrix},$

in which the new hopping parameters $\tau_{i}$ for this ladder geometry are
defined as
$\tau_{1}=t,\tau_{2}=i\lambda_{so},\tau_{3}=te^{ik_{x}/2},\tau_{4}=2\lambda_{so}\cos(k_{x}/2)$.
Our effective Hamiltonian is now constructed. However, for the reason which
will become clear later, we make a unitary transformation with the following
unitary matrix $U=e^{i\frac{\pi}{4}\sigma_{x}}$ which satisfies
$U\sigma_{y}U^{T}=\sigma_{z}$ where $T$ stands for transpose. When applying
this unitary transformation, we can obtain the resulting transformed operators
$a^{\prime},b^{\prime},c^{\prime}$ and $d^{\prime}$ as follows:

$\displaystyle\begin{bmatrix}a_{i}^{\prime}(c_{i}^{\prime})\\\
b_{i}^{\prime}(d_{i}^{\prime})\end{bmatrix}=U\begin{bmatrix}a_{i}(c_{i})\\\
b_{i}(d_{i})\end{bmatrix}=\frac{\sqrt{2}}{2}\begin{bmatrix}1&i\\\
i&1\end{bmatrix}\begin{bmatrix}a_{i}(c_{i})\\\ b_{i}(d_{i})\end{bmatrix}.$ (8)

After performing the unitary transformation, it becomes straightforward to
devide the Hamiltonian in Eq. (7) into hopping term $H_{0}$ and on-site term
$H_{on}$ as the following

$\displaystyle\hat{H}=H_{0}+H_{on},$ (9)

where

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\tau_{1}\begin{bmatrix}a_{i}^{\dagger\prime}&d_{i}^{\dagger\prime}\end{bmatrix}\hat{I}_{\sigma}\begin{bmatrix}b_{i}^{\prime}\\\
c_{i}^{\prime}\end{bmatrix}+\tau_{2}^{\prime}\begin{bmatrix}a_{i}^{\dagger\prime}&d_{i}^{\dagger\prime}\end{bmatrix}\hat{\sigma}_{y}\begin{bmatrix}a_{i+1}^{\prime}\\\
d_{i+1}^{\prime}\end{bmatrix}+(\tau_{2}^{\prime})^{*}\begin{bmatrix}b_{i}^{\dagger\prime}&c_{i}^{\dagger\prime}\end{bmatrix}\hat{\sigma}_{y}\begin{bmatrix}b_{i+1}^{\prime}\\\
c_{i+1}^{\prime}\end{bmatrix}+\tau_{4}^{\prime}\begin{bmatrix}a_{i}^{\dagger\prime}&d_{i}^{\dagger\prime}\end{bmatrix}\hat{\sigma}_{z}\begin{bmatrix}b_{i}^{\prime}\\\
c_{i}^{\prime}\end{bmatrix}$ (10)
$\displaystyle-\tau_{4}^{\prime}\begin{bmatrix}b_{i}^{\dagger\prime}&c_{i}^{\dagger\prime}\end{bmatrix}\hat{\sigma}_{z}\begin{bmatrix}a_{i+1}^{\prime}\\\
d_{i+1}^{\prime}\end{bmatrix}+H.C+\tau_{3}^{\prime}\begin{bmatrix}a_{i}^{\dagger\prime}&d_{i}^{\dagger\prime}\end{bmatrix}\hat{\sigma}_{x}\begin{bmatrix}a_{i}^{\prime}\\\
d_{i}^{\prime}\end{bmatrix}+\tau_{3}^{\prime}\begin{bmatrix}b_{i}^{\dagger\prime}&c_{i}^{\dagger\prime}\end{bmatrix}\hat{\sigma}_{x}\begin{bmatrix}b_{i}^{\prime}\\\
c_{i}^{\prime}\end{bmatrix},$

and

$\displaystyle
H_{on}=-\Delta(k_{x})\begin{bmatrix}a_{i}^{\dagger\prime}&d_{i}^{\dagger\prime}\end{bmatrix}\hat{\sigma}_{z}\begin{bmatrix}a_{i}^{\prime}\\\
d_{i}^{\prime}\end{bmatrix}+\Delta(k_{x})\begin{bmatrix}b_{i}^{\dagger\prime}&c_{i}^{\dagger\prime}\end{bmatrix}\hat{\sigma}_{z}\begin{bmatrix}b_{i}^{\prime}\\\
c_{i}^{\prime}\end{bmatrix}.$ (11)

Here, the new hopping parameters of the transformed Hamiltonian are defined as
$\tau_{2}^{\prime}=\lambda_{so},\tau_{3}^{\prime}=t\cos(k_{x}/2),\tau_{4}^{\prime}=\tau_{4}$
and the on-site energies $\Delta(k_{x})=t\sin(k_{x}/2)$. A graphical
representation of the new parameters in the transformed Hamiltonian are shown
in Fig 1(c). In the upcoming sections, we will utilize the transformed
Hamiltonian derived in this section to thoroughly investigate various
properties of the system.

## Results and discussion

To present the findings of our study, this section is divided into three
subsections, each focusing on a different aspect of the results. Firstly, we
analyze the edge states of the system and discuss their properties for
sufficiently wide ribbons. Secondly, we investigate the identification of the
topological phase transition in the system by exclusively examining the
properties of the edge state wave functions. Our investigation focuses on
determining the conditions under which the system undergoes a transition,
solely relying on the analysis of these edge states. Lastly, we delve into the
intriguing phenomenon of bulk-edge correspondence violation in relatively
small width ribbons and provide insights into its origin analitically. Through
these subsections, we aim to provide a comprehensive understanding of the
system’s topological properties and shed light on its unique characteristics.

### Edge stat analysis

Our initial focus is on determining the edge states of the Hamiltonian (9).
Considering that the presence of $H_{on}$ does not alter the edge state wave
function [28], we will confine ourselves to the edge states of $H_{0}$. Thus,
our objective is to find $|\psi_{\text{edge}}\rangle$ that satisfies the
following Schrödinger equation:

$\displaystyle\hat{H_{0}}|\psi_{edge}\rangle=0$ (12)

The equation mentioned above holds when there exists a subspace on which the
wave function vanishes. Our numerical analysis reveals that this subspace is
formed by the basis states $a_{i}^{\dagger}|0\rangle$ and
$c_{i}^{\dagger}|0\rangle$. This implies that our desired edge states have
zero amplitudes on the lattice sites marked in red in Fig. 1(c). In other
words, the wave function of the edge states vanishes at these particular
lattice sites, that is

$\displaystyle\langle
0|a_{i}|\psi_{edge}\rangle=0,\;\;\;\text{and}\;\;\;\langle
0|c_{i}|\psi_{edge}\rangle=0.$ (13)

Therefore, the general form of the edge state wave function is given by

$\displaystyle|\psi_{edge}\rangle=\sum_{i=1}\alpha_{2i-1}|b_{i}\rangle+\alpha_{2i}|d_{i}\rangle,$
(14)

where the coefficients $\alpha_{i}$ are the wave function amplitudes on $b$
and $d$ sites. By inserting the wave function Eq. (14) in Eq. (12) and
choosing initial coefficients $\alpha_{1}$ and $\alpha_{2}$ one gets the
following recursive relations:

$\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle\frac{\tau_{3}^{\prime}}{\tau_{2}^{\prime}}\alpha_{1}+\frac{(\tau_{1}+\tau_{4}^{\prime})}{\tau_{2}^{\prime}}\alpha_{2},$
(15) $\displaystyle\alpha_{4}$ $\displaystyle=$
$\displaystyle\frac{(\tau_{1}-\tau_{4}^{\prime})}{\tau_{2}^{\prime}}\alpha_{1}+\frac{\tau_{3}^{\prime}}{\tau_{2}^{\prime}}\alpha_{2}+\frac{(\tau_{1}+\tau_{4}^{\prime})}{\tau_{2}^{\prime}}\alpha_{3},$
(16) $\displaystyle\alpha_{5}$ $\displaystyle=$
$\displaystyle\alpha_{1}+\frac{(\tau_{1}-\tau_{4}^{\prime})}{\tau_{2}^{\prime}}\alpha_{2}+\frac{\tau_{3}^{\prime}}{\tau_{2}^{\prime}}\alpha_{3}+\frac{(\tau_{1}+\tau_{4}^{\prime})}{\tau_{2}^{\prime}}\alpha_{4},$
(17) $\displaystyle\alpha_{i+4}$ $\displaystyle=$
$\displaystyle\beta_{1}\alpha_{i}+\beta_{2}\alpha_{i+1}+\beta_{3}\alpha_{i+2}+\beta_{4}\alpha_{i+3},\;\;\;\text{for}\;\;\;i\geq
1,$ (18)

in which the coefficients $\beta_{i}$ reads as
$\beta_{1}=1,\beta_{2}=\frac{(\tau_{1}-\tau_{4}^{\prime})}{\tau_{2}^{\prime}},\beta_{3}=\frac{\tau_{3}^{\prime}}{\tau_{2}^{\prime}},$
and $\beta_{4}=\frac{(\tau_{1}+\tau_{4}^{\prime})}{\tau_{2}^{\prime}}$.

To obtain a solution that satisfies all of the aforementioned equations, we
can start by introducing a general solution in the form:

$\displaystyle\alpha_{i+n}=x^{n}\alpha_{i}.$ (19)

We then substitute this expression into Eq. (18), which results in the
following equation for $x$:

$\displaystyle x^{4}-\beta_{4}x^{3}-\beta_{3}x^{2}-\beta_{2}x-\beta_{1}=0.$
(20)

Our objective is to solve this equation to determine the proper values of $x$.
Considering these solutions, we need to make a combination of them in a way
that satisfies Eqs. (15) and Eq. (16) (Eq. (17) in not an independent
equation.). In principle, the fourth-order equation of Eq. (17) has four
potential solutions. The largest solution of this equation, denoted as
$x_{4}$, always satisfies the condition $x_{4}>1$ and should be excluded from
the final combination due to its divergence [29]. Thus, our desired solution
can be expressed in the following form:

$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\mathcal{N}}}(c_{1}+c_{2}+c_{3})$
$\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\mathcal{N}}}(c_{1}x_{1}+c_{2}x_{2}+c_{3}x_{3})$
$\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\mathcal{N}}}(c_{1}x_{1}^{2}+c_{2}x_{2}^{2}+c_{3}x_{3}^{2})$
$\displaystyle\alpha_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\mathcal{N}}}(c_{1}x_{1}^{3}+c_{2}x_{2}^{3}+c_{3}x_{3}^{3}),$
(21)

in which $\mathcal{N}=\sum_{i,j}\frac{1}{1-x_{i}x_{j}}c_{i}c_{j}$ is the
normalization factor and $c_{i}$ should be determined in such a way that
satisfies the given conditions in Eqs. (15) and (16) which results in the
following equations:

$\displaystyle\frac{c_{1}\beta_{1}}{x_{1}}+\frac{c_{2}\beta_{1}}{x_{2}}+\frac{c_{3}\beta_{1}}{x_{3}}=0,$
(22)

and

$\displaystyle
c_{1}(\frac{\beta_{2}}{x_{1}}+\frac{\beta_{1}}{x_{1}^{2}})+c_{2}(\frac{\beta_{2}}{x_{2}}+\frac{\beta_{1}}{x_{2}^{2}})+c_{3}(\frac{\beta_{2}}{x_{3}}+\frac{\beta_{1}}{x_{3}^{2}})=0,$
(23)

which should be solved with the additional condition

$\displaystyle\frac{1}{\sqrt{\mathcal{N}}}(c_{1}+c_{2}+c_{3})=1.$ (24)

Once the solution of Eqs.(22), (23), and (24) is known[30], we can utilize it
in Eq.(Edge stat analysis) to obtain the edge state wave function as defined
in Eq.(14). In Fig. 3, we have visually depicted this wave function for
$\lambda=0.2t_{1}$ and $k_{x}=0$. The energy spectrum corresponding to this
wave function can be obtained by evaluating [28] the expression
$\langle\psi_{\text{edge}}|H_{\text{on}}|\psi_{\text{edge}}\rangle$, and it
can be simplified as follows:

$\displaystyle E(k_{x})=t_{1}\sin(k_{x}/2),$ (25)

This energy spectrum is depicted in Fig. 2, and it precisely coincides with
the numerical edge band obtained for the same parameters.

Figure 3: Plot of the amplitudes of the edge states wave function versus site
index $i$ for the Kane-Mele model in the QSH phase. The wave function is
calculated using Eq. (14) for $\lambda_{SO}=0.2t$ and $M=0$ at $k_{x}=0$.

### Identification of the topological transition

In this section, our goal is to identify the topological transition point in
our system by analyzing the edge state wave function that we obtained
previously. To achieve this, we introduce the staggered on-site potential term
in the Kane-Mele Hamiltonian of Eq. (1), which facilitates the crossing of the
transition point. In this regime ($M\neq 0$), the only modification is to
adjust the hopping parameter $\tau_{3}^{\prime}$ as follows:

$\displaystyle\tau_{3}^{\prime}(M)\equiv\tau_{3}^{\prime}+iM=t_{1}\cos(k_{x}/2)+iM.$
(26)

To obtain the new edge states in the presence of $M$, we need to repeat the
analysis outlined in the previous subsection. The only difference is that now
the recursion relation of Eq. (18) should be modified as follows:

$\displaystyle\alpha_{i+4}=\beta_{1}\alpha_{i}+\beta_{2}\alpha_{i+1}+\beta_{3}^{\prime}(M)\alpha_{i+2}+\beta_{4}\alpha_{i+3},$
(27)

where the modified parameter $\beta_{3}^{\prime}(M)$ is given by:

$\displaystyle\beta_{3}^{\prime}(M)=\frac{\tau_{3}^{\prime}(M)}{\tau_{2}^{\prime}}.$
(28)

By performing this analysis and considering the general solution of Eq. (19)
we obtain the following equation for $x$:

$\displaystyle
x^{4}-\beta_{4}x^{3}-\beta_{3}^{\prime}(M)x^{2}-\beta_{2}x-\beta_{1}=0.$ (29)

In our previous discussion, we noted that in the absence of $M$, one solution
of the equation is always greater than one ($x_{4}>1$), while the remaining
three solutions are not ($x_{1},x_{2},x_{3}<1$). However, the presence of $M$
introduces a change in the conditions, allowing for the possibility of another
solution to be greater than one, depending on the value of $M$ (particularly
for $M\gg t_{1}$ as shown in [27]). This additional solution is referred to as
$x_{3}$. Consequently, under certain conditions, it is possible to have
$x_{1}$ and $x_{2}$ less than one, while $x_{3}$ and $x_{4}$ are greater than
one, rendering it impossible to find a solution for the edge states. This
indicates that there exists a criterion which distinguishes between the
regimes with and without a solution for the edge states. This criterion is
therefore defined by $x_{3}=1$. To determine which values of $M$ (which we
call $M^{*}$) satisfy this criterion, we can consider $x_{3}=e^{i\theta}$ and
substitute it into Eq. (29) and after performing some simple algebra, this
leads to the following equation:

$\displaystyle\lambda_{so}e^{2i\theta}-te^{i\theta}-2\lambda_{so}e^{i\theta}-te^{-i\theta}+2\lambda_{so}e^{-i\theta}-\lambda_{so}e^{-2i\theta}-(t+iM^{*})=0.$
(30)

If we equate the real and imaginary parts of the left side of the above
equation to zero, we obtain the following equations

$\displaystyle 2\cos(\theta)=-1$
$\displaystyle-4\lambda_{so}\sin(2\theta)+2\lambda_{so}\sin(2\theta)=M^{*}$
(31)

which results in

$\displaystyle M^{*}=3\sqrt{3}\lambda_{so}.$ (32)

Therefore, for values of $M$ less than $M^{*}$, a solution for the edge states
can be obtained. However, when $M$ reaches the critical value $M^{*}$, the
edge states become unstable and disappear, leading to the emergence of a gap
in the spectrum at $k_{x}=0$. This is a fascinating observation as it
demonstrates that the transition point can be determined solely by analyzing
the wave function of the edge states. By studying the behavior of the edge
states, we were able to identify the critical value of $M$ where the
transition occurs. This highlights the significance of edge state analysis in
understanding the topological properties and phase transitions in the system.

Before ending this subsection, it is worth mentioning that the presence of $M$
does not alter the energy spectrum of the edge states while we are in the
topological phase. This is due to the fact that the parameter $\Delta(k_{x})$,
which describes the on-site energies of the Hamiltonian $H_{0}$ in Eq.(11),
remains unaffected by the presence of $M$. As a result, the energy spectrum of
the edge states remains unchanged and can still be described by Eq.(25).

### Bulk-edge correspondence violation in relatively small width ribbons

So far, our analysis has focused on sufficiently wide ribbons where the wave
functions localized at the top and bottom edges, denoted as
$|\psi_{\text{edge}}^{\text{top}}\rangle$ and
$|\psi_{\text{edge}}^{\text{bottom}}\rangle$ respectively, do not overlap.
However, it is an intriguing question to investigate the behavior when the
ribbon width becomes small enough for these localized edge modes to overlap.
For this purpose, we start by considering a simple SSH chain of length $N$
that can be described by the following tight-binding Hamiltonian,

$\displaystyle\hat{H}_{SSH}=\upsilon a_{i}^{\dagger}b_{i}+\omega
b_{i}^{\dagger}a_{i+1}+H.C.$ (33)

The corresponding end mode of this chain can be described as [28]:

$\displaystyle|\psi_{L}\rangle=\sum_{i}a_{i}|a_{i}\rangle,$ (34)

where $a_{i}=x_{SSH}^{i}a_{1}$ and $x_{SSH}=-\frac{\upsilon}{\omega}$ with
$a_{1}=\frac{1}{\sqrt{\mathcal{N}_{SSH}}}$. Here, $\mathcal{N}_{SSH}$ is the
normalization factor associated to the edge states of this SSH chain and the
subscript $L$ refers to the left side of the chain. In a similar manner, the
localized edge state at the right side of the chain is given by,

$\displaystyle|\psi_{R}\rangle=\sum_{i}b_{i}|b_{i}\rangle,$ (35)
$\displaystyle b_{i}=\sum_{i}x_{SSH}^{(N-i)}b_{N}.$ (36)

In the case where the length of the chain is finite, such that the amplitudes
of the left edge mode near the right side of the chain are not close to zero,
it can be demonstrated that there exists an energy gap denoted as
$2\Delta^{\prime}$. The value of this mid gap energy can be expressed as
follows:

$\displaystyle\Delta^{\prime}=\langle\psi_{L}|\hat{H}_{SSH}|\psi_{R}\rangle=\frac{1}{\mathcal{N}_{SSH}}\upsilon\mid
x_{SSH}\mid^{N}.$ (37)

Figure 4: The behavior of the energy gap $\frac{2\Delta}{t_{1}}$ for a narrow
ribbon versus the ribbon width $N$ at $k_{x}=0$. The analytical expression
obtained in Eq. (40) reveals the dependence of the energy gap on the ribbon
width, highlighting the impact of confinement effects on the edge bands of the
system in the QSH phase.

Let us now replace our SSH chain with the narrow ribbon of width $N$. So we
need to consider $|\psi_{\text{edge}}^{\text{top}}\rangle$ and
$|\psi_{\text{edge}}^{\text{bottom}}\rangle$ instead of $|\psi_{L}\rangle$ and
$|\psi_{R}\rangle$, that is:

$\displaystyle|\psi_{top}\rangle=\sum_{i}a_{i}|a_{i}\rangle\;\;\;\text{with}\;\;\;a_{i}=c_{1}\alpha_{1}^{i}+c_{2}\alpha_{2}^{i}+c_{3}\alpha_{3}^{i}$
(38)

and

$\displaystyle|\psi_{bottom}\rangle=\sum_{i}b_{i}|b_{i}\rangle\;\;\;\text{with}\;\;\;b_{i}=c_{1}\alpha_{1}^{N-i}+c_{2}\alpha_{2}^{N-i}+c_{3}\alpha_{3}^{N-i}.$
(39)

The only remain step is to replace $\hat{H}_{SSH}$ in Eq. (37) with $H_{0}$ of
Eq. (10) which results in the following band gap:

$\displaystyle
2\Delta=2\langle\psi_{top}|H_{0}|\psi_{bottom}\rangle\approx\frac{1}{\mathcal{N}}4\tau_{1}c_{1}^{2}\mid
x_{3}\mid^{2N}.$ (40)

A graphical representation of this results is plotted in Fig. 4 which shows
the dependence of the energy gap $2\Delta$ on the width of ribbon $N$ for
$k_{x}=0$ and $\lambda_{SO}=0.2$.

Based on our analysis, it can be concluded that the bulk-edge correspondence,
which states the presence of edge states in a topological system, does not
hold for relatively narrow ribbons. In such cases, where the width of the
ribbon is small enough for the edge modes to overlap, the system may not
exhibit distinct edge states within the bulk energy gap. This deviation from
the bulk-edge correspondence highlights the importance of considering the size
and geometry of the system when studying topological properties which was
studied before numerically [26] and is addressed here analytically.

## Concluding remarks

In conclusion, our study focuses on the investigation of the edge states and
their topological properties in the Kane-Mele model within a ribbon geometry
with armchair boundaries. We have presented a novel analytical approach that
utilizes a mapping procedure to transform the system into an extended SSH
model, specifically a two-leg ladder, in the momentum space. This approach has
allowed us to derive explicit expressions for the edge states, including their
wave functions and energy dispersion. By analyzing the edge states, we have
determined the condition for topological transition, providing insights into
the unique characteristics of the edge states in the quantum spin Hall phase
of the Kane-Mele model.

Furthermore, our study sheds light on the violation of the bulk-edge
correspondence in relatively narrow ribbons. We have elucidated the underlying
reasons for this violation, highlighting the importance of considering the
size of the system in understanding its topological properties. The analytical
properties obtained through our approach contribute to a comprehensive
understanding of the edge states in the Kane-Mele model and provide valuable
insights into the topological properties of such systems.

Overall, our research provides a new analytical framework for studying edge
states in topological systems and contributes to the growing field of
topological condensed matter physics. Further investigations can explore the
implications of our findings in other topological models and the potential
applications of edge states in quantum information processing and device
design [9, 31].

## Acknowledgements

MA acknowledges the associateship award from the International Centre for
Theoretical Physics (ICTP), Trieste, Italy. This support greatly facilitated
the progress of the research and is greatly appreciated. The first author
would also like to express gratitude to the office of graduate studies at the
University of Isfahan for their generous support and provision of research
facilities. Their assistance has been instrumental in conducting this study.

## Author contributions statement

All of the authors have equally contributed to all aspects of the research
presented here.

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