††thanks: Contributed equally††thanks: Contributed equally

# Phase classification in the long-range Harper model using machine learning

Abee Nelson  Department of Physics, Indian Institute of Science Education and
Research, Bhopal, Madhya Pradesh 462066, India
Aamna Ahmed  Department of Physics, Indian Institute of Science Education and
Research, Bhopal, Madhya Pradesh 462066, India
Ankur Raina  Department of EECS, Indian Institute of Science Education and
Research, Bhopal, Madhya Pradesh 462066, India
Auditya Sharma  Department of Physics, Indian Institute of Science Education
and Research, Bhopal, Madhya Pradesh 462066, India

###### Abstract

In this work, we map the phase diagrams of one-dimensional quasiperiodic
models using artificial neural networks. We observe that the multi-class
classifier precisely distinguishes the various phases, namely the delocalized,
multifractal, and localized phases when trained on the eigenstates of the
long-range Aubry-André Harper (LRH) model. Additionally, when this trained
multi-layer perceptron is fed with the eigenstates of the Aubry-André Harper
(AAH) model, it identifies various phases with reasonable accuracy. We examine
the resulting phase diagrams produced using a single disorder realization and
demonstrate that they are consistent with those obtained from the conventional
method of fractal dimension analysis which employs a large number of disorder
samples. Interestingly, when the neural network is trained using the
eigenstates of the AAH model, the resulting phase diagrams for the LRH model
are less exemplary than those previously obtained. Further, we study binary
classification by training the neural network on the probability density
corresponding to the delocalized and localized eigenstates of the AAH model.
We are able to pinpoint the critical transition point by examining the metric
“accuracy” for the central eigenstate. The effectiveness of the binary
classifier in identifying a previously unknown multifractal phase is then
evaluated by applying it to the LRH model.

††preprint: APS/123-QED

##  I INTRODUCTION

The application of machine learning (ML) techniques in the field of condensed
matter physics has been a subject of great interest in recent times [1]. This
interdisciplinary field has rapidly gained popularity owing to the advantages
presented by machine learning [2, 3, 4]. As an actively evolving area of
research, ML techniques have been used to detect classical and quantum phase
transitions [5, 6, 7, 8, 9, 10, 11, 12], for the acceleration of Monte Carlo
simulations [13, 14, 15], and for representation of states of quantum many-
body systems [16, 17, 18, 19, 20, 21, 22, 23, 24, 21, 25]. A popular
subcategory of machine learning is supervised learning [26], where one trains
algorithms to utilize labelled training datasets in order to classify new data
or make predictions accurately. This motivates its application in condensed
matter physics for classifying various phases in strongly correlated systems
[5, 27], topological systems [28, 29, 30] and quantum many-body systems [16,
9].

Disordered quantum systems [31, 32, 33, 34, 35, 36] have been a fascinating
topic of study for a long time. In the 1-D Anderson model [31], the tiniest of
disorder is known to localize single-particle states exponentially. In
contrast, the Aubry-Andrè-Harper (AAH) [37, 38] model, governed by a
quasiperiodic disorder, exhibits a delocalization-localization transition even
in one dimension. Interestingly, instead of the exponential localization
observed in the short-ranged model, the eigenstates show algebraic
localization [39, 40, 41] in the case of long-range hopping. This has
naturally led to exciting studies exploring the effects of quasiperiodic
disorder in long-range systems [42, 43].

The interplay of power law hopping $\left(1/r^{\sigma}\right)$ and the
quasiperiodic potential results in a rich structure [42] in the single
particle eigenstates. The self-duality of the quasiperiodic AAH model is
broken, and mobility edges are observed when the hopping is no longer
restricted to nearest neighbours only. While multifractal eigenstates are
observed to coexist with delocalized eigenstates for $\sigma<1$, localized
states exist together with delocalized eigenstates for $\sigma>1$ [42]. The
localization characteristics of these single particle eigenstates have been
determined [42, 44] with the aid of several well-known measures, such as
fractal dimension [45, 46], inverse participation ratio [47, 48], level
spacing [49] and many more. These methods depend on a physical understanding
of the nature of the regime and require averaging over several disorder
realizations.

In the present work, we construct a neural network for characterizing the
phase diagram beyond established methods in single-particle Hamiltonian
models. We study the one-dimensional long-range AAH (LRH) model, where
multifractal (localized) eigenstates can coexist with delocalized eigenstates
for the long-range hopping parameter $\sigma<1(\sigma>1)$. The information
required for classifying the delocalized, localized and multifractal regimes
is obtained from the eigenstates which are used as inputs to the neural
network. The trained network is first used to classify all the eigenstates of
the LRH model for various values of the hopping parameter $\sigma$. We then
consider the case of $\sigma\rightarrow\infty$, i.e., the standard AAH model
and classify its eigenstates as delocalized/localized/multifractal. We show
that the phase diagram obtained by feeding a single disorder realization in
the neural network is in good agreement with results obtained using
conventional methods that require disorder averaging. We next train the same
network using the eigenstates of the AAH model. In this case, although the
network can identify the localized phase accurately, some discrepancy is seen
in the case of multifractal and delocalized states. We also utilize binary
classification to identify the transition point of the AAH model by training
the neural network on its delocalized and localized eigenstates. While this
network precisely identifies the phase diagram for the AAH model, it can also
identify the multifractal phase (for which no explicit training is given) of
the LRH model, although this is achieved only in a coarse manner.

This paper is organized as follows. Section II discusses the details of the
Hamiltonian model and carries an introduction to the general setup of the
neural network. In Section III, we discuss the network architecture used for
the classification as well as the details of the input data. In Section IV, we
present our multi-class and binary classification results and compare them
against the results obtained through a conventional method. We then summarize
our results in Section V.

##  II Model and Methods

###  II.1 Hamiltonian

We consider the one-dimensional long-range Harper (LRH) model given by the
Hamiltonian:

$H=-\sum_{i<j}^{N}\left(\frac{J}{r_{ij}^{\sigma}}\hat{c}_{i}^{\dagger}\hat{c}_{j}+\text{
H.c. }\right)+\lambda\sum_{i=1}^{N}\cos\left(2\pi\alpha
i+\theta_{p}\right)\hat{c}_{i}^{\dagger}\hat{c}_{i},$ (1)

where $\hat{c}_{i}^{\dagger}\left(\hat{c}_{i}\right)$ represents the single
particle creation (destruction) operator at site $i$. The first term describes
hopping, where $r_{ij}=\left(N/\pi\right)\sin\left(\pi|i-j|/N\right)$ is the
geometric distance between the sites $i$ and $j$ in a ring. The strength of
the long-range hopping is controlled by $J$, which we set to unity and the
long-range parameter in the hopping, namely $\sigma$. The second term
describes the quasiperiodic on-site energy, where the strength of the
quasiperiodic potential is $\lambda$, and the quasiperiodicity parameter
$\alpha$ is taken to be an irrational number, set as the golden mean
$(\sqrt{5}-1)/2$ [50]. $\theta_{p}$ is an arbitrary global phase chosen
randomly from a uniform distribution in the $[0,2\pi]$ range. The total number
of sites is $N$, and periodic boundary conditions are assumed. In the limit
$\sigma\rightarrow\infty$, the hopping is effectively nearest-neighbour, and
we recover the standard AAH model: [37, 38]

$H=-J\sum_{i=1}^{N}(\hat{c}_{i}^{\dagger}\hat{c}_{i+1}+\text{H.c.})+\sum_{i=1}^{N}\lambda\cos(2\pi\alpha
i+\theta_{p})\hat{c}_{i}^{\dagger}\hat{c}_{i}.$ (2)

It is well known [50] that all the energy eigenstates are delocalized when
$\lambda<2$, and all the energy eigenstates are localized when $\lambda>2$.
$\lambda=2$ is the critical point where all the eigenstates are multifractal
[51]. It is known that the AAH model is self-dual [37, 52]; at the critical
point $\lambda=2$, the AAH model in position space maps to itself in momentum
space. When long-range hopping is introduced ($\sigma$ is finite), the self-
duality condition is broken.

###  II.2 Artificial neural network

The artificial neural network (ANN) is inspired by the neuronal network in the
biological brain [53, 54]. The ANN combines a series of linear maps and non-
linear functions [55] that are successively applied to the input data in order
to obtain the final output. A mapping defines each layer of the network, and
the dimension of a layer corresponds to the number of neurons in it. In this
work, in the case of multi-class classification, we utilize vectors
$\left\\{x\right\\}$ of dimension $m_{1}$ and train a map $f(x)$ to map them
to the target set $\left\\{(0,0,1),(0,1,0),(1,0,0)\right\\}$. One-hot vectors
represent this target set and deliver the final output in the form of entries
of the neurons of the output layer. The network is trained on a data set
called the training set, and its performance is gradually improved by
adjusting its parameters. In supervised learning, the training set consists of
labelled data, i.e., for each input ${x}$, the output is already known to map
to one of the outcomes of the target set. The trained network then classifies
previously unseen data from the testing set.

We next describe the functioning of the neural network. The ‘input layer’ maps
the initial vectors of dimension $m_{1}$ to a space of dimension $m_{2}$ with
the help of a linear map and a non-linear activation function $A$ [56].
Subsequently, the mapping between the $n^{\text{th}}$ and the
$n+1^{\text{th}}$ layer is defined as:

$x^{(n)}\mapsto x^{(n+1)}=A\left(W^{(n+1,n)}x^{(n)}+B^{(n+1)}\right).$ (3)

Here matrix-vector multiplication is implied between $W^{(n+1,n)}$ (matrix of
dimension $m_{n+1}\times m_{n}$) and $x^{(n)}$ (vector of dimension $m_{n}$).
The resulting vector $x^{(n+1)}$ is a vector of dimension $m_{n+1}$. The
elements of the matrix $W^{(n+1,n)}$ are called the weights, and the
corresponding elements of the vector $B^{(n+1)}$ are called the biases. The
final layer of the ANN is referred to as the ‘output layer’, with the number
of nodes equal to the number of expected outputs that the network is trained
for. The layers between the input and output layers are called hidden layers,
which can range from one to several. The method of training neural networks
with multiple hidden layers is referred to as deep learning [57]. Also, if the
number of neurons in a layer is huge, the network learns non-universal
features and unnecessary details for classification. This can lead to
overfitting of data. Thus in this study we employ dropout regularisation [58]
to avoid the gradual accumulation of neuronal weight configurations.

In the case of multi-class classification, the inputs to the network consist
of the amplitudes of an eigenstate of the Hamiltonian of interest. For the
hidden layers, we utilize the Leaky Rectified Linear Unit (ReLU) activation
function [59] defined as:

$\text{Leaky ReLU:\quad}f(x_{i})=\begin{cases}0.01x_{i}&x_{i}<0\\\
x_{i}&x_{i}\geq 0.\end{cases}$ (4)

In the output layer, we consider the Softmax activation function [60] defined
as:

$\text{Softmax:\quad}f(x_{i})=\frac{e^{-x_{i}}}{\sum_{j}e^{-x_{j}}}.$ (5)

The Softmax function estimates the output corresponding to each target set
vector, which sums up to unity. These projections can be interpreted as the
confidence of the network to assign a class to the input data.

In the case of binary classification, since the probability densities are
considered as inputs, we utilize the ReLU activation function [61]:

$\text{ReLU:\quad}f(x_{i})=\begin{cases}0&x_{i}<0\\\ x_{i}&x_{i}\geq
0,\end{cases}$ (6)

in the hidden layers. In the output layer, we employ the Sigmoid activation
function [60] defined as:

$\text{Sigmoid:\quad}f(x_{i})=\frac{1}{1+e^{-x_{i}}}$ (7)

to obtain a single output. There are two main advantages of the ReLU function:
(a) it is computationally efficient as it only involves a simple comparison
and (b) it introduces non-linearity in the model. On the other hand, the
Sigmoid function has a smooth and continuous output, which makes it easier to
compute gradients during backpropagation and to optimize the model using
gradient-based methods such as the stochastic gradient descent.

In order to correctly estimate various parameters such as the weights and
biases, the network utilizes a loss/cost function [62] as well as an optimizer
during the training process. The cost function measures the distance between
the predicted outputs and their actual values. In general, cross-entropy is
widely used as a loss function when optimizing classification models. For
classification problems, we utilize the cross-entropy loss function:

$L_{\text{CE}}=\sum_{i=1}^{c_{n}}T_{i}\log S_{i},$ (8)

where $c_{n}$ denotes the number of classes, $\\{S_{i}\\}$ are the
probabilities obtained from the Softmax/Sigmoid output layer, and
$\\{T_{i}\\}$ are the true values namely $0$ or $1$.

Another step involved in network training is back-propagation [63], where the
weights and biases are adjusted in successive iterations to reduce the output
of the loss function. The updating rate of the parameters is called the
learning rate. This is done through the gradient descent optimization
algorithm, which is computationally expensive. To overcome storage issues, the
training data is broken down into small batches, which can be easily fed to
the model. When the entire training data is fed to the model in batches, it is
called an epoch. Thus, every batch in the training data set can update the
internal model parameters once during an epoch. In order for the model to
learn the gradient or the direction it should take to minimize the loss
function, we employ an adaptive learning rate optimizer Adam [64] which
incorporates adaptive estimates of the gradients and their squares. The neural
network evaluates the loss function on the training dataset at the end of each
epoch - this is known as the training loss [65]. An additional data set for
validation is considered for every epoch to determine whether or not the
network is fully trained, that is, if it can generalise its knowledge to sets
of previously unseen data. The neural network is then made to evaluate the
loss function on the validation dataset - this is known as the validation
loss. The training and validation loss for a good fit should gradually reduce
and converge to $0$ as the number of epochs increases.

##  III Neural network approach

We perform exact diagonalization on the Hamiltonian to obtain the single-
particle eigenstates. Our goal is to be able to build an effective neural
network that can classify these eigenstates according to their localization
properties. We build and analyze two neural networks: a multi-class and a
binary classification network. While the amplitudes of the eigenstates are
utilized as inputs for the multi-class classification network, the on-site
probability densities drawn from the eigenstates are taken to be the inputs in
the case of the binary classification network. We discuss both cases in detail
below.

###  III.1 Multi-class classification

We consider a system with $N=510$ sites, and obtain the single-particle
eigenstates of the LRH model [44] given by Eq. 1. We then generate the
training data by varying the disorder strength $\lambda$ in small steps
($0.02$) in the range of $0-5$ for the long-range hopping parameter strengths
$\sigma=0.5,1.5$ and $3$ for several disorder samples. As mentioned earlier,
the multifractal (localized) eigenstates coexist with delocalized eigenstates
for $\sigma<1\left(\sigma>1\right)$ in the LRH model; the chosen parameters
help us to obtain data corresponding to all the three classes, i.e.,
delocalized, multifractal and localized.

Figure 1: A schematic diagram of the neural network architecture for multi-
class classification of eigenstates. Here the input layer is equal to system
size $N$, and the output layer has $3$ neurons, which gives the confidence of
the state being delocalized, multifractal and localized. Nonlinearities are
introduced by the Leaky rectified linear unit (Leaky ReLU). Dropout is
included to increase classification accuracy.

(a) (b)

Figure 2: The training loss and training accuracy along with the validation
loss and validation accuracy versus the number of epochs for the neural
network trained using eigenstates of the (a) LRH model and (b) AAH model. The
stabilization in loss occurs around $50$ epochs and beyond, indicating that
the ANN is trained.

In order to label the eigenstates required for network training, we calculate
a well-known quantity called fractal dimension $D_{q}$ [51, 66, 67]:

$D_{q}=\frac{S_{q}}{\ln(N)},$ (9)

where $N$ is the dimension of the Hilbert space and $S_{q}$ is the
participation entropy, which is defined for the $k^{\text{th}}$ eigenstate
$\left|\psi_{k}\right\rangle=$ $\sum_{i}\psi_{k}(i)|i\rangle$ and
$q^{\text{th}}$ order moment as:

$S_{q}=\frac{1}{1-q}\ln\sum_{i=1}^{N}\left|\psi_{k}(i)\right|^{2q}.$ (10)

The fractal dimension in the limit $N\rightarrow\infty$ is given by [51]:

$D_{q}^{\infty}=\text{lim}_{N\rightarrow\infty}D_{q}.$ (11)

For a perfectly delocalized state, $D_{q}^{\infty}=1$ while for a localized
state, $S_{q}$ is a constant, as observed for Anderson localization and
results in vanishing $D_{q}^{\infty}$ for all $q>0$. For intermediate cases,
$0<D_{q}^{\infty}<1$, which is a sign that the state is multifractal.

Here for each eigenstate, we calculate the fractal dimension $D_{2}$ and label
it as follows:

$\displaystyle D_{2}<0.2$ Localized, $\displaystyle 0.2\leq D_{2}\leq 0.8$
Multifractal, $\displaystyle D_{2}>0.8$
$\displaystyle\qquad\text{Delocalized}.$ (12)

Figure 3:  Neural network architecture for binary classification of
delocalized and localized eigenstates. Here nonlinearities are introduced by
the rectified linear unit (ReLU).

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4: Fractal dimension $D_{2}$ (whose value is represented by a colour
according to the code shown) as a function of disorder strength $\lambda$ and
for increasing fractional eigenstate index $i/N$ starting from the ground
state for the long-range hopping parameter $\sigma$ equal to (a) $0.5$,
(b)$1.5$, and (c) $3.0$. Here averaging has been performed over $100$ disorder
realizations. Classification of the single disorder realization of the LRH
model using the network trained on the eigenstates of the LRH model for
hopping parameter $\sigma$ equal to (d) $0.5$, (e)$1.5$, and (f) $3.0$.
Classification of the single disorder realization of the LRH model using the
network trained on the eigenstates of the AAH Model for hopping parameter
$\sigma$ (g) $0.5$, (h)$1.5$, and (i) $3.0$. The colour of each point
$(\sigma,\lambda)$ represents the confidence of the network for the
delocalized (red), multifractal (yellow), and localized (purple) phases. Here
system size is $N=510$ in all cases.

We first generate a training data set comprising of $300000$ eigenstates of
the LRH model belonging to each of the three classes, by considering various
values of $\sigma$, $\theta$, and $\lambda$. A schematic flowchart to describe
the complete architecture of the neural network performing multi-class
classification is shown in Fig. 1. The network comprises of an input layer,
several hidden layers and an output layer coupled to Linear, Leaky ReLU and
Softmax activation functions, respectively. We have also added dropout layers
to the model to avoid overfitting. The cost function is cross-entropy, and the
neural network weights are optimized using the Adam optimizer. We consider
batch sizes of $500$ samples, which add up to $900000$ samples per epoch, with
an $80\%-20\%$ split between training and validation. By a process of trial-
and-error we find that fixing the number of training epochs to $60$ allows us
to obtain an accuracy of $\approx 95\%$ as shown in Fig. 2(a). Here accuracy
is defined as the ratio of eigenstates correctly classified to the total
number of eigenstates. Once the network is trained, the eigenvectors from each
point in the phase space of $\left(\lambda,\sigma\right)$ are fed to the
network. We obtain three values each of which lies between $0$ and $1$ and
correspond to the neural network’s confidence of classifying the given input
in each phase/class.

We next generate another training data set comprising of eigenstates of the
AAH model given by Eq. 2. Once again the training set consists of $300000$
eigenstates belonging to each class classified as localized, multifractal and
delocalized using $D_{2}$ (see Eq. 12). The network architecture is the same
as before (see Fig. 1). The neural network weights are tuned using an Adam
optimizer, and the cost function is cross-entropy as before. Here again $60$
training epochs have been considered, and the batch size is $500$. The
$80\%-20\%$ split between training and validation is followed once again. As
demonstrated in Fig. 2(b), the network is trained to an accuracy of $\approx
95\%$.

(a) (b) (c)

Figure 5: Fractal dimension $D_{2}$ (whose value is represented by a colour
according to the code shown) as a function of disorder strength $\lambda$ and
increasing fractional eigenstate index $i/N$ starting from the ground state.
Here averaging has been performed over $100$ disorder realizations.
Classification of the single disorder realization of the eigenstates using the
network trained on the eigenstates of the (b) LRH Model and (c) AAH model. The
color of each point indicates the confidence for the delocalized (red),
multifractal (yellow), and localized (purple) phases. Here system size $N=510$
in all cases.

###  III.2 Binary classification

For binary classification, we train the network using the on-site probability
densities (PDs) of the delocalized and localized eigenstates of the AAH model.
As mentioned earlier, all the eigenstates are delocalized (localized) below
(above) a critical disorder strength $\lambda=2$, at which the eigenstates
show multifractal behaviour. For a system size of $N=510$, we consider $100$
disorder realizations ($\theta_{i}$) generating $51000$ samples separately for
the delocalized and localized classes by choosing disorder strengths
$\lambda=0.5$ and $4$, respectively. We construct a neural network which
consists of an input layer of size $N$ and utilizes on-site probability
densities of the eigenstates as inputs. This is followed by a single hidden
layer with the number of neurons equal to the greatest integer less than
$\left(N\times(2/3)\right)+1$. The neurons are coupled to the ReLU activation
function and an output layer coupled to the Sigmoid activation function. The
single output obtained is a number that lies between $0$ and $1$ \- a value
close to $0$ indicates that the state is delocalized while a value close to
$1$ indicates that it is localized. The scheme of the neural network
architecture is shown in Fig. 3. Here the cost function is binary cross-
entropy, and the optimizer is Adam. The batch size is $300$ samples, and $5$
training epochs are considered - we find that this is sufficient to obtain an
accuracy of $\approx 100\%$, when the training and validation data set is
split into an $80:20$ ratio. We use the metric “accuracy” to identify the
phase transitions. By training on two phases only, we also examine the ability
of the neural network to identify a previously unknown phase.

##  IV Results

In this section, we perform multi-class and binary classification using the
eigenvectors and probability densities, respectively. The results obtained
using the neural network approach are compared to the ones acquired from the
conventional multifractal analysis of $D_{q}$.

###  IV.1 Multi-class classification

We begin our analysis by calculating the multifractal dimension $D_{q}$ as a
function of disorder strength $\lambda$ for all the single particle
eigenstates of the LRH Hamiltonian for $\sigma=0.5,1.5$ and $3.0$ as shown in
Fig. 4(a)–4(c) respectively. In the case of $\sigma=0.5$, with an increase in
disorder strength, the fraction of delocalized eigenstates decreases, and
multifractal states are observed to appear in blocks, as shown in Fig. 4(a).
Thus a delocalized-to-multifractal (DM) edge is observed in the eigenstate
spectrum, which changes in a step-like fashion with increasing disorder
strength $\lambda$. In the case of $\sigma=1.5$ (see Fig. 4(b)) and
$\sigma=3.0$ (see Fig. 4(c)), we observe that the delocalized eigenstates are
separated from the localized eigenstates, with a delocalized-to-localized (DL)
edge which changes in a step-like fashion as the number of delocalized states
decreases with increasing disorder strength $\lambda$. We also observe the
presence of multifractal states in the vicinity of the DM / DL edges.

We next employ the multi-class classification algorithm for a single disorder
realization and compare its performance with the multifractal analysis, which
utilizes averaging over several disorder samples. First, the network (see Fig.
1) is trained using the eigenstates of the LRH model generated over multiple
disorder realizations for various values of $(\sigma,\lambda)$. We assign the
class to the vectors in the training data set with the help of $D_{2}$ using
Eq. 12. The phase diagram obtained as a function of disorder strength
$\lambda$ using the neural network for a single disorder realization is shown
in Fig. 4(d)–4(f) corresponding to $\sigma=0.5,1.5$ and $3$ respectively. The
states are classified as delocalized, multifractal and localized with
confidence $p_{1},p_{2}$ and $p_{3}$, respectively. In all figures, we have
plotted the $max\left(p_{1},p_{2},p_{3}\right)$ where the red, yellow and
purple colour codes are used to represent $p_{1}$, $p_{2}$, and $p_{3}$
respectively. Comparing Figs. 4(a)–4(c) with Figs. 4(d)–4(f), we observe that
the shape and location of the transitions agree very well at all values of
$\sigma$.

Next, we compare the neural network-based transition characterization when the
same network (see Fig. 1) is trained using the eigenstates of the AAH model.
Although the existence of a critical disorder strength at which all
eigenstates are multifractal and separate the delocalized and localized
eigenstates is well-known, we still assign the class to the vectors in the
training data set with the help of $D_{2}$ using Eq. 12. Once the network is
trained, we utilize it to obtain the phase diagrams of the LRH model as a
function of disorder strength $\lambda$ using the neural network for a single
disorder realization as shown in Fig. 4(g)–4(i) corresponding to
$\sigma=0.5,1.5$ and $3$ respectively. While the network can precisely predict
the location of the transitions/steps, the distinction between the delocalized
and multifractal states is not very sharp. Nevertheless, we still observe the
step-like features and the distinct phases.

We next compute the multifractal dimension $D_{q}$ as a function of disorder
strength $\lambda$ for all the single particle eigenstates of the AAH model
(see Fig. 5(a)). All the eigenstates are delocalized for $\lambda<2$ while all
eigenstates are localized for $\lambda>2$. At $\lambda=2$, the eigenstates are
multifractal [51, 50]. We obtain the phase diagram of the AAH model using the
neural network trained on the eigenstates of the LRH model (see Fig. 5(b)) and
the eigenstates of the AAH model (see Fig. 5(c)). The neural network can
accurately predict the phase diagram using a single disorder realization in
both cases.

Figure 6: Neural Network classification accuracy with increasing disorder
strength $\lambda$ for the central eigenstate of the AAH model for system size
$N=510$ averaged over $25$ disorder realizations.

Figure 7:  Classification of the eigenstates for a single disorder realization
of the AAH model as a function of disorder strength $\lambda$ and for
increasing fractional eigenstate index $i/N$ starting from the ground state.
Here the network is trained on the delocalized ($\lambda=0.5$) and localized
eigenstates ($\lambda=4$) of the AAH model. The probability of the state being
localized is given by the single output $P$, whose value is represented by a
colour according to the code shown. The corresponding plot using fractal
dimension $D_{q}$ is shown in Fig. 5(a). Here system size is $N=510$.

(a) (b) (c)

Figure 8:  Classification of the eigenstates for a single disorder realization
of the LRH model as a function of disorder strength $\lambda$ and with
increasing fractional eigenstate index $i/N$ starting from the ground state
for the long-range hopping parameter $\sigma$ equal to (a) $0.5$, (b) $1.5$
and (c) $3.0$. Here the network is trained on the delocalized ($\lambda=0.5$)
and localized eigenstates ($\lambda=4$) of the AAH model. The probability of
the state being localized is given by the single output $P$, whose value is
represented by a colour according to the code shown. The corresponding plot
using fractal dimension $D_{q}$ is shown in Fig. 4(a)–4(c). Here system size
is $N=510$ in all cases.

###  IV.2 Binary classification

In this subsection, we discuss the results obtained with a binary classifier
using the neural network shown in Fig. 3. The training data set consists of
the probability densities (PDs) corresponding to the eigenstates of the two
classes, i.e., delocalized ($\lambda=0.5$) and localized ($\lambda=4$). Since
we have a single neuron in the output layer, the output represents the
probability $P$ of the state being localized, implying that $1-P$ is the
probability of the state being delocalized. We may infer that intermediate
values of $P$ indicate that the state exhibits multifractal nature.

The information about the transition from the delocalized to localized regimes
is incorporated in the properties of the eigenstates. We feed the central
eigenstate (corresponding to the energy $E_{N/2}$ of the spectrum) to the
trained network to determine the transition point. In Fig. 6, we have plotted
the classification prediction as a function of disorder strength $\lambda$.
The transition point is revealed to be at $\lambda=2$ as the network learns
the difference between the localized and delocalized eigenstates. The
transition point obtained agrees with the one shown using multifractal
dimension $D_{2}$ in Fig. 5(a).

Next, we test this trained neural network, by inputting the probability
densities drawn from all the eigenstates of the AAH model, as $\lambda$ is
varied across the transition. The neural network gives a single output $P$,
which signifies the confidence of the network to classify the state as
localized, indicating that $1-P$ is the probability of classifying the state
as delocalized. For a single disorder realization, the phase diagram obtained
from the neural network shown in Fig. 7 is consistent with the one obtained
using multifractal dimension $D_{2}$ as shown in Fig. 5(a). We also note that
at the transition point where the multifractal states exist, the prediction
$P$ lies roughly midway between $1$ and $0$, consistent with theoretical
results.

Next, we implement the binary classification algorithm on the LRH model to
investigate how well our neural network can identify a previously unknown
phase. Since the network is trained exclusively on the delocalized and
localized phases, it is unfamiliar with the multifractal regime. In Fig.
8(a)–8(c), we have plotted the phase diagram by feeding the eigenstates of a
single disorder realization of the LRH model to the binary classifier
corresponding to $\sigma=0.5,1.5$ and $3.0$ respectively. We observe that our
network can indicate the presence of a new phase (multifractal phase). In Fig.
8(a), for $\sigma=0.5$, while the network shows confusion in the multifractal
phase, the step-like features that distinctly separate the delocalized states
can still be observed. In Fig. 8(b)–(c), for $\sigma=1.5$ and $3.0$, we
observe that the delocalized and localized states are classified and separated
by step-like edges along which multifractal states are observed with $0<P<1$.
In other words, while the network identifies the delocalized and localized
phases, in the case of the multifractal phase, it cannot clearly distinguish
it from the localized phase. This inability of the network to produce results
similar to multi-class classification is expected; as a matter of fact, the
close connection between the phase diagram obtained by the binary classifer to
the actual one is quite remarkable.

##  V Conclusion

In this work, we explore the ability of artificial neural networks to extract
information about various phases from single-particle states. We build a
multi-layer perceptron network and employ it to classify the delocalized,
multifractal and localized phases of the quasiperiodic long-range Harper
model. Our neural network produces phase diagrams that align with theoretical
predictions after being trained using the data associated with the three
phases. We establish that the machine successfully learns each phase’s
property from the eigenstates. The machine has high confidence throughout all
the phases, which is an indication of good feature extraction.

We also build a binary classifier with a single hidden layer having
probability densities of the eigenstates of the AAH model as inputs to
identify the delocalized and localized regimes. When we test this neural
network with the AAH model, we find that the resulting phase diagram agrees
very closely with the theoretically known phase diagram. Even though the
network is not trained with multifractal states, the phase diagram obtained
indicates the presence of multifractal states, in a manner very similar to the
actual phase diagram. Subsequently, the network is applied to the LRH model
for detecting an unlearned phase. The neural network shows confusion, whenever
multifractal states are involved; however the phase diagram obtained is
remarkably close to the actual one. The significance of our results from the
neural network-based approach is that by feeding just a single disorder
realization, we can obtain phase diagrams that are typically obtained by
averaging over several disorder realizations in the conventional approach.

The phase classification problems in the literature have primarily focused on
binary classification. In this context, investigating multiple phase
transitions and training partially blind networks to recognize unknown phases
remains unexplored. The value of this multi-neuron output strategy will be
much more significant when dealing with novel phases for which acceptable
order parameters are not known in advance.

##  VI Acknowledgement

A.A. is grateful to the Council of Scientific and Industrial Research (CSIR),
India, for her PhD fellowship. A.S. acknowledges financial support from SERB
via the grant (File Number: CRG/2019/003447) and DST via the DST-INSPIRE
Faculty Award [DST/INSPIRE/04/2014/002461]. All neural networks used in this
work were implemented in Python using TENSORFLOW [68].

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