# Bag of Tricks and A Strong Baseline for Image Copy Detection

Wenhao Wang  Weipu Zhang  Yifan Sun  Yi Yang
Baidu Research
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Image copy detection is of great importance in real-life social media. In this
paper, a bag of tricks and a strong baseline are proposed for image copy
detection. Unsupervised pre-training substitutes the commonly-used supervised
one. Beyond that, we design a descriptor stretching strategy to stabilize the
scores of different queries. Experiments demonstrate that the proposed method
is effective. The proposed baseline ranks third out of $526$ participants on
the Facebook AI Image Similarity Challenge: Descriptor Track. The code and
trained models are available here.

##  1 Introduction

The goal of image copy detection is to determine whether a query image is a
modified copy of any images in a reference dataset. It has extensive numbers
of applications, such as checking integrity-related problems in social media.
Although this topic has been researched for decades, and it has been deemed as
a solved problem, most state-of-the-art solutions cannot deliver satisfying
results under real-life scenarios [8]. There are two main reasons. First,
real-life cases in social media involve billions to trillions of images, which
introduces many “distractor” images to degrade the performance. Second, the
transformations used to edit images are countless. It is very challenging for
an algorithm to be robust to unseen scenarios.

In this paper, we provide a bag of tricks and a strong baseline to compete for
Image Similarity Challenge: Descriptor Track at NeurIPS’21 (ISC2021) [8]. This
competition builds a benchmark that features a variety of image
transformations to mimic real-life cases in social media. To mimic a needle-
in-haystack setting, both the query and reference set contain a majority of
“distractor” images that do not match. The evaluation metric adopted is micro
Average Precision, which penalizes any detected pair for a distractor query.

The approach consists of three parts, pre-training, training, and test. In
pre-training, unsupervised pre-training on ImageNet [6] instead of the
commonly-used supervised pre-training is performed. Specifically, we
empirically find that Barlow-Twins [36] pre-training is superior to some other
unsupervised pre-training methods. In training, a strong deep metric learning
baseline is designed by combining a bag of tricks. Moreover, to make the
sample pairs more informative, some image augmentations are employed to
generate training images. The diversity of augmentations promotes learning
robust representation. During testing, the Descriptor Track setting directly
uses the Euclidean distance as the final matching score and thus prohibits the
score normalization operation in Matching Track. In response to this
characteristic, we propose a novel descriptor stretching strategy. It rescales
(stretches) the descriptors to stabilize the scores of different queries. In
another word, it approximates the score normalization effect by directly
stretching the descriptors. Empirically, this stretching strategy
significantly improves retrieval accuracy. The illustration of the proposed
approach is shown in Fig. 1.

Figure 1: The designed strong baseline. Unsupervised pre-training is used to
substitute supervised one. The strong baseline is trained with a bag of tricks
on augmented images. Descriptor stretching, which stabilizes the scores of
different queries, is proposed for testing. Figure 2: The bag of tricks and
the strong baseline. A recent self-supervised learning method, Barlow-Twins
[36], is performed for pre-training. During training, the built baseline
includes GeM [26], WaveBlock [34], a high-dimension projector, a learnable
matrix, two commonly-used losses, and a warming up with cosine annealing
learning rate.

In summary, the main contributions of this paper are:

  1. 1.

The paper provides a bag of tricks and a strong baseline for image copy
detection.

  2. 2.

The proposed baseline handles real-life copy detection scenarios in social
media well.

  3. 3.

Our baseline ranks third out of $526$ participants in the Image Similarity
Challenge: Descriptor Track at NeurIPS’21.

##  2 Related Work

###  2.1 Copy Detection

Although copy detection plays an important role in social media, the
publication of copy detection is not well known because organizations want to
keep copy detection techniques obscure, and the researchers often consider
that the task is easy [8]. For classical approaches, some methods are used to
extract global [17, 30, 12] and local descriptors [1, 3, 16]. For deep
learning methods, a descriptor, or known as a feature, is extracted by
convolutional neural networks [22, 38]. However, none of them gives a
satisfactory performance on challenging, large benchmarks like ISC2021.

###  2.2 Unsupervised Pre-training

The unsupervised pre-trained models are from recent self-supervised learning
methods. They are trained to extract image embedding unsupervisedly. MoCo [9]
builds a dynamic dictionary with a queue and a moving-averaged encoder to
conduct contrastive learning. BYOL [13] and its Momentum2 Teacher version
achieve a new state-of-the-art without negative pairs. Other self-supervised
methods, such as Barlow Twins [36], SimSiam [5], and SwAV [4], also show
promising performance.

For copy detection in ISC2021, we find that unsupervised pre-trained models
show superior performance to their fully-supervised counterparts. That may be
because the category defined in unsupervised pre-training is much more similar
to that in copy detection than in fully-supervised one.

Figure 3: The set of basic augmentations. It includes random resized cropping,
random rotation, random pixelization, random pixels shuffling, random
perspective transformation, random padding, random image underlay, random
color jitter, random blurring, random grayscale, random horizontal flipping,
random Emoji overlay, random text overlay, random image overlay, and resizing.
The first image is the resized original image.

###  2.3 Deep Metric Learning

Learning a discriminative feature is a crucial component for many tasks, such
as image retrieval [25, 18], face verification [28, 31], and object re-
identification [29, 37]. The commonly-used loss functions can be divided into
two classes: pair-based [11, 27, 23] and proxy-based [21, 32, 7] losses.
Circle loss [28] gives a unified formula for two paradigms.

In our solution to ISC2021, we use two common losses, triplet loss with hard
sample mining and cross-entropy loss. The two losses are proved to be simple
but effective.

##  3 Proposed Method

In this section, we introduce each important component in the proposed
baseline. In the training part, we discuss the tricks to train the baseline
and the designed augmentations. In the test part, we discuss the proposed
descriptor stretching. The whole baseline is shown in Fig. 2.

###  3.1 Unsupervised Pre-training

In the ISC2021, a category is defined at a very “tight” level, images
generated by exact duplication, near-exact duplication, and edited copy are
considered in the same category while two images that belong to the same
instance or object are not in the same category. However, when using fully-
supervised pre-training on the ImageNet, two images share the same instance or
object are in one class. Therefore, the definition of the category is
contradicted between the ISC2021 and ImageNet.

Fortunately, the recent research on self-supervised learning provides a new
direction. It defines every image as a category and uses invariance to data
augmentation as their main training criterion [8]. Although it seems natural
to directly adopt a self-supervised learning method to train a model for copy
detection, we choose to give up this solution for two reasons. First, the
self-supervised learning methods often cost a lot of days to get convergence
results. For instance, on ImageNet, BYOL [13] or its Momentum2 Teacher version
[20] takes about two weeks to pre-train $300$ epochs with ResNet-152 [10]
backbone using $8$ NVIDIA Tesla V100 32GB GPUs. In addition, when using
sophisticated transformations online, the time becomes much longer, which is
unaffordable. Second, even if the resource is unlimited, we do not get a
satisfying performance only trained by self-supervised learning methods, which
may be because that the hyper-parameters are not adjusted carefully.

As a result, we choose to use Barlow-Twins [36] to get the pre-trained model
on ImageNet. The used augmentations follow the default setting in the original
implementation.

###  3.2 Training Tricks

The original version of the baseline is from DomainMix [33]. To be more
powerful, we introduce some recent methods and adapt them to the Image
Similarity Challenge: Descriptor Track.

Generalized mean pooling. Generalized mean (GeM) pooling [26] is proposed to
compute the generalized mean of a three-dim tensor:

$\mathbf{f}^{(g)}=\left[\mathrm{f}_{1}^{(g)}\ldots\mathrm{f}_{k}^{(g)}\ldots\mathrm{f}_{K}^{(g)}\right]^{\top},$
(1)
$\mathrm{f}_{k}^{(g)}=\left(\frac{1}{\left|\mathcal{X}_{k}\right|}\sum_{x\in\mathcal{X}_{k}}x^{p_{k}}\right)^{\frac{1}{p_{k}}},$
(2)

where: $p_{k}$ is a learnable pooling parameter. The GeM pooling degenerates
to maxing pooling when $p_{k}\rightarrow\infty$, and mean pooling when
$p_{k}=1$. This can be easily proved by mathematical analysis. We
experimentally set the initialization of the pooling parameter as $3$. The
network focuses on the salient area with a larger $p_{k}$, while it learns
globally with a smaller $p_{k}$. For image copy detection, we find that
$p_{k}$ tends to increase, which shows that the matching of salient areas
matters.

WaveBlock. The WaveBlock is proposed in AWB [34] to deal with the noise in
pseudo-labels. We further discover WaveBlock can be regarded as a kind of
augmentation method at the feature level. Following the original paper, the
WaveBlocks are arranged after Stage $2$ and Stage $3$ when using ResNet50
[10]. The same hyper-parameters are also used. We argue that WaveBlocks
enhance the model ability for unseen augmentations.

High-dimension projector. We design a projector to project a learned 2048-dim
feature into 8192-dim via some linear and non-linear connections.
Specifically, the projector adopts the structure of FC-BN-LeakyReLU-FC where
the input dimension, hidden dimension, and output dimension are set to $2048$,
$4096$, and $8192$, respectively. Empirically, we find that when the IDs in
the training set are small, learning from high dimension space is not very
useful. However, when the IDs become larger, such as $100,000$, the projector
makes a difference.

Learnable Matrix. Per the official rule for Track 2, the maximum dimension of
a descriptor is $256$. Therefore, an $8192\times 256$ matrix is learned during
training. By using this matrix, the learned $8192$-dim features can be
transformed to $256$-dim ones. This matrix is initialized randomly.

Losses. In this baseline, we use two commonly-used losses, triplet loss with
hard sample mining and cross-entropy loss. The cross-entropy loss is used
three times. First, it is used for the classification of $8192$-dim features.
Second, it is used for the classification of $256$-dim features. Third, we
also design a soft cross-entropy loss. The prediction results of $8192$-dim
features are regarded as labels to conduct the learning process of $256$-dim
ones. The $2048$-dim feature is used for triplet loss with hard sample mining.

Warming up with cosine annealing learning rate. The adopted learning rate is
with warming up scheduling and cosine annealing. The basic learning rate is
$3.5\times 10^{-4}$. We train the baseline for $25$ epochs. The ratio for
scheduling is

$ratio=\left\\{\begin{array}[]{l}0.99\cdot epoch/5+0.01,0\leq epoch<5\\\
1,5\leq epoch<10\\\
0.5\cdot\left({\cos\left({{\textstyle{{epoch-10}\over{25-10}}}\cdot\pi}\right)+1}\right),10\leq
epoch<25\end{array}\right.$ (3)

###  3.3 Augmentation

We design a set of augmentations to transform images. The set of basic
augmentations includes random resized cropping, random rotation, random
pixelization, random pixels shuffling, random perspective transformation,
random padding, random image underlay, random color jitter, random blurring,
random grayscale, random horizontal flipping, random Emoji overlay, random
text overlay, random image overlay, and resizing. A display of using all the
basic augmentations on one image is shown in Fig. 3. The first image is the
resized original image.

###  3.4 Descriptor Stretching

Figure 4: The illustration of the proposed descriptor stretching strategy. The
$L_{2}$ norms of all the features except for the stretched ones are all equal
to $1$.

In the Descriptor Track, the Euclidean distance is used as a matching score
directly while the scores of different queries are not comparable. In Matching
Track, this phenomenon also exists, and thus the score normalization [14] is
used. However, in the Descriptor Track, score normalization, which applies to
similarity scores, is no longer appropriate. To stabilize the scores of
different queries, we propose a descriptor stretching strategy. The
illustration of the strategy is shown in Fig. 4.

Given the feature of a query image $\vec{q_{1}}$, and a reference image
$\vec{r_{1}}$, the original score $s_{1}$ is defined as

$s_{1}=\left|\vec{q_{1}}-\vec{r_{1}}\right|.$ (4)

Similarly, we have:

$s_{2}=\left|\vec{q_{2}}-\vec{r_{1}}\right|.$ (5)

If $s_{1}>s_{2}$, $\vec{q_{2}}$ is more similar to $\vec{r_{1}}$ than
$\vec{q_{1}}$, and vice versa. The definition of descriptor stretching is

$\vec{\hat{q_{1}}}=\alpha\cdot s_{n_{1}}\cdot\vec{q_{1}},$ (6)

where: $\alpha$ is a hyper-parameter, and $s_{n_{1}}$ is the mean of the top
$n$ inner product scores between $\vec{q_{1}}$ and the features of images from
the training set. Then the stretched score $\hat{s_{1}}$ is defined as:

$\hat{s_{1}}=\left|\vec{\hat{q_{1}}}-\vec{r_{1}}\right|.$ (7)

Similarly, we have:

$\vec{\hat{q_{2}}}=\alpha\cdot s_{n_{2}}\cdot\vec{q_{2}},$ (8)
$\hat{s_{2}}=\left|\vec{\hat{q_{2}}}-\vec{r_{1}}\right|.$ (9)

Therefore, after stretching, if $\hat{s_{1}}>\hat{s_{2}}$, $\vec{\hat{q_{2}}}$
is more similar to $\vec{\hat{r_{1}}}$ than $\vec{\hat{q_{1}}}$, and vice
versa. We use the stretched feature of a query image as its final descriptor.

Empirically, $\alpha$ equals $2.5$, and $n$ equals $5$. Although from theory,
it is still unclear that why the descriptor stretching strategy works so well
on stabilizing the scores of different queries, its outstanding performance is
crucial to our final ranking.

##  4 Experiments

##  5 Experimental Settings

In the unsupervised pre-training part, we choose Barlow-Twins [36], and the
selected backbone is ResNet50 [10]. We do not re-train the model, and just use
the model supplied by the official implement.

In training, we select $100,000$ out of $1000,000$ images from the official
training set. The criterion is selecting one every $10$. Each image is
augmented $19$ times, and thus for training, each ID has $20$ images. The
training process takes less than one day on $4$ NVIDIA Tesla V100 32GB GPUs.
Each training batch includes $128$ images of $32$ IDs. Adam optimizer [19] is
used to optimize the networks. The image size is $256\times 256$. The model is
trained for $25$ epochs, and in each epoch, we have $8000$ iterations. The
basic learning rate is set to $3.5\times 10^{-4}$.

During testing, we use YOLOv5 [15] to detect overlay automatically. The
detected overlays substitute the original images. For training YOLOv5 [15], we
just generate some images with overlay augmentations and corresponding
bounding boxes automatically from the training dataset. We test the trained
models for four scales, $200\times 200$, $256\times 256$, $320\times 320$, and
$400\times 400$. To fuse these multi-scale features, we firstly normalize them
such that their $L_{2}$ norm equals $1$, then the normalized features are
averaged, and finally, an $L_{2}$ normalization is applied to produce the
final descriptor [35]. Then, we use descriptor stretching to rescale all the
fused query features. The fused training and reference features remain the
same.

###  5.1 Comparison with State-of-the-Arts

To prove the superiority of the strong baseline, we compare the proposed model
with state-of-the-art methods from the leaderboard in Phase $2$. The
comparison results are shown in Table 1. In the ISC2021, there are $526$
participants, and $21$ teams have submitted their final results. With a bag of
tricks, the strong baseline shows promising performance.

Table 1: Comparison with state-of-the-art methods from the leaderboard in Phase $2$. Recall@Precision $90$ is a secondary metric provided for information purposes only as an indicator of a model’s recall at a reasonably good precision level but is not used for ranking purposes. Our results are highlighted in bold. Team | Score
---|---
Micro-average Precision |  Recall@Precision 90
lyakaap |  $0.6354$ |  $0.6354$
S-square |  $0.5905$ |  $0.5086$
Ours |  0.5788 |  0.4886
forthedream2 |  $0.5736$ |  $0.4980$
Zihao |  $0.5461$ |  $0.4813$
separate |  $0.5312$ |  $0.3169$
AITechnology |  $0.5253$ |  $0.4191$
… |  … |  …
GIST [24] |  $0.0526$ |  $-$

###  5.2 Ablation Studies

The ablation studies are conducted on $25,000$ query images in Phase $1$ to
prove the effectiveness of each component in our method. The evaluation metric
is Micro-average Precision. The experimental results are displayed in Table 2.

Comparison between supervised pre-training and unsupervised pre-training.
First, we discuss the improvement brought by unsupervised pre-training. The
experimental results are denoted as “Supervised” and “Unsupervised” in Table
2, respectively. We can find that, instead of supervised pre-training, over
$14\%$ improvement has been obtained by using unsupervised pre-trained models.
This phenomenon may be because the definition of category in unsupervised pre-
training is similar to that in the task of copy detection.

The improvement from descriptor stretching. By using descriptor stretching,
the Micro-average Precision can be improved by $17.26\%$ significantly. To
prove the significant performance indeed from the comparable scores, we also
adopt two other metrics, Recall@Rank $1$ and Recall@Rank $10$. The two metrics
are calculated according to per query separately. Therefore, the incomparable
scores of queries cannot influence it. Before stretching, the two scores are
$76.18\%$ and $79.72\%$, respectively. After stretching, the two scores are
$76.10\%$ and $79.98\%$, respectively. Therefore, there is no improvement on
these two scores, which shows the performance per query is not improved, and
the overall performance improvement comes from comparable scores of queries.

The improvement from overlay detection and multi-scale testing. We discover
that when image $A$ overlays on another image $B$, image $A$ has a true match
in the reference dataset while image $B$ is the background. Therefore, when
cropping the image from the background, about $1\%$ improvement can be
observed. Multi-scale testing is a common practice in computer vision, and it
also shows the effectiveness in the ISC2021. By using multi-scale testing, the
performance can be improved from $71.49\%$ to $73.02\%$.

Table 2: The ablation study about the proposed baseline. “Supervised” or “Unsupervised” denotes the supervised pre-training or unsupervised pre-training is used, respectively. “$+$ Des-Str” denotes adding descriptor stretching strategy. “$+$ Det” denotes adding overlay detection (YOLOv5 [15]). “$+$ Multi” denotes using multi-scale testing. The best results are highlighted in bold. Method | Score
---|---
Micro-average Precision |  Recall@Precision 90
Supervised |  $0.39089$ |  $0.18133$
Unsupervised |  $0.53218$ |  $0.29693$
$+$ Des-Str |  $0.70481$ |  $0.61631$
$+$ Det |  $0.71487$ |  $0.62913$
$+$ Multi |  0.73017 |  0.63975

##  6 Conclusion

In this paper, we introduce our winning solution to the Image Similarity
Challenge at NeurIPS’21. The proposed strong baseline uses recent self-
supervised learning methods for pre-training instead of traditional supervised
ones. Further, a descriptor stretching strategy is proposed to stabilize the
scores of queries. We hope the proposed solution is beneficial for real-life
applications including content tracing, copyright infringement, and
misinformation.

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# Transport through the network of topological channels in HgTe based quantum
well

G. M. Gusev, 1 Z. D. Kvon, 2,3 D. A. Kozlov, 2 E. B. Olshanetsky, 2 M. V.
Entin, 2 and N. N. Mikhailov 2,3 1Instituto de Física da Universidade de São
Paulo, 135960-170, São Paulo, SP, Brazil  2Institute of Semiconductor Physics,
Novosibirsk 630090, Russia  3Novosibirsk State University, Novosibirsk 630090,
Russia

###### Abstract

Topological insulators represent a new quantum state of matter which is
characterized by edge or surface states and an insulating band gap in the
bulk. In a two dimensional (2D) system based on the HgTe quantum well (QW) of
critical width random deviations of the well width from its average value
result in local crossovers from zero gap 2D Dirac fermion system to either the
2D topological insulator or the ordinary insulator, forming a complicated in-
plane network of helical channels along the zero-gap lines. We have studied
experimentally the transport properties of the critical width HgTe quantum
wells near the Dirac point, where the conductance is determined by a
percolation along the zero-gap lines. The experimental results confirm the
presence of percolating conducting channels of a finite width. Our work
establishes the critical width HgTe QW as a promising platform for the study
of the interplay between topology and localization.

##  I Introduction

Topological states of matter have attracted a lot of attention due to their
numerous intriguing transport properties. In particular, in two-dimensional
topological insulators (2D TI) there are gapless conducting helical edge
channels, that are protected against backscattering kane ; hasan ; qi ; moore
; moore2 . Many experiments have been performed to investigate the transport
properties of the 2DTI edge states in several materials where the helical edge
states are due to different physical mechanisms. For example the main factor
responsible for the existence of a nontrivial topological phase in HgTe/CdTe
quantum wells (QW) is a strong spin–orbit interaction konig ; buhman ; roth ;
gusev ; olshanetsky ; rahim , whereas in InAs/GaSb double QWs the
topologically protected helical edge states in the inverted phase emerge as a
consequence of the coupling between the bands leading to the opening of a
hybridization gap knez ; knez2 ; du ; nichele ; suzuki . As has been verified
on many occasions (for review see bernevig ) generally two basic
experimentally observed features indicate the presence of ballistic helical
edge channels in submicron 2DTI samples: the conductance of the order of the
universal value $e^{2}/h$ konig ; olshanetsky  and a strong nonlocal signal
due to the helical edge states current circulating along the sample perimeter
roth ; gusev ; olshanetsky ; rahim .

Figure 1: (Color online) Schematic of zero energy topological channels in the
slab shaped sample based on HgTe quantum well of a critical width
$w_{c}\approx 6.3nm$. Yellow regions are topological insulator domains, blue
regions correspond to ordinary insulator domains. The Dirac Fermions motion
direction at a fixed spin projection is shown by arrows.

The interplay between topology and localization is another challenging object
of study both for theoreticians and experimentalists. To gain a deeper insight
into the critical behaviour of matter, theoreticians often employ network
models. Such is the case for the metal-insulator transition and also for the
transition between different phases of topological insulator onoda ; obuse ;
obuse2 ; bondesan ; yamakage ; bhardwaj . Critical phenomena related to the
integer and fractional quantum Hall effects have been successfully described
by a chiral Chalker-Coddington-like network representation of bulk transport
in the high magnetic field limit chalker . In contrast to the quantum Hall
states, the two-dimensional topological insulator-metal transition could be
represented by uncoupled counter propagating channels with opposite spin, the
so called $Z_{2}$ network model onoda ; obuse ; bondesan ; yamakage ; bhardwaj
.

It was recognized that for topological metallic states a well-defined mobility
edge, i.e. a specific energy separating the region of extended states from
that of the localized states, is expected, while the extended states in
quantum Hall effect are located at particular distinct energy values ( no
mobility edge).

As already mentioned, a network of conducting channels occurs naturally in
HgTe quantum wells of critical width $w_{c}\approx 6.3nm$. It has been shown
konig ; buhman  that if the HgTe QW width is below the critical value, the
system is an ordinary insulator with a normal energy band structure. If, on
the other hand, the QW width is above the critical, then it is a 2D
topological insulator with an inverted energy spectrum. Finally, the QW width
$w=w_{c}$ corresponds to the 2D Dirac fermion system with a gapless, single-
cone spectrum (Figure 1). In-plane fluctuations of the QW width about its
average value $w=w_{c}$, that cannot be completely avoided during the QW
growth, lead to spacial gap variations (random gap sign changes), and
therefore, to transitions between the mentioned topological phases. This
results in a network of zero energy channels running along the boundaries
separating the normal insulator and the 2D topological insulator phases
(Figure 1). In addition to the gap fluctuations there will also be variations
of the electrostatic potential due to random distribution of charged
impurities. For the Fermi energy located near the Dirac point the system
conductivity is attributed to the percolation along zero energy channels
mahmoodian ; mahmoodian2 .

In the present paper we study the transport properties of the zero gap HgTe
quantum well (QW of critical width) both for $B=0$ and in the presence of
magnetic field. We find that the conductivity of the samples lies in the
interval $(1.5-4)e^{2}/h$, as is expected for the topological network formed
by finite width conducting channels. In order to estimate the width of the
channels we performed additional measurements of the Hall effect at low
magnetic fields. Surprisingly, we find that the Hall effect near the
percolation threshold is completely suppressed in a narrow interval of carrier
densities in agreement with the percolation model mahmoodian ; mahmoodian2 .

##  II RESULTS AND DISCUSSION

###  II.1 Structural properties

Quantum wells $Cd_{0.65}Hg_{0.35}Te/HgTe/Cd_{0.65}Hg_{0.35}Te$ with (013)
surface orientations and a nominal well thickness of (6.3-6.6) nm were
prepared by molecular beam epitaxy ( see Figures 2,3). Transmission electron
microscopy (TEM) of cross sectional sample allows to study the interfaces of
layered structures directly with a spatial resolution down to an nanometer
scale. Figure 2a shows TEM images of the cross sections of the specimen with
designations of various layers constituting the structures, which allow
assessment of abruptness of interfaces as well as lateral uniformity of layer
growth yahniuk . The difference of contrasts in TEM images is due to the
difference in the chemical compositions of the layers, which allows to
determine the fluctuation of the width of the individual layers in the image
of a multilayer structure. Both images indicate that HgTe/HgCdTe interfaces
are reasonably abrupt. A histogram in Figure 2b is a display of statistical
information on HgTe well width fluctuations. The histogram follows a normal
Gaussian distribution with mean with $d_{c}\approx 6.1nm$ with standard
deviation of 0.6 nm. As we mentioned above, the width fluctuations near
topological transition lead to spacial gap variations, and therefore, to
transitions between the topological phases.

Figure 2: (Color online) (a) TEM images of cross sections of sample. (b) Histogram displaying the distribution of well widths across the whole image. sample |  $w$ (nm) |  $V_{CNP}$ (V) | $\rho_{max}(h/e^{2})$ | $\mu(V/cm^{2}s)$
---|---|---|---|---
1 | 6,3 | -0.6 | 0.35 | 56.000
2 | 6.4 | -1.28 | 0.27 | 90.000
3 | 6.3 | -3 | 0.27 | 59.600
4 | 6.3 | -4.3 | 0.31 | 58.600
5 | 6.3 | -4.7 | 0.29 | 46.400
Table 1:  Some of the typical parameters of the electron system in HgTe
quantum well at T=4.2K.

###  II.2 Gapless Dirac fermions and Drude conductivity

Recently it has been demonstrated that HgTe quantum well with a critical width
$w_{c}=6.3nm$ constitutes a system of 2D fermions with a single Dirac cone
spectrum buttner ; kvon ; kvon2 ; gusev2 . The fluctuations of the QW width
result in a random potential in the bulk of the QW. In QW of critical width
these fluctuations lead to the formation of a random network of zero energy
lines, as shown in Figure 1. We believe that the physical properties of this
network are described by $Z_{2}$ quantum network model, and below we present
experimental evidence that supports this assumption.

Figure 3: (Color online) Resistance as a function of the gate voltage for
different samples, T=4.2K. Left bottom- schematic structure of the sample.
Right bottom - top view of the sample.  Figure 4: (Color online)(a) The
minimum conductivity and mobility at $n_{s}=10^{11}cm^{-2}$ for 24 samples.
Insert- Histogram displaying the distribution of the conductivity minimum. (b)
Conductivity of the five representative samples near the CNP as a function of
$V_{g}$. Insert- conductivity of sample 1 in a wider range of the gate
voltages. Red line-linear $n_{s}$ dependence.

Devices for transport measurements ( see chapter method) are specially
designed for multi-terminal measurements and consists of three narrow ( $50\mu
m$ wide) consecutive segments of different length (100, 250, 100 $\mu m$ )and
seven voltage probes (see Figure 3, left bottom panel). A dielectric layer was
deposited (100 nm of $SiO_{2}$ and 100 nm of $Si_{3}Ni_{4}$) on the sample
surface and then covered by a TiAu gate (see Figure 3, left top panel). In
conventional transport measurements the current is applied between contacts
1–6 and the potential difference is measured between contacts 2–3, 3–4 and 4–5
of the sample. Figure 3 shows the resistivity $\rho(Vg)$ at zero magnetic
field for five samples fabricated from different wafers and at different time.
The table 1 lists the typical parameters of the devices, such as the well
width $w$, the gate voltage corresponding to the Dirac point position
$V_{CNP}$ , the resistivity value at the CNP $\rho_{max}$ and the electron
mobility $\mu=\sigma/n_{s}e$ for the density $n_{s}=10^{11}cm^{-2}$. The
Figure 4a shows the conductivity distribution for 24 samples, grown during a
five years period. One can see that the conductivity values lie in the
interval $(1.5-4)e^{2}/h$. Figure 4b shows the conductivity as a function of
gate voltage $V_{g}$ for five representative samples with parameters listed in
table 1. The $\sigma(V_{g})$ dependence is symmetric and nearly parabolic
close to the CNP for low electron densities $(n_{s}<2\times 10^{10}cm^{-2})$
and approximately linear in electron density for higher density values. It is
convenient to consider the total conductivity as a sum of the network
conductivity $\sigma_{nw}$ and the bulk conductivity $\sigma_{2D}$:
$\sigma_{tot}=\sigma_{nw}+\sigma_{2D}$.

Let us start our analysis with the bulk contribution to the conductivity. The
unscreened Coulomb disorder induced by randomly distributed charge impurities
has been considered as the dominant mechanism of scattering in HgTe quantum
well because of a very large HgTe dielectric constant and small Dirac fermions
effective mass. Two distinct transport regimes can be indicated. For large
carrier density the Boltzmann transport theory is valid kvon3 . On approaching
the Dirac point (low carrier density), at a certain energy there will be an
Anderson transition from the higher energy delocalized states to the localized
states in narrow impurity bands located in small direct or inverted energy
gaps (see Figure 1). It would be reasonable to suppose that transport via the
delocalized states near the boundary with the localized bands can be described
using the ordinary expression for mobility of carriers with a parabolic
spectrum in the presence of impurity scattering. In this case the mobility is
given by $\mu=A(n_{s}/n_{i})$, where $n_{i}$ is the impurity density,
$A=(\varepsilon_{s}/e^{3})(16\pi\hbar^{3}/m_{DF}^{2})$, $m_{DF}$ is the
effective mass at the Fermi level davies . We get $\sigma_{2D}=e\mu n_{s}$
with quadratic dependence on electron density for $n_{s}<2\times
10^{10}cm^{-2}$, as shown in Figure 4b. At higher electron densities the
electron energy spectrum becomes linear and one has to use the transport time
expression calculated in kvon3 , which leads to a conductivity $\sigma(n_{s})$
nearly linear with the electron density. Such conductivity behaviour is shown
in Figure 4b.

###  II.3 Comparison with the network model

Transport in HgTe quantum wells of a critical width $w_{0}$ is believed to be
governed by the energy gap fluctuations leading to the formation of the
topological channels network (Figure 1). One can assume that the QW width
fluctuations $\delta w$ can be described by the Gaussian distribution around
the medium width value $w$, which follows from TEM images of the sample
(Figure 2).

The percolation network can be characterized by a dimensionless parameter
$\xi=\mbox{erf}((w-w_{0})/\sqrt{2}\delta w)$, where $\mbox{erf}(x)$ is the
complementary error function. The unavoidable QW width fluctuations lead to a
sample breaking-up into domains with positive (the ordinary insulator (OI))
and negative (the topological insulator (TI)) gap signs. If
$|w-w_{0}|\gg\delta w$ and $w>w_{0}$ ($\xi\to 1$), then rare OI domains are
embedded in TI domains. If, on the other hand, $w<w_{0}$, $\xi\to 1$, then
rare TI domains are embedded in OI domains. For $w\to w_{0}$
($\xi\to\xi_{c}=0.5$) these domains are mixed in approximately equal
proportion.

At the CNP the low-temperature the bulk conductivity of the OI or the TI
domains should be close to zero. However, in the immediate vicinity of the
lines separating the OI and TI domains (the zero gap line (ZGL)) the QW is a
nearly gapless conductor. In fact, any such ZGL can be considered as a quantum
wire. The sample conductance will be non-zero if such a line runs from one end
of the sample to the other. On approaching the percolation threshold there
forms a whole network of ZGLs covering the entire sample.

If the gap fluctuations are smooth, the wire associated with the ZGL will have
many 1D subbands occupied. The important point is that two of these subbands
will be the Weil states with a linear dispersion topologically protected from
electron backscattering. The Weil states are similar to the edge states in
Bernevig-Hughes-Zhang model of the 2D TI edge. In a small ballistic sample the
edge state yields a dissipationless conductance equal to the conductance
quantum.

In a large random system with a size exceeding the QW width fluctuation
correlation length the ZGLs form a dense network. In this case the 2D
conductivity is a combination of the ideal conductance of a Weil state and the
conductance associated with tunnelling between different ZGLs lying close to
each other.

The hopping between the ZGLs is realized via intermediate Dirac states, which
lowers the hopping energy (as compared to the total gap) and raises the
hopping probability amplitude. In any case one can use the characteristic
hopping length between different ZGLs as some given quantity. Our
consideration is based on geometric fractal properties of ZGLs.

We suppose that in our sample $|\xi-\xi_{c}|<<1$ so that a network of OI and
TI domains is formed and the ZGLs cover the entire sample. It is worth noting,
however, that in a realistic random network of channels a low temperature non-
zero 2D conductivity may occur only if electrons can tunnel between the
adjacent channels, for which a finite channel width is required mahmoodian2 .
One can estimate that :

$\frac{|w-w_{c}|}{w_{c}}\sim\left(\frac{\hbar v}{a\alpha w_{c}}\right)^{r}$
(1)

where $\alpha=\partial\Delta(w)/\partial w|_{w=w_{c}}$, $\Delta$ is the rms
energy gap fluctuation value, $v$ is the electron velocity, $r$ is parameter,
which is close to 0.45 mahmoodian2 . The conductivity of the network can be
estimated from the equation mahmoodian2 :

$\sigma=\frac{e^{2}D}{2\pi h\hbar va}\sim\frac{e^{2}}{h}\left(\frac{\Delta
a}{hv}\right)^{p}$ (2)

where $D\simeq av(\Delta a/\hbar v)^{p}$ is the diffusion coefficient , $p$ is
a coefficient close to 0.15. We estimate the corresponding HgTe QW parameters
from mahmoodian2  : $hv=560meV$;
$\alpha\approx-8.75meV/nm$,$\Delta\approx|\alpha|\sqrt{\overline{w^{2}}-\overline{w}^{2}}\approx
4meV$, $a=30$ nm.

Combining all parameters we obtain $\sigma_{(}n_{w})=(e^{2}/h)(1\pm 0.1)$,
which is smaller than the average experimental value
$\sigma_{exp}=(e^{2}/h)(2.5\pm 1)$. This disagreement can be diminished if one
takes into account the edge states nonzero width. This results in new
percolation paths and, correspondingly, in higher $\sigma_{nw}$ values and
also in a much stronger $\sigma_{nw}$ dispersion in agreement with the
experiment.

The percolation model mahmoodian2  predicts a non-zero conductivity in a
narrow width interval near the critical value $w_{c}$ due to the suppression
of backscattering of electrons propagating alone ZGLs and the growing hopping
between adjacent ZGLs as $\overline{w}\rightarrow w_{c}$. It is expected, that
the network conductivity vanishes outside of this percolation threshold. One
can estimate the energy and the charge density interval where percolation
conductivity occurs. However, in the experiment the total conductivity does
not show a crossover from $\sim e^{2}/h$ values to zero beyond the percolation
threshold, expected from the network model. Instead we observe a smooth
conductivity growth with density near the CNP (Figure 3 and Figure 4b). As we
already discussed above, the percolation conductivity is short-circuited by
the conductivity of the bulk electrons.

###  II.4 The Hall effect in topological channel system

The percolation model also predicts other characteristic peculiarities in the
transport coefficients behavior as well. One of the key effects proving the
presence of a network structure in the system studied is the quenching of the
Hall effect at low magnetic field.

The Figures 5a,b show the Hall resistivity as a function of gate voltage and
magnetic field. Figure LABEL:fig:5b shows a number of representative
$\rho_{xy}(B)$ curves for $V_{g}=V_{CNP}$, $V_{g}<V_{CNP}$ and
$V_{g}>V_{CNP}$. One can see a very narrow $\Delta V_{g}\simeq 0.05V$ region
near the CNP, where $\rho_{xy}\approx 0$. Figures 5 a,b show that the Hall
resistivity is flat and close to zero in the interval of magnetic field
$-0.1T<B<0.1T$. So, according to experiment there is a quenching of the Hall
effect at the CNP in the energy interval corresponding to the density
variation $\delta n_{s}\sim 5\times 10^{9}cm^{-2}$.

From the theoretical point of view one can treat this situation as follows. A
curved 1D ZGL state can be mapped on the QW plane using the coordinate
$\textbf{r}(s)$ , where $s$ is the length along the edge and
$\textbf{s}=dr(s)/ds$ is the tangent to the curve. If we take the magnetic
field $\textbf{B}=(0,0,B)$ and the corresponding vector potential
$\textbf{A}=\textbf{B}\times\textbf{s}$ then the Hamiltonian of an electron in
the 1D ZGL state will be
$H=\sigma\nu_{F}(p-e[\textbf{B}\times\textbf{r}(s)]\textbf{s}/c)$ , where
$\sigma$ is a spin index and $\nu_{F}$ is the Fermi velocity. The gauge
transform $U=exp\left(i(e/c)\int[\textbf{B}\times\textbf{r}(s)]ds\right)$
converts the Hamiltonian $H$ to $\sigma p$ . In this way, as one can see, the
magnetic field can be effectively eliminated from the Hamiltonian for all
open-ended ZDLs which define the character of the low-temperature transport.
And this cancels the Hall current as well. Indeed, the above gauge transform
shows that the effect of magnetic field on such a system will be null unless
the ZGL are closed or are not single-connected. But in general the ZGL do not
branch out. In fact, the presence of a branching point $(x_{0},y_{0})$ means a
simultaneous fulfillment of three conditions: $\triangle(x_{0},y_{0})=0$,
$\partial_{x_{0}}(x_{0},y_{0})=0$ and $\partial_{y_{0}}(x_{0},y_{0})=0$, which
is practically impossible. Hence, the theory predicts a zero Hall current in
the presence of magnetic field, in agreement with the experimental.

Figure 5: (Color online) (a) The Hall resistivity as a function of the gate
voltage and magnetic field near the CNP. (b)The Hall resistivity near the CNP
for different values of gate voltage.

It worth noting that the bulk contribution to the Hall current will also be
absent. Indeed, under these conditions the Fermi energy will be located in a
narrow band of localized states. One can estimate this band width $\delta_{b}$
from the density interval $\delta n_{s}=5\times 10^{9}cm^{-2}$ corresponding
to the quenching of the Hall effect (see above). Using the value of DoS
obtained from capacitance measurements kozlov  yields $\delta_{b}$ = 3 meV.
This estimate means that the band width of localized states is close to the
characteristic disorder amplitude in HgTe quantum well.

##  III Methods

Quantum wells $Cd_{0.65}Hg_{0.35}Te/HgTe/Cd_{0.65}Hg_{0.35}Te$ with (013)
surface orientations and a nominal well thickness of (6.3-6.6) nm were
prepared by molecular beam epitaxy. As shown in the previous publications
gusev , the use of substrates inclined to the singular orientations
facilitates the growth of more perfect films. Therefore, the growth of alloys
is performed predominantly on the substrates with surface orientation [013],
which deviates from the singular orientation by approximately $19^{\circ}$.
Fabrication of ohmic contact to HgTe quantum well is similar to that for other
2D systems, such as GaAs quantum wells, for example: the contacts were formed
by the burning-in of indium directly on the surface of large contact pads.
Modulation-doped HgTe/CdHgTe quantum wells are typically grown at 180o C,
which is relatively low compared to III-V compounds. On each contact pad the
indium diffuses vertically down, providing ohmic contact to the underlying
quantum well, with the contact resistance in the range of 0.1-1 kOhm. During
the AC measurements we made sure that the Y-component of the impedance did not
exceed 5% of the total impedance, which is the indication of good ohmicity of
the contacts. The sample is a Hall bar device with eight voltage probes. The
bar has the width $W$ of $50\mu m$ and three consecutive segments of different
lengths $L$ $(100,250,100\mu m)$ (Figure 3, left bottom panel). A dielectric
layer was deposited (100 nm of $SiO_{2}$ and 100 nm of $Si_{3}Ni_{4}$) on the
sample surface and then covered by a TiAu gate. The density variation with
gate voltage was $1\times 10^{11}cm^{-2}V^{-1}$. The magnetotransport
measurements were performed in the temperature $4.2K$ using a standard four
point circuit with a $1-13Hz$ ac current of $1-10nA$ through the sample, which
is sufficiently low to avoid overheating effects.

##  IV Summary and conclusion

In conclusion, our results provide an opportunity to take a fresh look at the
nature of electron transport in HgTe quantum wells of critical thickness. The
description of transport in this system in terms of percolation through a
network of one-dimensional conducting channels makes it possible to study the
effects caused by the interplay of topology and localization. More generally,
the study of transport in zero gap HgTe quantum wells can improve our
understanding of the disorder induced topological insulator-to-metal
transition and may be important for a wider class of disordered 2D electron
systems, than was considered previously.

##  V Acknowledgment

G.M. Gusev acknowledges for financial support FAPESP (Brazil) and CNPq
(Brazil); Z. D. Kvon, D. A. Kozlov, E.B.Olshanetsky, M.V. Entin and N. N.
Mikhailov acknowledge for financial support Ministry of Science and Higher
Education of the Russian Federation, Grant No. 075-15-2020-
797(13.1902.21.0024).

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