# A Deep Learning Approach for Dynamic Balance Sheet Stress Testing

Anastasios Petropoulos<EMAIL_ADDRESS>Vassilis Siakoulis,
Konstantinos P. Panousis, Theodoros Christophides, and Sotirios Chatzis
Department of Electrical Engineering, Computer Engineering and Informatics
Cyprus University of Technology, Limassol, Cyprus

###### Abstract

In the aftermath of the financial crisis, supervisory authorities have
considerably improved their approaches in performing financial stress testing.
However, they have received significant criticism by the market participants
due to the methodological assumptions and simplifications employed, which are
considered as not accurately reflecting real conditions. First and foremost,
current stress testing methodologies attempt to simulate the risks underlying
a financial institution’s balance sheet by using several satellite models,
making their integration a really challenging task with significant estimation
errors. Secondly, they still suffer from not employing advanced statistical
techniques, like machine learning, which capture better the nonlinear nature
of adverse shocks. Finally, the static balance sheet assumption, that is often
employed, implies that the management of a bank passively monitors the
realization of the adverse scenario, but does nothing to mitigate its impact.
To address the above mentioned criticism, we introduce in this study a novel
approach utilizing deep learning approach for dynamic balance sheet stress
testing. Experimental results give strong evidence that deep learning applied
in big financial/supervisory datasets create a state of the art paradigm,
which is capable of simulating real world scenarios in a more efficient way.

###### keywords:

Stress Testing, Deep Learning, Bayesian Model Averaging, Capital Adequacy
Ratio, Forecasting, Neural Networks, Dynamic balance sheet, Constant balance
sheet assumption

††journal: Journal of Forecasting

##  1 Introduction & Motivation

Financial Stability is a core component for economic prosperity of countries
and individuals. The recent financial crises had significant impact in the
life of many individuals across the globe through the realization of
significant income reduction, rising unemployment and economic slowdown [1].
The recent experience from methods of risk management employed so far shows
that they fail to provide early warnings to central governments and central
banks in order to proactively intervene and prevent such adverse financial
events. Banks, regulatory authorities, and international organizations (like
IMF) performed stress testing exercises long before the financial crisis of
2007. Nevertheless, stress testing exercises before Lehman’s default failed to
predict the unprecedented economic turmoil, because they disregard the
propagation channels of a default event through the whole micro macro dynamics
of the global interconnected financial system. Additionally, the non-linear
relationships that realised between the macro economy and the financial
balance sheets was not adequately captured due to the broadly use of linear
regression models. Weaknesses also in the validation function of stress
testing frameworks also decreased the confidence in the quantification of the
impact an adverse scenario in the banking system. Since then, market
participants and regulators have performed rigorous stress testing exercises
by expanding the scenarios to be assessed, using more granular data, and in
some cases attempting to quantify second round effects stemming from a
liquidity shock or from the default of a counterparty.

Another aspect in the current regime, i.e. post crisis, of supervisory
regulation is the collection of a significant amount of granular information
as a response to a more proactive supervision. Although, the creation of big
data sets is the new era in banking supervision, regulators haven’t yet
explored statistical techniques from other fields, like machine learning, to
extract more information regarding the risks in their banking systems.
Segmentation, classification and data mining functionalities are important
tools for regulators to spot weaknesses in the supervised financial entities,
which can be further enhanced by using machine learning techniques.

Machine learning algorithms has dramatically improved the capabilities of
performing pattern recognition (like speech recognition, image recognition)
and forecasting, so that they offer state of the art performance in various
scientific fields like biology, engineering etc. Their structure offers the
ability to adjust in streaming sequences using continuous learning algorithms,
and recognize new and evolving patterns in time series data. In addition, deep
learning is proven to effectively deal with high dimensional data. Recent
studies suggest that due to their complicated and non-parametric structure,
machine learning techniques could lead to better predictive performance in
financial time series modelling problems [2, 3, 4]. This can be attributed to
their capacity to learn and adapt to new data. Thus, improve their performance
over time, offer increased capabilities to capture non-linear relationships,
and decompose the noise that often exist in financial data. Furthermore, this
new generation of statistical algorithms offer the necessary flexibility in
modelling multivariate time series, as its structure includes a cascade of
many layers with non-linear processing agents. Deep learning networks base
their functionality in the interaction of layers that simulate the abstraction
and composition functionalities of the human brain. Therefore, via capturing
the full spectrum of information contained in financial datasets, they are
capable of exploring in depth the inherent complexity of the underlying
dynamics in big and high dimensional time series data.

Motivated by the recent trends in the machine learning literature, this
empirical study introduces a new statistical technique for stress testing
using deep learning algorithms to model banks financial data in a holistic
way. In particular, shocks are propagated to banks’ balance sheets by
simultaneously training deep neural networks with macro and financial
variables, thus, taking advantage of their capabilities to capture more
information hidden in big datasets. We develop inference algorithms for our
networks, suitable for learning financial time series data on a multivariate
forecasting setup.

The main contribution of this study is that it proposes a holistic framework
for balance sheet stress testing, which overcomes the limitations of the
currently approaches and yields more robust and close to reality results by
loosening the static balance sheet assumption. Our research analysis lies at
the intersection of computational finance and statistical machine learning,
leveraging the unique properties and capabilities of deep learning networks to
increase the prediction efficacy and minimizing the modelling error. Under the
proposed approach, forecasting of balance sheet items can be heavily supported
by artificial intelligence algorithms, better simulating the propagation
channels of the macro economy into the financial institutions business models.
Our vision will be to provide a stress testing framework that succeeds to
perform as an early warning system for financial shocks on individual banks’
balance sheets.

This study is organized as follows. In section 2, we focus on the related
literature review on financial institutions stress testing. Section 3
describes the data collection and processing. In section 4 we provide details
regarding the estimation process of the various stress testing frameworks
examined in this study. In section 5 we compare across methodologies and
provide experimental results using a test dataset of financial balance sheet
sequences of data. This way, we assess not only the applicability of the
proposed approach but also its generalization capacity. Finally, in the
concluding section 6 we summarize the performance superiority of the proposed
methodology, we identify any potential weaknesses and limitations, while we
also discuss areas for future research extensions.

Figure 1: Feedforward architecture of currently established stress testing
frameworks.

A typical stress testing engine is composed by four elements: the perimeter of
risks subjected to stress, the scenario design, the calculation engine that
transforms the shocks into an outcome in Banks balance sheet, and a measure of
the outcome [5]. Reviewing current stress testing frameworks performed by
regulatory authorities they follow broadly this structure. In particular the
most famous stress testing exercises currently publicly available are: EBA
[6], CCAR (FED) [7], PRA - Bank of England [8], ECB (top down) [9, 10], Bank
of Canada [11], Central Bank of Austria (ARNIE) [12], IMF [13], Bank of Greece
(Diagnostic Exercise) [14]. The structure of all these exercises follows a
left to right flow to estimate the impact of an adverse shock in the economy.
One of the basic components of all these exercises is the time horizon they
estimate future losses for the banks which ranges between 2-5 years. During
this period the macro economic scenario is given. This set of macro scenarios
is passed through to financial institutions to project their P&L and Risk
Weighted Assets (RWA) and eventually estimate capital using regulatory hurdle
rates. Some of these exercises include in their structure a second round
effects mechanism for the banking system to account for contagion risk.
Macroeconomic feedback effects i.e for example the impact of a significant
institution becoming insolvent in the macro economy, usually they are not
considered in these frameworks. Stress tests under this structure can mainly
serve as a tool to challenge the recovery plans of banks and to assess their
viability. But their role as an early warning system is questionable.

As Drehmann [15] aptly points, systemic banking crises are reflected in the
performance of credit and property prices and usually they appear at the high
point of the medium-term financial cycle. Therefore, crisis starts before it’s
depicted in macro scenarios. According to Borio [5] a system is not fragile
when a large financial shock materializes but when even a small negative
change in financial and macro variables is amplified through the different
dynamic system relationships and can lead to a systemic shock. For example
after the default of Lehman the financial market crashed and the US GDP
exhibited a sharp decrease causing a structural break in the macro data time
series.

Current versions of stress tests possess a macro scenario over time in a
static way without modelling or tracking in a path dependent nature the
multistep decision process and financial behaviour that in reality takes place
from all economic participants. [16] Furthermore, non-linearity is not
modelled adequately in the statistical techniques currently employed. Risks
under the current globalized market tend to be amplified when a stress event
occurs. Non-linear relationship kicks in through channels of amplification
leading to a chain of events unpredictable from the static nature of stress
test. Under stressed conditions the relationships between modelled variables
are non-linear [17] [18] and exhibit structural breaks [19]. Stress testing
frameworks are composed by standalone models usually combined in a
qualitatively manner. A small single-step prediction error at the beginning
could accumulate and propagate when combined without taking the correlation of
the financial variables, often resulting in poor prediction accuracy.
Furthermore standalone models can lead to double counting effects or
overestimating the impact stemming from the changes of the predefined macro
variables. Finally univariate setups are not able to model adequately complex
distributed variables with non-linear behaviour.

In addition regular stress testing frameworks exhibit simplification
assumptions that may affect the reliability of the final estimation. EBA EU
wide stress testing is a bottom up exercise covering only specific risk on
banks individuals balance sheet based on a macro scenario usually based on
simplified assumptions. One of the weaknesses in EBA methodology is the static
balance sheet assumption i.e assets and liabilities remain constant over the
horizon without acknowledging for management actions and new generation of
loans. In addition mitigation actions are taken into account after the stress
testing are finalized through a strong qualitative overlay and not in a
dynamic way. [6]

On the other hand System wide stress testing exercises on micro prudential
level are heavily relied upon on the interaction with individual banks with
respect to data analytics and propagating the macro scenarios to their balance
sheet. Thus estimations are not performed in a uniform statistical process but
inherit the model deficiencies and forecasts errors embedded in banks
individual models. The heterogeneity in the results increases significantly
the estimation errors and there is no robust process for regulators to account
for it. Thus the need for independent central modelling for simulating the
financial system is of great concern [20]. Furthermore the stress testing
process involves the disclosure of the methodological framework to all market
participants which in turn there are evidence of second round effects
regarding the accounting treatment of banks. Specifically based on a study
[21] banks participating in regulatory exercises tend to manipulate their
provisions for credit risk to absorb the impact of the upcoming stress test.

Finally Stress Testing outcomes in the current regulatory exercises heavily
rely on regulatory ratios like capital adequacy ratio which in turn is highly
dependable on the estimation of RWA. Evidence in the literature [22] indicates
that relaying in the risk weights applied internally by the financial
institutions under the Basel Framework can lead to underestimation in capital
needs. This is driven by the significant variability stemming from internal
models of the banks when applying internal model methods. Furthermore the
regulatory framework currently employed for assessing the RWA cannot capture
the hidden risk in banks complex portfolio structure. In the current
literature [23] there evidence regarding especially more sophisticated banks
(A-IRB) that they may perform regulatory arbitrage and manipulates their true
risks to lower their capital requirements. Thus robust macro modelling of the
RWA using an independent top down model is important to account for these
cases.

Although a significance progress in designing stress testing has been realized
in recent years, even today there are concerns that this type of exercises
cannot be used as early warning systems for financial distress [1]. By
analysing the publications regarding stress testing exercises either performed
by regulators or individuals banks we outlined a series of weaknesses and
inefficiencies to provide a clear and concise view on the nature and on the
way how the proposed approach in this study, DeepStress, attempts to address
part of the aforementioned weaknesses.

Deep neural networks architecture is one of the main innovations in our
proposed approach for dynamic balance sheet stress testing. Our approach is
putting all the components together in a multivariate structure. We identify
the main channels of risk propagation in a recurrent form to account of all
the existing evidence of feedback effects in a financial institutions’ balance
sheet. The current architectures is constrained by the use classical
econometric techniques which offer limited capabilities for simulating complex
systems. Our approach accounting for temporal patterns in banks’ balance
sheets provides a dynamic modelling approach. This is achieved through the
multivariate training of deep neural networks taking account the dynamic
nature of banks metrics and the whole structure of the bank’s balance sheet.
The approach proposed is composed by multivariate input and output layers able
to capture the cross correlation between balance sheet items and the macro
economy. Training is performed as one big complex network minimizing
estimation errors and double counting effects among various financial
variables.

To account for non-linear relationships that materialize under adverse
macroeconomic conditions machine learning techniques like deep learning can
provide more efficient estimations. Deep Neural networks based on academic
literature are capable of simulating real life phenomena where relationships
are complex. Therefore, our proposed framework using multilayer deep networks
envisages in capturing the dynamics inherent in a financial distress. In
addition the architecture of aims to capture the amplifications channels
leading to structural breaks.

The methodology applied relies only on publicly available data and models are
developed in a uniform way thus making the process of validation and error
correction more feasible to be performed centrally. In addition offers the
opportunity to experiment on advanced statistical machine learning techniques
a need recognized also in the academic literature [24]. To sum up, our
modelling approach is balanced between capturing the determinants that
strongly affect the health of a financial institution, while at the same time
developing a dynamic balance sheet simulator engine for establishing an early
warning system to predict bank failures under an adverse scenario. The
modelling framework that we implement captures temporal dependencies in a
bank’s financial indicators and the macro economy. At the same time, it
explores up to 3 years of lagged observations, which are assumed to carry all
the necessary information to describe and predict the financial soundness of a
bank, and combines their evolution with the relevant macroeconomic indicators.

##  2 Data Collection and Processing

The dataset supporting this is study refers to the United States banking
system. Specifically, we have collected information on non-failed, failed and
assisted entities from the database of the Federal Deposit Insurance
Corporation (FDIC), an independent agency created by the US Congress in order
to maintain the stability and the public confidence in the financial system.
The collected information is related to all US banks, while the adopted
definition of a default event in this dataset includes all bank failures and
assistance transactions of all FDIC-insured institutions. Under the proposed
framework, each entity is categorized either as solvent or as insolvent based
on the indicators provided by FDIC. Observations referring the failed banks
are excluded from the analysis since stress testing is performed on healthy
financial entities.

The dataset covers the 2007-2015 period; a 9 years’ period with quarterly
information resulting in dataset with more than 175,000 records. The selected
time period, seems to approximate a full economic cycle, in terms of the
Default Rate evolution. Fig. 2, shows the number of records included in each
observation quarter and the corresponding default rate. From a supervisory
perspective, most of the financial institutions in the sample apply the
standardized approach for measuring the Credit risk weights assets based on
the United States adaptation of the Basel regulatory framework [25].

Figure 2: USA financial institutions in the sample. Historical overview for
the period 2008-2014 of the failed entities (source: FDIC)

The dataset was split into three parts (Fig. 2). An in-sample dataset (Full in
sample) that is comprised of the data pertaining to the 80% of the examined
companies over the observation period 2008-2013 amounting to 101.641
observations. For performing hyper parameter tuning of deep neural networks we
define an out-of-sample dataset (validation sample), including the rest 20% of
the observations for the period 2008-2013 amounting to 25.252 observations.
This is useful for deep learning models, in which the training sample is used
to train various candidate models with different architectures and
specifications, while the validation set is used for selecting the best
parameter setup and avoid overfitting in the training dataset. This way the
generalization capabilities in other datasets of the final selected model
increases substantially.

Finally performance evaluation is performed on an out-of-time dataset (test
sample) that spans over the 2014-2015 observation period reaching 48.756
observations. In all cases, the dependent (target) variable is the Capital
Adequacy Ratio (CAR) of each bank in the end of the one year forecast horizon.
To summarize, we performed model fitting using exclusively the available
training sample prescribed above. To perform model selection, we employed
five-fold cross-validation, using predictive accuracy as our model selection
criterion (CAR prediction error). Performance evaluation results are assessed
on the available test sample, to allow for evaluating the generalization
capacity of the developed models.

In developing our model specifications, we examine an extended set of
variables that fully describe the financial status of each bank in the sample.
In addition to the above-mentioned variables, we have also included in the
dataset quarterly observations of the most commonly used macro-economic
variables. Macro variables are the main input in the models developed since
they are important for scenario analysis under a stress testing framework. The
current model setup includes contemporaneous macro variables along with 3 year
lags. The intuition for this approach is to build models for scenario
prediction which is the main methodology for Stress Testing modelling. The
macro variables included in the development are:

  * 1.

GDP: Gross Domestic Product growth

  * 2.

EXPORT: US Total Exports growth

  * 3.

GOVCREDIT: Government Credit to GDP

  * 4.

DEBT: US public debt to GDP

  * 5.

GOVEXP: US government expenditure to GDP

  * 6.

INFLAT: US inflation

  * 7.

RRE: House Price Index growth

  * 8.

UNR: Unemployment Rate

  * 9.

YIELD10Y: 10Y US sovereign bonds yields

  * 10.

STOCKS: US Stock index – S&P 500 returns

The relevant stress financial variables for simulating the profitability and
the risk weighted assets of each financial institution are:

  * 1.

NLOAN: Net loans exposure

  * 2.

DEP: Total Deposits

  * 3.

DDEP: Total domestic deposits

  * 4.

ASSET: Average Total Assets

  * 5.

EASSET: Average Total Earning Assets

  * 6.

EQUITY: Average Total Equity

  * 7.

LOAN: Average total loans

  * 8.

CFD: Deposits Cost of funding

  * 9.

YEA: Yield on earning assets

  * 10.

NFIA: Noninterest income to average assets

  * 11.

RW: Risk Weight Density

  * 12.

LOSS_LOAN: Loss allowance to loans

  * 13.

RWA: Total risk weighted assets

  * 14.

CAR$\%$: Total risk based capital ratio

Modelling for the evolution of the balance sheet is performed on the growth
rate of 4 key financial items: Deposits, Total Earning Assets, Total Loans and
Total Assets. In order to capture the idiosyncratic characteristics of each
financial entity, 3 year lags are included in the training process for each
financial variable. In the final model setup the use of multiple years
financial and macroeconomic variables allows for capturing internal trends of
key items of a bank’ balance sheet and also the degree each entity is affected
by the status of US economy.

##  3 Model Development

The success of the stress testing exercises performed in the past by
regulatory authorities was put under scrutiny by all market participants and
the research community. In order to investigate the capabilities of the
proposed approach for stress testing against broadly used frameworks we
simulate two additional methods for balance sheet forecasting to benchmark its
performance. Specifically we developed a constant balance sheet approach
following the framework adapted by EBA to perform EU wide stress tests [6] and
a dynamic balance sheet approach support by a group of satellite models to
forecast individual financial variables used by other regulatory authorities
like ECB for macro prudential stress testing. In this section we provide an
overview of the overall setup of the study and technical details of the three
individual approaches employed.

###  3.1 General Setup of the Study

The main component of a micro prudential solvency stress testing framework is
the projection of a financial institution capital adequacy ratio or recently
the CET 1 ratio (Core Equity Tier I ratio). In this study, we develop a Deep
Neural Network structure which receives as input the Macro variables and
Balance sheet components mentioned in Section 2 and provides as output the
balance sheet and profitability structure of the bank on one year horizon as
measured by 9 core variables namely Net loans, Deposits, Assets, Earning
Assets, Deposits, Cost of funding, Yield on earning assets, Noninterest income
to assets, Risk Weight Density and Cost of Risk Loss allowance to loans).

Figure 3: Stress Test Deep Neural Network architecture.

We focus on the forecasting of the CAR ratio since CET-1 ratio was introduced
under Basel III and is not available throughout our dataset. Specifically, our
aim is to project the in a one-year-ahead the CAR ratio of each financial
institution in the sample. CAR ratio by definition is the ratio of a bank’s
capital over the risk weighted assets in each time point t. In order to
simulate the core mechanics of a stress testing framework we simulate the
evolution of the key financial variables of a financial institutions balance
sheet. The main setup is that we project one year ahead the evolution of the
capital and the the risk weighted assets in order to forecast the one year
ahead CAR. The approach followed to adjust the capital in time t is given by
the formula:

$\displaystyle\begin{split}\text{Capital}_{t}=&\text{Earnings from
Assets}_{t}-\text{loans loss provisions}_{t}\\\ &+\text{Net fees and
commisions}_{t}\\\ &-\text{cost of funding from
deposits}_{t}+\text{Capital}_{t-1}\end{split}$

In order to adjust the capital of each entity we model 8 key financial
variables. The first four refer to the dynamic evolution of the balance sheet
i.e the growth of the asset and liability side: the growth rate of Deposits,
Total loans, Total Assets, Total Earning Assets. The remaining 4 refer to the
yield in the next year of each item from the asset or liability side: cost of
risk of loans, yield on earning assets, yield on deposits and yield of net
fees and commissions of total assets.

The RWA are adjusted in 3 different ways depending on the ST methodology.
Specifically for deep learning we project the growth of the RWA, for satellite
modelling a dedicated model is trained to project the RW density of each
financial institution in the sample, while for the constant balance sheet
approach we assume RWA remain constant for one year.

###  3.2 Deep Learning

We implement a Deep Neural Network (henceforth DNN) to address the issue of
dynamic balance sheet forecasting. Deep learning has been an active field of
research in the recent years, as it has achieved significant breakthroughs in
the fields of computer vision and language understanding. In particular they
have been extremely successful in diverse time-series modelling tasks as
machine translation [26, 27] machine summarization and recommendation engines
[28]. However, their application in the field of finance is rather limited.
Specifically, our paper constitutes one of the first works presented in the
literature that considers application of deep learning to address the
challenging financial modelling task of financial balance sheet stress
testing. Deep Neural Networks differ from Shallow Neural Networks (one layer)
on the multiple internal layers employed between the input values and the
predicted result (Fig. 3).

Constructing a DNN without nonlinear activation functions is impossible, as
without these the deep architecture collapses to an equivalent shallow one.
Typical choices are logistic sigmoid, hyperbolic tangent and rectified linear
unit (ReLU). The logistic sigmoid and hyperbolic tangent activation functions
are closely related; both belong to the sigmoid family. A disadvantage of the
sigmoid activation function is that it must be kept small due to their
tendency to saturate with large positive or negative values. To alleviate this
problem, practitioners have derived piecewise linear units like the popular
ReLU, which are now the standard choice in deep learning research ReLU. The
activation layers increase the ability and flexibility of a DNN to capture
non-linear relationships in the training dataset. On a different perspective,
since DNNs comprise a huge number of trainable parameters, it is key that
appropriate techniques be employed to prevent them from overfitting.

Indeed, it is now widely understood that one of the main reasons behind the
explosive success and popularity of DNNs consists in the availability of
simple, effective, and efficient regularization techniques, developed in the
last few years. Dropout has been the first and the most popular regularization
technique for DNNs [29]. In essence, it consists in randomly dropping
different units of the network on each iteration of the training algorithm.
This way, only the parameters related to a subset of the network units are
trained during each iteration. This ameliorates the associated network
overfitting tendency, and it does so in a way that ensures that all network
parameters are effectively trained. Inspired from these merits, we employ
Dropout DNNs with ReLU activations to train and deploy feed forward deep
neural networks. More precisely we employ the Apache MXNET toolbox of R1. We
postulated deep networks that are up to five hidden layers deep and comprise
various numbers of neurons. Model selection using cross-validation was
performed by maximizing the RMSE metric on the projected CAR.

In our setup multivariate deep learning networks will learn the balance sheet
of financial institutions separately generating yearly forecasts by the
interactions of layered neurons after receiving historical values of banks
previous economic states. This hierarchical transmission of observed data
between cascading layers of abstraction can decompose the structure of a bank
balance sheet and foster the multivariate representation of the financial
variables for capturing the correlations between various assets and
liabilities. This provides the functionality of simultaneously modelling the
balance sheet as a whole instead of using satellite models of regular stress
testing frameworks. This is feasible based on the fact that DNN are composed
of multiple features for input and output complex representations. Deep
learning can facilitate the dynamic balance sheet projection approach through
the non-linear relationships representations of each layer offering a more
realistic approach for stress testing. Information flows through the system as
a vector of macro and financial variables describing the state of both the
bank and the macro economy at any time stamp during the forecast period.
Specifically the input vector contains around 60 variables and the output
vector is composed of 9 variables. The DNN architecture employed is capable to
model the lead lag relationships between macro variables banks variables
financial variables and sovereign variables. Finally in the DeepStress engine
using the aforementioned multivariate forecasting setup on individual balance
sheet we model simultaneously the RWA evolution of the bank and connect it to
the macro environment.

####  3.2.1 Bayesian Deep Learning

Conventional architectures compute point estimates of the unknown values
(e.g., layers’ weights) without taking into consideration any prior
information and without any uncertainty estimation of the produced values. The
Bayesian framework offers a flexible and mathematically solid approach to
incorporate prior information and uncertainty estimation by explicitly
employing model averaging. The Bayesian treatment of particular model has been
shown to increase its capacity and potential, while offering a natural way to
assess the uncertainty of the resulting estimates. To this end, we augment the
conventional architectures of the previous sections, by relying on the
Bayesian framework. Specifically, we impose a prior Normal distribution over
network weights, seeking to infer their posterior distribution given the data.
Since the marginal likelihood is intractable for the considered architectures,
from the existing Bayesian methods, we rely on approximate inference and
specifically on Variational Inference.

Since the true posterior of the model cannot be computed, in Variational
Inference, we introduce an auxiliary variational posterior distribution of a
family of distributions; we then try and find the optimal member of the
considered family to match the true posterior. This matching is achieved
through the minimization of the Kullback-Leibler divergence between the true
and the introduced variational posterior. The KL divergence is a metric of
similarity between two distributions and is a non-negative value; KL is zero,
if and only if, the two considered distributions match exactly. Minimizing the
KL divergence is equal to the maximization of the Evidence Lower Bound, a well
known bound on the marginal likelihood derived using Jensen’s inequality. Thus
for training the following architectures, we resort to ELBO maximization.

##### Local-Winner-Takes-All Mechanism

The commonly employed nonlinearities such as ReLUs are a mathematically
convenient tool for training deep networks but nevertheless do not come with a
biologic plausibility. Research has shown that in the mammal brain, neurons
with similar functionality and structure tend to group and compete with each
other for their output. To this end, researchers have devoted significant
effort to explore this type of competition between neuron and apply it in
existing models. The resulting procedure is referred to as Local Winner-Takes-
All and has been shown to provide competitive, or even better results in
benchmarks architectures in the image recognition domain [30]. Thus, apart
from the conventional ReLU activations of the previous section, we
additionally explore the potency of the LWTA to our domain. The linear units
after the affine transformation in each layer are grouped together and compete
from their outputs. This competition is performed in a probabilistic way, by
employing a softmax nonlinearity, obtaining the probability of activation of
each unit in each block.

Specifically, let us consider input data $X\in\mathbb{R}^{N\times D}$,
containing $N$ observations, with $D$ features each. In a traditional hidden
layer, we compute an inner product between the input and a weight matrix
$W\in\mathbb{R}^{D\times K}$, and the resulting activation is then passed
through a non-linear function; the output of each layer is denoted as
$Y=\\{\boldsymbol{y}_{n}\\}_{n=1}^{N}$, with
$\boldsymbol{y}_{n}\in\mathbb{R}^{K}$. Thus, the corresponding procedure can
be described by the following computation:

$\displaystyle y_{nk}=\sigma\left(\sum_{d=1}^{d}w_{dk}x_{nd}\right)$ (1)

The most widely used non-linearity $\sigma(\cdot)$ is the Rectified Linear
Unit (ReLU), such that for an input $x$:

$\displaystyle relu(x)=\max(0,x)$ (2)

In the LWTA approach, this mechanism is replaced by introducing blocks of
units in each layer; therein, each unit is competing with the rest for its
activation. The winner unit gets to pass its output to the next layer, while
the rest are zeroed out. Now, the layer input is presented to each block; the
weights of the layer are reorganized in a three dimensional matrix, such that,
$W\in\mathbb{R}^{D\times B\times U}$, where $B$ denotes the number of blocks
and $U$ the units in each block. Assuming an input $x\in\mathbb{R}$ and
weights $W\in\mathbb{R}^{D\times U}$ the output of each block and each unit
therein, reads:

$\displaystyle y_{ku}=\begin{cases}h_{ku},&\text{if }h_{ku}\geq h_{ki},\forall
i\neq u\\\ 0,&\text{otherwise}\end{cases}$ (3)

where $h_{ku}=\sigma\left(w_{ku}x\right)$. A graphical illustration of the
adopted approach is presented in Fig. 4.

$x_{1}$$x_{D}$$\dots$$\dots$$\dots$$\dots$ LWTA layer $1$ $1$ $K$ $K$ LWTA
layer Input layer Output layer

Figure 4: The LWTA architecture: Each layer comprises blocks of competing
units; therein each unit computes its activation and competes with the rest
for its output. The winner get to pass its output to the next layer, while the
rest are zeroed out.

We use the same architectures as in the previous experiments for comparability
and in order to assess any potential gains from the use of a biologically
inspired mechanism.

###  3.3 Satellite Modelling - Bayesian Model Averaging

Satellite models are used for univariate estimations of the impacts of
standalone balance sheet items in current stress testing frameworks [9]. A
usual statistical technique employed from regulators and by the banking
industry is the Bayesian Model Averaging. The main intuition behind the use of
BMA econometric technique is to account for the uncertainty surrounding the
main determinants of risk dynamics especially in a period of recession. This
approach is able to handle a short time series of balance sheet realizations
which is usually the case for stress testing. Thus BMA offers the possibility
to perform multivariate modelling including all potential predictors with
different weight while the output of each trained model remains univariate.

Using BMA, a pool of equations is generated using a random selection subgroup
of determinants. Subsequently a weight is assigned to each model that reflects
their relative forecasting performance. Aggregating all equations using the
corresponding weights produces a posterior model probability. The number of
equations estimated in the first step is large enough to capture all possible
combinations of a predetermined number of independent variables. Thus Bayesian
Model Averaging addresses model uncertainty and misspecification in selected
explanatory variables in a simple linear regression problem.

To further illustrate BMA, suppose a linear model structure, with $Y_{t}$
being the dependent variable, $X_{t}$ the explanatory variables, $\alpha$
constant, $\beta$ the coefficients, and $\epsilon_{t}$ a normal error term
with variance $\sigma$.

$\displaystyle Y_{t}$
$\displaystyle=\alpha_{\gamma}+\beta_{\gamma}X_{\gamma,t}+\epsilon_{t}$ (4)
$\displaystyle\epsilon_{t}$ $\displaystyle\sim\mathcal{N}(0,\sigma^{2}I)$ (5)

A problem arises when there are many potential explanatory variables in a
matrix $X_{t}$ which transforms the task of selecting the correct combination
quite burdensome. The direct approach to inference in a single linear model
that includes all variables is inefficient or even infeasible with a limited
number of observations. It can lead to overfitting, multicollinearity and
increased manual re-estimations to account for non-significant determinants.
BMA tackles the problemy estimating models for all possible combinations of
$\\{X\\}$ and constructing a weighted average over all of them.

Under the assumption that $X$ contains $K$ potential explanatory variables,
BMA estimates $2^{K}$ combinations, and thus, $2^{K}$ models. Applying Bayes’
Theorem, model averaging is based on the posterior model probabilities:

$\displaystyle\begin{split}p(M_{\gamma}\cup Y,X)&=\frac{p(Y\cup
M_{\gamma},X)p(M_{\gamma})}{p(Y\cup X)}\\\ &=\frac{p(Y\cup
M_{\gamma},X)p(M_{\gamma})}{\sum_{s=1}^{2^{K}}p(Y\cup
M_{s},X)p(M_{s})}\end{split}$ (6)

In Equation (6), $p(Y,X)$ denotes the integrated likelihood which is constant
over all models and is thus simply a multiplicative term. Therefore, the
posterior model probability (PMP) is proportional to the integrated likelihood
$p(Y\cup M,X)$ which reflects the probability of the data given the model $M$.
Thus, the corresponding weight assigned to each model is measured by using
$p(M_{\gamma}\cup Y,X)$ in Eq. (6).

In equation (6), $p(M)$ denotes the prior belief of how probable model $M$ is
before analyzing the data. Furthermore, to estimate $p(Y,X)$ integration is
performed across all models in the model space and to estimate the probability
$p(Y\cup M,X)$ integration is performed given model $M$ across all parameter
space. By performing renormalization of the product in equation (6), PMPs can
be inferred and subsequently the model’s weighted posterior distribution for
estimator $\beta$ is given by

$\displaystyle p(\beta\cup Y,X)=\sum_{\gamma=1}^{2^{K}}p(\beta\cup
M_{\gamma},Y,X)p(M_{\gamma}\cup X,Y)$ (7)

The priors, posteriors and the marginal likelihood employed in the estimation
are described analytically in Appendix A. For model development, the same
train set used for DNN is employed. Before applying the Bayesian Averaging
algorithm we remove and linearly interpolate the outliers. In Bayesian Model
Averaging estimation we employ unit information prior (UIP), which sets g=N
commonly for all models. We use also a birth/death MCMC algorithm (20000
draws) due to the large number of covariates included since using the entire
model space would lead to a large number of iterations. We fix the number of
burn-in draws for the MCMC sampler to 10000. Finally the models prior employed
is the “random theta”’ prior by Ley and Steel [31], who suggest a binomial-
beta hyper prior on the a priori inclusion probability. This has the advantage
that is less tight around prior expected model size (i.e. the average number
of included regressors) so it reflects prior uncertainty about model size more
efficiently. For robustness purposes we varied the used prior employing the
Fernandez [32] propositions but the results were not substantially different.
In order to develop all the satellite models for this approach we employ the
utilities of BMS R
package111https://cran.r-project.org/web/packages/BMS/index.html. After the
training process 9 BMS models are developed: 4 for the growth of balance sheet
items, 4 models are forecasting the yields of a various assets and liabilities
and one model for forecasting the RW assets density.

###  3.4 Constant Balance Sheet Modelling Setup

For the constant balance all balance sheet items are assumed constant along
with the RWA metric for one year. Thus we combine the respective univariate
satellite models BMA to project yields of assets and liabilities while assume
zero growth in the balance sheet in order to project the CAR ratio one year
ahead.

Table 1: Comparison of the predicted one year ahead CAR by ST approach for all
banks and only for Large financial institutions (more than 200 billion in
assets).

All banks in the dataset | Out of Sample CAR | In Sample CAR
---|---|---
Satellite Modelling (BMS) | 20.61 | 17.07
Deep Learning (MXNET) | 18.01 | 17.89
Deep Learning (Bayesian ReLU) | 18.80 | 17.83
Deep Learning (Bayesian LWTA) | $\mathbf{19.23}$ | $\mathbf{18.53}$
Constant Balance Sheet | 20.03 | 17.49
Actual | 19.33 | 18.73

Large Banks ($>$ 200 bl) | Out of Sample CAR | In Sample CAR
---|---|---
Satellite Modelling (BMS) | 15.07 | 11.04
Deep Learning (MXNET) | 12.7 | 11.12
Deep Learning (Bayesian ReLU) | $13.2$ | $11.72$
Deep Learning (Bayesian LWTA) | $\mathbf{13.43}$ | $\mathbf{12.13}$
Constant Balance Sheet | 15.11 | 11.48
Actual | 13.75 | 14.16

##  4 Model validation - Experimental Evaluation

No thorough and consistent framework exists for validating the results of a
stress testing exercise since the adverse scenario used in their design never
materialize. Back testing methods is an important process to recognize
modelling inefficiencies and fine tune the estimations taking into account
specificities in the time series data that were not capture in the initial
calibration and development phase. Thus in order to improve the quality of
stress testing rigorous validation procedures of actual vs predicted financial
variables are important. Furthermore, according to academic literature the
success of the stress testing exercises after the financial crises maybe be
circumstantial [20] since no robust methods are applied to quantify their
estimation error.

Figure 5: Out of sample back testing results of the Capital of the three
balance sheet approaches compared with the actual figures (Whole Sample).
Figure 6: In sample back testing results of CAR ratio of the balance sheet
approaches (Whole Sample).

Following a different venue in this study we perform a thorough validation
procedure in order to assess the robustness of our approach. In this section
we summarize the results of the 5 approaches. More precisely, we report the
performance results obtained from the experimental evaluation of our methods,
in terms of in-sample fit (train dataset) and out-of-time performance (test
sample). To sum up, after developing our Stress testing frameworks in the “In-
sample” datase spanning the years 2010 to 2013 (16 quarters), we assess its
performance under the “Out-of-time” dataset in which the performance of each
model is evaluated during a future time period for evaluating their
generalization capacity. More precisely, we report performance results
obtained by evaluating our method over a two year (8 quarters) out of sample
time-period spanning from 2014 – 2015. Validation is performed with respect
the one year ahead forecast of the CAR ratio. Note that the last 2 two years
of the dataset were not used for model development. Prediction accuracy of the
CAR ratio, as measured by the deviation between the forecast of each framework
against the actual CAR ratio of each financial institution, is the main
criterion to assess the efficacy of each method and to select the most robust
one. In this section, we present a series of metrics that are broadly used for
quantitatively estimating the forecasting accuracy on continuous outcomes. We
evaluate the stress testing methods with the usual forecast metrics of Root
Mean Square Error (RMSE), Mean Absolute Error (MAE) and the Mean Absolute
Percentage Error (MAPE). These metrics are used so as to derive a full-
spectrum conclusion regarding the relative forecasting power of each
framework.

Figure 7: Out of sample back testing results of CAR ratio of the three balance
sheet approaches (Large Banks in the out of Sample)

As we observe in Table 1, Deep Learning algorithms provide the best empirical
fit both in-sample and out-of sample terms. Deep learning offers a more
efficient and holistic way to simulate the CAR ratio under a specific set of
macro scenarios of key macroeconomic variables as the predicted average CAR is
closer to the actual one compared to Satellite Modelling and Constant Balance
sheet stress testing assumptions. Table 2 summarizes the results of all
aforementioned samples with respect the CAR ratio and the prediction error
validation metrics (RMSE, MAE, MAPE). Based on the figures reported in the
test sample Deep Learning algorithms provide more accurate estimation of the
CAR ratio exhibiting a significant decrease in the forecasting error. Another
remark based on the experimental results is that, by moving from simple neural
networks to Bayesian networks, we are able to infer richer and subtler
dynamics from the data, thus increasing our capacity in modelling
nonlinearities and cross-correlations among balance sheet P&L items. This is
also evident from Figs. 5 and 7 where the out of sample performance of
constant balance sheet and satellite modelling diverge significantly from the
actual evolution of Regulatory Capital (Fig. 5) and the CAR ratio (Fig. 8) in
the dataset even though they provide adequate fit in the developent sample
(Fig. 6). Contrary average CAR ratios estimated based on Deep Learning methods
depict in an appropriate manner the CAR dymanics in the projection period.

Figure 8: Out of sample back testing results of the Capital of the three
balance sheet approaches compared with the actual figures (Whole Sample).
Table 2: Comparison of the predicted one year ahead CAR by ST approach for all
banks and only for Large financial institutions (more than 200billions in
assets)

All banks | Out of Sample (2014Q1-2015Q4)
---|---
| RMSE | MAPE | MAE
Satellite Modelling (BMS) | 11.32 | 2.88 | 0.15
Deep Learning (MXNET) | 11.18 | 2.36 | 0.12
Deep Learning (Bayesian ReLU) | 15.36 | 2.12 | 0.10
Deep Learning (Bayesian LWTA) | $\mathbf{10.75}$ | $\mathbf{1.77}$ | $\mathbf{0.09}$
Constant Balance Sheet | 11.15 | 2.85 | 0.15
| In Sample (2010Q1-2013Q4)
Satellite Modelling (BMS) | 13.46 | 2.58 | 0.16
Deep Learning (MXNET) | 13.49 | 2.55 | 0.15
Deep Learning (Bayesian ReLU) | 16.58 | 2.41 | 0.15
Deep Learning (Bayesian LWTA) | 18.70 | 2.16 | 0.14
Constant Balance Sheet | 17.25 | 2.56 | 0.15

Large Banks ($>$ 200 bl) | Out of Sample (2014Q1-2015Q4)
---|---
| RMSE | MAPE | MAE
Satellite Modelling (BMS) | 3.21 | 2.31 | 0.17
Deep Learning (MXNET) | 2.28 | 1.97 | 0.15
Deep Learning (Bayesian ReLU) | $\mathbf{1.96}$ | 1.56 | 0.12
Deep Learning (Bayesian LWTA) | 2.04 | $\mathbf{1.51}$ | $\mathbf{0.11}$
Constant Balance Sheet | 3.56 | 2.58 | 0.19
| In Sample (2010Q1-2013Q4)
Satellite Modelling (BMS) | 3.44 | 3.14 | 0.23
Deep Learning (MXNET) | 3.46 | 3.13 | 0.22
Deep Learning (Bayesian ReLU) | 3.07 | 2.78 | 0.20
Deep Learning (Bayesian LWTA) | 2.76 | 2.42 | 0.18
Constant Balance Sheet | 3.27 | 2.94 | 0.21

To further investigate the performance of Deep Stress approach we narrow down
the results on a subset of large financial institutions where performance of a
robust stress testing methodology is more important due to their size and
social-economic impact. Big financial institutions are defined as entities
with more than 200 billion in assets for the purpose of this study. Tables 1
and 2 also shows that both in terms of fit and terms of error metrics the
superiority of deep neural network is confirmed with significant drops in the
forecasting error in the test sample. Another worth mentioning results is the
fact that although satellite univariate modelling in the sample dataset was
expected to provide a better fitting against the DNN this is not the case. DNN
is trained in a multivariate setup attempting to model 9 variables at the same
time and still exhibits a rather comparable in sample error against the other
two methods. The same pattern also holds in Fig. 7 where the projected CAR is
graphed only for this category of large banks (more than 200 billion in
assets).

Summarizing the results across all metrics in the test sample, it is evident
that Deep Learning Algorithms exhibits higher predicting power compared to all
the considered benchmark approaches. Among the other two approaches it is
evident that the constant balance assumption although easier to implement
exhibits the highest error. Hence, it is crucial for supervisory authorities
to rethink current stress testing exercises that are based on the constant
balance sheet assumption and move towards a dynamic balance sheet approach.

##  5 Conclusions and Future Work

In this study we propose a new approach to be utilized in regulatory stress
testing exercises called Deep Stress that utilizes the properties of deep
learning. The main novel contribution of this empirical research to the
literature of forecasting economic and financial crisis events is that we
explore this new statistical technique to tackle the problem of dynamic
balance stress testing. Deep learning is utilized to provide a holistic
modelling approach for a bank’s key financial items. We perform thorough
testing and validation of the proposed technique. Experimental results provide
strong evidence to further be explored in the future by regulators and
financial institutions in order to produce a new generation of stress testing.
Deep stress is compared with two broadly accepted stress testing frameworks:
constant balance sheet and satellite dynamic modelling.

Summarizing our experimental results, we have found that Deep Neural Networks
consistently outperform the benchmark approaches, Our analysis provided strong
evidence of increased forecasting accuracy with respect to the CAR ratio and
performance consistency, which implies a much stronger generalization capacity
compared to alternative benchmark frameworks. Validation measures RMSE, MAE
and MAPE significantly decrease in the test sample using DeepStress providing
better simulation of the CAR ratio. Hence, these findings render our approach
much more attractive to researchers and practitioners working in real-world
financial institutions. The main driver for this higher forecasting accuracy
is the potential to model the balance sheet intercorrelation of P&L items
providing better simulation of the banks one-year-ahead activities.
Summarizing, Deep Stress offers a better dynamic balance sheet simulator which
is a major component in any stress testing framework by better capturing that
small macro and financial changes that can be amplified exponentially under a
crisis event. The holistic and dynamic nature of our framework leads to
significant decrease in the forecasting error by modelling better the feedback
loops and the interdependence of various items of a financial institution
balance sheet with the macro economy.

The aforementioned cascading layers structure of deep learning algorithms will
open up new horizons for financial system simulation combining brain inspired
computation and statistical machine learning. Of course our initial endeavour
is concentrated on the banking system which the backbone of the global economy
but is scalable to other entities as well like large corporate, insurances and
shadow banking. The system can be used by policy makers to test various
measures and to monitor the system in a forward looking manner. Our aim is to
provide innovation in the way regulatory authorities monitor the system and
increase awareness for possible future financial shocks. By simulating the
complex nexus of the current financial establishment governments can
proactively take steps to mitigate any forthcoming adverse events. DeepStress
can finally be used to measure the social impact of a possible financial or
systemic shock through the adjusted projections of various key macro variables
like unemployment, wealth, credit expansion etc.

An aspect this work has not considered concerns developing deep learning
models that can be continuously retrained in a moving window (online learning)
setup. Another possible way forward is the exploration of deep neural networks
under a broad datasets referring to multiple jurisdictions. Finally, it is
evident that the postulated Deep Learning networks can effectively capture
nonlinearities in the relationship between the input variable and the output
variable. Although, the validation framework cannot fully capture the
estimation error in a Stress testing exercise due to the fact the dataset does
not include crisis years. The results though provide evidence for the
forecasting efficacy of DeepStress for several years simulating a baseline
scenario of a Stress Testing exercise. To enhance the validation framework of
our approach we will intensify the data collection process to gather
information referring to several years before the financial crises in order to
use DeepStress to simulate and predict failed entities that took place during
this period. The value of such novel developments remains to be examined in
our future research endeavours.

## Appendix A

###  A.1 Prior Selection In BMA models

It is a popular choice to set a uniform prior probability for each model to
represent the lack of prior knowledge. It is often the case in BMA to assume
no prior knowledge for each model and assign a uniform prior probability i.e.
$p(M_{\gamma})\propto 1$. Regarding the marginal likelihoods $p(M_{\gamma}\cup
Y,X)$ and the posterior distributions $p(\beta\cup M_{\gamma},Y,X)$, the
literature standard is to use a specific prior structure called Zellner’s g
prior in order to estimate posterior distributions in an efficient
mathematical way. In this setup the prior knowledge for the coefficients is
assumed to be a normal distribution with pre-specified mean and variance.
Specifically the parametric formulation is given by Eq. (8):

$\displaystyle\beta_{\gamma}\cup
g\mathcal{N}\left(0,\sigma^{2}\left(\frac{1}{g}X_{\gamma}^{\prime}X_{\gamma}\right)\right)$
(8)

According to Eq. (8), coefficients are assumed to have zero mean and a
variance-covariance structure which is broadly in line with that of the data
$X_{\gamma}$. The hyper-parameter g denotes the prior level of confidence that
the coefficients are zero. The posterior distribution of the coefficients
follows a t-distribution with expected value
$\frac{g}{1+g}\hat{\beta}_{\gamma}$, where $\hat{\beta}_{\gamma}$ is the
standard OLS estimator for model $\gamma$. Thus, as $g\rightarrow\infty$ the
coefficient estimator approaches the OLS estimator. Similarly, the posterior
variance of $\beta_{\gamma}$ is affected by the value of $g$:

$\displaystyle\begin{split}\text{Cov}(\beta_{\gamma}\cup
Y,X,g,M_{\gamma})&=\frac{(Y-Y^{\prime})^{\prime}(Y-Y^{\prime})}{N-3}\frac{g}{1+g}\times\\\
&\quad\left(1-\frac{g}{1+g}R_{\gamma}^{2}\right)\left(X_{\gamma}^{\prime}X_{\gamma}\right)^{\prime}\end{split}$
(9)

The posterior covariance is similar to that of the OLS estimator, times a
factor that includes $g$ and $R_{\gamma}^{2}$.(OLS R squared for model
$\gamma$). For BMA, this prior framework results in a marginal likelihood
which includes a size penalty factor adjusting for model size $k_{\gamma}$
given by:

$\displaystyle\begin{split}p(Y\cup
M_{\gamma},X,g)&\propto(Y-y^{\prime}Y)^{\prime}(Y-Y^{\prime}y)^{\frac{-N-1}{2}}(1+g)^{-\frac{k_{\gamma}}{2}}\times\\\
&\quad\left(1-\frac{g}{1+g}\right)^{\frac{-N-1}{2}}\end{split}$ (10)

The “default” approach for hyper-parameter g is the “unit information prior”
(UIP), which sets $g=N$ for all models.

## Acknowledgment

This work was co-funded by the European Regional Development Fund and the
Republic of Cyprus through the Research and Innovation Foundation (Project:
POST-DOC/0718/196).

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#  Aharonov-Bohm flux and dual gaps effects on energy levels in graphene
magnetic quantum dots

Fatima Belokda  Laboratory of Theoretical Physics, Faculty of Sciences,
Chouaïb Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Ahmed
Bouhlal  Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb
Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Ahmed Siari
Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb Doukkali
University, PO Box 20, 24000 El Jadida, Morocco  Ahmed Jellal
<EMAIL_ADDRESS>Laboratory of Theoretical Physics, Faculty of Sciences,
Chouaïb Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Canadian
Quantum Research Center, 204-3002 32 Ave Vernon, BC V1T 2L7, Canada

###### Abstract

We address the question of how the Aharonov-Bohm flux $\Phi_{AB}$ can affect
the energy levels of graphene magnetic quantum dots (GMQDs) of radius $R$. To
answer this question, we consider GMQDs induced by a magnetic field $B$ and
subjected to two different gaps - an internal gap $\Delta_{1}$ and an external
gap $\Delta_{2}$. After determining the eigenspinors and ensuring continuity
at the boundary of the GMQDs, we formulate an analytical equation describing
the corresponding energy levels. Our results show that the energy levels can
exhibit either a symmetric or an asymmetric behavior depending on the valleys
$K$ and $K^{\prime}$ together with the quantum angular momentum $m$. In
addition, we find that $\Phi_{AB}$ causes an increase in the band gap width
when $\Delta_{1}$ is present inside the GMQDs. This effect is less significant
when a gap is present outside, resulting in a longer lifetime of the confined
electronic states. Further increases in $\Phi_{AB}$ reduce the number of
levels between the conduction and valence bands, thereby increasing the band
gap. These results demonstrate that the electronic properties of graphene can
be tuned by the presence of the AB flux, offering the potential to control the
behavior of graphene-based quantum devices.

###### pacs:

73.22.Pr, 72.80.Vp, 73.63.-b
Keywords: Graphene, magnetic field, quantum dot, Aharonov-Bohm flux, dual
gaps, energy levels, band gap, symmetry.

##  1 Introduction

It is the hexagonal arrangement that gives graphene its unique properties [1].
These include exceptional electronic conductivity, record thermal
conductivity, high mechanical strength, flexibility, and transparency [2, 3,
4], making it one of the most robust materials [5, 6]. Graphene has a peculiar
electronic band structure and is characterized by the presence of singular
points called Dirac points, which are at the Fermi level. This means that
there is no direct gap between the valence and conduction bands [7, 8]. The
Dirac equation provides valuable insights into the properties of relativistic
massless particles that underlie the linear behavior observed in graphene [9,
10].

The need for a band gap has led to the development of several techniques to
remotely control the conductance in graphene. Among these techniques is the
confinement of fermions in graphene to create quantum dots (QDs). Numerous
methods for achieving this confinement have been documented in the literature,
one see the review paper [11]. These include the application of inhomogeneous
magnetic fields [12, 13], cylindrical potential symmetry [14], and spatial
modulation of the Dirac band gap [15]. The confinement can also be realized by
carving the flake into small nanostructures [16, 17], inducing a band gap by
substrate interaction [18, 19], and using a co-precipitation method [20, 21].
We recall that QDs offer innovative applications in electronic devices [22],
valves [23], photovoltaics [24], qubits [25], and gas detection [26], among
others. The electronic properties of graphene quantum dots has been studied in
several occasions.

We are interested in studying how the Aharonov-Bohm (AB) flux $\Phi_{AB}$
affects the energy levels of graphene magnetic quantum dots (GMQDs) of radius
$R$ and thus modifies the band gap. More specifically, we want to understand
how $\Phi_{AB}$ together with both the external band gap $\Delta_{2}$ and the
internal band gap $\Delta_{1}$ affect the electronic bands. By solving the
Dirac equation, we obtain the eigenspinors as a function of $\Phi_{AB}$,
$\Delta_{1}$, and $\Delta_{2}$ for the valleys $\tau=\pm 1$. Then we determine
the corresponding energy levels by ensuring the continuity of the eigenspinors
at the boundary of the quantum dots. Among the results, we observe both
symmetric and asymmetric behavior in the energy levels. In particular, the
flux changes the electronic band structure of the quantum dots. Specifically,
we show that the incorporation of $\Phi_{AB}$ widens the band gap, creating a
distinction between the valence and conduction bands while maintaining the
symmetry of the energy levels. This broadening extends the trapping time of
electrons within the GMQDs. Furthermore, the introduction of an energy band
further widens the gap and introduces new energy levels, albeit breaking the
energy symmetry. In summary, we argue that the AB flux serves as an external
source to control the electronic properties of our system.

The paper is organized as follows: In Sec. 2, we present the theoretical
framework relevant to the present system and solve the eigenvalue equation to
obtain the eigenspinors. Matching these at the boundary of the GMADs yields
the formula for the energy levels as a function of $B$, $R$, $\Phi_{AB}$,
$\Delta_{1}$, and $\Delta_{2}$. In Sec. 3, due to various dependencies, we
perform a numerical analysis of the energy levels under appropriate
conditions. Finally, we conclude our results.

##  2 Theoretical Model

We consider a system of graphene magnetic quantum dots (GMQDs) subjected to
the Aharonov-Bohm (AB) flux and dual gaps, as shown in Fig. 1. The
introduction of these two quantities can have significant effects on the
electronic properties of MGQDs. The AB effect is a quantum mechanical
phenomenon that occurs when charged particles are affected by a magnetic
field, even in regions where the magnetic field strength is zero. This effect
occurs due to the phase acquired by the wave function when charged particles
traverse a closed path in the presence of a magnetic flux. In our case, the AB
effect is produced by the magnetic flux passing outside the MGQD [27]. The
combination of the AB flux and dual gaps introduces additional complexity into
the system, resulting in unique electronic properties. The AB effect can
modify the energy spectrum, wave functions, and transport properties of the
fermions in the graphene quantum dot. The dual gaps also contribute to the
spatial variation of the energy levels and can lead to localized states or
resonances within the dot [28, 29]. The present study contributes to our
understanding of novel electronic phenomena and their potential applications
in areas such as in bioimaging, biosensing, and therapy [30].

Figure 1: (color online) Schematic profile of a graphene quantum dot with
radius $R$ generated by a magnetic field $B$ that is influenced by Aharonov-
Bohm flux fields $\Phi_{\text{AB}}$ and dual gaps $\Delta_{i},i=1,2$.

The magnetic field $\vec{B}=B(r)\vec{e}_{z}$, which ensures the confinement of
the fermions, is chosen to be perpendicular to the $(x,y)$-plane and is given
by

$B(r)=\begin{cases}B,&r\leq R\\\ 0,&r>R\end{cases}$ (1)

where $\vec{e}_{z}$ is the unit vector. Actually, the resulting vector
potential is composed of two components: $\vec{A}=\vec{A_{1}}+\vec{A_{2}}$
[31], where $\vec{A_{1}}=G(r)\vec{e_{\theta}}$ and
$\vec{A_{2}}=\frac{\phi_{AB}}{2\pi r}\vec{e_{\theta}}$ is related to the
applied AB flux, such as

$G(r)=\begin{cases}\frac{B}{2r}\left(r^{2}-R^{2}\right),&r\leq R\\\
0,&r>R\end{cases},\quad\phi_{i}=\frac{\phi_{AB}}{\phi_{0}}$ (2)

and $\phi_{0}=h/e$ is unit flux. We study the energy spectrum of Dirac
fermions in the honeycomb lattice of covalently bonded carbon atoms in
graphene, subjected to both a magnetic field and an AB flux, by considering
the Hamiltonian

$H_{\tau}=v_{F}\left(\pi_{x}\sigma_{x}+\tau\pi_{y}\sigma_{y}\right)+\tau\Delta_{i}(r)\sigma_{z}$
(3)

where $v_{F}=10^{6}\mathrm{~{}m}/\mathrm{s}$ denotes the Fermi velocity,
$\tau=\pm$ denotes the valleys $K$ and $K^{\prime}$, $\pi_{i}=p_{i}+eA_{i}$
represents the kinetic momentum operator, and $\sigma_{i}(i=x,y,z)$ are the
Pauli matrices in the basis of the two sublattices of $A$ and $B$ atoms. The
non-uniform gap $\Delta_{i}(r)$ is given by

$\Delta_{i}(r)=\begin{cases}\Delta_{1}=\hbar v_{F}\delta_{1},&r<R\\\
\Delta_{2}=\hbar v_{F}\delta_{2},&r>R.\end{cases}$ (4)

The system under consideration has a spherical geometry (see Fig.1), then we
introduce the polar coordinates $(r,\theta)$. Consequently, we can express (3)
as

$H^{\tau}=\hbar v_{F}\begin{pmatrix}\tau\delta_{i}&\pi^{+}\\\
\pi^{-}&-\tau\delta_{i}\end{pmatrix}$ (5)

where we have define the operators

$\pi^{\pm}=e^{\mp i\tau\theta}\left[-i{\partial_{r}}\mp
i\tau\left(\frac{i}{r}{\partial_{\theta}}+\frac{eBr}{2}+\frac{e\phi_{AB}}{2\pi
r}\right)\right].$ (6)

As long the total momentum operator $J_{z}=L_{z}+\hbar\sigma_{z}/2$ commute
with the Hamiltonian (3), the eigenstates $\Psi^{\tau}(r,\theta)$ ca be
represented by

$\Psi^{\tau}(r,\theta)=e^{im\theta}\binom{\varphi^{\tau}_{A}(r)}{ie^{i\tau\theta}\varphi^{\tau}_{B}(r)}$
(7)

where $m\in\mathbb{Z}$ is a quantum number related to $J_{z}$. Using the
eigenvalues $H^{\tau}\Psi^{\tau}(r,\theta)=E\Psi^{\tau}(r,\theta)$, we derive
two coupled equations within the region $r<R$ (inside the GMQDs)

$\displaystyle\left[{\partial_{\rho}}+\frac{\tau}{\rho}\left(m+\tau\right)+\tau\beta\rho\right]\varphi^{\tau}_{B}(\rho)=(\epsilon-\tau\delta_{1})\varphi^{\tau}_{A}(\rho)$
(8) $\displaystyle\left[-{\partial_{\rho}}+\frac{\tau
m}{\rho}+\tau\beta\rho\right]\varphi^{\tau}_{A}(\rho)=(\epsilon+\tau\delta_{1})\varphi^{\tau}_{B}(\rho)$
(9)

where we have introduced dimensionless parameter dimensinless parameters
$\rho=\frac{r}{R}$, $\delta_{i}={\frac{\Delta_{i}R}{\hbar\nu_{F}}}$, and
$\epsilon={\frac{E}{\hbar\nu_{F}}}$, with $\beta=\frac{eBR^{2}}{2\hbar}$. By
injecting (9) into (8) to find a second order differential equation for
$\varphi^{\tau}_{A}(r)$

$\left[{\partial^{2}_{\rho}}+\frac{1}{\rho}{\partial_{\rho}}-\frac{m^{2}}{\rho^{2}}-\rho^{2}{\beta}-{2(m+\tau)}{\beta}+k_{1}^{2}\right]\varphi^{\tau}_{A}(\rho)=0$
(10)

and $k_{1}=\sqrt{|\varepsilon^{2}-\delta_{1}^{2}|}$ is the wave vector. To
solve (10), we first examine the asymptotic limits that describe the required
physical behavior depending on the value of $\rho$. Since
$\rho\rightarrow\infty$, we can approximate (10) as

$\left(\partial_{\rho}^{2}+\frac{1}{\rho}\partial_{\rho}-\rho^{2}\beta\right)\varphi^{\tau}_{A}(\rho)=0$
(11)

which is of zero-order Bessel type. Thus, the solution is expressed as

$\varphi^{\tau}_{A}(\rho)=c_{1}I_{0}\left(\rho^{2}\beta\right)+c_{2}K_{0}\left(\rho^{2}\beta\right)$
(12)

where $I_{0}(x)$ and $K_{0}(x)$ represent the modified zero-order Bessel
functions of the first and second types, respectively. We choose $c_{1}=0$ and
$c_{2}=1$ to prevent the function $I_{0}(x)$ from diverging as $x$ approaches
infinity. Using the asymptotic behavior of $K_{0}(x)\underset{x\gg
1}{\sim}\frac{e^{-x}}{\sqrt{x}}$, we approximate $\Phi^{A}(r)$ as

$\varphi^{\tau}_{A}(\rho)\sim e^{\frac{-\beta\rho^{2}}{2}}.$ (13)

In the limit $\rho\rightarrow 0$, (10) reduces to

$\left(\partial_{\rho}^{2}+\frac{1}{\rho}\partial_{\rho}-\frac{m^{2}}{\rho^{2}}\right)\varphi^{\tau}_{A}(\rho)=0$
(14)

which has the following solution

$\varphi^{\tau}_{A}(\rho)=\frac{c_{3}}{2}(\rho^{m}+\rho^{-m})+\frac{ic_{4}}{2}(\rho^{m}-\rho^{-m})$
(15)

To ensure that the solution satisfies the physical constraints, $c_{3}$ and
$c_{4}$ must be chosen appropriately. Therefore, we consider the cases of
positive and negative values of $m$ separately. For $m\geq 0$, where
$\sim\rho^{-m}$ must vanish, we set $c_{4}=-ic_{3}$, and by convention we
assign $c_{3}=1$. Conversely, for $m<0$, where $\sim\rho^{m}$ must vanish, we
replace $c_{4}=ic_{3}$ with $c_{3}=1$. Combining these considerations, we
express the asymptotic behavior of $\varphi^{\tau}_{A}(\rho)$ as

$\displaystyle\varphi^{\tau}_{A}(\rho)\sim\rho^{m},\qquad m\geq 0,$ (16a)
$\displaystyle\varphi^{\tau}_{A}(\rho)\sim\rho^{-m},\qquad m<0.$ (16b)

Now it is clear that the solution of (10) can be written as

$\varphi_{A}(\rho)=\rho^{|m|}e^{-\frac{\beta\rho^{2}}{2}}\xi(\beta\rho^{2})$
(17)

and define a new variable $\zeta=\beta\rho^{2}$. Substituting (17) into (10),
we obtain the confluent hepergeometric ordinary differential equation

$\left[\zeta\partial^{2}_{\zeta}+(p-\zeta)\partial_{\zeta}-q\right]\xi(\zeta)=0$
(18)

where we have defined

$p=1+|m|,\quad q=-\frac{k^{2}_{1}}{4\beta}+\frac{|m|+m+\tau+1}{2}$ (19)

Consequently the solution are the confluent hypergeometric functions [32]. We
combine all to get the eigenspinors inside the GMQDs

$\psi^{\tau}_{1}(\rho,\theta)=C^{\tau}_{1}\rho^{|m|}e^{\frac{\beta\rho^{2}}{2}}e^{im\theta}\begin{pmatrix}U(q,p,\beta\rho^{2})\\\
\frac{ie^{i\tau\theta}}{\varepsilon+\tau\delta_{1}}\left[\frac{-|m|+\tau
m}{\rho}+(\tau+1)\beta\rho\ U(q,p,\beta\rho^{2})-2\beta\rho\frac{q}{p}\
U(q+1,p+1,\beta\rho^{2})\right]\end{pmatrix}.$ (20)

In region $r>R$ (outside the GMQDs), (10) can be represented by

$\left[{\rho^{2}\partial\rho^{2}}+\rho\partial\rho-(m+\phi_{i})^{2}-k_{2}^{2}\rho^{2}\right]\varphi^{\tau}_{A}(\rho)=0$
(21)

with the wave vector $k_{2}=\sqrt{|\varepsilon^{2}-\delta_{2}^{2}|}$. This
differential equation has the Bessel function of the first kind as solution

$\varphi^{\tau}_{A}(\rho)=C^{\tau}_{2}J_{m+\phi_{i}}(k_{2}\rho)$ (22)

and $C^{\tau}_{2}$ is the normalization constant. Now, we can use (8) to
derive the second component. By combing all, we write the eigenspinors as

$\displaystyle\psi^{\tau}_{2}{(\rho,\theta)}=C^{\tau}_{2}e^{im\theta}\begin{pmatrix}J_{m+\phi_{i}}(k_{2}\rho)\\\
\frac{ie^{i\tau\theta}}{\varepsilon+\tau\delta_{2}}[-k_{2}\
J_{m+\phi_{i}-1}(k_{2}\rho)+\frac{(m+\phi_{i})(\tau+1)}{\rho}\
J_{m+\phi_{i}}(k_{2}\rho)]\end{pmatrix}.$ (23)

Since we are unable to obtain the eigenvalues explicitly, we proceed by
formulating an equation that dictates their behavior. Specifically, we use the
continuity of both solutions (inside and outside the point) at the boundary
$r=R$ or $\rho=1$, denoted as $\psi^{\tau}_{1}(1)=\psi^{\tau}_{2}(1)$. This
approach results in the following equation:

$\displaystyle\begin{split}\frac{J_{m+\phi_{i}}(k_{2})}{\varepsilon+\tau\delta_{1}}\left[-|m|+\tau
m+(\tau+1)\beta\ U(q,p,\beta)-2\beta\frac{q}{p}\ U(q+1,p+1,\beta)\right]-\\\
\frac{U(q,p,\beta)}{\varepsilon+\tau\delta_{2}}\left[-k_{2}\
J_{m+\phi_{i}-1}(k_{2})+(m+\phi_{i})(\tau+1)\
J_{m+\phi_{i}}(k_{2})\right]=0.\end{split}$ (24)

The analytical solution of equation (24) is complex and requires the use of a
numerical approach. We will plot the dependence of the energy spectrum on the
physical parameters characterizing the GMQDs. This will help us to explore the
fundamental properties of the GMQDs and underline their basic characteristics.

##  3 Discussions

Fig. 2 shows the energy levels versus the magnetic field $B$ for a radius of
$R=70$ nm and dual gaps, $\Delta_{1}=50$ meV and $\Delta_{2}=2\Delta_{1}=100$
meV. In Figs. 2(a,b,c) for zero AB flux $\phi_{i}=0$, there is emergence of
continuous energy bands for three quantum numbers $m=0,\pm 1$ as $B$
approaches zero ($B\rightarrow 0$). As a consequence, the energy states in the
valleys $K$ ($\tau=+$) and $K^{\prime}$ ($\tau=-$) become degenerate. We
observe that the increase in $B$ removes the degeneracy and breaks the
symmetry $E(m,\tau)\neq E(m,-\tau)$ due to the appearance of an induced band
gap. This is consistent with previous results reported in [29, 34, 33, 28].
This symmetry breaking is analogous to the Zeeman effect, where energy levels
are split based on spin orientation relative to the magnetic field. At low
energies, minor energy levels appear in addition to the main energy bands, and
as $B$ grows, linear levels appear. This increases the band gap between the
valence and conduction bands. Figs. 2(d,e,f) show that the symmetry of energy
levels remains broken even in the presence of $\phi_{i}$. Furthermore, as
$\phi_{i}$ increases, one sees that emergence of an additional band gap. This
later causes a further separation between the energy levels occupied by the
electrons and those occupied by the corresponding holes. This trend becomes
more pronounced in Figs. 2(g,h,i) for $\phi_{i}=1.5$.

Figure 2: (color online) Energy levels plotted against the magnetic field $B$
for a radius $R=70$ nm, dual gaps: $\Delta_{1}=50$ meV and $\Delta_{2}=10$
meV, quantum number $m$: (a,d,g): -1, (b,e,h): 0, (c,f,i): 1, AB flux
$\phi_{i}$: (a,b,c):0, (d,e,f): 0.5, (g,h,i): 1.5, with green curves for
$\tau=-1$, and dotted red curves for $\tau=1$.

With the same conditions as in Fig. 2, we simply invert the dual gap values
$\Delta_{1}=100$ meV and $\Delta_{2}=50$ meV in Fig. 3. We observe that the
energy levels behave similarly to Fig. 2, but there are notable differences.
For small $B$, new energy levels with vertical parabolic shapes appear, which
are approximately confined in the energy range between $-\Delta_{1}$ and
$\Delta_{1}$. However, as $B$ increases, these additional levels gradually
disappear, the energy degeneracy decreases, and the emerging energy levels
break the symmetry. This leads to a significant widening of the band gap
between the valence and conduction bands, as shown in Figs. 3(a,b,c) for
$\phi_{i}=0$. Now, in the presence of $\phi_{i}$, we find that the symmetry of
the system remains broken, creating an additional band gap, which is clearly
visible in Figs. 3(d,e,f,g,h,i). Contrary to previous results [29], the
introduction of $\phi_{i}$ has a significant effect on the quantum confinement
in GMQDs. It reduces the presence of inter-band linear levels and increases
the width of the band gap. Consequently, the AB flux affects the electronic
properties of the fermions in the GMQDs.

Figure 3: (color online) The same as in Fig. 2 but by inverting dual gaps:
$\Delta_{1}=100$ meV and $\Delta_{2}=50$ meV, with magenta curves for
$\tau=-1$, and dashed blue curves for $\tau=1$.

Figure 4: (color online) The energy levels as a function of the radius $R$
with the same conditions as in Fig. 2. Here green curves for $\tau=-1$ and
dashed red curves for $\tau=1$.

Fig. 4 shows the energy spectrum versus radius $R$ for three AB flux values
($\phi_{i}=0,0.5,1.5$) and quantum numbers $m$. When $R$ is very small, we
observe that the energy levels of the two valleys $K$ and $K^{\prime}$ are not
degenerate, especially when $R\rightarrow 0$, i.e., the GMQDs disappear. As
$R$ is increased, it becomes clear that the energy levels have a linear shape,
demonstrating the symmetry $E(m,\tau)=E(m,-\tau)$, similar to that obtained in
[35, 34]. The band gap also decreases, suggesting a trend where larger GMQDs
tend to have a smaller disparity between the occupied electron states and the
corresponding hole states [35, 34]. Furthermore, especially for higher values
of $R$, all figures show numerous additional levels between the two bands. In
Figs. 4(g,h,i), an increase in the number of energy levels is shown in the
region bounded by the dual gaps $-\Delta_{2}$ and $\Delta_{2}$, in contrast to
the results reported in [35, 34]. We note that the introduction of $\phi_{i}$
reduces the number of energy levels present in the band gap. This decrease
becomes more significant as $\phi_{i}$ increases.

In Fig. 5, we analyze the same scenario as in Fig. 4 by inverting the dual
gaps, $\Delta_{1}=100$ meV and $\Delta_{2}=50$ meV. It is noticeable that the
space between the two bands decreases, along with the levels they encompass,
as the inner gap $\Delta_{2}$ decreases. These results are consistent with
those reported in [29]. As the AB flux increases to $\phi_{i}=0.5$ in Fig.
5(d,e,f) and $\phi_{i}=1.5$ in Fig. 5(g,h,i), a reduction in the number of
energy levels generated within the band gap is observed. There is also a
noticeable decrease in the width of the band gap as $\phi_{i}$ increases, as
shown in Fig. 5(d,e,f). This decrease is consistent for all states of the
quantum number $m$. Our analysis leads to the conclusion that the AB flux
influences both the number of energy levels present and the size of the band
gap.

Figure 5: (color online) The energy levels as a function of the radius $R$
with the same conditions as in Fig. 3. Here orange curves for $\tau=-1$ and
dashed black curves for $\tau=1$.

Figure 6: (color online) The energy levels as a function of the internal gap
$\Delta_{1}$ for $\Delta_{2}=50$ meV and with the same conditions as in Fig.
2. Here blue curves for $\tau=-1$ and dashed red curves for $\tau=1$.

In Fig. 6, the energy levels are plotted against the gap $\Delta_{1}$ within
the GMQDs for three values of the quantum number $m=0,\pm 1$, with the gap
$\Delta_{2}=50$ meV. The AB flux is varied as $\phi_{i}=0$ for the first row,
$\phi_{i}=0.5$ for the second row, and $\phi_{i}=1.5$ for the third row. In
Figs. 6(a,b,c) it is noticeable that the energy levels show a linear variation
as long as $\Delta_{1}$ increases, the symmetry $E(m,\tau)=E(m,-\tau)$ is
still valid. A distinct band gap is observed between the valence and
conduction bands, the magnitude of which increases with the AB flux
$\phi_{i}$. This trend is particularly pronounced in Figs. 6(d,e,f) and Figs.
6(g,h,i), highlighting how the AB flux increases the band gap except when
$m=0$ and $\phi_{i}=0.5$. Such an observation indicates the influence of the
AB flux on the electronic properties of the GMQDs.

Fig. 7 presents a comprehensive analysis of the energy levels versus the gap
$\Delta_{2}$ outside the GMQDs, with $\Delta_{1}=50$ meV and
$\phi_{i}=0,0.5,1.5$. Figs. 7(a,b,c) clearly show that the energy levels
consist of levels with a horizontal parabolic shape, reaching a minimum at
$\Delta_{2}=0$. We note the appearance of additional vertical parabolic lines
in the range $-2\Delta_{1}$ to $2\Delta_{1}$ ($-2\Delta_{1}\leq E\leq
2\Delta_{1}$). These show symmetrical parallels centered around the energy
point $E(m,\tau)=E(m,-\tau)$. Notably, the study reveals the emergence of
additional levels within the two existing bands, confirming previous findings
in research articles [29, 36, 37]. Furthermore, under the influence of the AB
flux $\phi_{i}$, an expansion phenomenon occurs between the bands, leading to
exciting new observations. Fig. 7(d,e,f) and Fig. 7(g,h,i) visually illustrate
the effect of varying the flux on the energy levels. As $\phi_{i}$ increases,
the number of energy levels between the bands decreases significantly, leading
to a noticeable increase in the band gap, especially for certain quantum
numbers $m=0,\pm 1$. It is therefore clear that the AB flux plays a key role
in controlling the band gap. Consequently, it affects the distribution of
accessible energy states in the conduction and valence bands, contrary to the
results presented in the previous study [29].

Fig. 8 shows the energy levels versus the AB flux $\phi_{i}$ for three quantum
numbers $m=-1,0,1$. For the case of zero dual gaps in Figs. 8(a,b,c), we
observe that the energy levels exhibit a linear dependence on $\phi_{i}$. In
addition, these energy levels exhibit the symmetry $E(m,\tau)=E(m,-\tau)$.
There is also a significant band gap between the valence and conduction bands.
For $\Delta_{1}=50$ meV and $\Delta_{2}=100$ meV in Figs. 8(d,e,f), parabolic
levels with a vertical shape appear within the band gap for different quantum
number states. This effect is particularly pronounced for AB flux values
$\phi_{i}\leq 6$. Interestingly, at the critical energy value $E=\Delta_{1}$,
the energy levels are only present for $\tau=1$, leading to a symmetry
breaking. When the dual gaps are reversed to $\Delta_{1}=100$ meV and
$\Delta_{2}=50$ meV in Figs. 8(g,h,i), the energy levels show a similar
pattern to the two previous scenarios. However, there is a noticeable
reduction in the number of parabolic levels within the band gap. By comparing
these results with those previously reported in [36], it can be concluded that
the AB flux can be effectively used as a tool to tune both the band gap and
the number of energy levels in the GMQDs.

.

Figure 7: The energy levels as a function of the external gap $\Delta_{2}$ for
$\Delta_{1}=50$ meV and with the same conditions as in Fig. 2. Here green
curves for $\tau=-1$ and dashed red curves for $\tau=1$.

Figure 8: (color online) The energy levels as a function of the AB-flux
$\phi_{i}$ for $R=70$ nm , B=$10$ T and different values of $m$ and dual gaps:
(a,b,c) for ($\Delta_{1}=0$ meV, $\Delta_{2}=0$ meV), (d,e,f) for
($\Delta_{1}=50$ meV, $\Delta_{2}=100$ meV), (g,h,i) for ($\Delta_{1}=100$
meV, $\Delta_{2}=50$ meV). Different panels are also represented by changing
the value of $m$, such as (a,d,g) for $m=-1$, (b,e,h) for $m=0$ and (c,f,i)
for $m=1$. The magenta curves represent $\tau=-1$ and the dashed black curves
represent $\tau=1$.

##  4 Conclusion

We have studied the behavior of Dirac fermions in a graphene magnetic quantum
dots (GMQDs) subjected to both an Aharonov-Bohm (AB) flux $\Phi_{AB}$ and dual
gaps ($\Delta_{1}$ inside and $\Delta_{2}$ outside the GMQDs). In particular,
we have studied how $\Phi_{AB}$ affects the energy levels of the GMQDs. To
achieve this goal, we first solved the Dirac equation to determine the
eigenspinors within two different regions bounded by the radius $R$ of the
GMQDs. By ensuring the continuity of the eigenspinors at the boundary, we were
able to derive an equation that describes the relationship between the energy
levels and the physical parameters that define the GMQDs.

Recognizing the complexity involved in obtaining an explicit solution for the
energy spectrum, we resorted to a numerical approach to elucidate the
fundamental features of the system. This allowed us to study the correlation
between the energy levels and several parameters, including the magnetic field
$B$, the angular momentum $m$, the radius $R$, the AB flux $\phi_{i}$, and
dual gaps ($\Delta_{1}$, $\Delta_{2}$). More precisely, we have shown how
$\phi_{i}$, $\Delta_{1}$, and $\Delta_{2}$ can affect the energy levels of the
GMQDs. Indeed, under weak $B$ we observed a degeneracy in the energy levels
along with the appearance of parabolic levels within the band gap. As $B$
increased, new energy levels associated with Landau levels, typical of systems
exposed to a magnetic field, began to appear. As a result, we observed that
the AB flux increases the separation between the valence and conduction bands,
while the energy gap breaks the symmetry of the system.

In summary, the AB flux and the dual gaps are proving to be powerful tools for
modifying the electronic properties of graphene. This controllability opens
promising avenues for the design of optoelectronic, spintronic, and other
quantum devices. Our study underscores the critical importance of these
parameters in manipulating the electronic properties of graphene, paving the
way for further fundamental and applied research in this field.

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