# TSUP Speaker Diarization System for Conversational Short-phrase Speaker
Diarization Challenge

###### Abstract

This paper describes the TSUP team’s submission to the ISCSLP 2022
conversational short-phrase speaker diarization (CSSD) challenge which
particularly focuses on short-phrase conversations with a new evaluation
metric called conversational diarization error rate (CDER). In this challenge,
we explore three kinds of typical speaker diarization systems, which are
spectral clustering (SC) based diarization, target-speaker voice activity
detection (TS-VAD) and end-to-end neural diarization (EEND) respectively. Our
major findings are summarized as follows. First, the SC approach is more
favored over the other two approaches under the new CDER metric. Second,
tuning on hyperparameters is essential to CDER for all three types of speaker
diarization systems. Specifically, CDER becomes smaller when the length of
sub-segments setting longer. Finally, multi-system fusion through DOVER-LAP
will worsen the CDER metric on the challenge data. Our submitted SC system
eventually ranks the third place in the challenge.

Index Terms: speaker diarization, spectral clustering, TS-VAD, EEND

##  1 Introduction

Speaker diarization is to determine “who spoke when” in an audio stream that
may contain an unknown number of speakers [2, 22]. It is an indispensable task
in multimedia information retrieval, speaker turn analysis and audio
processing [32, 3]. In particular, speaker diarization has the potential to
significantly improve automatic speech recognition (ASR) accuracy in multi-
speaker conversation scenarios [36, 40].

Clustering-based methods have dominated speaker diarization for many years,
which is composed of multiple, independently-optimized modules including voice
activity detection (VAD), speech segmentation, speaker embedding extraction
and speaker clustering [19, 15]. Although these systems have shown superior
performance in several speaker diarization challenges [36, 35], there are
apparently two defects in such clustering-based methods. Specifically, they
cannot properly deal with speaker overlap and the independent optimization of
different sub-modules may lead to sub-optimal performance [25, 31, 21].

To deal with the overlapping speech problem particularly, speech separation
can be used as a pre-processing step [36] and target-speaker voice activity
detection (TS-VAD) can be adopted as a post-processing step [16].
Particularly, the use of TS-VAD has led the speaker diarization system to
achieve state-of-the-art (SOTA) performance in several open challenges [35,
33]. However, TS-VAD has a drawback that the oracle information about the
maximum number of speakers in an audio stream has to be known in advance.

Recently, end-to-end neural diarization (EEND) [9, 10] has been proposed to
deal with both overlapping speech as well as to directly optimize a
diarization system via diarization error minimization. Specifically, EEND
treats speaker diarization as a classification task and estimates speech
activities of all speakers jointly frame-by-frame. To solve the permutation
problem [1], Fujita et al. introduced a permutation-free scheme [14, 39]
particularly into the training objective function. Thus the EEND system is
trained in an end-to-end fashion under the objective function that provides
minimal diarization errors.

To advance the speaker diarization performance in conversational short-phrase
scenario, ISCSLP2022 specifically held a CSSD challenge with a new
conversation dataset. In conversions like agent-customer telephone calls, the
speech utterance from each side is usually very brief and sometimes it is
limited to only a short phrase, which results in frequent speaker changes.
Speaker diarization in such a scenario poses particular challenges to both
current systems as well as evaluation metrics. Different from the previous
speaker diarization challenges [26, 4], the evaluation metric of the CSSD
challenge is the so-called conversational diarization error rate (CDER) [5]
rather than the typical diarization error rate (DER). Although DER has been
used as a standard metric for speaker diarization for a long time, it fails to
give enough emphasis to the short conversational phrases which last for a
short time but accurate discrimination on them play a vital role to the
downstream tasks such as speech recognition and understanding. Different from
DER calculated on the time duration level, CDER is regardless of the length of
the utterance, and all types of mistakes are equally reflected in the
evaluation metric [5]. Under this measurement, short-phrase and long sentences
have identical importance.

Considering the specific task and the new evaluation metric, in this paper, we
make a comparative study on the three typical speaker diairzation approaches –
spectral clustering (SC), TS-VAD and EEND. Our major findings are twofold.
First, SC is more preferred by the new CDER metric because frame-level
prediction nature of TS-VAD and EEND may lead to more errors on short
segments, which makes small impact on DER but significant impact on CDER which
equally treats long and short utterances. Second, tuning on hyperparameters is
essential to CDER for all three types of speaker diarization system.
Specifically, CDER will be reduced when the length of sub-segments setting
longer. Finally our submitted spectral clustering system manages to win the
third place in the challenge.

Figure 1: The overview of three kinds of speaker diarization systems and their
fusion

##  2 System Description

In the CSSD challenge, we explore several systems including SC, TS-VAD and
EEND, which are summarized in Figure 1. In addition, we also explore DOVER-LAP
[24] to fuse the RTTM outputs inferred from the above three systems to get the
final speaker diarization results.

###  2.1 Spectral Clustering System

A clustering-based speaker diarization system generally consists of VAD,
speaker embedding extractor and clustering modules. In general, the VAD module
can be simple energy-based or neural network based. Thanks to the recent
advances in neural speaker verificatoin, various types of speaker embedding
extractors, including x-vectors [28], ResNet [13] and ECAPA-TDNN [7], can be
considered to extract discriminative speaker embeddings from audio segments.
As for the clustering step, agglomerative hierarchical clustering (AHC) [17]
and spectral clustering [30] are the two typical methods.

In this challenge, we implement a ResNet-LSTM based VAD module similar to [33]
using the allowed training data and its performance on the dev and test sets
is comparable with that achieved by the TDNN-based VAD in Kaldi [23] which is
trained using more data. The timestamp labels during training are generated
from the transcripts of the training set. In our VAD module, the ResNet
structure is first to extract frame-level feature map. Then the current frame
feature map is concatenated with the feature maps from both the previous and
next frames and the concatenated feature map then goes through a statistics
pooling layer. Finally, two BLSTM layers and a linear layer are used to obtain
the probabilities of speech for the current frame.

The structure of our speaker embedding extractor is ResNet34 [34], which is
used to extract utterance-level embeddings after speaker segmentation.
Finally, we adopt the spectral clustering algorithm to cluster the speaker
identity of the embeddings. Spectral clustering has been widely adopted in
speaker diarization. The conventional AHC approach is highly time-consuming
where the processing time for an audio stream depends on the number of the
initial segments. SC is thus introduced to mitigate this problem.

In details, spectral clustering is a graph-based clustering algorithm. After
scoring pairs of sub-segment embeddings after speaker segmentation, we can get
a similarity matrix. Given the similarity matrix $S$, SC finds a partition of
the graph such that different groups edges have very low weights. Our
implementation on SC is summarized in Algorithm 1.

Algorithm 1 Algorithm of spectral clustering

0: The similarity matrix $S\in\mathbb{R}^{n\times n}$

0: The clustering label of every segment embedding.

1: Set similarity matrix $S$ diagonal elements to 0.

2: Compute the normalized Laplacian $L_{norm}$ of matrix $S$.

3: Compute the eigenvalues and eigenvectors of $L_{norm}$.

4: Compute the number of cluster $k$. In our implementation, we set a
threshold $\alpha$ and count the number of eigenvalues which is below $\alpha$
as $k$.

5: Concatenate the eigenvectors $\lambda_{1},\cdots,\lambda_{k}$ as columns
corresponding to the $k$ smallest eigenvalues into a matrix
$P\in\mathbb{R}^{n\times k}$.

6: For $i=1,\cdots,n$, let $r_{i}$ be the $i$-th row of $P$. Cluster row
vectors $r_{1},\cdots,r_{k}$ with the $k$-means algorithm.

7: return The label of each embedding $A_{1},A_{2},\cdots,A_{n}$ where
$A_{i}=\\{j|j=1,\cdots,k\\}$

###  2.2 Target-speaker Voice Activity Detection

A TS-VAD system [16] aims to handle the overlapped speech in speaker
diarization, which adopts the target-speaker embeddings to identify the
specific speakers within the overlapped speech. The target-speaker embeddings
are estimated based on the initial diarization results from the clustering-
based system.

Our TS-VAD system follows the structure in [34], which consists of a ResNet
front-end and a detection back-end. We use a deep speaker embedding model to
extract frame-level embeddings instead of the original TS-VAD [16] taking four
CNN layers to process the acoustic features. For an audio, we expect to detect
speech segments for each speaker. First, we use a ResNet34 network which has
the same structure with the speaker embedding model to extract frame-level
embeddings. Unlike speaker embedding model that applies statistics pooling to
project variable-length utterances into fixed-length speaker representation
embeddings, our TS-VAD employs the same pooling on each frame which is
combined with its adjacent frames to obtain frame-level embeddings. Then, the
frame-level embeddings are concatenated with the target-speaker embeddings.
Next, a Transformer encoder is used to extract detection state of each
speaker. After that a BLSTM layer processes these detection states which are
concatenated together to find relationship between each speaker. Finally, a
linear layer with a sigmoid function are used to determine the probabilities
of each speaker at every time step. By analyzing and processing the speech
probabilities, we can obtain speech segments of each speaker.

###  2.3 End-to-End Neural Diarization

EEND adopts a single neural network to obtian the final diarization result
directly from input audio feature. EEND is composed of an encoder block and a
binary classification block. Given a sequence generated from an audio signal,
encoder block extracts the feature containing diarization information and
classification block estimates a two-dimensional sequence to express a
probability of whether a speaker speaks at a frame. We employ permutation
invariant training (PIT) [9] to traverse all orders and optimize the minimal
one because changing speakers order does not affect the final result in
diarization task. Eventually, each frame is classified into one of the
following three cases, non-speaker, one-speaker and overlap.

In this challenge, we attempt two kinds of model structures, which are
residual auxiliary EEND (RX-EEND) [41] and speaker-wise chain EEND (SC-EEND)
[11]. In RX-EEND, each encoder block is enriched by a residual connection to
restrict gradient to a reasonable range. In addition, the output tensor of the
encoder block with the exception of the last block is aggregated to calculate
an extra auxiliary loss to gather more diarization information. In SC-EEND,
there is an LSTM layer to model the relation that whether a speaker spoken in
the past will affect other speakers between different speakers since we will
indeed be temporarily silent when the other person is talking during the real
conversation. RX-EEND improves the structure of the network and SC-EEND models
a more appropriate structure in the application scene.

##  3 Experiments

In this section, we first describe the model configuration and training
details. Then the score metric used in the challenge is introduced. Finally we
report the experimental results and major findings.

###  3.1 Model Configuration and Training

####  3.1.1 Speaker Embedding Model

Following the challenge rules, our experiments are conducted on VoxCeleb [18],
CN-Celeb [8] and MagicData-RAMC [38] datasets. We use ResNet34 as the speaker
embedding model which has {32, 64, 128, 256} channels of residual blocks. The
pooling layer is statistics pooling and the embedding size is 128. AAMSoftmax
[6] with a margin of 0.2 and scale of 32 is used to train the ResNet34 model.
The input of model is 1.5s fix-chunked wave and the acousic feature is 80-dim
log Mel filter bank feature with 25ms frame length and 10ms frame shift.
Specifically, we use a two-stage training strategy to optimize the speaker
embedding extractor. First, we pre-train the speaker embedding extractor for
10 epochs with the VoxCeleb and CN-Celeb datasets and the learning rate is set
to 0.01. To mitigate the mismatch between the pre-training datasets and the
MagicData-RAMC data, we adopt fine-tuning to get the in-domain speaker
embedding extractor. Specifcially, the model is fine-tuned for 30 epochs using
the non-overlapped speech segments of MagicData-RAMC training set and the
learning rate is set to 0.001. The optimizer is Adam. We also perform online
data augmentation during the training of the speaker embedding model.
Specifically, frequency-domain SpecAug [20], additive noise augmentation [27]
and reverberation augmentation [12] are adopted.

####  3.1.2 VAD

We implement a ResNet-LSTM based VAD model which follows the configuration in
[33]. The input is 16s chunked waveform, and the acoustic feature is 80-dim
log Mel filter bank with a frame size of 25ms and a frame shift of 10ms. The
model is trained on the MagicData-RAMC training set for 30 epochs with a
learning rate of 0.0001. We optimize the model with binary cross-entropy (BCE)
loss and Adam optimizer.

####  3.1.3 TS-VAD

We build a simulated dataset from the MagicData-RAMC training set for training
the TS-VAD model. The data simulation method is similar to that described in
[34]. First, we extract all non-overlapped speech segments from the MagicData-
RAMC training set for each speaker. Then, the corresponding labels are
extracted from the transcripts associated with the MagicData-RAMC training
set. The difference is that we do not remove the silence region, which aims to
learn silence information so that TS-VAD can detect speaker change region
better. Finally, in training stage, we fill the activated region with non-
overlapped speech segments.

The ResNet front-end configuration is the same as the speaker embedding model
as well as the Transformer encoder includes four encoder layers with 256
attention units containing 2 heads and 1024 internal units. The TS-VAD model
is first trained on the simulated dataset for 10 epochs with freezed ResNet34
front-end. Next, we train the whole system for 5 epochs further on the
simulated dataset. Finally, we fine-tune the whole system using the MagicData-
RAMC training set. We apply the learning rate of 0.0001 when training on the
simulated dataset and 0.00001 during the fine-tuning stage respectively. The
model is optimized with binary cross-entroy (BCE) and Adam optimizer.

In the inference stage of TS-VAD model, we first extract all the non-
overlapped speech segments from the results of the SC system. Then we extract
the target-speaker embeddings. Next, we split one audio stream into 16s chunk
waves with a 4s shift as the input of the TS-VAD model. After getting the
probabilities of each speaker at the frame level, we use median filtering with
a window size of 5 to smooth the probabilities. Then we binarize the
probabilities with the threshold of 0.9 and delete speech segments shorter
than 0.1s.

####  3.1.4 EEND

In the RX-EEND and SC-EEND models, we use four encoder blocks with 256
attention units containing four heads. We use 2048 internal units in a
position-wise feed-forward layer. The input feature is 23-dim log Mel filter
bank with total of 15 frames context and sub-sampling ratio is 10. The loss
function is BCE with the permutation-free scheme. The Adam optimizer is used,
where the learning rate schedule with warm-up steps of 100,000 is applied
[29].

EEND has two stages in training. Initially, we simulate 100,000 two-speakers
conversations as training data and reserve 500 for evaluation. The simulation
algorithm is described in [9]. We train a pre-train model using these simu-
data for 100 epochs. Afterwards, we fine-tune the pre-train model using
MagicData-RAMC training set for 30 epochs with 0.00001 learning-rate. In the
inference stage, we prepare a 50 frames buffer [37] to save partial input
sequence and corresponding prediction from model to prevent memory overflow,
because the average duration of MagicData is 30 minutes which is too large to
process the entire audio in GPU.

###  3.2 Score Metric

The speaker embedding models are evaluated by equal error rate (EER) and min
detection cost function (minDCF) with $P_{target}=0.01$ and
$C_{FA}=C_{Miss}=1$.

The speaker diarization task generally employs diarization error rate (DER) as
an evaluation metric on time duration level. In order to make all types of
mistakes equally reflect in the final evaluation metric, this challenge adopts
conversational-DER (CDER) [5] to evaluate the performance of the speaker
diarization system on the sentence level.

###  3.3 Experimental Results

####  3.3.1 Speaker Embedding Model

Speaker embedding extractor contributes a lot to the performance of the
speaker diarization system. So we first evaluate the EER/minDCF of our speaker
embedding extractor and results which are shown in Table 1. After fine-tuning,
the EER/minDCF on MagicData-RAMC dev and test set are absolutely reduced by
0.45%/0.04 and 0.17%/0.02 respectively.

Table 1: The EER (%) and $minDCF_{0.01}$ of speaker embedding model.

Training | Vox-O | CN-Eval | MagicData-Dev | MagicData-Test
---|---|---|---|---
EER | minDCF | EER | minDCF | EER | minDCF | EER | minDCF
Pre-training | 1.27 | 0.14 | 7.13 | 0.45 | 3.40 | 0.56 | 3.32 | 0.26
Fine-tuning | - | - | - | - | 2.95 | 0.52 | 3.15 | 0.24

####  3.3.2 VAD

Table 2 shows the false alarm (FA), miss detection (MISS) and accuracy(ACC) on
MagicData-RAMC dev set and test set. We report the performance of Kaldi VAD
model and our ResNet-LSTM VAD model at different thresholds. As presented in
Table 2, the ResNet-LSTM VAD model and Kaldi VAD model have comparable
performance. When we evaluate CDER performance after each VAD model is
integrated into the overall speaker diarization system, we find ResNet-LSTM
VAD(0.5) model achieves the lowest CDER. Therefore, ResNet-LSTM is selected as
the VAD module in our submitted speaker diarization system.

Table 2: The false alarm (FA), miss detection (MISS) and accuracy (%) of VAD
models at different threshold.

Model | MagicData-Dev | MagicData-Test
---|---|---
FA | MISS | ACC | FA | MISS | ACC
Kaldi VAD [23] | 8.19 | 4.31 | 95.69 | 8.35 | 4.84 | 95.16
ResNet-LSTM (0.5) | 0.37 | 7.89 | 92.11 | 0.80 | 6.95 | 93.05
ResNet-LSTM (0.4) | 0.46 | 6.87 | 93.13 | 0.96 | 5.95 | 94.05
ResNet-LSTM (0.3) | 0.58 | 5.98 | 94.02 | 1.14 | 5.00 | 95.00

####  3.3.3 Speaker Diarization

Table 3 shows the DER (collar=0.25) and CDER results of all the systems on the
MagicData-RAMC dev, test set and final blind test set. Results show that all
the performance of our models are better than those of baseline [5] on CDER.
For the TS-VAD model, we obtain the best CDER of 9.70% on the dev set. In
addition, the SC system achieves the lowest CDER of 9.50% on the test set. On
blind-test set, the TS-VAD model does not achieve the acceptable result.
Finally, we obtain a CDER of 9.10% in SC system, which is our submitted
results on the leadboard of CSSD challenge.

Table 3: The DER (%) and CDER (%) of different speaker diarization systems on
MagicData-RAMC dev, test and binld test sets.

Model | MagicData-Dev | MagicData-Test | Blind-Test
---|---|---|---
DER | CDER | DER | CDER | CDER
Baseline | 5.57 | 26.90 | 7.96 | 28.20 | -
SC | 14.33 | 12.00 | 18.19 | 9.50 | 9.10
TS-VAD | 12.79 | 9.70 | 17.52 | 10.50 | 16.40
RX-EEND | 12.05 | 16.30 | 12.01 | 19.20 | -
SC-EEND | 11.29 | 17.80 | 13.36 | 24.20 | -
SC&TS-VAD Fusion | 18.22 | 11.40 | 16.82 | 13.40 | -

After manual tuning hyperparameters, we find that CDER is reduced as the
length of sub-segments setting longer. Figure 2 shows the trend of the
influence of sub-segments duration to the DER/CDER. Since the penalty of
mistakes caused by different lengths in the calculation of CDER is the same,
the expanse of the mistake penalty caused by short sub-segments is relatively
larger. We observe that longer sub-segment duration can help to reduce this
kind of mistake, which improves the performance of CDER at the same time. This
means that a sentence from one speaker is divided into several sub-segments
with short duration, and these sub-segments may be recognized as two or more
speakers, which leads to serious error. As a result, longer sub-segment can
help to avoid this problem.

Figure 2: The influence of different sub-segment duration to DER (%) and CDER
(%)

Furthermore, the DOVER-LAP [24] fusion strategy is used to fuse the SC and TS-
VAD systems. However, the CDER of the fusion result decreases to 11.40%/13.40%
on MagicData-RAMC dev and test sets respectively. This is mainly because there
is few speaker overlaps in the MagicData-RAMC dataset while DOVER-LAP is
originally designed to better handle overlapped speech via voting.

##  4 Conclusion

In the ISCSLP2022 CSSD challenge, we have explored three different speaker
diarization systems – spectral clustering (SC), target speaker VAD (TS-VAD) as
well as end-to-end neural diarization (EEND). Our study shows that spectral
clustering based speaker diarization is still competitive in the current
challenge setup, i.e., accessing speaker diarization performance via the new
CDER metric for conversations composed of short-phrases. With the submitted SC
system, we achieve our lowest CDER of 12.0$\%$ and 9.5$\%$ on the dev set and
test set respectively, eventually leading our system to the 3rd rank on the
blind test set.

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# Observation of type-III corner states induced by long-range interactions

Huanhuan Yang  Lingling Song  Yunshan Cao  Peng Yan<EMAIL_ADDRESS>School
of Electronic Science and Engineering and State Key Laboratory of Electronic
Thin Films and Integrated Devices, University of Electronic Science and
Technology of China, Chengdu 610054, China

###### Abstract

Long-range interactions (LRIs) are ubiquitous in nature. Higher-order
topological (HOT) insulator represents a new phase of matter. A critical issue
is how LRI dictates the HOT phases. In this work, we discover four
topologically distinct phases, i.e., HOT phase, the bound state in the
continuum, Dirac semimetal (DSM) phase, and trivial insulator phase in a
breathing kagome circuit with tunable LRIs. We find an emerging type-III
corner state in the HOT phase, which splits from the edge state continuum and
originates from the strong couplings between nodes at different edges. We
experimentally detect this novel state by impedance and voltage measurements.
In the DSM phase, the Dirac cone exhibits an itinerant feature with a tunable
position that depends on the LRI strength. Our findings provide a deeper
understanding of the LRI effect on exotic topological states and pave the way
for regulating interactions in topolectrical circuits.

##  I introduction

Long-range interactions (LRIs) typically represent the two-body potential
decaying algebraically at large distances with a power smaller than the
spatial dimension. Paradigms of LRIs include electromagnetism, gravity, 2D
vortices, etc. Physical systems with LRIs can exhibit many peculiar properties
in their dynamics and statistics, such as negative specific heat and
temperature jumps Campa2009 ; French2010 . Recently, the LRIs have been shown
to impose certain effects on the topological states Varney2010 ; Beugeling2012
; Li2019 ; Shen2021 ; Olekhno2022 ; Yu2017 ; Leykam2018 . For instance, Li _et
al._ observed a type-II corner state in the photonic breathing kagome crystal
in the presence of a weak LRI Li2019 . An open question is how a strong enough
LRI dictates the topological phase. Quantum-optical technology may be a choice
to address this issue with controllable spin-spin interactions, but it is
difficult to implement in experiments Sandvik2010 ; Britton2012 .

Recently, electrical circuit manifests as an exceptional platform to study
topological physics Olekhno2022 ; Jia2015 ; Albert2015 ; Imhof2018 ; Ezawa2018
; Lee2018 ; Hadad2018 ; Hofmann2019 ; JBao2019 ; WZhang2020 ; Helbig2020 ;
Liu2020 ; Olekhno2020 ; Song2020 ; Yang2020 ; SLiu2020 ; RChen2020 ; Zhang2020
; Yang2021 ; Yangyt2021 ; Yang2022 ; Ventra2022 ; Wu2022 ; NAO2022 ;
Galeano2022 ; Pei2022 ; Zhang2022 , which can imitate the tight-binding (TB)
model in condensed matter physics by the inductor-capacitor ($LC$) network.
Conveniently, one can introduce an arbitrary hopping term between different
nodes in the circuit. For example, the higher-order topological (HOT)
insulators have been widely explored in circuits Imhof2018 ; Ezawa2018 ;
JBao2019 ; WZhang2020 ; Song2020 ; Yang2020 ; SLiu2020 ; RChen2020 , which
however focused on short-range interactions or hoppings.

It is thus interesting to establish a fruitful connection between these two
different topical areas of physics, i.e., LRIs and HOT phases. In this work,
we investigate how LRIs affect the HOT insulators both in theory and
experiment. To this end, we first derive the full phase diagram which includes
four topologically distinct phases in the breathing kagome circuit with LRIs,
specifically, a HOT phase, a bound state in the continuum (BIC, higher-order
topology supporting corner-localized bound states in the continuum
WABenalcazar2020 ), a Dirac semimetal (DSM) phase, and a trivial insulator
phase. Interestingly, the HOT phase can be further classified into three types
of corner states. The type-I corner states are pinned to zero admittance due
to the protection of the generalized chiral symmetry, and they are localized
at the three vertexes of the lattice. The type-II and type-III corner states
are at finite admittances and split from the edge state continuum, which are
akin to the edge states but exponentially decaying away from the 2nd and 3rd
cells along the boundary, respectively [see the 2nd and 3rd subfigures in Fig.
3(d)]. Moreover, the type-II corner states separate from the lowest and
highest edge modes and appear for an arbitrary LRI. By contrast, the type-III
ones split from the edge state continuum only in the strong-coupling region,
namely, a critical LRI value is required to move them out of the edge band. By
measuring the impedance distribution and the voltage signal propagation, we
directly observe all three types of corner states in experiments. In addition,
we numerically identify the BIC and DSM phases. We show that the position of
the Dirac points depends on the strength of LRI, which may help to control the
transport behaviors of Dirac states.

The paper is organized as follows. In Sec. II, we present the circuit model
and derive the full phase diagram. The emerging corner states are analyzed in
Sec. III. We perform experimental measurements in Sec. IV. Conclusions are
drawn in Sec. V. Technical details are given in Appendixes.

Figure 1: Breathing kagome circuit with LRIs supporting four topologically
distinct phases. (a) Schematic plot of an infinite breathing kagome circuit
with NNN hoppings. The breathing means alternatively arranged intracell and
intercell hoppings $C_{A}$ and $C_{B}$. The black, green, and gray line
segments represent the capacitors $C_{A}$, $C_{B}$, and $C_{N}$, respectively,
and each node is grounded by an inductor $L$. The basic vectors ${\bf
a}_{1}=(1,0)$ and ${\bf a}_{2}=(1/2,\sqrt{3}/2)$. Inset: the first BZ. (b)
Phase diagram of $\mathbb{Z}_{3}$ Berry phase by tuning the ratio
$C_{A}/C_{B}$ and $C_{N}/C_{B}$. The blue, green, white, and gray regions
represent the HOT phase, BIC phase, DSM phase, and trivial insulator phase,
respectively. The red curve indicates the critical NNN hopping strength for
the emergence of type-III corner states. Insets: schematic plots of three
types of corner states. (c) At $C_{A}/C_{B}=0.1$ ($C_{B}=1$ nF), the band
structures for $C_{N}/C_{B}=0$, 0.25, 0.5, 0.75, 1, and 1.5, respectively,
corresponding to the blue symbols (from left to right) in (b).
Figure 2: Finite-size circuit model and the photograph of experimental setup.
(a) Illustration of a finite-size kagome circuit with 360 nodes. The gray,
green, and gray line segments represent the capacitors $C_{A}$, $C_{B}$, and
$C_{N}$, respectively, and each node is grounded by capacitors and inductors
with configurations shown in the insets. (b) Photograph of the printed circuit
broad used in experiments. The inset zooms in the local details of the
circuit.
Figure 3: Admittance spectrums, wave functions, and impedances. (a) The
admittance spectrum as a function of the capacitance ratio $C_{N}/C_{B}$ at
$\omega=\omega_{c}$. The red, magenta, green, blue, and gray curves represent
the type-I corner, type-II corner, type-III corner, edge, and bulk states,
respectively. (b) The details of the admittance spectrum [gray rectangle in
(a)]. (c) Admittance spectrum at $C_{N}/C_{B}=0.275$ [dashed black line in
(b)]. (d) The wave-function profiles of the type-I corner, type-II corner,
type-III corner, and edge states. (e) The impedance of the 1st, 3rd, 9th,
84th, and 78th node as a function of the driving frequencies. (f) The
distributions of impedance at different frequencies. Colored pentagrams on the
top right corner indicate the measured frequencies marked in (e). Insets: The
distributions of impedance of type-III corner and edge state with higher
$Q$-factor inductors ($Q=200$).

##  II Circuit model and phase diagram

As shown in Fig. 1(a), we consider an infinite breathing kagome $LC$ circuit
with LRIs, by introducing next-nearest-neighbor (NNN) hopping terms
(capacitors $C_{N}$) to the lattice. The NNN interactions can be regarded as
LRIs for two reasons. In most topological systems with true LRIs, one can
consider the NNN hopping terms without missing the key physics, because
higher-order terms beyond NNN can only contribute negligible effects Li2019 ;
Shen2021 . Besides, the concept of LRI very recently has been adopted in
systems with NNN interactions Olekhno2022 . Labelling the nodes of the circuit
by $a=1,2,...$ the response at frequency $\omega$ follows Kirchhoff’s law:
$I_{a}(\omega)=\sum_{b}J_{ab}(\omega)V_{b}(\omega)$, with $I_{a}$ the external
current flowing into node $a$, $V_{b}$ the voltage at node $b$, and
$J_{ab}(\omega)$ being the circuit Laplacian:
$J_{ab}(\omega)=i\omega\left[-C_{ab}+\delta_{ab}\left(\sum_{n}C_{an}-\frac{1}{\omega^{2}L_{a}}\right)\right]$
with $C_{ab}$ the capacitance between nodes $a$ and $b$, and $L_{a}$ being the
grounded inductance at node $a$. The sum is taken over all connected nodes.
The dashed red triangle represents the unit cell including three nodes, with
the first Brillouin zone (BZ) plotted in the inset of Fig. 1(a). For
convenience, we express $J(\omega)$ as $i\mathcal{H}(\omega)$, with
$\mathcal{H}(\omega)$ akin to the Hermitian TB Hamiltonian expressed as

$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{\text{NNN}},$ (1)

where $\mathcal{H}_{0}$ and $\mathcal{H}_{\rm NNN}$ represent the NN and NNN
coupling Hamiltonian, respectively (see Appendix A for details). Next, as
suggested in Refs. Hatsugai2011 ; Kariyado2018 ; Kudo2019 ; Araki2020 , we use
$\mathbb{Z}_{3}$ Berry phase to distinguish these phases in the parameter
space (here is the $C_{A}$-$C_{N}$ plane), which is computed by

$\mathcal{\theta}=\int_{L_{1}}\text{Tr}[\textbf{A}(\textbf{k})]\cdot
d\textbf{k}\ \ (\text{mod}\ 2\pi),$ (2)

where $\textbf{A}(\textbf{k})$ is the Berry connection:
$\textbf{A}(\textbf{k})=i\Psi^{{\dagger}}(\textbf{k})\frac{\partial}{\partial\textbf{k}}\Psi(\textbf{k})$
with $\Psi(\textbf{k})$ being the eigenvector of (1) for the lowest band.
$L_{1}$ is an integral path in BZ: $K^{\prime}\rightarrow\Gamma\rightarrow
K^{\prime\prime}$, shown by the green line segment in the inset of Fig. 1(a)
(see Appendix A for discussion). The computation of $\mathbb{Z}_{3}$ Berry
phase is displayed in Fig. 1(b). Here, we obtain three different
$\mathbb{Z}_{3}$ Berry phases, i.e., $\mathbb{Z}_{3}=2\pi/3$, $-2\pi/3$, $0$,
and an ill-defined value, filled by blue, green, gray, and white colors,
respectively. Specifically, $\mathbb{Z}_{3}=2\pi/3$ denotes the HOT phase,
$\mathbb{Z}_{3}=-2\pi/3$ indicates the BIC phase (see Appendix B), and
$\mathbb{Z}_{3}=0$ is the trivial insulator phase. The ill-defined
$\mathbb{Z}_{3}$ belongs to the DSM phase owing to the gapless band structures
($\mathbb{Z}_{N}$ Berry phase can only be rigorously defined for a gapped
system. In the presence of strong NNN couplings, the lowest two bands touch
each other. One thus cannot have a well-defined $\mathbb{Z}_{3}$). It is noted
that, in the absence of the NNN coupling, one can only observe the type-I
corner state and trivial phase by tuning the ratio $C_{A}/C_{B}$ Yang2020 .
However, by adjusting certain capacitances of $C_{A}$ or $C_{B}$ in this
lattice by breaking the crystal symmetry (such as $C_{3}$ symmetry), one may
expect some peculiar edge states as illustrated in Ref. Yang2022 .

Figures 1(c) displays the band structures for different capacitance ratio
$C_{N}/C_{B}$ at $C_{A}/C_{B}=0.1$ [blue symbols in Fig. 1(b)]. For
$0<C_{A}/C_{B}<1$, in the absence of NNN hopping term ($C_{N}=0$), the band
structures are gapped, which belongs to the HOT phase Yang2020 . As we
increase the capacitance $C_{N}$, the lowest two bands firstly converge at $K$
point at $C_{N}/C_{B}=0.5$ and reopen beyond it, and then close again at $M$
point when $C_{N}/C_{B}=1$ (see Appendix A). The degenerate point subsequently
moves along the $M\rightarrow\Gamma$ trace, forming the DSM state (see
Appendix C). This phenomenon may provide a feasible method to manage the
unique transport properties of itinerant Dirac states, in exploring the
physics of anomalous quantum Hall effect Zhang2005 , minimum conductivity
Ando2002 ; Tworzydlo2006 , and Klein tunneling Katsnelson2006 ; Stander2009 .
For $C_{A}/C_{B}>1$, the two gapless thresholds appear at
$C_{A}/C_{B}=2C_{N}/C_{B}$ and $C_{A}/C_{B}=C_{N}/C_{B}$, and the behaviors of
band structures are similar to the foregoing ones (not shown). Below, we focus
on the HOT phase, i.e., $0<C_{A}/C_{B}<1$ and $0<C_{N}/C_{B}<0.5$ [blue region
in Fig. 1(b)].

Figure 4: Experimental measurements of the impedance. (a) Experimental
impedance dependence on the driving frequency. The red, magenta, green, and
blue pentagrams mark the positions of the impedance peak of type-I corner,
type-II corner, type-III corner, and edge states ($f_{1}=450$ kHz, $f_{2}=503$
kHz, $f_{3}=537$ kHz, and $f_{4}=528$ kHz). (b) The distributions of
experimental impedance at the four frequencies $f_{1}$, $f_{2}$, $f_{3}$, and
$f_{4}$. (c) The negative resistance is connected to each boundary node inside
the green shadow. The schematic diagram (d) and the experimental realization
(e) of the negative resistance (here, we adopt $R_{n}=R_{a}=R_{b}=4.99$
k$\Omega$). (f) Experimental impedance as a function of the driving frequency
with negative resistances, with $f_{1}^{\prime}=447$ kHz, $f_{2}^{\prime}=499$
kHz, $f_{3}^{\prime}=534$ kHz, and $f_{4}^{\prime}=523$ kHz. The impedance
distribution of (g) the type-III corner state and (h) edge state with negative
resistances measured at frequency $f_{3}^{\prime}$ and $f_{4}^{\prime}$,
respectively.

##  III Corner states

To observe the corner states, we consider a finite-size circuit network with
$\mathcal{N}=360$ nodes with the configuration being shown in Fig. 2(a).
Diagonalizing the circuit Laplacian $J(\omega)$, we obtain both the admittance
spectrum and wave functions. In the following calculations, we consider
$C_{A}/C_{B}=0.1$ with $C_{B}=1$ nF. Figure 3(a) shows the admittance spectrum
for different NNN hopping strengths. Except for the isolated zero-admittance
mode (red segment), the other two modes (magenta and green segments) escape
from the edge spectrums, which represent the type-II and type-III corner
states, see Fig. 3(b) for details. We demonstrate that the type-II corner
states appear for any $C_{N}>0$, while the type-III corner states emerge only
when $C_{N}>C_{Nc}$ with $C_{Nc}$ the critical value, as shown by the dark
blue region in Fig. 1(a) (see Appendix D for details). In Ref. Li2019 , the
same lattice is constructed in a photonic system but with a rather weak LRI.
Therefore, only the type-II corner states were observed there, and they split
from both the lowest and highest boundary modes. In order to find the type-III
corner states, one needs a stronger LRI, and the circuit then exhibits its
superiority for such a purpose. The strong NNN interactions induce strong
couplings between nodes at different edges, so the internal edge modes can
separate from the edge state continuum and form the type-III corner states.

Next, we analyze a representative example at $C_{N}/C_{B}=0.275$ [the dashed
black line in 3(b)]. Specifically, we choose $C_{N}=0.275$ nF and $L=39$
$\mu$H. The admittances $j_{n}$ are depicted in Fig. 3(c), with five kinds of
eigenmodes donated by red, magenta, green, blue, and gray circles,
corresponding to type-I corner, type-II corner, type-III corner, edge, and
bulk states, respectively. It is noted that the type-I corner states are
pinned to zero admittance, while type-II and type-III corner states are at
finite admittances. Figure 3(d) shows the wave functions of the corner states
and edge states, from which one can distinguish them clearly. The wave
functions of the type-II (type-III) corner states are resembling those of the
edge states, which have been proved to be topological Ni2019 ; Li2019 , but
exponentially decaying away from the 2nd (3rd) cell along the boundary. This
behavior can be viewed as the boundary states of the topological edge modes,
i.e., the corner states.

In an electrical circuit, one can use the impedance between the node and
ground to reflect the mode near the zero admittance Lee2018 ; Yang2020 ;
Yang2021 . In order to observe the different states mentioned above, one can
shift the corresponding modes to zero admittance by changing the driving
frequency without modifying the wave functions. Here, we adopt the impedance
between the 1st, 3rd, 9th, 84th, 78th nodes and the ground to quantify the
signals from the type-I corner, type-II corner, type-III corner, edge, and
bulk states [see numbered nodes in Fig. 2(a)], respectively. Figure 3(e) shows
the theoretical impedance as a function of the driving frequency, in which the
red, magenta, green, blue, and gray curves denote the signal from the
aforementioned five nodes, respectively. It is noted that we consider the
quality factor ($Q$ factor) of inductor here with the value $Q=45$ measured in
the experiment. One can see that the impedance peaks of the type-I, type-II,
type-III, and edge states appear at $f_{1}=444$ kHz, $f_{2}=497$ kHz,
$f_{3}=530$ kHz, and $f_{4}=520$ kHz, respectively, marked by four colored
pentagrams, which means the four states close to zero admittance at these
frequencies.

We calculate the distributions of the impedance at $f_{1},f_{2},f_{3}$ and
$f_{4}$, with the results plotted in Fig. 3(f). At frequency $f_{1}=444$ kHz,
the type-I corner modes are pinned at zero admittance. We depict the
distribution of impedance at the resonant frequency in the first subfigure of
Fig. 3(f). The impedances concentrate on the three vertexes of the sample,
which confirms the existence of the type-I corner states. For type-II corner
states, we find the distributions of the impedance are identical to the
corresponding eigenmode profiles at frequency $f_{2}$. However, the impedance
curve of the edge state behaves as an extension and submerges the type-III
corner states due to the finite $Q$ factor of inductor. Therefore, the
distributions of the impedance for type-III corner states seem to sprinkle
throughout the edge. To distinguish the type-III corner states from edge
states, one needs to improve the $Q$ factor of inductors. We plot the
distributions of impedance with higher $Q$-factor inductors ($Q=200$) in the
insets of Fig. 3(f), then one can distinguish the type-III corner states from
the edge states. Next, we verify these numerical results by experiments.

##  IV Experiment

As shown in Fig. 2(b), we manufacture a printed circuit broad (PCB) to observe
three kinds of corner states. To directly compare with the numerical
calculations, we choose the same electric elements as those in theoretical
analysis, but each circuit element has a natural 2% tolerance. The impedance
of circuit is measured by the impedance analyzer (Keysight E4990A). Figure
4(a) shows the experimental impedances at different driving frequencies.
Similarly, we adopt the impedances between the 1st, 3rd, 9th, 84th, 78th nodes
and the ground to characterize the type-I corner (red curve), type-II corner
(magenta curve), type-III corner (green curve), edge (blue curve), and bulk
(gray curve) states, respectively. The experimental results are fully
consistent with the numerical ones. Then, we measure the distribution of
impedance at four frequencies marked by red, magenta, green, and blue
pentagrams in Fig. 4(a), with the results plotted in Fig. 4(b). One can
observe the type-I corner states, type-II corner states clearly, but the type-
III corner states mix with the edge states due to the low $Q$ factor.

The $Q$ factor of an inductor is defined as $Q=\omega L/r$, where the $r$
comes from inevitable losses. To compensate for the dissipation, we introduce
active elements to the boundary nodes [light green region in Fig. 4(c)],
realized by connecting a series of negative resistances $-R$ with the
configuration shown in Figs. 4(d) and 4(e) (see Appendix E). We numerically
compare two cases that $-R$ in introduced to all nodes and to boundary nodes
only, which gives almost identical results. We therefore only add negative
resistances to edge nodes to reduce the experimental complexity. As a
consequence, the effective $Q$-factor of inductors is improved to about 150,
so the impendence peak becomes very sharp, with the peak value being enhanced
by about 400%, as shown in Fig. 4(f). We then measure the distribution of
impedance at frequency $f_{3}^{\prime}$ and $f_{4}^{\prime}$ again and clearly
separate the type-III corner states from the edge states, see Figs. 4(g) and
4(h). From these experimental results, we find that all corner states are
robust against the disorder because the elements intrinsically have 2%
tolerance.

To directly observe the dynamic response of the circuit, we measure the
propagation of the voltage signals for all three corner states and edge
states. As shown in Fig. 5(a), we impose a voltage signal at the first node
(labeled by red arrow) by $v(t)=v_{0}\sin(2\pi ft)$ with the amplitude
$v_{0}=5$ V and $f=f_{1}^{\prime}$ by an arbitrary function generator (GW
AFG-3022), and measure the amplitudes of the steady voltage sign by the
oscilloscope (Keysight MSOX3024A) over the whole sample. We find that the
signal is only localized at the first node, indicating the type-I corner
state. With the same method, we futher observe the type-II corner, type-III
corner, and edge states at frequency at the $f_{2}^{\prime}$,
$f_{3}^{\prime}$, and $f_{4}^{\prime}$, with the results plotted in Figs.
5(b)-5(d), respectively. Meanwhile, we show theoretical results in the insets
of each subfigure for comparison, which agree well with experimental findings.

Figure 5: Experimental observation and theoretical calculation of the voltage
propagations. (a)-(d) The distributions of the amplitude of the voltage signal
in devices with the signal source imposed between the 1st, 3rd, 9th, 84th node
and the ground at frequency $f_{1}^{\prime}=447$ kHz, $f_{2}^{\prime}=499$
kHz, $f_{3}^{\prime}=534$ kHz, and $f_{4}^{\prime}=523$ kHz, respectively.
Insets: numerical results.

##  V Conclusion

To summarize, we investigated the effect of LRIs on the topological phases in
breathing kagome circuits. With appropriate LRIs, we discovered the type-III
corner state and presented a direct experimental observation. We showed that
the type-III corner states split from the edge modes, and originate from the
strong coupling between nodes at different edges. With stronger LRIs, the BIC
and DSM phases subsequently emerge in this system. Our results highlighted the
important role played by LRI in topological phases and phase transitions. Our
findings significantly advance the understanding of the localization behavior
of HOT states and may spur future studies in other systems, such as acoustic
lattices, photonic crystals, and cold atoms, and in higher dimensions allowing
peculiar hinge states and Weyl semimetal states. It is worth mentioning that
the NNN hopping terms in our system differ from the LRIs in the field of
strongly correlated electrons that are closely related to local and non-local
density interactions. LRIs beyond long-range hopping may exhibit other
appealing properties, which deserve thorough exploration in the future.

###### Acknowledgements.

## ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China
(Grants No. 12074057, No. 11604041, and No. 11704060).

## APPENDIX

##  Appendix A The generalized chiral symmetry, $\mathbb{Z}_{3}$ Berry phase
and phase transitions

As shown in Fig. 6, the NN coupling Hamiltonian is given by

$\mathcal{H}_{0}=\left(\begin{matrix}Q_{0}&Q_{1}&Q_{2}\\\
Q_{1}^{*}&Q_{0}&Q_{3}\\\ Q_{2}^{*}&Q_{3}^{*}&Q_{0}\end{matrix}\right),$ (3)

with the matrix elements

$\displaystyle Q_{0}$
$\displaystyle=\omega\left[2(C_{A}+C_{B})-\frac{1}{\omega^{2}L}\right],$ (4)
$\displaystyle Q_{1}$
$\displaystyle=-\omega[C_{A}+C_{B}e^{-i(k_{x}/2+\sqrt{3}k_{y}/2)}],$
$\displaystyle Q_{2}$ $\displaystyle=-\omega[C_{A}+C_{B}e^{-ik_{x}}],$
$\displaystyle Q_{3}$
$\displaystyle=-\omega[C_{A}+C_{B}e^{-i(k_{x}/2-\sqrt{3}k_{y}/2)}],$

where $k_{x(y)}$ is the wave vector in the $\hat{x}(\hat{y})$ directions, and

$\mathcal{H}_{\text{NNN}}=\left(\begin{matrix}Q_{N0}&Q_{N1}&Q_{N2}\\\
Q_{N1}^{*}&Q_{N0}&Q_{N3}\\\ Q_{N2}^{*}&Q_{N3}^{*}&Q_{N0}\end{matrix}\right),$
(5)

with the matrix elements

$\displaystyle Q_{N0}$ $\displaystyle=4\omega C_{N},$ (6) $\displaystyle
Q_{N1}$ $\displaystyle=-\omega
C_{N}[e^{-ik_{x}}+e^{i(k_{x}/2-\sqrt{3}k_{y}/2)}],$ $\displaystyle Q_{N2}$
$\displaystyle=-\omega
C_{N}[e^{-i(k_{x}/2-\sqrt{3}k_{y}/2)}+e^{-i(k_{x}/2+\sqrt{3}k_{y}/2)}],$
$\displaystyle Q_{N3}$ $\displaystyle=-\omega
C_{N}[e^{-ik_{x}}+e^{i(k_{x}/2+\sqrt{3}k_{y}/2)}].$

It can be verified that $\mathcal{H}_{0}$ satisfies the generalized chiral
symmetry at resonant frequency $\omega_{0}=1/[\sqrt{L(2C_{A}+2C_{B})}]$,
because of $\Gamma_{3}^{-1}\mathcal{H}_{0}\Gamma_{3}=\mathcal{H}_{1}$,
$\Gamma_{3}^{-1}\mathcal{H}_{1}\Gamma_{3}=\mathcal{H}_{2}$,
$\mathcal{H}_{0}+\mathcal{H}_{1}+\mathcal{H}_{2}=0$, with
$\Gamma_{3}=\text{diag}\left(1,e^{\frac{2\pi i}{3}},e^{\frac{4\pi
i}{3}}\right)$. If we add the diagonal element $Q_{N0}$ to $Q_{0}$, the
resonant frequency will shift from $\omega_{0}$ to
$\omega_{c}=1/[\sqrt{L(2C_{A}+2C_{B}+4C_{N})}]$. We find that the remainder of
$\mathcal{H}_{\text{NNN}}$ still holds the generalized chiral symmetry. It is
to say that the periodic NNN hopping terms do not break the generalized chiral
symmetry but only cause a shift of the resonant point, which provides
topological protection for the triply degenerate zero-admittance mode.

Figure 6: Schematic plot of an infinite-size kagome circuit composed of three
types of capacitors $C_{A}$, $C_{B}$, and $C_{N}$, and each node is grounded
by an inductor. Inset: the first BZ with three colored arrows indicating the
integral paths.

Next, we discuss the relationship between the $C_{3}$ symmetry and the
quantized $\mathbb{Z}_{3}$ Berry phase. The Berry phase is closely related to
the local twist of the Hamiltonian Hatsugai2011 , which is quantized due to
time-reversal symmetry and inversion symmetry, for instance. In the breathing
kagome lattice, it has $C_{3}$ symmetry, so the points $K$, $K^{\prime}$,
$K^{\prime\prime}$ in the Brillouin zone are equivalent, as shown in the inset
of Fig. 6. So, there are three equivalent paths for computing $\mathbb{Z}_{3}$
Berry phase: $L_{1}$: $K^{\prime}\rightarrow\Gamma\rightarrow
K^{\prime\prime}$, $L_{2}$: $K^{\prime\prime}\rightarrow\Gamma\rightarrow K$
and $L_{3}$: $K\rightarrow\Gamma\rightarrow K^{\prime}$. As a result, the
Berry phases $\theta(L_{i})$ should take the same value, i.e.,
$\theta(L_{1})=\theta(L_{2})=\theta(L_{3})$. In addition, the integral along
the path $L_{1}+L_{2}+L_{3}$ is equal to zero, i.e.,
$\sum_{i=1,2,3}\theta(L_{i})=0$. Therefore, we obtain the quantized
$\mathbb{Z}_{3}$ Berry phase $\theta\equiv\theta(L_{i})=2\pi k/3$ with
$k=0,1,2$.

To show the details of phase transition at $K$ and $M$ points, we analyze the
eigenvalues and wave functions at these two points. At $K$ point, the
admittance spectra are given by $j_{K}^{(1)}=C_{A}+C_{B}-4C_{N}$,
$j_{K}^{(2)}=C_{A}-2C_{B}+2C_{N}$, and $j_{K}^{(3)}=-2C_{A}+C_{B}+2C_{N}$ with
the corresponding wave functions expressed as
$\phi_{1}=\left[-\frac{\sqrt{3}}{2}i-\frac{1}{2},\frac{\sqrt{3}}{2}i-\frac{1}{2},1\right]^{T}$,
$\phi_{2}=\left[\frac{\sqrt{3}}{2}i-\frac{1}{2},-\frac{\sqrt{3}}{2}i-\frac{1}{2},1\right]^{T}$
and $\phi_{3}=[1,1,1]^{T}$. We find that the lowest band exchanges from
$j_{K}^{(2)}$ to $j_{K}^{(1)}$ at $C_{N}/C_{B}=0.5$ [the first phase
transition in Fig. 1(b)]. Meanwhile, the expectation value of the phase of
wave functions for the lowest band $\varphi_{C_{3}}={\rm
arg}(\phi^{\dagger}R_{3}\phi$) Ni2019  changes from $\frac{2\pi}{3}$ to
$-\frac{2\pi}{3}$ (same as the $\mathbb{Z}_{3}$ Berry phase), where
$R_{3}=\left(\begin{array}[]{ccc}0&0&1\\\ 1&0&0\\\ 0&1&0\\\
\end{array}\right)$ is the three-fold rotational operator.

At $M$ point, the admittance spectra are $j_{M}^{(1)}=C_{A}+C_{B}-2C_{N}$,
$j_{M}^{(2)}=C_{N}-C_{B}/2-C_{A}/2-\Delta/2$, and
$j_{M}^{(3)}=C_{N}-C_{B}/2-C_{A}/2+\Delta/2$ with the eigenmodes
$\phi_{1}=[0,-1,1]^{T}$,
$\phi_{2}=\left[-(C_{A}+C_{B}-2C_{N}-\Delta)/(2C_{A}-2C_{B}),1,1\right]^{T}$,
and
$\phi_{3}=\left[-(C_{A}+C_{B}-2C_{N}+\Delta)/(2C_{A}-2C_{B}),1,1\right]^{T}$.
Here
$\Delta=\sqrt{9C_{A}^{2}-14C_{A}C_{B}-4C_{A}C_{N}+9C_{N}^{2}-4C_{B}C_{N}+4C_{N}^{2}}$.
The lowest two bands close at $C_{N}/C_{B}=1$, accompanied by the exchange of
the lowest band from $j_{M}^{(2)}$ to $j_{M}^{(1)}$ and the wave functions
from $\phi_{2}$ to $\phi_{1}$ [the second phase transition in Fig. 1(b)]. In
the case of $C_{N}/C_{B}>1$, the lowest two bands touch each other and form
the semimetal states. For the trivial state, the wave-function phase
$\varphi_{C_{3}}$ is always equal to 0.

##  Appendix B Bound state in the continuum

In Fig. 3(a), one cannot find any isolated mode in the range of
$0.5<C_{N}/C_{B}<1$ [green region in Fig. 1(b)]. However, if we evaluate the
inverse participation ratio (IPR)

$p=\sum_{i}|\phi_{n}|^{4}$ (7)

of the system, where $\phi_{n}$ is the normalized wave function with
$\sum_{i}|\phi_{n}|^{2}=1$, we find localized states buried in the bulk and
edge continuum, as shown in Fig. 7(a). The IPR has been widely adopted to
study Anderson localization in disordered systems Janssen1998 , and to confirm
the existence of HOT states recently HAraki2019 ; Wakao2020 . Taking
$C_{N}/C_{B}=0.75$ as an example, we find two groups of localized states
appearing in the continuum, as shown in Fig. 7(b). We plot their wave
functions in Figs. 7(c) and 7(d), respectively. We confirm that both localized
states actually are corner states.

Figure 7: Admittance spectrums and eigenmodes of BIC. (a) The admittance
spectrum as a function of $C_{N}/C_{B}$ at $C_{A}/C_{B}=0.1$, with the IPR
indicated by the color map. The magenta dots represent the localized states,
while gray dots mean extended states. (b) The admittance spectrum at
$C_{N}/C_{B}=0.75$ with the inset showing the details. (c)(d) The profile of
the wave functions of the localized states with the state number $n=180$ and
194, respectively.

##  Appendix C Dirac semimetal state

To illustrate the DSM state, we plot the band structure of an infinite kagome
circuit in Fig. 8(a). The lowest two bands linearly touch each other at six
points, which corresponds to the DSM state. Besides, by tuning the capacitance
$C_{N}$, we find that the positions of the six Dirac points continuously move
(one of them moves along the path $M\rightarrow\Gamma$), see Fig. 8(b).

Figure 8: Tunable Dirac points. (a) Illustration of the band structures of an
infinite system with $C_{A}/C_{B}=0.1$ and $C_{A}/C_{N}=1.5$. (b) The
positions of the six Dirac points (green dots) move in the first BZ as
$C_{N}/C_{B}$ ranges from 1 to 20. The blue hexagon shows the first BZ and the
gray arrows indicate the moving direction of the Dirac points.

To obtain an analytical understanding, we begin with Hamiltonian (1), and
rewrite it as a concise form

$\mathcal{H}=\left(\begin{array}[]{ccc}0&a&b\\\ a^{*}&0&c\\\ b^{*}&c^{*}&0\\\
\end{array}\right),$ (8)

with $a=Q_{1}+Q_{N1}$, $b=Q_{2}+Q_{N2}$, and $c=Q_{3}+Q_{N3}$ [see Eq. (4) and
Eq. (6) for details]. The secular equation is given by

$\lambda^{3}-(aa^{*}+bb^{*}+cc^{*})\lambda-ab^{*}c-a^{*}bc^{*}=0.$ (9)

For a cubic equation, if the discriminant
$27(ab^{*}c+a^{*}bc^{*})^{2}=4(aa^{*}+bb^{*}+cc^{*})^{3}$ is satisfied, it
must have three real roots with two degenerate ones. Imposing
$k_{y}=\frac{\sqrt{3}}{3}k_{x}$, the matrix elements $a,b,$ and $c$ can be
simplified as

$\displaystyle a$
$\displaystyle=b=-\omega(C_{A}+C_{B}e^{-ik_{x}}+C_{N}e^{-ik_{x}}),$ (10)
$\displaystyle c$ $\displaystyle=-\omega(C_{A}+2C_{N}\cos k_{x}).$

Divided by $-\omega C_{B}$, the above equations are expressed as

$\displaystyle a$ $\displaystyle=b=t_{a}+e^{-ik_{x}}+t_{n}e^{-ik_{x}},$ (11)
$\displaystyle c$ $\displaystyle=t_{a}+2t_{n}\cos k_{x},$

with $t_{a}=C_{A}/C_{B}$ and $t_{n}=C_{N}/C_{B}$. The discriminant is
simplified as $8(aa^{*})^{3}-15(aa^{*})^{2}c^{2}+6aa^{*}c^{4}+c^{6}=0$.
Substituting the expressions of $a,c$ to this equation, we get

$2t_{a}(t_{n}-1)\cos k_{x}+2t_{n}^{2}\cos(2k_{x})+t_{n}^{2}-2t_{n}-1=0.$ (12)

The solution is given by

$\cos(k_{x})=-\frac{t_{a}-t_{a}t_{n}+\sqrt{t_{a}^{2}t_{n}^{2}-2t_{a}^{2}t_{n}+t_{a}^{2}+4t_{n}^{4}+8t_{n}^{3}+4t_{n}^{2}}}{4t_{n}^{2}}.$
(13)

At $t_{n}=1$, we have $\cos(k_{x})=-1$ ($k_{x}=\pm\pi$). In the case of
$t_{n}<1$ ($t_{n}>1$), the value of the right part of Eq. (13) is greater
(less) than 1, so we can obtain a real solution of $k_{x}$ only for $t_{n}>1$.
In the limit of $t_{n}\rightarrow\infty$, we obtain
$\lim_{t_{n}\rightarrow\infty}\cos{k_{x}}=-1/2$ and $k_{x}=\pm\frac{2\pi}{3}$,
which is the terminal of the Dirac points. We therefore prove that the
solution $k_{y}=\frac{\sqrt{3}}{3}k_{x}$ is indeed the trajectory of the Dirac
points. Due to the $C_{3}$ symmetry of the lattice, the other two equivalent
traces of Dirac points exist along the lines of
$k_{y}=-\frac{\sqrt{3}}{3}k_{x}$ and $k_{x}=0$, respectively. To show the
linear touching of lowest two bands, we consider the circumstance of
$C_{N}\rightarrow\infty$ with Dirac points at $(k_{x}^{\rm DP},k_{y}^{\rm
DP})=(\pm\frac{2\pi}{3},\pm\frac{2\sqrt{3}\pi}{9})$ and
$(0,\pm\frac{4\sqrt{3}\pi}{9})$, marked by the blue dots in Fig. 8(b). In such
a case, we can analytically solve the band structure as

$\displaystyle j^{(1)}_{n}=2\omega C_{N},$ (14) $\displaystyle
j^{(2)}_{n}=\omega
C_{N}\left[-1-\sqrt{3+4\cos\frac{3k_{x}}{2}\cos\frac{\sqrt{3}k_{y}}{2}+2\cos(\sqrt{3}k_{y})}\right],$
$\displaystyle j^{(3)}_{n}=\omega
C_{N}\left[-1+\sqrt{3+4\cos\frac{3k_{x}}{2}\cos\frac{\sqrt{3}k_{y}}{2}+2\cos(\sqrt{3}k_{y})}\right].$

In the vicinity of the Dirac points, one can expand the lowest two bands
[$j^{(2),(3)}_{n}$] as

$j_{n}(p_{x},p_{y})=\omega
C_{N}[-1+\frac{3\sqrt{2}}{2}p_{x}+3p_{y}+\mathscr{O}({\bf p^{2}})],$ (15)

with $p_{x}=k_{x}-k_{x}^{\rm DP}$, $p_{y}=k_{y}-k_{y}^{\rm DP}$ and $|{\bf
p}|\ll 1$. One can clearly see a linear band crossings near the Dirac points.

##  Appendix D Origin of the type-II and type-III corner states

To explain the origin of the type-II corner states, we consider the first-
order perturbation theory as shown in Fig. 9(a). The perturbation Hamiltonian
is given by

$V=-\omega\left(\begin{array}[]{cccccc}0&\frac{C_{A}}{2}&0&0&0&0\\\
\frac{C_{A}}{2}&0&\frac{C_{A}}{2}&0&0&0\\\
0&\frac{C_{A}}{2}&0&\frac{C_{A}}{2}+C_{N}&0&0\\\
0&0&\frac{C_{A}}{2}+C_{N}&0&\frac{C_{A}}{2}&0\\\
0&0&0&\frac{C_{A}}{2}&0&\frac{C_{A}}{2}\\\ 0&0&0&0&\frac{C_{A}}{2}&0\\\
\end{array}\right).$ (16)

Diagonalizing the above matrix, we obtain the perturbation solution

$\delta j_{n}=\pm\omega\frac{(C_{A}+2C_{N})^{2}+C_{A}^{2}}{2C_{A}+4C_{N}},$
(17)

with the results plotted by dashed red lines in Fig. 9(c). The two solutions
are type-II corner states. The NNN couplings cause an effective interaction
($\omega C_{N}$) between two dimers across the corner, which splits the type-
II corner states from the edge spectrums.

Figure 9: Simplified models explaining the origin of type-II and type-III
corner states. (a)(b) The first-order and second-order perturbation models.
(c)(d) The numerical admittance spectrums (gray lines) and the perturbed ones
(dashed red lines).

For type-III corner states, we have to consider the second-order perturbation
theory as shown in Fig. 9(b). The system is described by a $15\times 15$
Hamiltonian matrix $V^{\prime}$ (not shown). By diagonalizing it, we obtain
the dashed red spectrums in Fig. 9(d), which can explain the type-III corner
states well. The NNN couplings induce extra interactions to edge nodes by
connecting to the bulk nodes, which yields the type-III corner states.

To study the band splittings, we consider three Su-Schrieffer-Heeger (SSH)
models Ventra2022  as shown in Fig. 10(a), described by the following
Hamiltonian $\mathcal{H}_{i}$ ($i=1,2,3$, we set $C_{B}=1$)

$\mathcal{H}_{1}=\omega\left(\begin{array}[]{cc}0&C_{N}\\\ C_{N}&0\\\
\end{array}\right),$ (18)

$\mathcal{H}_{2}=\omega\left(\begin{array}[]{cccc}0&C_{N}&0&0\\\
C_{N}&0&1&0\\\ 0&1&0&C_{N}\\\ 0&0&C_{N}&0\\\ \end{array}\right),$ (19)

$\mathcal{H}_{3}=\omega\left(\begin{array}[]{cccccc}0&C_{N}&0&0&0&0\\\
C_{N}&0&1&0&0&0\\\ 0&1&0&C_{N}&0&0\\\ 0&0&C_{N}&0&1&0\\\ 0&0&0&1&0&C_{N}\\\
0&0&0&0&C_{N}&0\\\ \end{array}\right).$ (20)

The hopping terms $C_{N}$ cause the splitting of the boundary modes as
$\delta\omega_{1}=2\omega
C_{N},~{}\delta\omega_{2}=2\omega(C_{N}^{2}-C_{N}^{4}),~{}\delta\omega_{3}=2\omega(C_{N}^{3}-C_{N}^{5})$,
which can be approximatively regarded as an effective capacitances $C_{N}$,
$C_{N}^{2}$ and $C_{N}^{3}$ to directly connect the two boundary nodes, as the
red arrows shown in Fig. 10(a). Next, we will show how these connections
induce the type-II and type-III corner states.

Figure 10: Effective interactions between edge nodes. (a) SSH model with
different bulk nodes. The red arrows represent the effective capacitance for
energy splitting. (b) Effective interactions between edge nodes. (c) The
admittance spectrum of the two effective models in (b) (black lines for left
panel and blue lines for right panel). (d) Critical curve for the emergence of
type-III corner states.

As shown in Fig. 10(b), one can separate the nodes of the breathing kagome
lattice into three types: single, dimer, and trimer. The edge spectra are
formed by the boundary dimers. Then, we reduce to the effective one-
dimensional problem with two edges (green sectors) meeting at the corner, as
shown in Fig. 10(b). The effective hopping term $\omega C_{N}$ connects the
3rd node to the 4th node, and as a result, the 1st and 6th bands split from
the edge state continuum and form the type-II corner state [see black line in
Fig. 10(c)]. To explain the energy splitting of the type-III corner state, we
consider the interaction $\omega C_{N}^{2}$ between the 2nd and 5th, and the
result is plotted in Fig. 10(c). We find the 2nd and 5th blue spectrums first
submerge in the edge spectrums and then diverge from them at a critical point,
and result in the emergence of the type-III corner states. Therefore, a
threshold exists for the appearance of type-III corner states, as labeled by
the orange dot in Fig. 9(d). In Fig. 10(d), we plot the critical points of
emergence of type-III corner state for different capacitance $C_{A}$, and one
can find the type-III corner state in the blue region. The critical points can
be fitted by the following curve

$\frac{C_{A}}{C_{B}}=0.0058\exp\left(21.11\frac{C_{N}}{C_{B}}\right),$ (21)

and different $C_{A}$ gives different threshold $C_{Nc}$. Furthermore, by
taking the effective coupling $\omega C_{N}^{3}$ between 1st and 6th nodes
into account, we find the 3rd and 4th can deviate from the edge state
continuum a little but cannot escape from them. Therefore, one can only
observe two types of corner states splitting from the edge spectrums in the
present model.

##  Appendix E The realization of the negative resistance

As shown in Figs. 4(d) and 4(e), the negative resistance is realized by an
integrated operational amplifier XL5532 and three resistances $R_{a},R_{b}$,
and $R_{n}$.

For an ideal operational amplitude, $i_{+}=0$, $i_{-}=0$, and $u_{d}=0$ (most
modern amplifiers have large gains and input impedances, so the analysis is
feasible in a real circuit). According to Ohm’s law, we obtain

$\displaystyle v_{\rm in}-v_{\rm o}=iR_{n},$ (22) $\displaystyle v_{\rm
o}=\frac{R_{a}+R_{b}}{R_{b}}v_{\rm in}.$

Combining these two equations, we have

$\frac{v_{\rm in}}{i}=-\frac{R_{a}}{R_{b}}R_{n}.$ (23)

Here, we choose $R_{a}=R_{b}=R_{n}=4.99$ k$\Omega$, so the network is
equivalent to a negative resistance due to $v_{\rm in}=i(-R)$.

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# Accretion rate in AGN and X-ray-to-optical flux ratio at z $\leq$ 0.2

Asrate Gaulle1,2 Mirjana Pović2,3 and Dejene Zewdie4,5 1 Dilla University,
Department of Physics, Dilla, Ethiopia; email<EMAIL_ADDRESS>and Astrophysics Research and Development Division, Ethiopian Space Science
and Technology Institute, Addis Ababa, Ethiopia; 3Institute of Astrophysics of
Andalucía (IAA-CISC), Department of Extragalactic Astronomy, Granada, Spain; 4
Núcleo de Astronomía, Universidad Diego Portales, Santiago, Chile; 5
Department of Physics, Debre Berhan University, Debre Berhan, Ethiopia

(2019)

###### Abstract

We explored a sample of 545 local galaxies using data from the 3XMM-DR7 and
SDSS-DR8 surveys. We carried out all analyses up to z $\sim$ 0.2, and we
studied the relation between X/O flux ratio and accretion rate for different
classes of active galaxies such as LINERs and Seyfert 2. We obtained a slight
correlation between the two parameters if the whole sample of AGN is used.
However, LINERs and Sy2 galaxies show different properties, slight correlation
and slight anti-correlation, respectively. This could confirm that LINERs and
Sy2 galaxies have different accretion efficiencies and maybe different
accretion disc properties, as has been suggested previously.

###### keywords:

galaxies - active; AGN - accretion rate; AGN - black hole masses; AGN - X-ray
properties; AGN - optical properties.

††volume: 356††journal: Nuclear Activity in Galaxies Across Cosmic
Time††editors: M. Pović, P. Marziani, J. Masegosa, H. Netzer,
S. H. Negu, & S. B. Tessema, eds.

##  1 Introduction

Active galactic nuclei (AGN) are powerful sources of radiation in a wide
spectral range, from gamma-rays to radio waves (Netzer 2015). In particular,
AGN are strong X-ray sources. X-ray emission is shown to be a powerful tool of
AGN detection and a study of the growth of supermassive black holes (SMBHs)
and AGN properties (Brandt and Hasinger 2005). On the other side, optical data
are very important for AGN classification and for studying the properties of
AGN host galaxies (Pović et al. 2009a,b, 2012). Therefore, the combination of
X-ray and optical data allows the successful study of the connection between
the AGN and their host galaxies. Furthermore, the accretion rate (AR) in
galaxies remains a prerequisite for understanding the physics behind SMBHs and
AGN, the evolution and growth of galaxies, and the connection between active
and non-active galaxies. However, AR measurements are still not easy since
they mainly depend on the availability of spectroscopic data, which contain
smaller data sets and poorer statistics than photometric data. In previous
studies, Pović et al. (2009a, b) suggested that there might be a photometric
indicator of AR in galaxies, based on the ratio between the flux in X-rays
(0.5 - 4.5 keV) and optical flux. In this work, we used X-ray data, and
optical spectroscopic and photometric data to measure both X/O flux ratio and
AR in different types of active galaxies and to test the correlation between
the two parameters.

##  2 Data and Sample Selection

X-ray data were obtained from the 3XMM-DR7 catalogue (Rosen et al. 2016) which
contains 727,790 sources. Also, we used optical photometric and spectroscopic
data from the Sloan Digital Sky Survey (SDSS) data release 8 (DR8).
Spectroscopic measurements have been obtained from the MPA-JHU SDSS DR8
catalogue (Brinchmann et al. 2004) for 1,472,581 sources. Finally,
morphological classification has been obtained from the Galaxy Zoo catalogue
(Lintott et al. 2011). We cross-matched all three catalogues using a cross-
matched radius of 3 arcsec and we selected 2151 sources in total up to
redshift z $\leq$ 0.2. We stick to this redshift for avoiding the K-correction
in X-rays. Using the BPT-NII diagram (Baldwin, Phillips, & Terlevich, 1981),
and signal-to-noise ratio of emission lines of S/N $>$ 3, we selected 545 AGN
galaxies in total, of those 209 and 336 being Seyfert 2 and LINER sources,
respectively.

##  3 Analysis and Results

In this work, we analyzed the correlation between the X/O flux ratio and AR of
spectroscopically selected AGN up to z $\leq$ 0.2. After selecting LINERs and
Sy2 sources, we went through the following analysis: we first measured the X/O
flux ratio by using the (0.5 - 4.5keV) X-ray and optical r bands (Pović et al.
2009a, b). Secondly, we measured the mass of a black hole using the velocity
dispersion method (Tremaine et al. 2002) and velocity dispersion measurements
from the MPA-JHU catalogue (Brinchmann et al. 2004). Eddington luminosity was
then measured as Ledd = 1.5 $\times$ 1038 (Mbh/M)erg/s. Bolometric luminosity
was measured through the H$\beta$ and [OIII] emission lines using the results
of Netzer (2013). Emission lines were first corrected for extinction through
the H$\alpha$/H$\beta$ emission lines. Finally, the accretion rate was
measured as AR = Lbol /Ledd. We tested the correlation between the SMBH mass
and AR (see Fig. 1) and X/O flux ratio and AR (see Fig. 2) for the whole
sample of AGN, and also for LINERs and Sy2 galaxies. We finally analyzed the
same relations, but for different morphological types. We found anti-
correlation between the SMBH mass and AR independently on AGN type, as shown
in Fig. 1, confirming some of the previous results (e.g., Woo and Urry 2002).
Only mild correlation has been found between the X/O flux ratio and AR in
LINERs, and mild correlation in Seyfert 2 galaxies, as shown in Fig. 2.
between X/O flux ratio and AR as shown inFig. 3. When observing the previous
in relation to morphology, the same trends have been found in all cases for
both elliptical and spiral galaxies at redshifts z $\leq$ 0.2.

Figure 1: The correlation between BH mass and AR, left-right (LINER,Sy2)

Figure 2: The correlation between X/O flux ratio & AR, left- right: LINER,
Sy2.

##  4 Conclusion

The results obtained above could suggest previous findings that Sy2 and LINER
galaxies have different accretion properties and belong to two different
modes: radiatively efficient and radiatively inefficient advection dominated
accretion. Preliminary results obtained in this work do not discard the
possibility of using the X/O flux ratio as an AR indicator, however, more
detailed studies and larger statistical samples are needed for confirming
this.

## Acknowledgments

AG acknowledges the support from Dilla University. AG and MP acknowledge
financial support from the EORC under ESSTI. Mirjana Povic acknowledges the
support from the Spanish Ministry of Science, Innovation, and Universities
(MICIU) through project AYA2016-76682C3-1-P.

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