11institutetext: Zuminternet, Seoul, South Korea
11email<EMAIL_ADDRESS>
# Data-Efficient Deep Learning Method for Image Classification Using Data
Augmentation, Focal Cosine Loss, and Ensemble

Byeongjo Kim  Chanran Kim  Jaehoon Lee  Jein Song  Gyoungsoo Park

###### Abstract

In general, sufficient data is essential for the better performance and
generalization of deep-learning models. However, lots of limitations(cost,
resources, etc.) of data collection leads to lack of enough data in most of
the areas. In addition, various domains of each data sources and licenses also
lead to difficulties in collection of sufficient data. This situation makes us
hard to utilize not only the pre-trained model, but also the external
knowledge. Therefore, it is important to leverage small dataset effectively
for achieving the better performance. We applied some techniques in three
aspects: data, loss function, and prediction to enable training from scratch
with less data. With these methods, we obtain high accuracy by leveraging
ImageNet data which consist of only 50 images per class. Furthermore, our
model is ranked 4th in Visual Inductive Printers for Data-Effective Computer
Vision Challenge.

###### Keywords:

Data-Efficient, Deep-Learning, Image Classification, Data Augmentation, Cosine
Loss, Focal Loss, Ensemble

##  1 Introduction

There are limits to collecting a lot of data used to train deep-learning
models. A transfer learning and a pre-training technique trained with large
image dataset are used to achieve good performance even if there is small
dataset. However, most large image dataset consist of images collected from
web, which may be licensed or commercially prohibited [1]. Therefore, in some
cases, pre-trained weights trained with ImageNet [4] may not be applied for
commercial applications.

Nowadays it is known that self-training can replace pre-trained model and even
perform better [10, 6]. However, self-training requires a lot of data that is
not labeled. So data problems eventually arise again. Therefore, even if given
dataset is small, we should use that data very effectively.

Visual Inductive Printers for Data-Effective Computer Vision Challenge [5] is
a challenge improving performance using small dataset productively. There are
only 50 images per class, and as with the ImageNet dataset, there are total
1000 classes. This challenge also restricts the use of external data such as
pre-trained weights, ontology knowledge. Due to the small dataset, it is
difficult to perform well with general training.

We solve this problem by experimenting with various techniques. First, we
propose LSB Swap data augmentation technique not only to make various data
available but also to improve the robustness of the model. Then, we apply
Focal Cosine Loss function to enable model training well with even a small
amount of data. Last, we apply voting ensemble to compensate for the wrong
answers. Through the model applied with these techniques, the Visual Inductive
Printers for Data-Effective Computer Vision Challenge recorded 0.67 top-1
accuracy and ranked 4th.

##  2 Method

###  2.1 LSB Swap

Figure 1: Exchange $k$ LSB from all pixels in the image with different images.
$k$ is set to 2 in the example.

We apply two data augmentation techniques in the two aspects. First,
RandAugment [3] is used to transform the data and to create the effect of
training with a lot of data. Secondly, LSB(Least Significant Bit) Swap is
proposed and applied to focus on structural features for classifying objects.
The process of LSB Swap can be seen in Figure 1.

Because changing a few last bits do not make a difference to be judged by
humans, this technique allows data augmentation by swapping non-critical parts
of the image classification task. In addition, unlike other data augmentations
which use random noise, the LSB Swap helps to use natural noise values present
in the images. In environments with less training data, LSB Swap keeps these
values from training as much as possible, and helps the model focus on
structural features.

###  2.2 Focal Cosine Loss

We used cosine loss, which is known as performing well when training without
pre-trained weights and with small dataset [1]. Let $x$ be image, and $y$ be
the class label of $x$. $\varphi(y)$ is a one-hot vector mapped each class
label. $f_{\theta}(x)$ denotes embedding(i.e., CNN) with learned parameters
$\theta$ from $x$ to feature space. $\psi(\cdot)$ denotes embedding(i.e.,
fully-connected layer) from features into prediction space. The cosine loss is
defined as

$\displaystyle L_{cos}(x,y)=1-<\varphi(y),\psi(f_{\theta}(x))>$ (1)

where $<\cdot,\cdot>$ denotes the dot product. For classification accuracy,
Barz et al. [1] used cosine loss with a categorical cross-entropy loss. In
addition, we add a focal loss [7] to focus more on classes which are difficult
to classify. The Focal Cosine Loss is defined as

$FL(p,\hat{p})=-\sum_{h}p^{(h)}(1-\hat{p}^{(h)})^{\gamma}log(\hat{p}^{(h)})$

$\displaystyle
L_{cos+focal}(x,y)=1-<\varphi(y),\psi(f_{\theta}(x))>-\lambda\cdot
FL(\varphi(y),g_{\theta}(\psi(f_{\theta}(x))))$ (2)

where $p^{(h)}$ and $\hat{p}^{(h)}$, respectively, refer to the probability of
label $h$ in correct answers and the predictions. Moreover, $g_{\theta}$
denotes an additional fully-connected layer with softmax activation function.
Furthermore, we use two hyper-parameters $\gamma$ for focusing and $\lambda$
for combination, following [1, 7].

###  2.3 Plurality Voting Ensemble

We use plurality voting technique of hard voting ensemble to compensate for
the difficult problems through various models and to produce good performance.
The plurality voting ensemble algorithm is shown in Algorithm 1. If the
predicted confidence score($s_{j}$) for a particular problem in the base
model($B$) is less than a certain $threshold$, it is replaced with voting
result($Mode(V_{j})$) of predictions for various models($K$).

Algorithm 1 Plurality Voting Ensemble

1:procedure PluralityVoting($B,K,T$)

2: $B$: Base classification model

3: $K$: $m$ Classification models$\triangleright$ $B$ also belongs to $K$

4: $T$: $n$ Test data

5: $V_{j}$: Votes from $m$ classifiers using $j$ th test data

6: $O$: results of plurality voting

7: for $j\leftarrow 1$ to $n$ do

8: $s_{j}\leftarrow$ Prediction result of $T_{j}$ from $B$

9: if confidence score of $s_{j}$ $<$ $threshold$ then

10: for $i\leftarrow 1$ to $m$ do

11: $v_{i}\leftarrow$ Prediction result of $T_{j}$ from $K_{i}$

12: append $v_{i}$ to $V_{j}$

13: end for

14: $O_{j}\leftarrow Mode(V_{j})$

15: else

16: $O_{j}\leftarrow s_{j}$

17: end if

18: end for

19: return $O$

20:end procedure

###  2.4 Other Techniques

Various techniques are also applied. First, EMA(Exponential Moving Average)
model is used to adjust parameters when the model is training, and those
parameters are used for testing. Second, drop out and drop connection are used
for preventing overfitting. Last, the TTA(Test Time Augmentation) technique is
used, which also have a single ensemble effect.

##  3 Experiments

###  3.1 Setup

We use open source named timm [8] for training. The code is developed by
PyTorch and is capable of applying the recent image classification models and
many techniques. All hyper-parameters such as optimizer, learning rate, etc.
are set as recommended by timm. For LSB Swap, Focal Cosine Loss, and plurality
voting ensemble described above, we developed using Python, NumPy, and
PyTorch. The hyper-parameters in Focal Cosine Loss, $\lambda$ and $\gamma$ are
set to 0.1 and 2.0 respectively. In addition, $threshold$ for plurality voting
ensemble is set to 0.7. When we apply TTA technique, we use the same
RandAugment used in training and soft voting ensemble with confidence scores
of 1000 classes. Last, k in LSB Swap is set to 2. We trained model using only
the data given, not the pre-trained weights and external data. The given
training data and validation data are combined to be used in training.

###  3.2 Results

Table 1: 10 models trained using proposed techniques and given data. Models
---
EfficientNet-B3 [9]
EfficientNet-B3 with TTA
EfficientNet-B3 with LSB Swap
EfficientNet-B3 with LSB Swap + TTA
EfficientNet-B4
EfficientNet-B4 with TTA
EfficientNet-B5
EfficientNet-B5 with TTA
EfficientNet-B7
EfficientNet-B7 with TTA

Table 1 shows that 10 models which is trained with given data and techniques.
The RandAugment, cosine loss, and other techniques are applied to all models,
and LSB Swap is applied only to EfficientNet-B3.

Because there is no answer for the test data, accuracy could not be
calculated. However, in the case of EfficientNet-B3, the accuracy could be
obtained by submitting it to the competition(EfficientNet-B3 in Table 2).
Because the size of EfficientNet-B3 model is still small, we use other larger
models as shown in Table 1. Then, we applied plurality voting ensemble as
shown in Figure 2 with 10 trained models using EfficientNet-B7 as the base
model(Ensemble model in Table 2). It has improved 0.09 accuracy more than the
EfficientNet-B3.

Table 2: The accuracy we got from submitting to the competition. Model | Accuracy
---|---
EfficientNet-B3 | 0.5823
Ensemble model | 0.6730
Figure 2: An example of applying plurality voting ensemble to 10 models. Base
model $B$ is set to EfficientNet-B7(orange box) and $threshold$ is set to 0.7.

##  4 Conclusions

We apply techniques such as LSB Swap and Focal Cosine Loss to train with only
a small amount of data. In addition, even if it is difficult to predict with
the best single model, the final result can be supplemented through the
proposed ensemble method.

Because the deadline is set for the competition, it is hard to test all the
techniques. For example, LSB Swap data augmentation technique apply only to
EfficientNet-B3. However, considering that the training loss of
EfficientNet-B3 applied with LSB Swap is lower than the one not applied, we
think it will be effective in other models as well. All source code used for
training and testing is provided [2], and we informs that pre-trained weights
and external data are not used.

## References

  * [1] Barz, B., Denzler, J.: Deep learning on small datasets without pre-training using cosine loss. In: The IEEE Winter Conference on Applications of Computer Vision. pp. 1371–1380 (2020)
  * [2] byeongjokim: VIPriors Image Classification Challenge Source Code (2020 (accessed July 14, 2020)), https://github.com/byeongjokim/VIPriors-Image-Classification-Challenge
  * [3] Cubuk, E.D., Zoph, B., Shlens, J., Le, Q.V.: Randaugment: Practical automated data augmentation with a reduced search space. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops. pp. 702–703 (2020)
  * [4] Deng, J., Dong, W., Socher, R., Li, L.J., Li, K., Fei-Fei, L.: Imagenet: A large-scale hierarchical image database. In: 2009 IEEE conference on computer vision and pattern recognition. pp. 248–255. Ieee (2009)
  * [5] vipriors ewi: 1st Visual Inductive Priors for Data-Efficient Deep Learning Workshop (2020 (accessed July 14, 2020)), https://vipriors.github.io/
  * [6] He, K., Girshick, R., Dollár, P.: Rethinking imagenet pre-training. In: Proceedings of the IEEE international conference on computer vision. pp. 4918–4927 (2019)
  * [7] Lin, T.Y., Goyal, P., Girshick, R., He, K., Dollár, P.: Focal loss for dense object detection. In: Proceedings of the IEEE international conference on computer vision. pp. 2980–2988 (2017)
  * [8] rwightman: pytorch-image-models (2020 (accessed July 14, 2020)), https://github.com/rwightman/pytorch-image-models
  * [9] Tan, M., Le, Q.V.: Efficientnet: Rethinking model scaling for convolutional neural networks. arXiv preprint arXiv:1905.11946 (2019)
  * [10] Zoph, B., Ghiasi, G., Lin, T.Y., Cui, Y., Liu, H., Cubuk, E.D., Le, Q.V.: Rethinking pre-training and self-training. arXiv preprint arXiv:2006.06882 (2020)


# Effect of spin-orbit coupling in one-dimensional quasicrystals with power-
law hopping

Deepak Kumar Sahu<EMAIL_ADDRESS>Department of Physics and Astronomy,
National Institute of Technology, Rourkela, Odisha, India  Sanjoy Datta
<EMAIL_ADDRESS>Department of Physics and Astronomy, National Institute of
Technology, Rourkela, Odisha, India  Center for Nanomaterials, National
Institute of Technology, Rourkela, Odisha, India

(August 27, 2024)

###### Abstract

In the one-dimensional quasiperiodic Aubry-André-Harper Hamiltonian with
nearest-neighbor hopping, all single-particle eigenstates undergo a phase
transition from ergodic to localized states at a critical disorder strength
$W_{c}/t=2.0$. There is no mobility edge in this system. However, in the
presence of power-law hopping having the form $1/r^{a}$, beyond a critical
disorder strength mobility edge appears for $a>1$, while, for $0<a\leq 1$, a
multifractal edge separates the extended and the multifractal states. In both
these limits, depending on the strength of the disorder, lowest $\beta^{s}L$
states are delocalized. We have found that, in the presence of the spin-orbit
coupling, the critical disorder strength is always larger irrespective of the
value of the parameter $a$. Furthermore, we demonstrate that for $a\leq 1$, in
the presence of spin-orbit coupling, there exists multiple multifractal edges,
and the energy spectrum splits up into alternative bands of delocalized and
multifractal states. Moreover, the location of the multifractal edges are
generally given by the fraction $(\beta^{s}\pm\beta^{m})$. The qualitative
behavior of the energy spectrum remains unaffected for $a>1$. However, in
contrast to the previously reported results, we find that in this limit,
similar to the other case, multiple mobility edges can exist with or without
the spin-orbit coupling.

##  I Introduction

Understanding the nature of the eignestates in disordered systems has
continued to attract the attention since the publication of the seminal
article of P. W. Anderson Anderson . It was argued that in three dimensions,
beyond a critical disorder strength all the single particle electronic
eigenstates fall of exponentially from the point of their maximum amplitude,
while for weak disorder a mobility edge separates the delocalized and
localized states. Furthermore, in one and two dimensions all the electronic
states are localized even for arbitrarily weak disorder Mott-Twose ; Borland ;
Berezinskii . However, the existence of delocalization-localization (DL)
transition has been predicted in two-dimensional system with random disorders
in the presence of spin-orbit coupling Hikami , and there are numerical
evidences in support of this assertion Evangelou ; Asada-2D-SOC-PRL02 ; Su-
Wang . In one-dimension, however, there is still no evidence of the DL
transition, in systems with random disorder.

On the other hand, quasiperiodic lattices are paradigmatic examples of systems
which hosts all the richness of randomly disordered systems, even in one-
dimension. The minimal Hamiltonian describing such systems is the Aubry-André-
Harper (AAH) model Harper ; Azbel ; Aubry-Andre  having nearest-neighbor
hopping and a cosine potential whose periodicity is modulated by the inverse
golden ratio. AAH Hamiltonian shows a DL transition at a critical strength of
the potential. All the energy eigenstates are extended below the critical
point, while all of them are localized above it. In contrast to the lattice
with random disorders, there is no mobility edge in the energy spectrum at any
value of the disorder strength in the original AAH Hamiltonian.

Recently, however, it has been demonstrated Deng  that mobility edge can exist
in the one-dimensional (1D) quasiperiodic lattice if the hopping amplitude $t$
is replaced by a power-law hopping of the form $t/{r^{a}}$, where $r$ is the
distance between two lattice sites $i$ and $j$ $(i\neq j)$. It is evident that
this generalized Aubry-André (GAA) Hamiltonian converges to the minimal AAH
Hamiltonian in the limit $a>>1.$ The spectrum shows much richer structure when
$``a^{\prime\prime}$ is not too large compare to unity. It is observed that
for $a>1$ and $a\leq 1$ there exists a critical disorder strength, below which
all the states delocalized. Interestingly, above the critical disorder
strength the energy spectrum shows a mobility edge when $a>1$, while a
multifractal edge appears for $a\leq 1$ which separates the extended and the
multifractal states.

Since spin-orbit coupling has dramatic effect on the randomly disordered
systems, it is natural to wonder what happens to the energy spectrum and to
the critical properties of the GAA Hamiltonian in the presence of spin-orbit
coupling. To address these questions, in this article, we have studied the 1D
GAA Hamiltonian in the presence of spin-orbit coupling of Rashba type. The
Rashba Hamiltonian consists of a spin preserving hopping and a spin-flip
hopping. It has recently been reported dsahu  that the qualitative structure
of the energy spectrum remains intact for the minimal AAH Hamiltonian ($a>>1$
limit of the GAA Hamiltonian) in the presence of the Rashba spin-orbit (RSO)
coupling, while the critical point moves to a higher strength of the disorder
potential. In this article, however, we demonstrate that in the presence of
the RSO coupling, the energy spectrum of the GAA Hamiltonian shows distinctly
different behaviour compared to the original GAA Hamiltonian, especially when
$a\leq 1$. We have found that, in the presence of the RSO coupling, beyond the
critical disorder strength the energy spectrum becomes fragmented and multiple
multifractal edges appear instead of a single multifractal edge observed in
the pure GAA Hamiltonian. On the other hand, for $a>1$, we have found that
above the critical point there are certain stretches of disorder strengths in
which a single mobility edge exists in the pure GAA Hamiltonian. These windows
of single mobility edge gradually disappear with an increase in the RSO
coupling strength. Furthermore, in all the limits the critical disorder
strength increases in the presence of the RSO coupling.

Figure 1: (i) Projection of IPR as a function of $W/t$ and eigen-energies
$(index)$ for the GAA Hamiltonian ($L=1597$) in the limit $a\gg 1$. (ii) and
(iii): IPR $(I_{n})$ vs eigen-energies $(index)$ at $W/t=1.0$ and $W/t=2.5$
respectively.

Figure 2: (i) Projection of IPR as a function of $W/t$ and eigen-energies
$(index)$ in the limit of the short-range ($a=1.5$) GAA Hamiltonian
($L=1597$). Dotted lines show the presence of single mobility edge regions.
(ii) and (iii): IPR $(I_{n})$ vs eigen-energies $(index)$ at $W/t=0.5$ and
$W/t=1.2$ respectively.

Figure 3: (i) Projection of IPR as function of $W/t$ and eigen-energies
$(index)$ in the limit of long-range ($a=0.5$) GAA Hamiltonian ($L=1597$).
Dotted lines show the presence of single multifractal edge regions. (ii) and
(iii): IPR $(I_{n})$ vs eigen-energies $(index)$ at $W/t=0.2$ and $W/t=0.5$
respectively.

This article is organized as follows. In Sec. II, we have discussed the model
Hamiltonian, and different parameters are used in this ambiance. In Sec. III,
we have shown the presence of mobility and multifractal edges with the help of
the Inverse Participation Ratio (IPR). We have done the multifractal analysis
of the RSO coupled GAA model to analyze the multifractal edges in Sec. IV.

##  II Generalized Aubry-André Hamiltonian With RSO coupling

The Hamiltonian considered in this work is given by,

$H=H_{G}+H_{R},$ (1)

where, $H_{G}$ is the GAA model considered in Ref. Deng , and it is given as,

$H_{G}=-t\displaystyle\sum_{\langle i,j\rangle,\sigma}^{L}\frac{1}{\lvert
i-j\rvert^{a}}(c^{\dagger}_{{i},{\sigma}}c_{{j},{\sigma}}+c^{\dagger}_{{j},{\sigma}}c_{{i},{\sigma}})+\\\
W\displaystyle\sum_{{i=1},{\sigma}}^{L}cos(2\pi\beta
i+\phi)c^{\dagger}_{{i},{\sigma}}c_{{i},{\sigma}}.$ (2)

In the above Hamiltonian $H_{G}$, $t/\lvert i-j\rvert^{a}$ is the rate of
hopping from site $i$ to site $j$ and $L$ is the size of the quasiperiodic
lattice, $c^{\dagger}_{{i},{\sigma}}$ and $c_{{i},{\sigma}}$ are the fermionic
creation and annihilation operators respectively at the lattice site $i$ with
the spin $\sigma(\uparrow/\downarrow)$.$W$ is the strength of the
quasiperiodic potential, and $\phi$ is an arbitrary phase varying from
$(0,2\pi)$. Our conclusions are independent of the value of $\phi$.
Henceforth, we set $\phi=0$ for all our calcuations. The quasiperiodicity is
introduced via the irrational constant quantity $\beta$ which has been set to
$(\sqrt{5}-1)/2$.

The RSO Hamiltonian $H_{R}$ is given by Birkholz ,

$H_{R}=-\alpha_{z}\displaystyle\sum_{{i=1},{\sigma},{\sigma^{\prime}}}^{L-1}t(c^{\dagger}_{i+1,\sigma}(i\sigma_{y})_{\sigma,\sigma^{\prime}}c_{i,\sigma^{\prime}}+h.c.)+\\\
\alpha_{y}\displaystyle\sum_{{i=1},{\sigma},{\sigma^{\prime}}}^{L-1}t(c^{\dagger}_{i+1,\sigma}(i\sigma_{z})_{\sigma,\sigma^{\prime}}c_{i,\sigma^{\prime}}+h.c.),$
(3)

where $\sigma_{y}$ and $\sigma_{z}$ are Pauli’s spin matrices in $y$\- and
$z$-direction respectively. $\alpha_{y}$ is a complex spin-conserving hopping
due to the confinement in $y$-direction and $\alpha_{z}$ is a spin-flip
hopping due to the confinement in $z$-direction. The RSO Hamiltonian $H_{R}$
considered here, has been studied in the context of quantum nanowires Ando-
PRB89 ; Janssen-Huckestein . Also, recently, localization property of
attractive fermions have been studied by taking only the spin-flip component
$(\alpha_{z})$ of the RSO HamiltonianMinakuchi . In this article, we mainly
focus on the two limits of the Hamiltonian: long-range $(0<a\leq 1)$ and
short-range $(a>1)$ hoppings. In the limit $a\gg 1$, $H_{G}$ converges to the
pure AAH Hamiltonian. The pure AAH Hamiltonian in the presence of the RSO
coupling has been studied in Ref. dsahu .

##  III Inverse Participation Ratio

The Inverse participation ratio (IPR) is an useful indicator to distinguish
localized and delocalized states in the energy spectrum. It gives a measure of
the number of sites contributing to a given stateKinnon . Also, it is very
convenient to pinpoint the mobility-edge. If a mobility edge exists, IPR value
is expected to jump from system size independent higher value to a lower value
that scales inversely with the system size. For $n$-th quasiparticle
eigenstate the usual definition of IPR ADMirlin  can be generalized as
follows;

$I_{n}(q)=\sum_{\sigma}\sum_{i=1}^{N}\left|\psi_{n,\sigma}(i)\right|^{2q},\leavevmode\nobreak\
\leavevmode\nobreak\ q=2.$ (4)

Here, the $n$th single quasiparticle eigenstate is given by
$\ket{\Psi_{n}}=\sum_{\sigma}\sum_{i=1}^{N}\psi_{n,\sigma}(i)\ket{1,\sigma}_{i},$
where $\ket{1,\sigma}_{i}=\ket{0,0,\cdots,\sigma_{i},\cdots,0,0}$ represents
the localized basis state having one particle with spin $\sigma$ at site $i$,
and $N$ is the size of the lattice. Since, this is a non-interacting problem,
usual scaling properties with respect to the system size is also expected to
hold for the quasiparticle states, i.e., for a perfectly extended metallic
state $I_{n}(q)=1/N$, while for a state localized at a single site
$I_{n}(q)\simeq 1$. The IPR has similar qualitative features for the
multifractal states as the localized states. However, the IPR values show
large fluctuations compared to a localized state in this case. The distinct
scaling behavior of IPR helps to identify delocalized states unambiguously. It
can also be used for preliminary identification of localized/multifractal
states.

###  III.1 GAA Hamiltonian without RSO coupling

Before presenting our results for the GAA Hamiltonian in the presence of the
RSO coupling, in this section, we first review the results of the pure GAA
Hamiltonian and discuss some new observations that were not addressed earlier.
In Fig. 1(i), we have presented the projection of IPR as a function of $W/t$
for the GAA Hamiltonian for the limit $a\gg 1$. As mentioned earlier, the GAA
Hamiltonian coincides with the pure AAH Hamiltonian in this limit. In the pure
AAH Hamiltonian, the DL transition takes place at a critical disorder strength
$W_{c}/t=2$. This is evident from the sharp change in the IPR values in Fig.
1(i). Below the critical disorder strength all the eigenstates are
delocalized, while all of them are localized above the critical value. To
verify the behavior of IPR, we have shown the corresponding results in Figs.
1(ii) and (iii) for $W=1.0$ (delocalized region) and $W=2.5$ (localized
region) respectively.

In Figs. 2 and 3, we have shown the IPR results for the short-range $(a>1)$
and long-range $(a\leq 1)$ limits of the GAA model, respectively. From Fig.
2(i) it is evident that in the short-range hopping limit ($a=1.5$) there is a
critical disorder strength (lower than that in the $a\gg 1$ limit) below which
all the eigenstates are delocalized. Interestingly, however, beyond this
critical point a mobility edge appears in the energy spectrum which separates
the localized and delocalized states. A closer look at Fig. 2(i) reveals that
there are certain windows of disorder strength (marked with horizontal dotted
lines) in which all the states are localized above the mobility edge.
Interestingly, however, at the beginning of each of these windows delocalized
states can exists after the appearance of the first mobility edge. From Fig.
2(iii) one can observe that for $W/t=1.2$ a small band of delocalized states
appear in the spectrum after the appearance of the mobility edge at $\beta$.
This observation of the delocalized states in the energy spectrum beyond the
critical point is in contrast with the conclusion of the Ref. Deng . In Ref.
Deng , it was pointed out that beyond the critical point, all the states are
localized above the mobility edge, and only a certain fraction of the lowest
energy eigenstates are delocalized. This is indeed true only within the above
mentioned windows of disorder strength. Within these windows, all the states
are localized beyond the mobility edge, and such mobility edges appear at
$n/L=\beta$, next at $n/L=\beta^{2}$, and so on. Depending on the window of
the disorder strength, the lowest $\beta^{s}L$ amount of states, where
$s=1,2,3,\cdots$, remain delocalized. From Fig. 2(i), it can be seen that the
location of the mobility edge remains unchanged at $\beta$ for a range of a
disorder strength. Next, such mobility edge appears at $\beta^{2}$, then at
$\beta^{3}$, and so on. In Figs. 2(ii) and (iii), we have plotted the IPR
results for different system sizes below and above the critical disorder
strength respectively. The existence of a mobility edge beyond the critical
point is quite clear.

Figure 4: Projection of IPR as function of $W/t$ and eigen-energies $(index)$
in the limit of the short-range $(a=1.5)$ GAA model with RSO coupling . (i)
$\alpha_{y}=0.8,$ $\alpha_{z}=0.0$, (ii) $\alpha_{y}=0.0,$ $\alpha_{z}=0.8$,
and (iii) $\alpha_{y}=0.6,$ $\alpha_{z}=0.8$. Dotted lines show the presence
of single mobilty edge regions.

Figure 5: IPR $(I_{n})$ vs eigen-energies $(index)$ in the limit of the short-
range ($a=1.5$) GAA model with RSO coupling at, (i) $W/t=1.2$,
$\alpha_{y}=0.0,$ $\alpha_{z}=0.8$, (ii) $W/t=3.0$, $\alpha_{y}=0.0,$
$\alpha_{z}=0.8$, and (iii) $W/t=5.0$, $\alpha_{y}=0.0,$ $\alpha_{z}=0.8$.

The energy spectrum changes dramatically in the limit of long-range hopping.
Similar to the other limits, there exists a critical disorder strength below
which all the eigenstates are delocalized. However, above the critical
disorder strength a multifractal-edge separates delocalized and multifractal
states. These results have been summarized in Fig. 3(i). Here, the value of is
$a$ is set to $0.5$. Apart from the appearance of the multifractal edge, the
other crucial difference from the short-range hopping model is that all the
states are multifractal type beyond the multifractal-edge. Furthermore, in
contrast to the short range limit, there are no delocalized states in the
energy spectrum above the multifractal edge. Interestingly, similar to the the
short range hopping limit, in this case as well there exist disorder strength
dependent sharp mobility edges in the spectrum. Furthermore, the location of
the multifractal-edge follows exactly the same pattern as $a>1$ limit that is
the lowest $\beta^{s}L$ amount of states are delocalized depending on the
strength of the disorder. To reveal the true nature of the states, we have
plotted the IPR results with multiple system sizes in Figs. 3(ii) and (iii).
In Fig. 3(ii), we have presented the IPR results for $W<W_{c}$, where
$W_{c}/t\approx 0.4$. As expected, the IPR values scale inversely with the
system size. The signature of multifractal states and multifractal edges can
be seen in the IPR results presented in Fig. 3(iii).

Figure 6: Projection of IPR as function of $W/t$ and eigen-energies $(index)$
in the limit of the long-range $(a=0.5)$ GAA model with RSO coupling . (i)
$\alpha_{y}=0.8,$ $\alpha_{z}=0.0$, (ii) $\alpha_{y}=0.0,$ $\alpha_{z}=0.8$,
and (iii) $\alpha_{y}=0.6,$ $\alpha_{z}=0.8$.

Figure 7: IPR $(I_{n})$ vs eigen-energies $(index)$ in the limit of the long-
range ($a=0.5$) GAA model with RSO coupling at, (i) $W/t=0.5$,
$\alpha_{y}=0.0,$ $\alpha_{z}=0.8$, (ii) $W/t=4.0$, $\alpha_{y}=0.0,$
$\alpha_{z}=0.8$, and (iii) $W/t=5.0$, $\alpha_{y}=0.0,$ $\alpha_{z}=0.8$.

###  III.2 GAA Hamiltonian with the RSO coupling

We now discuss the effect of the RSO coupling on the energy spectrum of the
GAA Hamiltonian. We are going to demonstrate that in the presence of RSO
coupling, the energy spectrum of the GAA Hamiltonian dramatically changes,
especially in the long range hopping limit. At first we discuss the effect of
RSO coupling in the short-range hopping limit. In general, depending on the
strength of the RSO coupling, the critical point shifts to a higher disorder
strength irrespective of the range of the hopping. The effect of RSO coupling
on the $a>>1$ limit has already been studied in Ref. dsahu .

In Fig. 4, we have analyzed the IPR result for the short-range $(a>1)$ GAA
model with RSO coupling. The value of $a$ is set to $1.5$, which is same as in
Fig. 2. It is quite evident that when the RSO coupling is introduced into the
system the critical point shifts to a higher disorder strength. Without the
RSO coupling, the critical disorder strength is $W_{c}/t\approx 1.1$. From
Fig. 4(i), we can observe that when only the spin-conserving part of the RSO
Hamiltonian is turned on ($\alpha_{y}/t=0.8,\alpha_{z}/t=0$) the critical
disorder strength turns out to be $W_{c}/t\approx 2.4$. Interestingly, the
spin-flip hopping has an identical effect on the critical disorder strength.
In Fig. 4(ii) we have presented the IPR results as a function of $W/t$ for the
case when only the spin-flip hopping part of the RSO Hamiltonian is turned on.
In this case $\alpha_{y}/t=0$ while $\alpha_{z}/t=0.8$. It is clear that the
critical disorder strength turns out to be $W_{c}/t\approx 2.4$ in this case
as well. Moreover, the individual effect of the two types of hopping of the
RSO Hamiltonian on the energy spectrum is also identical. A comparison between
the IPR results of Fig. 2(i) and Fig. 4(i) reveal that in the presence of the
RSO coupling the disorder windows at $\beta$ and $\beta^{2}$ of the GAA
Hamiltonian have now disappeared. The mobility edge, where all the states are
localized above it, now first appears at $\beta^{3}$, then at $\beta^{4}$ with
an increase in the disorder strength. Outsidethese windows, in the presence of
either of the RSO couplings, multiple mobility edges can be found where small
bands of delocalized states separate bands of localized states. The effect of
a weaker RSO coupling strength is qualitatively similar, and in Appendix A we
have presented some results to demonstrate this.

To demonstrate the existence of multiple mobility edges beyond the critical
point, in Fig. 5 we have plotted the IPR results as a function of the
eigeneneries below the critical point, and for two different disorder
strengths above the critical value. From Fig. 5(i) ($W/t=1.2$), it is clear
that the IPR values across the entire energy spectrum scale inversely with the
system size indicating that all the states are delocalized. The most
interesting features can be observed at $W/t=3$ (Fig. 5(ii)). It is evident
from this figure that a mobility edge first appears approximately at
$n/L=\beta^{2}-\beta^{5}$, below which all the states are delocalized, and
above which a narrow band of localized states appear. These localized states
survives till $n/L=\beta^{3}+\beta^{5}$ where another mobility edge appears
beyond which another narrow band of delocalized states can be observed till
$n/L=\beta^{2}$. After this point, the states are localized till the
appearance of a tiny set of delocalized states close to highest eigenenergies
of the spectrum. When the disorder strength is set at $W/t=5.0$ in Fig.
5(iii), only a single mobility edge at $n/L=\beta^{4}$, originally present in
the pure GAA Hamiltonian, can be seen in the energy spectrum.

As quite expected, when both the RSO coupling is present, the critical
disorder strength is pushed towards a higher disorder strength compared to the
case when only one of the RSO coupling is considered. In Fig. 4(iii), the
strength of the spin-flip hopping amplitude $\alpha_{z}/t$ has been fixed at
$0.8$ as in Fig. 4(ii) while $\alpha_{y}/t$ is set to value of $0.6$. It is
quite evident that the critical disorder strength in this case is
$W_{c}/t\approx 2.7$. We would like to point out that the critical point and
the spectrum remain unchanged if the strengths of $\alpha_{y}/t$ and
$\alpha_{z}/t$ are interchanged. The presence of both the RSO couplings do not
introduce any significant change in the qualitative behaviour of the energy
spectrum, except that the window of disorder strengths at which only a single
mobility edge exists is now appear first at $n/L=\beta^{4}$.

The effect of the RSO coupling on the energy spectrum is more dramatic in the
case of the long-range hopping limit $(a\leq 1)$ of the GAA Hamiltonian. In
this limit as well, for a given value of $a$, the critical disorder strength
increases in the presence of the RSO coupling compared to the pure GAA
Hamiltonian. Similar to the short-range hopping limit case, when either of the
two hopping processes of the RSO Hamiltonian is considered, the critical
disorder strength turns out to be same for the same strength of $\alpha_{y}/t$
or $\alpha_{z}/t$. This can be observed from the results presented in Figs.
6(i) and 6(ii). However, the effect of the RSO couplings on the spectrum is
remarkably different in the case of long-range hopping limit. It can be
observed from Fig. 6(i) (Fig. 6(ii)) that when $\alpha_{y}/t=0.8$ and
$\alpha_{z}/t=0.0$ ($\alpha_{y}/t=0.0$ and $\alpha_{z}/t=0.8$) the spectrum no
longer has a single multifractal edge at any disorder strength, rather
multiple multifractal edges emerge beyond the critical point, which separates
successive bands of delocalized and multifractal states. The regular pattern
of the lowest $\beta^{s}L$ amount states being delocalized in the case of pure
GAA Hamiltonian is no longer true. However, the location of the multifractal
edges (separating the delocalized and multifractal states or vice versa) are
in general approximately given by $n/L=\beta^{s}\pm\beta^{m}$, where
$s,m=0,1,2,\cdots$.This is quite similar to the short-range GAA Hamiltonian in
the presence of the RSO coupling discussed in the previous paragraph.

To demonstrate the existence of multiple multifractal edges and their
locations, in Fig. 7 we have plotted the IPR values as a function of the
eigeneneries below and above the critical point. We have set $\alpha_{y}/t=0$
and $\alpha_{z}/t=0.8$. From Fig. 7(i), it is evident that below the critical
disorder strength all the eigenstates are delocalized. From Figs. 7(ii) and
(iii), however, we can see that above the critical disorder strength the
spectrum gets divided into alternative regions of multifractal and delocalized
states. For $W/t=4.0$, in the GAA Hamiltonian without the RSO coupling, a
single multifractal edge appears at $n/L=\beta^{4}$. From Fig. 7(ii), however,
it is clear that in the presence of the RSO coupling, bands of delocalized
states emerges between $n/L=\beta^{2}$ and $n/L=\beta-\beta^{4}$, and again at
$n/L=\beta$. From Fig. 7(iii) it is also quite evident that with an increase
in the disorder strength, the width of these delocalized bands shrink.

From the IPR results, the existence of multiple mobility or multifractal edges
in the GAA Hamiltonian in the presence of RSO coupling is quite clear.
Furthermore, we can identify the localized and multifractal states from these
results only qualitatively. A more quantitative approach is needed to analyze
the states to distinguish the localized and the multifractal states
unambiguously. In the next section, we present the multifractal analysis of
the quasiparticle eigenstates, which helps to analyze the states
quantitatively.

Figure 8: Multifractal analysis of the short-range ($a=1.5$) GAA model with
RSO coupling coefficients as $\alpha_{y}=0$,$\alpha_{z}=0.8$. (i) at $W/t=1.2$
(Pure delocalization region). (ii) at $W/t=3.0$ in the $ERG^{1}$ region of
Fig.5(ii). (iii) at $W/t=3.0$ in the $ERG^{2}$ region of Fig.5(ii). (iv) at
$W/t=3.0$ in the $LOC^{1}$ region of Fig.5(ii).

##  IV Multifractal Analysis of Eigenstates

The IPR results show the fragmentation of the energy spectrum into successive
windows of delocalized, localized, or multifractal states in the presence of
RSO coupling in the GAA model. In this section we carry out the multifractal
analysis, especially for $W>W_{c}$, to determine the true nature of the
quasiparticle eigenstates in the short and long range limits. In the
multifractal analysis Halsey ; Martin Janssen , one generally starts by
identifying that $P_{n}(q)=\sum_{i=1}^{N}\left|\psi_{n}(i)\right|^{2q}\propto
N^{-\tau(q)}$, where $\psi_{n}$ is the normalized wave function corresponding
to $n$-th eigenvalue and $i=1,2,\cdots,N$. The exponent $\tau(q)$ is
alternatively expressed in terms of $D(q)$ as

$\tau(q)=D(q)/(q-1),$ (5)

where $D(q)$ is called the fractal dimension.

Typically, to understand the nature of the eigenstates the most commonly used
quantity is $D(2)$, as its numerical value indicates the degree of
multifractality. It turns out that $\tau(q)=q-1$ in case of delocalized states
DeLuca , which effectively means that for that for delocalized states
$D(2)=1$. In the other extreme case, when the eigenstates are completely
localized $D(2)=0.$ For the multifractal states $D(2)$ lies in between these
two limits, that is $0<D(2)<1$. Hence, in principle if one can compute
$\tau(q)$ then in principle $D(q)$ can be estimated. In practice, however,
instead of directly computing $\tau(q)$ an equivalent parameter $f(\alpha)$ is
computed, which characterizes the multifractal property of the eigenstates
much more conveniently. These two parameters $f(\alpha)$ and $\tau(q)$ are
connected by the Legendre transformation as follows,

$f(\alpha(q))=q\alpha(q)-\tau(q),$ (6)

where $\alpha(q)=d\tau(q)/dq$.

Generally, $f(\alpha)$ is a smooth positive valued convex function of
$\alpha$. It has a global maximum but no local minima or maxima. It is
interesting to note that, $f_{max}=f(\alpha(q=0))=d$, where $d$ is the
Euclidean dimension of the system Martin Janssen . From the $f(\alpha)$ vs
$\alpha$ spectrum, it is easy to identify the nature of the eigenstates. For
delocalized states, when $\alpha\neq 1$, $f(\alpha)=-\infty$ and
$f(\alpha=1)=1$ in the limit of large system sizes. For multifractal states
$f(\alpha(q))$ is bounded and non-zero only between
$0<\alpha_{min}<\alpha(q)<\alpha_{max}$, while for the localized states
$0=\alpha_{min}<\alpha(q)<\alpha_{max}$, where $\alpha_{max}=\alpha(q=0)$.

Figure 9:  Multifractal analysis of the long-range ($a=0.5$) GAA model with
RSO coupling coefficients as $\alpha_{y}=0$,$\alpha_{z}=0.8$. (i) at $W/t=1.0$
(Pure delocalization region). (ii) at $W/t=4.0$ in the $ERG^{1}$ region of
Fig.7(ii). (iii) at $W/t=4.0$ in the $ERG^{2}$ region of Fig.7(ii). (iv) at
$W/t=4.0$ in the $MF^{1}$ region of Fig.7(ii).

###  IV.1 Calculation of Multifractal Spectrum

From the previous discussion we observe that to analyze the nature of the
eigenstates it is sufficient to compute the multifractal spectrum
$f(\alpha(q))$ for $0\leq\alpha(q)\leq\alpha(q=0)$. In this article, we are
going to follow the Ref.Martin Janssen  and Ref.Cuevas  to numerically compute
the multifractal function $f(\alpha(q))$. Before discussing our results, we
briefly review the key steps to determine $f(\alpha(q))$. Initially the
lattice is divided into small boxes of linear size $l<L$, where $L$ is the
length of the lattice, and then we compute the following normalized
probability,

$\mathcal{P}_{k}(l,q)=\frac{\mathcal{P}_{k}^{q}(l)}{\sum_{j=1}^{N_{b}}\mathcal{P}_{j}^{q}(l)},$
(7)

where $1\leq k\leq N_{b}=L/l$ represents the $k$-th box and
$\mathcal{P}_{k}^{q}(l)=\sum_{i\in
l_{k}}\left|\psi_{n}(i)\right|^{2},l_{k}=l\leavevmode\nobreak\
\forall\leavevmode\nobreak\ k,$ is the probability of the $n$-th eigenstate.
Once $\mathcal{P}_{k}^{q}(l)$ is computed, $\alpha(q,L)$ and $f(\alpha(q,L))$
can be obtained from the following relations;

$\displaystyle\alpha(q,L)$ $\displaystyle=$
$\displaystyle\underset{\delta\rightarrow
0}{\mathrm{lim}}\frac{\sum_{k=1}^{N_{b}}\mathcal{P}_{k}(l,q)\textrm{ln}(\mathcal{P}_{k}(l,1))}{\textrm{ln}\delta}$
(8) $\displaystyle f(\alpha(q,L))$ $\displaystyle=$
$\displaystyle\underset{\delta\rightarrow
0}{\mathrm{lim}}\frac{\sum_{k=1}^{N_{b}}\mathcal{P}_{k}(l,q)\textrm{ln}(\mathcal{P}_{k}(l,q))}{\textrm{ln}\delta},$
(9)

where $\delta=l/L$.

Here, we mainly focus on the multifractal analysis of the narrow band of
states that appear beyond the critical point in the presence of the RSO
coupling. We have presented our results for $\alpha_{y}=0$, and
$\alpha_{z}\neq 0$. In Fig. 8, we have presented the multifractal spectrum for
the short-range $(a>1)$ GAA Hamiltonian in the presence of RSO coupling. At
first, in Fig. 8(i), the multifractal spectrum is presented for the disorder
strength $W/t=1.2$, which is below the critical value of $W_{c}/t\approx
2.35$. From the results it is clear that as the system size increases to a
large value $f(\alpha=1)$ converges to the value $1$, which is the signature
of purely delocalized states. This result corroborates our observations made
from the IPR results of Fig. 4(i) and Fig. 5(i), which clearly shows the
presence of only delocalized states at $W/t=1.2$. The $D(2)$ value is
calculated as $0.99$ with the system size $L=6765$. In Figs. 8(ii)-(iv), we
have fixed the disorder strength beyond the critical point at $W/t=3.0$. In
these figures we have performed the multifractal analysis of the states
corresponding to the different regions, marked as $ERG^{1},ERG^{2}$ and
$LOC^{1}$ in the Fig.5(ii). We primarily focus on these three regions since it
is sufficient to prove the existence of the multiple mobility edges which
separate alternative bands of delocalized and localized states. In Figs.8(ii)
and (iii), we have shown the multifractal analysis of the eigenstates present
in the regions of $ERG^{1}$ and $ERG^{2}$. It is quite evident that as the
system size increases to a large value, $f(\alpha)$ is non-zero only for
$\alpha=1$, clearly indicating that these states are delocalized. The $D(2)$
value is calculated for both the regions and found to be $0.99$ with system
size $L=6765$. The multifractal spectrum is most crucial to determine the
nature of the states of the $LOC^{1}$ region. From the IPR results, we expect
the eigenstates of the $LOC^{1}$ to be localized. However, relatively large
fluctuations in the IPR values in this region, although they are system size
independent, makes it difficult to draw a definite conclusion. In Fig. 5(iv)
we have presented the multifractal spectrum for these states. In contrast to
the other regions, $f(\alpha)\rightarrow 0$ as $\alpha\rightarrow 0$ with an
increase in the system size, which shows that these states are indeed
localized. We have estimated the $D(2)$ value of $LOC^{1}$ for the system size
$L=6765$, and it is found to be $0.1$.

In Fig. 9, a similar analysis has been carried out in the long-range $(a\leq
1)$ limit of the GAA Hamiltonian in the presence of the RSO coupling. The
results are presented for two different disorder strengths $W/t=1.0$ and
$W/t=4.0$, where the critical value $W_{c}/t\approx 2.1$. Similar to the
previous analysis, we have set $\alpha_{y}=0$, and $\alpha_{z}\neq 0$ in this
case well. From Fig. .7(ii), we have concluded that above the critical point
the spectrum breaks into alternative bands of delocalized and multifractal
states, a few of which have been labelled as $ERG^{1}$, $ERG^{2}$, and
$MF^{1}$. For computing the multifractal spectrum of Fig. 9(i) we have
considered all the eigenstates. The disorder strength is set at $W/t=1.0$. It
is obvious that the quasipaticle eigenstates are delocalized. The $D(2)$ value
is calculated to be $0.99$ with the system size $L=6765$. In Figs. 9(ii)-(iv),
we have presented the multifractal spectrum corresponding to the states
$ERG^{1}$, $ERG^{2}$, and $MF^{1}$ respectively. The disorder strength is set
at $W/t=4.0$. The multifractal spectrum of Figs. 9(ii) and (iii) clearly
indicates that the states $ERG^{1}$ and $ERG^{2}$ are delocalized. For both
these regions, the estimated value $D(2)$ turns out to be $0.99$ for $L=6765$.
In Fig.9(iv), we have presented the multifractal spectrum of the $MF^{1}$
region. The difference in the spectrum compared to the delocalized and
localized states are quite self evident. In this case, as the system size
increases $f(\alpha)\rightarrow 0$ while $\alpha\neq 0$ which clearly
demonstrate that these states are of multifractal in nature. As expected, the
$D(2)$ value of this $MF^{1}$ region is found to be $0.48$.

Figure 10: Projection of IPR as function of $W/t$ and eigen-energies $(index)$
in the limit of the short-range $(a=1.5)$ GAA model with RSO coupling . (i)
$\alpha_{y}=0.1,$ $\alpha_{z}=0.0$, (ii) $\alpha_{y}=0.0,$ $\alpha_{z}=0.1$,
and (iii) $\alpha_{y}=0.1,$ $\alpha_{z}=0.1$. Dotted lines show the presence
of single mobilty edge regions.

##  V Conclusions

In conclusions, we have studied the effect of spin-orbit coupling of Rashba
type on the one-dimensional quasiperiodic lattice, described by the GAA
Hamiltonian, in the short-range and long-range hopping limits. In the original
GAA Hamiltonian, it had been reported earlier that there is a single mobility
edge in the short-range hopping limit beyond a critical disorder strength. In
contrast, a single multifractal edge separates delocalized and multifractal
states. In this work, at first, we have found that while this is true in the
long-range hopping limit, in the short-range hopping limit, there exist
certain windows of disorder strengths where a single edge separates
delocalized and localized states. Outside these windows, in the short-range
hopping limit, we have found, in contrast to previously reported results, that
a narrow band of delocalized states can exist after the appearance of the
first mobility edge. In the original GAA Hamiltonian, in both of these limits,
where single mobility (multifractal) states exist, the lowest $\beta^{s}L$
($s=1,2,\cdots$) states are delocalized, and all the states above it are
localized (multifractal type). This article demonstrates that spin-orbit
coupling has a dramatic effect on the energy spectrum, especially in the long-
range hopping limit. In general, in the presence of the spin-orbit coupling,
the critical disorder strength increases to a larger value compared to the
original GAA Hamiltonian in both these limits. In the short-range hopping
limit, the windows of disorder strengths where a single mobility edge
completely separates delocalized and localized states get successively
destroyed with an increase in the spin-orbit coupling strength, and multiple
mobility edges can be found outside these windows. The spin-orbit coupling
similarly affects the windows mentioned above in the long-range hopping limit.
In this case, they also get successively destroyed with an increase in the
spin-orbit coupling strength. Furthermore, in the presence of the spin-orbit
coupling, multiple multifractal edges appear in the spectrum that separate
alternative bands of delocalized and multifractal states beyond the critical
point. Interestingly, in contrast to the short-range hopping limit, these
bands of the delocalized states survive up to a very large value of disorder
strength. We have found that the multifractal edges roughly appear at
$n/L=\beta^{s}\pm\beta^{m}$, where $s,m=0,1,2,\cdots$.

###### Acknowledgements.

The authors would like to acknowledge the computational facility provided by
SERB (DST), India (EMR/2015/001227).

## Appendix A

In this section, we have discussed the behavior of mobility edges in the
energy spectrum for a short-range $(a>1)$ GAA model having a smaller amplitude
of RSO coefficients $\alpha_{y},\alpha_{z}$ (as shown in Fig.10). We can
observe that with weaker RSO coupling, only the first window (where only a
single mobility edge exists) just after the critical point gets destroyed,
while the rest of such windows remain unaffected. In Fig. 4 of the main text,
we have observed that as the strength of the RSO coupling is increased, such
windows appear only at a higher strength of the disorder.

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