11institutetext: Koji Inoue 22institutetext: Kyoto University, Japan, 22email:
<EMAIL_ADDRESS>33institutetext: Divesh Lala 44institutetext:
Kyoto University, Japan, 44email<EMAIL_ADDRESS>55institutetext: Keiko Ochi 66institutetext: Kyoto University, Japan, 66email:
<EMAIL_ADDRESS>77institutetext: Tatsuya Kawahara
88institutetext: Kyoto University, Japan, 88email<EMAIL_ADDRESS>99institutetext: Gabriel Skantze 1010institutetext: KTH Royal Institute of
Technology, Sweden, 1010email<EMAIL_ADDRESS>

This paper has been accepted for presentation at International Workshop on
Spoken Dialogue Systems Technology 2024 (IWSDS 2024) and represents the
author’s version of the work.

# An Analysis of User Behaviors for Objectively Evaluating Spoken Dialogue
Systems

Koji Inoue  Divesh Lala  Keiko Ochi  Tatsuya Kawahara and Gabriel Skantze

###### Abstract

Establishing evaluation schemes for spoken dialogue systems is important, but
it can also be challenging. While subjective evaluations are commonly used in
user experiments, objective evaluations are necessary for research comparison
and reproducibility. To address this issue, we propose a framework for
indirectly but objectively evaluating systems based on users’ behaviors. In
this paper, to this end, we investigate the relationship between user
behaviors and subjective evaluation scores in social dialogue tasks: attentive
listening, job interview, and first-meeting conversation. The results reveal
that in dialogue tasks where user utterances are primary, such as attentive
listening and job interview, indicators like the number of utterances and
words play a significant role in evaluation. Observing disfluency also can
indicate the effectiveness of formal tasks, such as job interview. On the
other hand, in dialogue tasks with high interactivity, such as first-meeting
conversation, behaviors related to turn-taking, like average switch pause
length, become more important. These findings suggest that selecting
appropriate user behaviors can provide valuable insights for objective
evaluation in each social dialogue task.

##  1 Introduction

In the research and development of spoken dialogue systems (SDSs),
establishing evaluation methods is a significant challenge walker1997paradise
; SIM2015305 ; abd2020technical ; deriu2021survey ; zhang2021automatic ;
ni2023survey . The performance of dialogue understanding and response
generation has seen remarkable progress in recent years, thanks to the
development of large language models (LLMs). The advancement of LLM research
and development has been supported by commonly used evaluation methods in the
field, along with extensive text dialogue datasets. Both objective evaluation
methods (automatic evaluation) and subjective ones (human evaluation) have
been used complementarily. Objective evaluation allows for efficient
scalability of evaluation data by enabling automated evaluation measures such
as BLEU papineni2002bleu  and Distinct li2016distinct . Additionally, using
the same evaluation data as other studies ensures comparability and research
reproducibility. On the other hand, subjective evaluation enables a more
detailed assessment of each generated system response, capturing aspects that
cannot be measured objectively. For instance, evaluating the empathy of
responses currently relies on subjective evaluation by humans. Therefore, in
SDSs, it is ideal to enhance the efficiency and reproducibility of research in
the field by appropriately utilizing both objective and subjective evaluation
methods.

For future research and development of speech dialogue systems, it is
important to establish objective evaluation criteria. For typical task-
oriented dialogues, such as restaurant searches, where the dialogue goal is
clear, objective evaluation criteria like the accuracy of slot-filling tasks
and the success rate of the dialogue have been utilized henderson2014dstc ;
budzianowski2018multiwoz . However, the dialogue tasks for SDSs do not always
have clearly defined goals. With the advent of conversational AI, SDSs are
becoming more realistic in their ability to handle everyday social
conversations. This includes brief exchanges like reception and information
guidance swartout2010ada ; iio2020human , as well as more extended
conversations like counseling devault2014simsensei ; laranjo2018conversational
; scoglio2019use ; rasouli2022potential  and interviews han-
etal-2013-counseling ; johnston-etal-2013-spoken ; yu2019open ; inoue2020job .
In these dialogues, while the purpose of the dialogue can be described
clearly, the goal itself is not always explicit and gradually becomes clear
through the dialogue, fostering mutual understanding and relationships.
Consequently, subjective evaluations have been more commonly employed than
objective evaluations in the past.

Figure 1: Overview of proposed evaluation scheme

In this study, our objective is to establish an objective evaluation method
for SDSs in social scenarios where the goals are not clearly defined. Instead
of analyzing system utterances or relying on subjective evaluations from
users, we aim to indirectly evaluate the system based on users’ spoken-
dialogue behaviors during the dialogue, as depicted in Fig. 1. Users’
behaviors in this context, such as the rate of speaking time or the number of
spoken words, are objectively observable. However, it is not clear which
behaviors in specific dialogue tasks can be used as clues for evaluation. To
address this, we analyze the relationship between users’ behaviors during the
dialogue and their subjective evaluations using dialogue data from several
social dialogue tasks, including attentive listening, job interview, and
first-meeting conversation. The goal is to identify the behaviors that serve
as clues for objective evaluation in different dialogue tasks, which will
enable the appropriate selection of users’ behaviors for objective evaluation
in future research and development. By measuring and comparing these
behaviors, we can achieve the objective evaluation of SDSs across multiple
studies, contributing to the overall expansion of the field, which is the
ultimate goal of this study. For example, when comparing two systems, it is
ideal to have a situation where not only traditional subjective evaluations
are conducted, but also numerical behavior data related to the task is
reported.

This paper positions itself as an initial analysis of the relationship between
user behavior and subjective evaluation toward the aforementioned goal. The
contributions of this paper are twofold:

  * •

Propose an objective evaluation scheme for spoken dialogue systems based on
users’ objective behaviors

  * •

Clarify the users’ behaviors related to subjective evaluation in social
dialogue tasks such as attentive listening, job interview, and first-meeting
conversations

The rest of this paper is organized as follows. The proposed evaluation scheme
is introduced in Section 2. The dialogue data used is explained in Section 3.
Then, the relationship between users’ behaviors and subjective evaluation is
analyzed in Section 4. Finally, this paper concludes in Section 5.

##  2 Proposed Evaluation Scheme

The evaluation method proposed in this study is designed to indirectly assess
SDSs by analyzing users’ behaviors during the dialogue. The focus is on
specific behaviors related to spoken dialogue, including speech, language, and
dialogue features. It is important to note that future research will explore
additional modalities, such as eye-gaze behaviors analyzed through image
processing.

User behaviors analyzed in this study are listed below.

  * •

Utterance time / min.

  * •

Number of utterances (IPU segments) / min.

  * •

Number of utterance words / min.

  * •

Number of unique utterance words / min.

  * •

Number of utterance content words / min.

  * •

Number of unique utterance content words / min.

  * •

Number of backchannels / min.

  * •

Number of fillers / min.

  * •

Number of laughs / min.

  * •

Number of disfluencies / min.

  * •

Average switching pause length

The criterion for dividing an IPU (inter pausal unit) is set as a silent
interval of 200 milliseconds or more. Since the dialogue data used in this
study is in the Japanese language, word segmentation is performed using
MeCab111https://taku910.github.io/mecab. Content words are defined as nouns,
verbs, adjectives, adverbs, and conjunctions. Backchannels are defined as
responsive interjections such as “yes” or “uh-huh”, and emotional
interjections such as “hmm” or “oh”. Fillers are expressions used to bridge
gaps in conversation, such as “um” or “well”, while speech disfluencies are
expressions used for self-correction, such as “spe, specifically”. The
linguistic behaviors described above were calculated based on manually
transcribed data in this study. Switching pause length refers to the duration
of the silent interval when the speaking floor transitions from the system to
the user.

Intuitively, the above-mentioned behaviors vary depending on how natural the
interaction with the system is. This can be understood by comparing human-
system dialogues and human-human dialogues. For example, in human-system
dialogues, users typically speak clearly and with a limited vocabulary. On the
other hand, in human-human dialogues, it is natural to speak fluently and with
a diverse vocabulary. Regarding backchannels, users in human-system dialogues
rarely use them, while in human-human ones, backchannels are used frequently.
Studies have shown that the average switching pause length in human-human
dialogues is almost zero seconds Levinson2015TurnTaking ; skantze2021turn ;
dingemanse2022text , while in human-system dialogues, it often takes around 1
to 3 seconds. Based on these characteristics, evaluating a spoken dialogue
system using the above-mentioned behaviors can be seen as evaluating its
naturalness and similarity to human behavior. In this study, we aim to
identify specific behaviors related to user subjective evaluation in three
different social dialogue tasks.

##  3 Dialogue Data

The dialogue data used in this study is explained below. These data were
recorded using android ERICA inoue-etal-2016-talking , as shown in Figure 2.
It is important to note that in order to introduce variation in the quality of
the dialogue, a mixture of dialogues between human-system and human-human
(referred to as WOZ: Wizard-of-OZ) was used, depending on the dialogue task.
Table 1 summarizes the number of dialogues used in this study based on the
type of setting. In the case of WOZ, there was an operator in a separate room,
and the operator’s voice was played through the android’s speaker. Non-verbal
expressions, such as the android’s gaze and gestures, were controlled by the
operator using a handheld controller. All the dialogues in this study were
conducted in Japanese.

In this study, we utilize dialogue data for three different social dialogue
tasks, each with its own unique characteristics. Table 2 provides a summary of
these characteristics, specifically in terms of implementing a spoken dialogue
system for each task. Given the variations in the initiative of dialogue, as
well as the frequency and clarity of turn-taking, it is anticipated that
users’ behaviors related to subjective evaluation will also differ.

Figure 2: Scene of dialogue data collection (Bottom: an operator in WOZ) Table 1: Number of dialogue in different settings (The names of dialogue tasks are represented as AL, JI, and FMC for Attentive Listening, Job Interview, and First-meeting Conversation, respectively.) Dialogue task | AL | JI | FMC
---|---|---|---
# Autonomous (human-system) | 19 | 86 | 0
# WOZ (human-human) | 50 | 0 | 50
Total | 69 | 86 | 50
Table 2: Characteristics of dialogue tasks targeted in this study Dialogue task | Attentive Listening | Job Interview | First-meeting Conversation
---|---|---|---
System role | Listening | Asking | All
Initiative | User | System | Mixed
Majority of utterances | User | User | Both
Majority of backchannels | System | System | Both
Turn-taking | Few | Explicit | Complicated

###  3.1 Attentive Listening

Attentive Listening involves the task of the listener (system) actively
listening to the user’s talk. The listener responds through various types of
utterances such as backchanneling and elaborating questions. The authors have
developed a real-time spoken dialogue system capable of generating listener
responses inoue2020sigdial . This system was used to record the dialogue data
for this study. We recruited 69 university students as users. They engaged in
an 8-minute dialogue with the attentive listening dialogue system, focusing on
the topic of “difficulties during the COVID-19 pandemic.” The dialogue task
also included human-human dialogue (WOZ). In total, there were 20 interactions
with the autonomous system and 50 WOZ dialogues, resulting in a total of 70
dialogues.

After each dialogue, a subjective evaluation was conducted on the 19 items
created in our previous study inoue2020sigdial . These items include
statements such as “The words uttered by the robot were natural,” “The robot
understood the talk,” and “The robot showed empathy towards me.” The
participants were asked to rate each item on a 7-point scale ranging from 1 to
7. For this study, we used 18 items, excluding one item that showed variation
in interpretation. We calculated the average value for each dialogue (per
participant), which serves as the dependent variable. In other words, the goal
is to predict this average rating value based on the user’s behaviors
mentioned in the previous section. Fig. 3 (a) shows the distribution of the
evaluation scores. Overall, the scores are somewhat high, but it can be
observed that there are also a certain number of participants who gave low
scores.

(a) Attentive Listening

(b) Job Interview

(c) First-meeting Conversation

Figure 3: Distribution of subjective evaluation scores in each dialogue task

###  3.2 Job Interview

Job interview is a dialogue between an interviewer (system) and an applicant
(user), where the applicant answers questions posed by the interviewer. For
this study, we utilized a job interview dialogue system developed by the
authors inoue2020job , and all interactions were conducted using this
autonomous system. In this task, we treated it as a practice job interview and
recruited 43 university students. Each participant engaged in two times of
dialogue with two different systems, which differed in the presence of follow-
up questions from the system. Follow-up questions are inquiries that delve
deeper into the applicant’s previous response, such as extracting keywords and
asking, “Could you please provide more details about (keyword)?” Before
starting the dialogue, participants were asked to select their desired
industry and company, and also prepare answers to several expected questions.

After each dialogue, subjective evaluations were conducted on 19 items created
in our previous study inoue2020job . These items include statements such as “I
was nervous during the interview,” “Thanks to the interview, I was able to
notice my weak points,” and “The interviewer understood my answers.”
Participants were asked to rate each item on a 7-point scale ranging from 1 to
7. The average value was calculated for each dialogue using the 18 items,
excluding the one item that seemed to have variation in interpretation. This
average value was then used as the dependent variable. Fig. 3 (b) shows the
distribution of the evaluation scores. It can be seen that there is slightly
more variation compared to those of Attentive Listening.

###  3.3 First-meeting Conversation

First-meeting conversation is a dialogue that allows both participants to get
to know each other and establish a relationship. For our study, we recruited
50 university and graduate students as users to engage in conversations with a
robot, simulating a first-meeting scenario. Note that the system was
completely operated and controlled by the WOZ setup. Furthermore, the
participants were given a list of commonly discussed topics in first-meeting
conversations prior to the interaction.

After each dialogue, a subjective evaluation was conducted on 18 items. These
items include “I was able to get to know the other person well,” “The
atmosphere of the conversation was pleasant,” “I had a favorable impression of
the other person.” Participants were asked to rate each item on a 7-point
scale, ranging from 1 to 7. The average value of these 18 items was calculated
for each dialogue (participant) and used as the dependent variable. Fig. 3 (c)
shows the distribution of the evaluation scores. It can be observed that the
scores are not as distributed as the other two tasks. The reasons for this
could be that all dialogues were performed by the WOZ operator, resulting in
overall high quality of the conversations. Additionally, subjective
evaluations of this task are often ambiguous, making it difficult to determine
superiority or inferiority.

##  4 Analysis

The relationship between the subjective evaluation scores mentioned in the
previous section and the user behavior mentioned in Section 2 was
investigated. To analyze this relationship, we utilized SHAP (SHapley Additive
exPlanations) lundberg2017shap 222https://pypi.org/project/shap/. SHAP
analysis calculates the contribution level (SHAP value) of each feature in the
output value of a trained model. Specifically, the SHAP value $\phi_{f}$ of a
feature (behavior) $f\in\bm{x}$ is calculated as follows:

$\displaystyle\phi_{f}=\sum_{A\subseteq\bm{x}\setminus\\{f\\}}\frac{|A|!(|\mathbf{x}|-|A|-1)!}{|\bm{x}|!}(y(A\cup\\{f\\})-y(A))$

Note that $A$ represents a subset that excludes the feature $f$, and
$y(\cdot)$ represents the output of the model when using the given set of
features. In other words, it is the average difference between the output
values when using the feature $f$ and when not using it, for all subsets of
the features. The larger the absolute value of this SHAP value, the greater
the interpretation that the corresponding feature has a larger impact on the
trained model. One advantage of this analysis method is that it calculates the
influence of each feature, taking into account the interaction among the
features. The behaviors described so far are not independent and co-occur
during the dialogue, so it is reasonable to analyze them considering the
interaction among them, just like SHAP does.

User behavior |  | Dialogue task
---|---|---
| AL |  | JI |  | FMC
Utterance time / min. |  | 0.098 | (07.69) |  | 0.212 | (15.07) |  | 0.043 | (05.42)
# utterances (IPU segments) / min. |  | 0.199 | (15.63) |  | 0.124 | (08.80) |  | 0.126 | (15.87)
# utterance words / min. |  | 0.162 | (12.72) |  | 0.145 | (10.31) |  | 0.070 | (08.84)
# unique utterance words / min. |  | 0.242 | (18.97) |  | 0.183 | (12.98) |  | 0.068 | (08.58)
# utterance content words / min. |  | 0.048 | (03.74) |  | 0.087 | (06.15) |  | 0.016 | (01.98)
# unique utterance content words / min. |  | 0.066 | (05.16) |  | 0.042 | (02.96) |  | 0.025 | (03.21)
# backchannels / min. |  | 0.050 | (03.91) |  | 0.067 | (04.73) |  | 0.062 | (07.83)
# fillers / min. |  | 0.111 | (08.66) |  | 0.187 | (13.26) |  | 0.047 | (05.99)
# laughs / min. |  | 0.093 | (07.28) |  | 0.042 | (02.98) |  | 0.016 | (02.08)
# disfluencies / min. |  | 0.129 | (10.13) |  | 0.224 | (15.88) |  | 0.192 | (24.23)
Average switching pause length |  | 0.078 | (06.10) |  | 0.097 | (06.87) |  | 0.126 | (15.95)
Table 3: Absolute mean of SHAP value in each behavior and dialogue task (The
numbers in parentheses represent the percentages in each dialogue task. The
bold numbers represent the absolute mean SHAP values exceeding 0.100. The
names of dialogue tasks are represented as AL, JI, and FMC for Attentive
Listening, Job Interview, and First-meeting Conversation, respectively.)

The procedure for applying SHAP is as follows: For each dialogue task, we
trained a regression model using the user’s behavioral features described in
Section 2 as explanatory variables and the subjective evaluation scores
described in Section 3 as the target variable. XGBoost was used as the
regression model in this case. Then, we computed the SHAP value for each
feature using the aforementioned method for the trained model.

(a) Attentive Listening

(b) Job Interview

(c) First-meeting Conversation

Figure 4: Distributions of SHAP value for each behavior and dialogue task

Table 3 presents the average absolute SHAP value for each behavior in the
dialogue tasks. Across all tasks, the number of utterances (IPU segments) and
the number of disfluencies remained consistently high. When comparing
Attentive Listening and Job Interview, similar trends were observed. Both
tasks emphasize the user’s speaking ability, leading to the importance of
metrics like the number of utterances, words, and those unique numbers.
However, the number of disfluencies was given greater significance in the job
interview task. This is because effective questioning is crucial in Job
Interview, so the behavior of disfluencies reflects the evaluation of the
system. For First-meeting Conversation, in addition to the number of
utterances and disfluencies, a higher SHAP value was observed for the average
switch pause length. This type of dialogue involves mixed initiative and
frequent exchanges, making smooth turn-taking vital for system evaluation. It
should also be noted that First-meeting Conversation, being conducted in the
WOZ setting, tends to be more dynamic compared to the autonomous setting so
that these interactive factors would be more indicative. In summary, in
dialogue tasks where user utterances are primary, such as Attentive Listening
and Job Interview, indicators like the number of utterances and words play a
significant role in evaluation. Additionally, in formal situations like Job
Interview, disfluencies are also effective. On the other hand, in dialogue
tasks with high interactivity, such as First-meeting Conversation, the
importance of behaviors related to turn-taking, like average switch pause
length, increases.

To analyze the specific tendencies of SHAP values, we present the distribution
of SHAP values per behavior in each dialogue task in Fig. 4. Each point
represents a dialogue, and the shading indicates the magnitude of the behavior
feature value. The correlation between the number of utterances in Attentive
Listening (Fig. 4 (a)) and the number of disfluencies in Job Interview (Fig. 4
(b)) with the magnitude of SHAP values suggests a positive correlation between
subjective evaluation scores and these behaviors. In contrast, the number of
unique utterance words in Job Interview (Fig. 4 (b)), and the number of
disfluencies and the average switching pause length in First-meeting
Conversation (Fig. 4 (c)) exhibit an inverse relationship with SHAP values.
Therefore, it can be concluded that there is a negative correlation between
subjective evaluation scores and these behaviors.

Finally, we investigated the predictive performance of the regression model
trained using leave-one-out cross-validation on subjective evaluation scores.
The input consisted of all the behavioral features (11 dimensions) mentioned
earlier, and the output was the subjective evaluation score of the system
shown in Figure 3. XGBoost was used for the model. When calculating the mean
absolute error for the test data, we obtained the following results: 0.970 for
Attentive Listening, 0.953 for Job Interview, and 0.683 for First-meeting
Conversation. In other words, for all dialogue tasks, the errors were smaller
than the granularity of the evaluation scores (1.000). This confirms the
feasibility of the proposed evaluation framework and the validity of users’
behaviors selected for this study.

##  5 Conclusion

In this paper, we proposed an objective evaluation method for spoken dialogue
systems used in social dialogue tasks. We examined the relationship between
users’ behaviors and subjective evaluation scores for three different dialogue
tasks: attentive listening, job interview, and first-meeting conversation. Our
findings revealed that the behaviors associated with the subjective
evaluations vary depending on the characteristics of the dialogue task.
Additionally, we evaluated the proposed evaluation method framework through
cross-validation and found that it can accurately predict subjective
evaluation scores from the users’ behaviors, with an absolute error smaller
than the evaluation score’s granularity. Moving forward, our future work aims
to expand the analysis by considering more target behaviors and dialogue
tasks. We also plan to further develop the proposed evaluation method
framework for other research and development purposes.

## Acknowledgement

This work was supported by JST ACT-X (JPMJAX2103), JST Moonshot R&D
(JPMJPS2011), and JSPS KAKENHI (JP19H05691 and JP23K16901). The authors also
express appreciation to Dr. Masato Komuro for his insightful comments on this
research.

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#  Chern Bands with Higher-Order Van Hove Singularities on Topological Moiré
Surface States

Lakshmi Pullasseri  Department of Physics, Emory University, 400 Dowman Drive,
Atlanta, GA 30322, USA  Luiz H. Santos  Department of Physics, Emory
University, 400 Dowman Drive, Atlanta, GA 30322, USA

###### Abstract

In two-dimensional electronic lattices, changes in the topology of the Fermi
surface (Lifshitz transitions) lead to Van Hove singularities characterized by
a divergence in the electronic density of states. Van Hove singularities can
enhance the effect of electronic interactions, providing a platform to explore
novel correlated electronic states. In this work, we investigate the emergence
of topological Chern bands on the surface of three-dimensional topological
insulators, which host higher-order Van Hove singularities that are
characterized by the power-law diverging density of states. These
singularities can arise from the interplay between a time-reversal breaking
Zeeman field induced by proximity to a ferromagnetic insulator and a time-
reversal invariant moiré potential on the surface electrons, created by
quintuple layer misalignment in a family of topological insulators such as
Bi2Se3 and Bi2Te3, which host a single surface Dirac fermion. We establish the
onset of Chern bands near charge neutrality with Chern numbers $C=\pm 1$ that
also possess a manifold of higher-order Van Hove singularities on the moiré
Brillouin zone valleys controlled by the Zeeman and moiré potential energy
scales, unveiling a new platform to realize exotic Lifshitz transitions in
topological bands. Furthermore, we show that the strong peaks in the density
of states in the vicinity of Lifshitz transitions give rise to characteristic
features in the low-temperature intrinsic anomalous Hall conductivity,
yielding a path to probe Van Hove singularities in Chern bands through
anomalous transport measurements.

##  I Introduction

In two-dimensional systems, topological changes of the Fermi surface, known as
Lifshitz transitions Lifshitz  _et al._ (1960), are marked by saddle points in
the energy dispersion, which give rise to Van Hove singularities (VHS) where
the density of states has a logarithmic divergence. Van Hove (1953) Van Hove
has demonstrated that these singularities are generic features of two-
dimensional periodic systems, and the existence of saddle points in the
periodic energy dispersion $\varepsilon({\bm{k}})$ is of topological origin
and can be understood within the framework of Morse theory. Recent progress in
the synthesis of new classes of two-dimensional materials along with the
possibility of engineering electronic band structures have renewed interest in
characterizing electronic states near VHS. In particular, the enhancement of
interaction effects due to the large accumulation of electronic states in the
vicinity of VHSs has been studied as a fruitful setting to characterize Fermi
liquid instabilities in cuprates Schulz (1987); Dzyaloshinski (1987);
Markiewicz (1997), doped graphene Nandkishore  _et al._ (2012); Wang  _et al._
(2012a); Kiesel  _et al._ (2012); González (2008), Hofstadter systems Shaffer
_et al._ (2022); Shaffer and Santos (2023) and moiré graphene superlattices Wu
_et al._ (2021); Isobe  _et al._ (2018); Sherkunov and Betouras (2018); Liu
_et al._ (2018); Kennes  _et al._ (2018); You and Vishwanath (2019); Lin and
Nandkishore (2020); Hsu  _et al._ (2020); Classen  _et al._ (2020);
Chichinadze  _et al._ (2020).

A physically rich scenario occurs when the energy dispersion of electronic
bands supports higher-order saddles, giving rise to higher-order Van Hove
singularities (HOVHS) characterized by stronger power-law divergence of the
density of states.Shtyk  _et al._ (2017); Yuan  _et al._ (2019) While previous
studies of HOVHS have mainly focused on the properties of time-reversal
invariant bands, the interplay between HOVHS and time-reversal broken Chern
bands has been recently highlighted as a path towards new electronic phases.
Castro  _et al._ (2023); Aksoy  _et al._ (2023) In particular, Ref. Castro
_et al._ , 2023 has mapped the landscape of VHS of Haldane Chern bands on the
honeycomb lattice Haldane (1988) and demonstrated that, under inversion
symmetry, the system supports a pair of HOVHS at the two valleys of the
Brillouin zone which, under repulsive interactions, give rise to a rich phase
diagram containing pair density wave superconductivity and a Chern super metal
state. These developments highlight the importance of identifying new
electronic platforms supporting Chern bands with HOVHS.

In this work, we provide a route to realize Chern bands supporting HOVHS on
the surface of three-dimensional topological insulators (TI). Moore (2010);
Hasan and Kane (2010); Qi and Zhang (2011) Our mechanism combines the breaking
of time-reversal symmetry on the TI surface via proximity to a ferromagnetic
insulator and the presence of a moiré pattern. Remarkably, in topological
insulators hosting a single Dirac fermion on the surface, such as Bi2Te3 and
Bi2Se3, a moiré pattern can emerge when the crystals are grown using the
technique of molecular beam epitaxy (MBE). In the case of Bi2Te3Chen  _et al._
(2009) a small in-plane rotation of the top layer (facilitated by Cu dopants
that reduce the interlayer coupling) results in the emergence of a triangular
moiré superlattice of constant $\approx 13nm$. Schouteden  _et al._ (2016)
Moreover, on Bi2Se3 Kuroda  _et al._ (2010) grown using MBE on a substrate,
superstructures with lattice constants $\approx 10nm$ can be constructed from
the direct lattice mismatch of the crystal with substrates like graphene, Song
_et al._ (2010) FeSe,Wang  _et al._ (2012b) Au$(111)$,Jeon  _et al._ (2011)
hBNXu  _et al._ (2015) and In2Se3. Wang  _et al._ (2011) The interplay of
Dirac surface states with a time reversal invariant moiré surface potential
reorganizes the surface states into a sequence of moiré bands hosting
satellite Dirac fermions.Wang  _et al._ (2021); Cano  _et al._ (2021).
Furthermore, near charge neutrality, one of these time-reversal invariant
bands can support a pair of HOVHS with cubic dispersion on each of the valley
points $\pm{\bm{K}}$ of the moiré Brillouin zone.Wang  _et al._ (2021).

We explore the effects of a Zeeman field on these HOVHS states near charge
neutrality. The addition of a Zeeman field significantly modifies the
characters of the HOVHS band near charge neutrality. First, the Zeeman field
gaps out this band resulting in a pair of topological bands with Chern numbers
$\pm 1$ near charge neutrality. Second, the Zeeman field behaves as a control
knob for the character of the Lifshtz transition. Specifically, while the
time-reversal symmetric pair of HOVHS occurs for a fine-tuned value of the
moiré potential $V$,Wang  _et al._ (2021) the Zeeman field gives rise to a
line HOVHS in the $(V,h)$ plane. Remarkably, we uncover simple expressions for
these lines of HOVHS in a Chern band for moiré potentials with C3 and C6
rotation symmetries, and we estimate that this HOVHS manifold could be within
experimental reach for certain TI systems. These results establish a promising
route to achieve higher-order Lifshitz transitions in time-reversal broken TI
surface states.

The existence of exotic Lifshtz transitions in topological Chern bands raises
the prospect of exploring new interplays between Van Hove singularities and
band topology. Along these lines, we establish a connection between Lifshitz
transitions and the quantum geometry of Chern bands through the intrinsic part
of the anomalous Hall conductivity $\sigma^{\textrm{int}}_{xy}$. Karplus and
Luttinger (1954); Jungwirth  _et al._ (2002); Nagaosa  _et al._ (2010) The
intrinsic Hall anomalous Hall contribution arises from non-zero Berry
curvature of the Bloch states, which leads to an anomalous Hall velocityChang
and Niu (1996); Sundaram and Niu (1999); Xiao  _et al._ (2010) when charges
are coupled to an electric field. In the low temperature limit, we show that
$d\sigma^{\textrm{int}}_{xy}/d\mu$ is proportional to the average Berry
curvature on the Fermi surface and the density of states at the Fermi energy.
In particular, this relationship implies a diverging behavior in
$d\sigma^{\textrm{int}}_{xy}/d\mu$ as the chemical potential $\mu$ is tuned
across the Lishshitz transition, a salient feature that could be observed in
low-temperature transport experiments when the anomalous Hall response is
dominated by Berry phase effects.

Figure 1: Schematic illustration of the experimental realization of $3$D TI
surface with a relative twist between the top quintuple layers, as shown in
the inset, in proximity with a ferromagnetic insulator that breaks TRS.

This work is organized as follows. In Sec. II, we present a model for the
realization of HOVHS on moiré surface states of $3$D TIs, relying on the
interplay of a moiré potential and a uniform Zeeman field. We analyze two
moiré hexagonal superlattices and discuss the effects of inversion symmetry in
Sec. II.1, and of inversion symmetry breaking in Sec. II.2 on the properties
of HOHVS. The low energy theory is described by Chern bands with Chern number
$\pm 1$ hosting HOVHS on the moiré Brillouin zone valleys. In Sec. III, we
describe a connection between Lifshitz transitions and the quantum geometric
properties of Chern bands, by showing that the divergence in the density of
states in Van Hove singularities imprints on the intrinsic quantum anomalous
Hall response, and we discuss this effect for the HOVHS Chern bands on the TI
surface, as well as other Chern bands supporting logarithmic VHSs. We conclude
with a discussion and outlook in Sec. IV.

##  II Higher Order Van Hove Singularities

We consider a description of the TI surface states hosting a single Dirac
fermion, which is under the effect of a time-reversal invariant periodic
superlattice potential $V({\bm{r}})=V({\bm{r}}+{\bm{R}})$ (where ${\bm{R}}$ is
a translation vector), and a time-reversal breaking uniform Zeeman field
$h_{z}$. The single-particle Hamiltonian is

$\hat{H}=\int
d^{2}{\bm{r}}\,\hat{\psi}^{\dagger}({\bm{r}})H\,\hat{\psi}({\bm{r}})$ (1a)
where
$\hat{\psi}^{T}({\bm{r}})=(\psi_{\uparrow}({\bm{r}}),\psi_{\downarrow}({\bm{r}}))$
is the two-component spinor with $\psi_{\sigma=\uparrow,\downarrow}({\bm{r}})$
the electron annihilation operator, and
$H=v_{F}\,\hat{z}\cdot(-i\,{\bm{\nabla}}\times{\bm{\sigma}})+V(\bm{r})\sigma_{0}+h_{z}\sigma_{z}\,,$
(1b)

where $v_{F}$ is the Fermi velocity,
${\bm{\sigma}}=\big{(}\sigma_{x},\sigma_{y},\sigma_{z}\big{)}$ are the Pauli
matrices, $\sigma_{0}$ is the identity matrix, and $\hbar=1$. The low energy
theory depends upon three energy scales: the effective bandwidth $v_{F}/a$
where $a$ is the superlattice constant, the strength of the moiré potential
$V$, and the Zeeman energy scale $h_{z}$. Henceforth, we work in rescaled
units where the low-energy physics is determined by the dimensionless
parameters $V\,a/v_{F}$ and $h_{z}\,a/v_{F}$.

In TIs such as Bi2Te3 and Bi2Se3, the moiré pattern can emerge when the
crystals are grown using MBE. For Bi2Te3 with a Fermi velocity of
approximately $0.3$ eV nm, Chen  _et al._ (2009) a small in-plane rotation of
the top layer, which is facilitated by Cu dopants that reduce the interlayer
coupling, results in the emergence of moiré triangular superlattice with
lattice constant $a\approx 13$ nm. Schouteden  _et al._ (2016) In Bi2Se3 with
a Fermi velocity of $0.4$ eV nm Kuroda  _et al._ (2010) grown using MBE on a
substrate, superlattices with periodicity $a$ ranging from $2$ to $7$ nm can
be constructed from the direct lattice mismatch of the crystal with the
substrates like graphene, Song  _et al._ (2010) FeSe,Wang  _et al._ (2012b)
Au(111),Jeon  _et al._ (2011) hBNXu  _et al._ (2015) and In2Se3. Wang  _et
al._ (2011)

Since the single particle Hamiltonian Eq. (1b) is invariant under translations
by a lattice vector ${\bm{R}}$ of the moiré superlattice, the eigenvalue
equation for the Bloch states reads

$H\Psi_{n,{\bm{k}}}=E_{n,{\bm{k}}}\Psi_{n,{\bm{k}}}\,,$ (2)

where the two-component spinor
$\Psi_{n,{\bm{k}}}({\bm{r}})=e^{i{\bm{k}}\cdot{\bm{r}}}U_{n,{\bm{k}}}({\bm{r}})$
with $U_{n,{\bm{k}}}({\bm{r}}+{\bm{R}})=U_{n,{\bm{k}}}({\bm{r}})$. From Eq.(2)
it follows that

$H_{{\bm{k}}}U_{n,{\bm{k}}}({\bm{r}})=E_{n,{\bm{k}}}U_{n,{\bm{k}}}({\bm{r}})\,,$
(3a) where
${H_{{\bm{k}}}=v_{F}\big{[}\hat{z}\cdot(-i\bm{\nabla}+{\bm{k}})\times\bm{\sigma}\big{]}+V({\bm{r}})\sigma_{0}+h_{z}\sigma_{z}}\,.$
(3b)

We numerically diagonalize Eq.(3) for a total number of $242$ bands. One of
our main findings is that the reconstructed Dirac spectrum obtained from
Eq.(3) contains a Chern band near charge neutrality supporting cubic HOVHS
with power law diverging density of states
$\rho(\varepsilon)\sim|\varepsilon|^{-1/3}$, where the energy scale of the
HOVHS is set to zero.

The appearance of HOVHS in this topological band is controlled by $h_{z}$ and
$V$. The energy scale $h_{z}$ corresponds to the Zeeman gap opened in the
Dirac spectrum, which can be experimentally realized in heterostructures where
the $3$D TI surface is coupled by the proximity effect to a ferromagnetic
insulator (FMI). Experimental realizations of the TI-FMI heterostructure of
the quintuple layered $3$D TIs such as Bi2Se3 and Bi2Te3, and FMIs with
comparable lattice constants, such as EuS,Wei  _et al._ (2013); Jiang  _et
al._ (2016) MnSe,Matetskiy  _et al._ (2015); Hirahara  _et al._ (2017)
MnTe,Rienks  _et al._ (2019) and Y3Fe5O12Lang  _et al._ (2014) present
evidence for broken TRS on the TI surface. Leveraging on these experimental
conditions, we propose a path to realizing moiré Chern bands on the TI surface
by coupling $3$D TIs where there is an in-plane rotation of the top quintuple
layers, with an FMI of comparable lattice constant, as shown in Fig. 1.

The point group symmetries of the moiré potential play an important role in
the structure of the higher-order Lifshitz transitions. To address the role of
point group symmetries, we consider two classes of superlattice potentials
with C6 and C3 symmetry. In the C6 symmetric case, the HOVHS appears as a pair
at the valley $\pm{\bm{K}}$, as shown in Section II.1. Breaking the C6
symmetry down to C3 results in a single HOVHS located either at ${\bm{K}}$ or
at $-{\bm{K}}$ valley, as discussed in Section II.2. In what follows, we study
each of these cases separately.

###  II.1 $C_{6}$ symmetric moiré potential

Figure 2:  (a) Contour plot of the $C_{6}$ symmetric potential
$V_{6}(\bm{r})$, given in Eq. (4), with superlattice constant $a$. (b) Band
structure of the model given in Eq. (1b) with the $C_{6}$ periodic potential
for $(h_{z},V)=(0.5,1.38)$ in units of $(v_{F}/a)$ to the left. The mini
Brillouin zone is shown in the center. The isolated band shown in red carries
the Chern number $C=1$, and it supports HOVHS at the $\pm{\bm{K}}$ points, as
indicated by the peak in the DOS shown on the right. The green dashed line
denotes the energy at the $\pm{\bm{K}}$ points, where the DOS diverges.

We consider Dirac electrons under the effect of the $C_{6}$ periodic moiré
potential as shown in Fig. 2a

$V_{6}(\bm{r})=2V\sum_{j=1}^{3}\cos(\bm{G}_{j}\cdot\bm{r})\,,$ (4)

where $\bm{G}_{j}=\frac{4\pi}{\sqrt{3}a}\Big{[}-\sin\big(\frac{2\pi
j}{3}\big{missing}),\cos\big(\frac{2\pi j}{3}\big{missing})\Big{]}$ are the
reciprocal lattice vectors, and $V$ is the strength of the lattice potential.
This potential has invariance under three-fold rotations and inversion
symmetry since $V({\bm{r}})=V(-{\bm{r}})$. Without a Zeeman field, the
potential given in Eq.(4) yields a gapless spectrum that supports a band with
a pair of HOVHS at the valleys $\pm{\bm{K}}$ of the moiré Brillouin zone when
the strength of the potential $V_{0}=1.36v_{F}/a$ Wang  _et al._ (2021).
Despite this higher-order Lifshtiz transition, these bands remain
topologically trivial due to time-reversal symmetry. At $V=V_{0}$, the higher-
order saddle point at each valley carries a topological index of $-2$
corresponding to the winding of the vector field
${\bm{\nabla}}_{{\bm{k}}}\varepsilon({\bm{k}})$ around the higher-order saddle
point.

To achieve a topological band supporting HOVHS, time-reversal symmetry is
broken with the Zeeman field induced by the proximity effect. On turning the
Zeeman field on for $V=V_{0}$, the higher-order saddle point splits into $4$
critical points, of which $3$ are ordinary saddle points with a topological
index of $-1$ each and the remaining one is a local extremum with a
topological index of $1$, such that the sum of the topological indices
$3\times(-1)+1=2$ remains conserved. At the same time, the lowest pair of
conduction bands acquire an energy gap and Chern numbers of $\pm 1$ Liu  _et
al._ (2022), as shown in Fig.2(b).

Remarkably, we notice that these split critical points (i.e., three
conventional VHS and one extremum) can be merged together again at
$\pm{\bm{K}}$ to generate a HOVHS in the Chern band marked in red in Fig.2(b)
by adjusting either $V$ or $h_{z}$. We uncover a line in $(h_{z},V)$ parameter
space, shown in Fig. 3(a), where the $C=1$ band supports pairs of HOVHS. This
line in $(h_{z},V)$ parameter space obeys the relation

$V_{6}(h_{z})=V_{6}^{(0)}+\Gamma\,h_{z}^{2}\,,$ (5)

where, $V_{6}^{(0)}=V_{6}(h_{z}=0)=1.36v_{F}/a$, and $\Gamma\approx 0.05$ is a
fitting parameter. The quadratic scaling between $h_{z}$ and $V$ is observed
for $h_{z}$ values ranging from $0$ to $\approx 2v_{F}/a$. As a result, for
moiré surface states with lattice constant $a\approx 10$ nm and Fermi velocity
$v_{F}\approx 300$ meV. nm, the parameters $h_{z}$ and $V$ that can be
employed to tune the HOVHS follow the scaling relation given in Eq. (5) for
$h_{z}$ as large as $60$ meV

To understand the relation given by Eq. (5), we study the energy dispersion
around the valley ${\bm{K}}$,

$\varepsilon({\bm{p}})\equiv E(\bm{p}+\bm{K})-E(\bm{K})=\alpha
p^{2}+\beta(p_{x}^{3}-3p_{x}p_{y}^{2})\,,$ (6)

where $\bm{p}=(p_{x},p_{y})$ is the momentum in the vicinity of the valley.
The momentum dependence of the dispersion Eq. (6), expanded up to the third
order, is dictated by the point group symmetries of the hexagonal lattice.
From this, we obtain the critical points
${\bm{\nabla}}_{{\bm{p}}}\varepsilon({\bm{p}})=0$ and characterize their
behavior by computing the Hessian
$\mathcal{H}=\mathrm{det}\Big{(}\partial_{p_{i}}\partial_{p_{j}}\varepsilon({\bm{p}})\Big{)}$.

Figure 3: (a) Parameter space plot showing the values of the Zeeman mass
($h_{z}$) and the lattice potential strength ($V$) that corresponds to HOVHS
located at the $\pm{\bm{K}}$ points of the BZ. The $(h_{z},V)$ pairs follow a
scaling relation given by Eq.(5). The contour plots of the Fermi surfaces
corresponding to the regions- $V>V_{6}(h_{z})$, $V=V_{6}(h_{z})$, and
$V<V_{6}(h_{z})$ are shown in the inset, where the dotted black lines indicate
the boundaries of the FBZ and the black lines show the Fermi surfaces passing
through VHS. (b) $\alpha(h_{z},V)$ obtained using Eq. (7) plotted in the
$h_{z}-V_{6}$ parameter space. The thick black curve indicates the
$\alpha(h_{z},V)=0$ line which, as expected, coincides with the $h_{z}-V$
curve shown in (a) that corresponds to the emergence of HOVHS at $\pm{\bm{K}}$
points.

While the gradient of the dispersion vanishes for
$\bm{p}\in\Big{\\{}(0,0),\big{(}\frac{-2\alpha}{3\beta},0\big{)},\big{(}\frac{\alpha}{3\beta},\pm\frac{\alpha}{\sqrt{3}\beta}\big{)}\Big{\\}}$,
the corresponding Hessian $\mathcal{H}=4(\alpha^{2}-9\beta^{2}p^{2})$
evaluated at each of the four aforementioned critical points vanishes only
when the coefficient $\alpha=0$, which implies the vanishing of the term
quadratic in momentum. Thus, the scaling relation Eq. (5) characterizes the
points in the $(h_{z},V)$ parameter space for which the quadratic in momentum
coefficient $\alpha(h_{z},V)=0$, so that the energy dispersion around
${\bm{K}}$ can be described by a third-order polynomial
$\varepsilon({\bm{p}})\approx\beta(p_{x}^{3}-3p_{x}p_{y}^{2})$, corresponding
to a HOVHS with diverging density of states
$\rho(\varepsilon)\sim\frac{1}{|\varepsilon|^{1/3}}$. Shtyk  _et al._ (2017)

Furthermore, in the range of $(h_{z},V)$ values where we observe the quadratic
scaling given in Eq. (5), the parameter $\alpha(h_{z},V)$ can be well
approximated by the empirical form,

$\alpha(h_{z},V)=\alpha_{0}\ln[\frac{V}{V_{6}(h_{z})}\Big{]}\,,$ (7)

where $\alpha_{0}\approx 0.4$, and $V_{6}(h_{z})$ is given by Eq. (5). The
manifold of $(h_{z},V)$ values for which $\alpha(h_{z},V)$ vanishes is denoted
by the black curve in Fig. 3(b). When the strengths of the potential and of
the Zeeman energy deviate from condition (5), the HOVHS splits into three
ordinary saddle point VHS for
$\bm{p}\in\Big{\\{}\big{(}\frac{-2\alpha}{3\beta},0\big{)},\big{(}\frac{\alpha}{3\beta},\pm\frac{\alpha}{\sqrt{3}\beta}\big{)}\Big{\\}}$
and a local extremum at $\bm{p}=(0,0)$. Under small deviations $\delta V$ of
the moiré potential and the Zeeman energy from $(h_{z}=0,V=V_{6})$, a
perturbative expansion of the coefficient $\alpha(h_{z},\delta V)$ to lowest
order yields $\alpha(h_{z},V)\approx c_{1}\,\delta V+c_{2}\,h^{2}_{z}$, where
a liner term in $h_{z}$ vanishes due to inversion symmetry. Then the condition
$\alpha(h_{z},\delta V)=0$ yields Eq. (5).

Apart from the high concentration of density of states, the bands supporting
HOVHS display another interesting feature. As shown in Fig. 4, the real space
wave functions for the energy states at the HOVHS and projected on the valleys
$\pm{\bm{K}}$ are particularly polarized on two sublattices that form a
honeycomb lattice localized between the maxima of the moiré potential. This
sublattice polarization behavior is observed along the HOVHS line given by Eq.
(5), and highlighted in Fig. 4 for $(h_{z},V)=(0.5,1.38)$. We note that this
sublattice polarization in the vicinity of HOVHS on the TI surface states is a
feature observed in other two-dimensional systems tight-binding systems such
as in Haldane Chern insulators tuned to higher-order Lifshitz transitions
Castro  _et al._ (2023) and kagomé bands. Kiesel and Thomale (2012); Wu  _et
al._ (2023)

Figure 4:  Real space wavefunction of the up (left) and down (right) spin
electrons with momentum ${\bm{K}}$ on the HOVHS band of the $C_{6}$ symmetric
model with $(h_{z},V)=(0.5,1.38)$ in units of $(v_{F}/a)$. The difference in
the maximum amplitude of the up and down spins can be attributed to the broken
TRS. The maxima of the lattice potential are denoted with black dots.

###  II.2 $C_{3}$ symmetric moiré potential

Figure 5:  (a) Contour plot of the $C_{3}$ symmetric potential
$V_{3}(\bm{r})$, given in Eq. (8), with superlattice constant $a$. (b) Band
structure of the model given in Eq. (1b) with the $C_{3}$ periodic potential
with $(h_{z},V)=(0.5,0.83)$ in units of $(v_{F}/a)$, to the left and the
corresponding DOS to the right. The sharp peak in the DOS at the ${\bm{K}}$
point indicates the presence of HOVHS while the shorter peak right above it in
energy points towards the conventional VHS around the ${\bm{K^{\prime}}}$
point. Here, the lowest conduction band (shown in red) that carries a Chern
number of $-1$, hosts HOVHS located at the ${\bm{K}}$ points.

We investigate the effects of breaking inversion symmetry by adding a
contribution to the $C_{6}$ symmetric lattice potential given in Eq. (4) that
yields a potential with $C_{3}$ symmetry

$V_{3}(\bm{r})=2V\sum_{j=1}^{3}\Big{[}\cos(\bm{G}_{j}\cdot\bm{r})-\cos(\bm{G}_{j}\cdot\bm{r}+\phi)\Big{]}\,,$
(8)

where $\phi=\frac{2\pi}{3}$, as depicted in Fig. 5a. In the absence of the
Zeeman field, this model supports a gapless spectrum of moiré bands, wherein
we uncover one band supporting a pair of HOVHS a the moiré Brillouin zone
valleys when the strength of the moiré potential
$V_{3}^{(0)}=V_{3}(h_{z}=0)=0.802(v_{F}/a)$. The existence of a pair of valley
HOVHS follows from TRS degenerate states at $\pm{\bm{K}}$ despite breaking of
inversion symmetry by the potential Eq. (8).

On turning the Zeeman field on with $V=V_{3}^{(0)}$, the HOVHS splits into 3
conventional VHS located around each of the ${\bm{K}}$ and $-{\bm{K}}$ points
of the lowest conduction band which carries a Chern number of $-1$, as shown
in Fig. 5b. By adjusting the lattice potential strength or the Zeeman field,
the conventional VHS can merge back into a HOVHS. However, due to the breaking
of inversion symmetry, this HOVHS is located in either one of the valleys.
This situation is analogous to the effect of an inversion-breaking sublattice
potential on the Haldane Chern insulator model that can lead to a single HOVHS
on one of the valleys. Aksoy  _et al._ (2023)

Similarly to the approach of Sec. II.1, we characterize the $(h_{z},V)$
parameter space for which the $C=-1$ band supports HOVHS in one of the
valleys. For the ${\bm{K}}$ valley HOVHS, we uncover the relation between
moiré potential and Zeeman field,

$V_{3}(h_{z})=V_{3}^{(0)}+\Gamma_{1}h_{z}+\Gamma_{2}h_{z}^{2}+\Gamma_{3}h_{z}^{3}\,,$
(9)

in terms of numerically fitted coefficients
$(\Gamma_{1},\Gamma_{2},\Gamma_{3})\approx(0.033,0.039,0.007)$, and $h_{z}$
and $V$are measured in units of $(v_{F}/a)$. We note that unlike in the case
discussed in Sec. II.1, the breaking of inversion symmetry allows for terms
with odd powers in $h_{z}$. Furthermore, while $(h_{z},V_{3}(h_{z}))$
corresponds to a higher-order saddle point at the ${\bm{K}}$ point,
$(h_{z},V_{3}(-h_{z}))$ corresponds to the one at the $-{\bm{K}}$ point. In
Fig. 6(a), we plot these curves and the character of the Fermi surface
contours at and in the vicinity of the higher-order Lifshitz transitions.

Figure 6:  (a)Parameter space plot showing the values of the Zeeman mass
($h_{z}$) and the lattice potential strength ($V$) that corresponds to HOVHS
located at the ${\bm{K}}$ (blue) and the ${\bm{K^{\prime}}}$ (green) points of
the BZ. The $(h_{z},V)$ pairs that support HOVHS located at K ($K^{\prime}$)
follow a scaling relation $V_{3}(h_{z})$ ($V_{3}(-h_{z})$) given by Eq. (9).
The contour plots of the Fermi surfaces corresponding to the regions-
$V<V_{3}(h_{z})$, $V=V_{3}(h_{z}\neq 0)$, and $V>V_{3}(h_{z})$ are shown
above, and those corresponding to the regions- $V<V_{3}(-h_{z})$,
$V=V_{3}(-h_{z}\neq 0)$, and $V>V_{3}(-h_{z})$ are shown below. When $h_{z}=0$
and $V=V_{3}^{(0)}$, we find HOVHS located at both ${\bm{K}}$ and
${\bm{K^{\prime}}}$ points, as in the case of the $C_{6}$ symmetric lattice
potential. Here, the dotted black lines indicate the boundaries of the first
moiré Brillouin zone and the black lines show the Fermi surfaces passing
through VHS. (b)$\alpha(h_{z},V)$ obtained numerically from the expression
given in Eq. (10) (top) and from the Taylor expansion of the energy dispersion
around the ${\bm{K}}$ points (bottom). The thick black curve indicates the
$\alpha(h_{z},V)=0$ line which, as expected, coincides with the $h_{z}-V$
curve corresponding to the emergence of HOVHS at the ${\bm{K}}$ point shown in
(a).

Eq. (9), which describes the condition for the coefficient $\alpha(h_{z},V)$
in Eq. (6) to vanish, can be well approximated by

$\alpha(h_{z},V)=\alpha_{0}\ln[\frac{V}{V_{3}(h_{z})}\Big{]}\,,$ (10)

where $\alpha_{0}\approx 0.4$ for $h_{z}a/v_{F}<2$. Similarly, the Hessian
vanishes at the $-{\bm{K}}$ points when
$\alpha(h_{z},V)=\alpha_{0}\ln(V/V_{3}(-h_{z}))$ with $\alpha_{0}\approx 0.4$.
Thus, we have established a mechanism to create and access a larger landscape
of HOVHS topological bands with Chern number $C=\pm 1$, which occur through
the interplay of a moiré potential and a uniform Zeeman field coupled to the
surface states a 3D topological insulator. In the next section, we discuss an
important feature in anomalous Hall response implied by the existence of Van
Hove singularities in Chern bands and apply it to the case of conventional and
higher-order VHS.

##  III Intrinsic anomalous Hall conductivity near Lifshitz transitions

In Chern bands, the presence of non-zero Berry curvature of the Bloch states
leads to an anomalous Hall velocityChang and Niu (1996); Sundaram and Niu
(1999); Xiao  _et al._ (2010) when charges couples to an electric field,
giving rise to the intrinsic anomalous Hall effect.Karplus and Luttinger
(1954); Jungwirth  _et al._ (2002); Nagaosa  _et al._ (2010) While for a
filled Chern band insulator, this leads to a quantized Hall conductivity in
units of $e^{2}/h$, in a partially filled Chern band, the intrinsic anomalous
Hall conductivity $\sigma^{\textrm{int}}_{xy}$ can be continuously tuned by
the electronic filling of the Chern band, and it probes the Berry phase
contribution of the occupied electronic states. In particular, when the Fermi
energy crosses a Lifshitz transition in a Chern band, the large peak in the
density of states entailed by the change in the topology of the Fermi surface
suggests that the anomalous Hall response may correspondingly manifest some
distinct property. Furthermore, this distinct behavior should be a generic
feature of Chern bands, provided the sharp features in the density of states
are not significantly rounded by disorder and thermal broadening. Thus, we
hereafter discuss the character of the intrinsic anomalous Hall response of
Chern bands at low temperatures and in cases where the anomalous Hall effect
is dominated by the intrinsic contributionNagaosa  _et al._ (2010), where we
uncover a characteristic feature relating the anomalous Hall conductivity and
the density of states. Specifically, in the $T\rightarrow 0$ limit, we show
that $d\,\sigma^{\textrm{int}}_{xy}/d\,\mu$ diverges when the Fermi energy
$\mu$ crosses the scale of logarithmic and power-law Van Hove singularities.
Also, in what follows, we assume that the system remains a Fermi liquid as the
Fermi energy crosses a Van Hove singularity.

To establish these connections, we consider an isolated Chern band described
by Bloch states $\ket{u_{{\bm{k}}}}$ with energy dispersion
$\varepsilon({\bm{k}})$ and non-vanishing Berry curvature
$\Omega({\bm{k}})=\nabla_{{\bm{k}}}\crossproduct{\bm{A}}_{{\bm{k}}}$, where
the Berry connection
${\bm{A}}_{{\bm{k}}}=i\,\bra{u_{{\bm{k}}}}\nabla_{{\bm{k}}}\,\ket{u_{{\bm{k}}}}$.
This system is characterized by the intrinsic anomalous Hall conductivity
Karplus and Luttinger (1954)

$\sigma^{\textrm{int}}_{xy}(\mu;T)=\frac{e^{2}}{h}\,\frac{1}{2\pi}\,\int\,d^{2}{\bm{k}}\,\Omega({\bm{k}})f_{\mu;T}(\varepsilon({\bm{k}}))\,,$
(11)

where the integral extends over the first Brillouin zone, $e$ is the charge of
the electron, $h$ is Planck’s constant and
$f_{\mu;T}(\varepsilon({\bm{k}}))=\Big{(}{e^{\frac{\varepsilon({\bm{k}})-\mu}{k_{B}T}}+1}\Big{)}^{-1}$
is the Fermi distribution at temperature $T$ and chemical potential $\mu$.

In the $T\rightarrow 0$ limit, Eq. (11) becomes

$\sigma_{xy}^{\textrm{int}}(\mu;0)=\frac{e^{2}}{h}\,\frac{1}{2\pi}\,\int\,d^{2}{\bm{k}}\,\Omega({\bm{k}})\Theta(\mu-\varepsilon({\bm{k}}))\,,$
(12)

where $\Theta(x)$ is the Heaviside step function signaling a sharp separation
between occupied and empty electronic states. This contribution varies
continually with the Fermi energy $\mu$ when the Fermi energy lies between the
minimum and maximum of the Chern band.

An insight can be obtained by considering differential anomalous Hall
conductivity, which measures the slope of $\sigma_{xy}^{\textrm{int}}$ as a
function of Fermi energy,

$\frac{d\sigma^{\textrm{int}}_{xy}(\mu;0)}{d\mu}=\frac{e^{2}}{h}\,\frac{1}{2\pi}\,\int\,d^{2}{\bm{k}}\,\Omega({\bm{k}})\delta(\mu-\varepsilon({\bm{k}}))\,.$
(13)

Eq. (13) shows that at $T=0$, the differential anomalous Hall conductivity
gets its contribution from states at the one-dimensional Fermi surface
manifold $\varepsilon({\bm{k}})=\mu$.

Figure 7: (a) Intrinsic anomalous Hall conductivity
($\sigma^{\textrm{int}}_{xy}$) and (b) differential anomalous Hall response
($\frac{d\sigma^{\textrm{int}}_{xy}}{d\mu}$), at zero temperature plotted as a
function of the Fermi energy $\mu$ for the Chern band supporting HOVHS at
$(h_{z},V)=(0.5,1.38)$ in units of $(v_{F}/a)$. The dashed red line denotes
the energy at which HOVHS occurs, $\mu_{*}$. The differential anomalous Hall
response exhibits a power law divergence around $\mu_{*}$, as given in Eq.
(17) (b), with $\nu=0.39\approx 1/3$, and $\kappa_{+}=\kappa_{-}=0.43$. (c)
Intrinsic anomalous Hall conductivity ($\sigma^{\textrm{int}}_{xy}$) and (d)
differential anomalous Hall response
($\frac{d\sigma^{\textrm{int}}_{xy}}{d\mu}$), at zero temperature plotted as a
function of the Fermi energy $\mu$ for the lower energy band of the Chern
insulator model defined on a square lattice of lattice constant $a$,
$H=(v_{F}/a)[\sin k_{x}\sigma_{x}+\sin k_{y}\sigma_{y}+(m-\cos k_{x}-\cos
k_{y})\sigma_{z}]$ at $m=1.5$ where it supports VHS at $\mu_{*}=-1.5(v_{F}/a)$
(denoted by the red dashed line) and carries a Chern number of $1$. The
differential anomalous Hall response exhibits a log divergence around
$\mu_{*}$, as given in Eq. (17) (a), with $\Lambda=1.35(v_{F}/a)=0.45\times$
bandwidth and $\rho_{0}=\frac{0.58}{v_{F}/a}$. Note that in (a) and (c), as
the Fermi reaches the top of the band, the total Hall conductivity approaches
$1$, the Chern number of the corresponding band.

Upon normalizing by the density of states at the Fermi energy,

$\rho(\mu)=\int\,d^{2}{\bm{k}}\,\delta(\mu-\varepsilon({\bm{k}}))\,,$ (14)

Eq. (13) reads

$\frac{d\sigma^{\textrm{int}}_{xy}(\mu;0)}{d\mu}=\frac{e^{2}}{2\pi
h}\,\langle\Omega\rangle_{FS}\,\rho(\mu)\,,$ (15)

where

$\langle\Omega\rangle(\mu)\equiv\frac{\int\,d^{2}{\bm{k}}\,\Omega({\bm{k}})\delta(\mu-\varepsilon({\bm{k}}))}{\int\,d^{2}{\bm{k}}\,\delta(\mu-\varepsilon({\bm{k}}))}$
(16)

defines the average of the Berry curvature on the Fermi surface.

Eq. (15) provides an insightful connection between the average Berry curvature
(which is the imaginary part of the quantum metric tensor), the density of
states, and the anomalous Hall transport, which can be in principle explored
to read out the Berry curvature by scanning the Fermi energy of the band.
Moreover, let us consider the behavior of Eq. (15) in the vicinity of a
Lifshitz transition occurring at energy $\mu^{*}$. In this case, if
$\langle\Omega\rangle(\mu^{*})\neq 0$ (which we expect to occur typically in a
Chern band), the differential anomalous Hall conductivity is dominated by the
Van Hove singularities in the density of states, resulting in the following
asymptotic forms for the conventional and higher-order Lifshitz transitions

$\begin{split}\frac{d\sigma_{xy}(\mu;0)}{d\mu}&\,\approx\frac{e^{2}}{2\pi
h}\,\times\langle\Omega\rangle(\mu)\,\times\underbrace{\rho_{0}\,\log(\Lambda/|\mu-\mu_{*}|)}_{\textrm{Conventional
Lifshitz transition}}\,,\end{split}$ (17a)
$\begin{split}\frac{d\sigma_{xy}(\mu;0)}{d\mu}&\,\approx\frac{e^{2}}{2\pi
h}\,\times\langle\Omega\rangle(\mu)\,\times\,\,\underbrace{\frac{{\Theta(\mu-\mu_{*})\kappa_{+}+\Theta(\mu_{*}-\mu)\kappa_{-}}}{|\mu-\mu_{*}|^{\nu}}}_{\textrm{Higher-
order Lifshitz transition}}\,,\end{split}$ (17b)

where $\nu$ is a positive exponent and $\kappa_{\pm}$ coefficients for
$\mu\geq\mu_{*}$ ($\mu\leq\mu_{*}$). Eq. (17) establishes a general connection
between differential anomalous Hall response and Lifshitz transitions in Chern
bands.

In Figure 7, we plot $d\sigma^{\textrm{int}}_{xy}/d\mu$ and
$\sigma^{\textrm{int}}(\mu)$ as a function of the Fermi energy $\mu$ for two
representative cases, namely, the Chern bands holding HOVHS discussed in
Sec.II and a two-band model Chern insulator model supporting logarithmic VHS
Qi  _et al._ (2006, 2008). In the top panels of Figure 7, we observe that the
cubic power-law divergence in the DOS for the topological insulator moiré
surface Chern bands gives rise to a pronounced peak in
$d\sigma^{\textrm{int}}_{xy}/d\mu$ (Figure 7b), which then reflects a
characteristic slope in $\sigma^{\textrm{int}}_{xy}$ (Figure 7a) as the system
undergoes a Lifshitz transition at $\mu=\mu^{*}$, which is indicated by the
red dotted lines. In the bottom panels of Fig. 7, we perform a similar
analysis for a Chern band with conventional logarithmic VHS. In both cases,
the characteristic behavior by Eq. (17) is confirmed despite some numerical
rounding introduced by the finite-size grid.

Figure 8: Differential anomalous Hall conductivity
$\frac{d\sigma^{\textrm{int}}_{xy}}{d\mu}$ plotted against the Fermi energy
$\mu$ for the HOVHS band with $(h_{z},V)=(0.5,1.38)$ in units of $(v_{F}/a)$,
for different values of temperature ($T$) that are labelled on the plot. The
temperature $T$ is scaled with $T_{0}=(v_{f}/a)/k_{B}$ where $k_{B}$ is the
Boltzmann constant and $v_{F}/a\approx 30$ meV. Schouteden  _et al._ (2016);
Chen  _et al._ (2009) The yellow dashed line denotes the energy at which the
HOVHS occurs, $\mu_{*}$. The peak in
$\frac{d\sigma^{\textrm{int}}_{xy}}{d\mu}$ associated with the HOVHS gets
progressively stronger as the temperature decreases.

To account for the effects of thermal broadening, we obtain from Eq. (11) the
temperature dependence of the differential anomalous Hall conductivity

$\begin{split}&\,\frac{d\,\sigma^{\textrm{int}}_{xy}}{d\,\mu}(\mu;T)=\\\
&\,\frac{e^{2}}{2\pi\,h}\,\int\,d^{2}\,{\bm{k}}\,\Omega({\bm{k}})\,\frac{1}{k_{B}\,T}\Big{[}2\,\cosh{\frac{(\varepsilon({\bm{k}})-\mu)}{2\,k_{B}\,T}}\Big{]}^{-2}\,,\end{split}$
(18)

where the last term in the integral accounts for the temperature dependence of
the Fermi distribution.

In Figure 8 we plot the temperature dependence of the differential anomalous
Hall conductivity Eq. (18) for the TI surface state discussed in Sec. II.1 for
the Chern band supporting a pair of HOVHS. As expected, for temperatures
compared with or greater than the bandwidth $T\gtrsim T_{0}\sim\Lambda/k_{B}$,
where $\Lambda$ is the bandwidth, thermal effects strongly destroy the effect
of the VHS. However, with decreasing of temperature substantially below the
bandwidth scale, the peak in $\frac{d\,\sigma^{\textrm{int}}_{xy}}{d\,\mu}$
becomes progressively more pronounced, allowing for the identification of a
VHS, and asymptotically tending to a sharp peak as in $T\rightarrow 0$ limit
described by Eq. (17).

##  IV Discussion and Outlook

In this work, we have identified a mechanism to create time-reversal broken
topological Chern bands that host higher-order Van Hove singularities on the
surface of a 3D topological insulator. We have shown that these HOVHS in
topologically nontrivial bands can emerge from the interplay of a time-
reversal breaking Zeeman field induced by the proximity to a ferromagnetic
insulator and a time-reversal invariant moiré potential induced on the surface
electrons of a three-dimensional topological insulator. The latter can
naturally occur through the misalignment of the quintuple layers in Bi2Se3 and
Bi2Te3, responsible for the onset of a nanometer scale moiré potential.
Employing exact diagonalization and symmetry analysis, we have demonstrated
that tuning of the Zeeman and moiré potential energy scales gives rise to a
manifold of higher order Lifshitz transitions on the moiré Brillouin zone
valleys. This setting opens a direction for future exploration of correlation
effects associated with the presence of strong density of state singularities
in moiré Chern bands on the surface of topological insulators.

Furthermore, we have identified a characteristic signature in the intrinsic
anomalous Hall response as the Fermi surface crosses a Lifshitz transition.
Specifically, the rate of change of the anomalous Hall conductivity as a
function of the Fermi energy, $d\sigma^{\textrm{int}}_{xy}/d\mu$, displays a
pronounced peak at low temperatures due to the large accumulation of states
near the VHS, which tracks the logarithmic or power-law divergences in
conventional and higher-order Lifshitz transitions, respectively. In the moiré
surface Chern bands studied in this work, this entails a power-law divergence
in $d\sigma^{\textrm{int}}_{xy}/d\mu$ as the temperature tends to zero. This
relationship opens a route to experimentally probe VHS and Chern bands through
transport measurements, and it would be not only applicable in the HOVHS on
the surface of a topological insulator but also to a wider class of
Hofstadter-Chern bands, Wang and Santos (2020); Herzog-Arbeitman  _et al._
(2020); Spanton  _et al._ (2018); Sharpe  _et al._ (2019); Serlin  _et al._
(2020); Saito  _et al._ (2021); Xie  _et al._ (2021); Nuckolls  _et al._
(2020); Wu  _et al._ (2021); Das  _et al._ (2021); Choi  _et al._ (2021); Park
_et al._ (2021); Stepanov  _et al._ (2021); Pierce  _et al._ (2021); Yu  _et
al._ (2022) and zero-field Chern bands in moiré heterostructures.Cai  _et al._
(2023); Park  _et al._ (2023); Zeng  _et al._ (2023); Xu  _et al._ (2023); Lu
_et al._ (2024)

## Acknowledgments

We thank Ben Feldman for the valuable discussions. This research was supported
by the U.S. Department of Energy, Office of Science, Basic Energy Sciences,
under Award DE-SC0023327. L. P. acknowledges funding from the Women in Natural
Science Fellowship of Emory University.

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