remarkRemark hypothesisHypothesis claimClaim MC is a good sampling strategyB.
Adcock and S. Brugiapaglia [][nocite]MCgood_supplement_Revision_decolorized

# Monte Carlo is a good sampling strategy for polynomial approximation in high
dimensions††thanks: Submitted to the editors DATE. BA acknowledges the support
of NSERC through grant RGPIN-2021-611675.SB acknowledges the support of NSERC
through grant RGPIN-2020-06766, the Faculty of Arts and Science of Concordia
University and the CRM Applied Math Lab.

Ben Adcock Department of Mathematics, Simon Fraser University, 8888 University
Drive, Burnaby BC, Canada, V5A 1S6 (, https://www.benadcock.ca).
<EMAIL_ADDRESS>Simone Brugiapaglia Department of Mathematics and
Statistics, Concordia University, J.W. McConnell Building, 1400 De Maisonneuve
Blvd. W., Montréal, QC, Canada, H3G 1M8 (, https://www.simonebrugiapaglia.ca).
<EMAIL_ADDRESS>
###### Abstract

This paper concerns the approximation of smooth, high-dimensional functions
from limited samples using polynomials. This task lies at the heart of many
applications in computational science and engineering – notably, some of those
arising from parametric modelling and computational uncertainty
quantification. It is common to use Monte Carlo sampling in such applications,
so as not to succumb to the curse of dimensionality. However, it is well known
that such a strategy is theoretically suboptimal. Specifically, there are many
polynomial spaces of dimension $n$ for which the sample complexity scales log-
quadratically, i.e., like $c\cdot n^{2}\cdot\log(n)$ as $n\rightarrow\infty$.
This well-documented phenomenon has led to a concerted effort over the last
decade to design improved, and moreover, near-optimal strategies, whose sample
complexities scale log-linearly, or even linearly in $n$.

In this work we demonstrate that Monte Carlo is actually a perfectly good
strategy in high dimensions, despite its apparent suboptimality. We first
document this phenomenon empirically via a systematic set of numerical
experiments. Next, we present a theoretical analysis that rigorously justifies
this fact in the case of holomorphic functions of infinitely-many variables.
We show that there is a least-squares approximation based on $m$ Monte Carlo
samples whose error decays algebraically fast in $m/\log(m)$, with a rate that
is the same as that of the best $n$-term polynomial approximation. This result
is non-constructive, since it assumes knowledge of a suitable polynomial
subspace in which to perform the approximation. We next present a compressed
sensing-based scheme that achieves the same rate, except for a larger
polylogarithmic factor. This scheme is practical, and numerically it performs
as well as or better than well-known adaptive least-squares schemes.

Overall, our findings in this paper demonstrate that Monte Carlo sampling is a
good choice for polynomial approximation in high dimensions and shed light on
why this is the case. Therefore, the benefits of improved sampling strategies
are generically limited to lower-dimensional settings.

###### keywords:

Monte Carlo sampling, optimal sampling, high-dimensional approximation,
polynomial approximation, holomorphic functions, parametric DEs

41A10, 41A63, 65C05

##  1 Introduction

Approximating a smooth function
$f:\mathcal{U}\subseteq\mathbb{R}^{d}\rightarrow\mathbb{C}$ from (noisy)
sample values

(1) $f(\bm{y}_{i})+e_{i},\quad i=1,\ldots,m,\qquad\text{where
}\bm{y}_{1},\ldots\bm{y}_{m}\in\mathcal{U},$

is a task of fundamental importance in computational science and engineering.
This task is well understood in low dimensions. But many modern applications
[smith2013uncertainty, sullivan2015introduction] call for the approximation of
functions depending on many (and potentially infinitely many) variables. Such
high-dimensional approximation problems occur in many fields, for example in
parametric modelling and computational Uncertainty Quantification (UQ). Here,
smooth, high-dimensional functions commonly arise as solution maps of
parametric Differential Equations (DEs).

Methods based on polynomials have proved to be effective tools for
approximating such functions. Both classical techniques such as Least Squares
(LS) and more recent tools such as Compressed Sensing (CS) have been
intensively investigated over the last several decades. See §1.5 for a
historical discussion. Although by no means the only choice, these tools have
been quite widely adopted in the aforementioned applications.

###  1.1 Monte Carlo sampling and sample complexity

This paper is about the choice of sampling strategy for polynomial
approximation in high dimensions. In any approximation scheme, the choice of
sample points $\bm{y}_{1},\ldots,\bm{y}_{m}$ is of singular importance.
Obtaining samples is often a key bottleneck in applications, since they may
require time- or resource-consuming numerical simulations or physical
experiments. Thus, it is vital to choose sample points in a judicious manner,
so as to facilitate accurate and stable approximations from as few samples as
possible.

Care is needed when selecting a sampling strategy for a high-dimensional
problem so as not to succumb to the curse of dimensionality. Taking
inspiration from high-dimensional quadrature, it is common to use Monte Carlo
(MC) sampling in polynomial approximation schemes, i.e., the points
$\bm{y}_{1},\ldots,\bm{y}_{m}$ are drawn randomly and independently from some
underlying probability measure. Such sampling strategies also occur naturally
in UQ settings, where the variables are stochastic.

MC sampling is a remarkably effective technique for high-dimensional
integration. Yet, in the context of high-dimensional polynomial approximation,
MC sampling suffers from a critical limitation. The sample complexity – the
number of samples $m$ required for a stable and accurate approximation –
usually scales poorly with the problem dimension $d$ or approximation space
dimension $n$. For example, if $\mathcal{U}=[-1,1]^{d}$ and the underlying
measure is the uniform measure, then there are polynomial subspaces of
dimension $n$ for which the sample complexity scales log-quadratically, i.e.,

(2) $m=c\cdot n^{2}\cdot\log(n)$

for some $c>0$. See, e.g., [chkifa2015discrete, migliorati2014analysis,
cohen2013stability]. Note that $n$ also scales exponentially with $d$ for many
classical approximation spaces, such as those corresponding to _tensor-
product_ , _total degree_ , or _hyperbolic cross_ index sets (see §2.3). As a
result, (2) implies that $m$ may need to grow twice as exponentially fast in
$d$ as $n$ for these choices.

###  1.2 Improved and (near-) near-optimal sampling strategies

This well-known limitation of MC sampling has led to a concerted effort in the
development of sampling strategies that offer either better theoretical sample
complexity, better practical performance or (ideally) both. One finds many
different approaches in the recent literature, including preconditioning
[rauhut2012sparse], asymptotic sampling [hampton2015coherence], coherence-
optimal sampling [hampton2015coherence], Christoffel sampling
[narayan2017christoffel], randomly subsampled quadratures [zhou2015weighted],
low-discrepancy points [migliorati2015analysis], Quasi Monte Carlo sampling,
Latin hypercube sampling, deterministically subsampled quadratures
[seshadri2017effectively], boosting techniques [haberstich2022boosted], and
methods based on optimal design of experiments [fajraoui2017sequential,
diaz2018sparse]. See [guo2020constructing, hadigol2018least] for further
discussion and references. However, perhaps most notably, in the last several
years random sampling strategies have been introduced that are provably near-
optimal [cohen2017optimal] (see also [hampton2015coherence] for earlier work
in this direction and [adcock2022towards, guo2020constructing] for reviews).
Specifically, for any fixed (polynomial or, in fact, nonpolynomial)
approximation space of dimension $n$, the corresponding sample complexity
scales log-linearly, i.e.,

(3) $m=c\cdot n\cdot\log(n).$

###  1.3 Aims of this paper

The previous discussion suggests that MC is a bad sampling strategy for
polynomial approximation. And indeed, almost all of the above strategies yield
significant benefits over MC sampling in low dimensions, where the dimension
$d$ may be on the order of five or less. However, such benefits have been
consistently observed to lessen as the dimension increases. When $d$ is on the
order of 10 or more, the performance gap between MC sampling and any improved
sampling strategy is often strikingly less. This is even the case for the
near-optimal sampling strategies, in spite of their theoretical optimality. A
typical example of effect is shown in Fig. 1, with further examples presented
in §LABEL:s:MC-bad. Here LS with MC sampling is unstable and potentially
divergent in low dimensions as the number of samples $m$ increases, but in
higher dimensions its performance is essentially the same as that LS with a
theoretically near-optimal sampling strategy.

|  |
---|---|---
|  |
$d=1$ | $d=4$ | $d=32$
Figure 1: In low dimensions, MC sampling from the uniform measure on
$\mathcal{U}=[-1,1]^{d}$ is a poor choice. However, it becomes far better as
the dimension increases. This figure plots the error (top row) and ($2$-norm)
condition number of the LS matrix (bottom row) versus $m$ for (adaptive) LS
polynomial approximation using MC samples (ALS-MC) and a near-optimal sampling
strategy (ALS-Opt) described in §1.2. The function considered is a quantity of
interest of a parametric DE with lognormal diffusion term (see
§LABEL:ss:lognormal-pDE). The scaling $m=\max\\{n+1,\lceil n\log(n)\rceil\\}$
is used in both cases. For MC sampling, the LS problem is highly ill-
conditioned when $d=1$, since $m$ scales log-linearly with $n$ rather than
log-quadratically, as in (2). This causes a dramatic loss of accuracy due to
round-off error. However, as $d$ increases, this LS problem becomes much
better conditioned, with both its condition number and the error closely
tracking those of the near-optimal strategy.

The aim of this paper is to shed light on this phenomenon. We first document
it through a series of numerical experiments on different smooth test
functions in both low and high dimensions (in the range $d=1$ to $d=32$). Then
we provide a theoretical explanation for why it occurs. We summarize these
theoretical results next.

###  1.4 Nontechnical summary of main theoretical results

Our theoretical analysis centres on polynomial approximation of holomorphic
functions of infinitely-many variables. Specifically, we combine the theory of
best $n$-term polynomial approximation of such functions with the theories of
LS and CS.

####  1.4.1 Polynomial approximation theory in infinite dimensions

The topic of polynomial approximation theory for infinite-dimensional
functions has developed over the last decade or so, motivated by applications
in parametric DEs and UQ. See [cohen2015approximation] and [adcock2022sparse,
Chpts. 3 & 4] for reviews. A key result in this area is the assertion of
certain algebraic rates of convergence for best $n$-term polynomial
approximations to certain classes of holomorphic, infinite-dimensional
functions.

Consider the infinite hypercube $\mathcal{U}=[-1,1]^{\mathbb{N}}$ equipped
with the uniform measure $\varrho$ and let $L^{2}_{\varrho}(\mathcal{U})$ be
the Lebesgue space of (complex-valued) square-integrable functions
$[-1,1]^{\mathbb{N}}\rightarrow\mathbb{C}$. (For the reader who is unfamiliar
with concepts of infinite-dimensional measures and function spaces, we refer
to §LABEL:s:polyapp-inf-dim). Any function $f\in L^{2}_{\varrho}(\mathcal{U})$
has an infinite expansion with respect to multivariate orthogonal polynomials,
which in this case are tensor products of the univariate Legendre polynomials.
The best $n$-term approximation to $f$ is the polynomial formed simply by
picking the largest $n$ terms of this expansion in absolute value. In this
paper, we denote this approximation as $f_{n}$.

Now let $0<p<1$, $\varepsilon>0$ and $\bm{b}\in\ell^{p}(\mathbb{N})$. Then a
key result in this area is that there is a class
$\mathcal{H}(\bm{b},\varepsilon)\subset L^{2}_{\varrho}(\mathcal{U})$ of
holomorphic functions within which the best $n$-term approximation converges
algebraically with rate depending on $1/p$. Specifically,

(4) ${\|f-f_{n}\|}_{L^{2}_{\varrho}(\mathcal{U})}\leq
C(\bm{b},\varepsilon,p)\cdot n^{\frac{1}{2}-\frac{1}{p}},\quad\forall
f\in\mathcal{H}(\bm{b},\varepsilon),\ n\in\mathbb{N}.$

In other words, there are classes of functions whose approximation is free
from the curse of dimensionality: despite depending on infinitely many
variables, they can be approximated with algebraic rate by a finite ($n$-term)
polynomial approximation.

The class $\mathcal{H}(\bm{b},\varepsilon)$, which is known as the class of
(unit-norm) $(\bm{b},\varepsilon)$-holomorphic functions [chkifa2015breaking,
schwab2019deep], is both theoretically interesting, since it defines a class
where algebraic rates are obtained, and practically relevant. It was first
introduced to describe the parametric regularity of the solutions of certain
parametric DEs. It is now known that several important families of parametric
DEs have parametric solution maps that belong to such classes (for suitable
$\bm{b}$, $\varepsilon$ depending on the DE). These include parametric
diffusion equations, parametric heat equations, PDEs over parametrized
domains, and parametric initial value problems with affine parametric
dependence. See §LABEL:s:pDE-background for some further discussion.

Note that $(\bm{b},\varepsilon)$-holomorphic functions are anisotropic. As we
see in §LABEL:s:polyapp-inf-dim, the $j$th component $b_{j}$ of the sequence
$\bm{b}$ effectively controls the degree of smoothness of functions in
$\mathcal{H}(\bm{b},\varepsilon)$ with respect to the $j$th variable $y_{j}$.

####  1.4.2 First contribution: sharpness of the rate (4)

In our first contribution, Theorem LABEL:t:alg-rate-sharp, we show that the
rate (4) is effectively sharp. In particular, there exist sequences
$\bm{b}\in\ell^{p}(\mathbb{N})$ and infinitely many functions
$f\in\mathcal{H}(\bm{b},\varepsilon)$ for which $n^{1/2-1/p}$ is the best
possible algebraic rate of convergence.

####  1.4.3 Second contribution: near-optimal approximation in the case of
known anisotropy

Our other two theoretical results concern constructing polynomial
approximations to functions in $\mathcal{H}(\bm{b},\varepsilon)$ from MC
samples, i.e., $\bm{y}_{1},\ldots,\bm{y}_{m}\sim_{\mathrm{i.i.d.}}\varrho$,
that attain this algebraic rate. In view of (2), one would generally expect to
be able to recover an $n$-term polynomial approximation from $m$ samples via
LS for $n$ at most $\sqrt{2m/(c\log m)}$. Combining this upper bound on $n$
with (4), it is therefore tempting to believe that the best one could hope to
achieve is

${\|f-\hat{f}\|}_{L^{2}_{\varrho}(\mathcal{U})}\leq
C\cdot\left(m/\log(m)\right)^{\frac{1}{2}(\frac{1}{2}-\frac{1}{p})},$

when $\bm{b}\in\ell^{p}(\mathbb{N})$. In other words, the algebraic rate is
halved, due to the log-quadratic sample complexity (2). However, this is not
the case. In Theorem LABEL:t:main-res we show that there is a LS approximation
for which the error behaves like

(5) ${\|f-\hat{f}\|}_{L^{2}_{\varrho}(\mathcal{U})}\leq
C\cdot\left(m/\log(m)\right)^{\frac{1}{2}-\frac{1}{p}}.$

Hence the approximation converges with the same rate as that of the best
$n$-term approximation, up to constants and the log term. We conclude that MC
sampling is not only a good sampling strategy in infinitely many dimensions,
but in view of the result in §1.4.2, it is in fact near optimal for the class
$\mathcal{H}(\bm{b},\varepsilon)$.

####  1.4.4 Second contribution: near-optimal approximation in the case of
unknown anisotropy

A limitation of any LS scheme is that it requires knowledge of a space in
which the function is well approximated. The above result is no different,
since it involves a judicious choice of polynomial subspace of dimension $\leq
n$ (depending on the parameters $\bm{b}$ and $\varepsilon$) in which one
simultaneously obtains the same algebraic rate of convergence (4) as the best
$n$-term approximation, while also not suffering from poor sample worst-case
complexity bound (3) for MC sampling. At the very least, constructing such a
subspace requires a priori knowledge of the parameters $\bm{b}$ and
$\varepsilon$. Unfortunately, these are generally unknown in practice.

Because of this limitation, LS is often used as part of an adaptive
approximation scheme [migliorati2019adaptive]. Here, rather than a single
approximation, one computes a sequence of approximations
$\hat{f}^{(1)},\hat{f}^{(2)},\ldots$ and (nested) polynomial subspaces
$\mathcal{P}^{(1)}\subseteq\mathcal{P}^{(2)}\subseteq\cdots$, in which the
approximation $\hat{f}^{(l)}$ is used to construct the next subspace
$\mathcal{P}^{(l+1)}$, typically in a greedy manner. As we see later in this
paper, so-called Adaptive Least-Squares (ALS) approximation is often quite
effective. However, it lacks theoretical guarantees. It is currently unknown
whether such an approximation achieves the algebraic rates of convergence (4)
of the best $n$-term approximation (up to constants and log factors).

With this in mind, in the final part of this paper we demonstrate that this
issue can be avoided by changing the approximation procedure. By using ideas
from CS, we show it is possible to compute a polynomial approximation from $m$
MC samples for which the error behaves like

(6) ${\|f-\hat{f}\|}_{L^{2}_{\varrho}(\mathcal{U})}\leq
C\cdot\left(m/\log^{4}(m)\right)^{\frac{1}{2}-\frac{1}{p}},$

subject to the slightly stricter requirement
$\bm{b}\in\ell^{p}_{\mathsf{M}}(\mathbb{N})$, where
$\ell^{p}_{\mathsf{M}}(\mathbb{N})$ is the monotone $\ell^{p}$-space. See
Theorem LABEL:t:main-res-2. Thus, for a slightly restricted class of
functions, it is possible to achieve the same algebraic rates of convergence
as the best $n$-approximation in terms of the number of samples $m$, up to a
polylogarithmic factor.

This scheme is also practical. We present a series of numerical experiments
comparing it to ALS. These show that the CS scheme offers consistently as good
as, or sometimes better, performance, while also having theoretical
guarantees.

####  1.4.5 The gap between theory and practice

Our theoretical analysis asserts that MC sampling is near-optimal for
polynomial approximation in infinite dimensions. Throughout this article we
present numerical experiments for functions of finitely many variables only.
Our analysis does apply in finite dimensions – since a function of $d$
variables can be viewed as a function of infinitely many variables that is
constant with respect to all but the first $d$ variables – but, as we discuss
later, it may not accurately predict the convergence rate when the dimension
is small. There is consequently a gap between the theoretical analysis and
practice, which leads one naturally to wonder ‘how high is high-dimensional’?
In our experiments, we consistently witness this effect in finite dimensions.
Most often, it occurs when $d$ is still relatively small (e.g., $d\leq 8$),
and certainly by $d=32$, which is the largest dimension considered in this
paper. However, we currently have no theoretical understanding of how large
$d$ should be for such behaviour to kick in. This is an interesting problem
that we leave for future work.

####  1.4.6 Why bother?

As noted, the development of improved sampling strategies for high-dimensional
approximation has been an active area of interest over the last few years. The
purpose of this work is to show that MC sampling is actually eminently
suitable (in fact, near-optimal) for certain high-dimensional approximation
tasks – in particular, those arising from parametric DE problems, which served
as the original motivation for much of this line of research.

However, the reader may justifiably be wondering why we bother. Why use MC
sampling when we know how to design sampling strategies that are near-optimal
regardless of the dimension? For this, we offer several arguments. First, it
is both academically interesting and practically relevant to understand the
limits of what one stands to gain by changing the sampling strategy. Our
results suggest that these gains are limited to lower-dimensional problems, at
least for smooth function approximation on bounded domains. Second, methods
based on MC sampling have the distinct advantage of allowing one to decouple
the function evaluation and function approximation phases. Thus, in the
parametric DE context, the major computational burden of evaluating the target
function at the sample points can be trivially parallelized. On the other
hand, when adaptive approximation schemes (e.g., the aforementioned ALS
method) are combined with near-optimal sampling, the sampling strategy also
becomes adaptive. Hence, the tasks of sampling the function (i.e., numerically
solving a DE) and constructing the polynomial approximation cannot be
decoupled. In practice, this might require the design of _ad hoc_ software
interfaces or data transfer strategies. Third and finally, MC samples are
exchangeable and usually easy to generate. They are found ubiquitously in UQ
and machine learning applications; in particular, applications involving
legacy data, where one is not afforded the luxury to adapt the samples to the
target function and/or approximation scheme.

In summary, there are ample reasons why MC samples may be preferred in
practice. Hence investigations into their theoretical and practical
performance in high dimensions are particularly relevant.

###  1.5 Historical context and discussion

Our primary interest is the polynomial approximation of smooth high- and
infinite-dimensional functions, with specific focus on Least Squares (LS) and
Compressed Sensing (CS) techniques. LS approximation is a classical topic,
with a genesis that can be traced back to very early work by Gauss and
Legendre at the turn of the 18th and 19th centuries [stigler1981gauss]. LS is
utilized in countless areas of applied mathematics and statistics, with
eminent examples such as function approximation (of direct interest in this
paper) and statistical regression.

Our work stems from a vigorous research stream started in the early 2010s,
characterized by renewed interest towards LS [cohen2013stability,
chkifa2015discrete, migliorati2014analysis] and the application of the CS
paradigm [doostan2011nonadapted, mathelin2012compressed, rauhut2012sparse] in
the context of high-dimensional function approximation. The fast development
of this research area is in tandem with progress in the fields of stochastic
and parametric DEs. For stochastic DEs, this is strongly related to
_polynomial chaos_ technique. This dates back to the 1930s
[wiener1938homogeneous], and has found tremendous success in computational
science and engineering since its inception in the 1990s
[ghanem1990polynomial] and thanks to a series of influential works in the
2000s, such as [xiu2002wiener, babuska2008stochastic], which contributed to
the development of the UQ field (see also [smith2013uncertainty,
sullivan2015introduction]). As noted, a signature result for parametric DEs
was the discovery that solution maps of a large class of parametric models
admit best $n$-term polynomial approximation rates that are independent of the
number of parameters, and therefore free from the curse of dimensionality. For
more details about parametric DEs and historical remarks we refer to
[cohen2015approximation] and [adcock2022sparse, Chpt. 4].

Although considerably younger than LS, MC sampling is also a classical
technique. The happenings that led to its widespread use in computational
science are intertwined with the construction of the first electronic
programmable digital computer (ENIAC), pioneering simulations of complex
physical systems, and – regrettably – the creation of the first nuclear
weapons in the 1940s. The history of the MC method features scientists of the
calibre of Fermi, von Neumann, Ulam and Metropolis [metropolis1990alamos,
metropolis1987beginning].

One of the most popular applications of MC sampling is numerical quadrature.
It is well known that the MC quadrature error scales proportionally to
$m^{-1/2}$, where $m$ is the number of MC samples of the function to be
integrated (see, e.g., [owen2013monte, Section 2.1]). Remarkably, this decay
rate is independent of the number of function’s variables, which – at least,
intuitively – motivates the use of MC in high dimensions. We recall, though,
that in this work we are concerned with high-dimensional function
approximation, as opposed to quadrature. In particular, it is worth noting
that best $n$-term approximation error rates of the smooth high-dimensional
functions of interest in this paper decay much faster than the MC quadrature
error (see §1.4.1).

Naturally, any discussion of LS and MC sampling would be incomplete without a
word on regression. We now warn the reader that this is _not_ a paper on
statistical regression. Statistical regression refers to the problem of
fitting a model to noisy data under some probabilistic assumptions on the
noise (e.g., homoscedasticity). This is not the perspective adopted in this
paper. In fact, our goal is the accurate and stable approximation of a ‘ground
truth’ function from pointwise samples, where the noise corrupting the samples
need not be modelled as a random variable. For a more in-depth discussion of
the differences between the function approximation and statistical regression
settings, we refer to [guo2020constructing, §1.1], of which we share the same
viewpoint.

###  1.6 Outline

In §2 we present an overview of polynomial approximation of multivariate
functions and (weighted) LS. In §3 we present relevant theory for (weighted)
LS, before discussing MC sampling and the near-optimal strategy mentioned in
§1.2. In §LABEL:s:MC-bad we present numerical experiments demonstrating the
main phenomenon considered in this work. We then turn our attention to its
theoretical explanation. In §LABEL:s:polyapp-inf-dim we introduce and review
polynomial approximation theory for $(\bm{b},\varepsilon)$-holomorphic
functions. We present our first result in §LABEL:s:main-res, i.e., the
existence of a LS approximation achieving the bound (5). Finally, we consider
CS schemes in §LABEL:s:polyappCS, including both the error bound (6) and a
numerical comparison between ALS and CS.

This paper also has supplementary materials. These contain background on
infinite-dimensional measures (§LABEL:s:inf_dim_measures), proofs of various
results (§LABEL:s:proofs), additional information on the experimental setup
(§LABEL:s:additional-info), background on parametric DEs (§LABEL:s:pDE-
background) and further experiments (§LABEL:s:further-experiments). MATLAB
code reproducing all the experiments is available at
https://github.com/benadcock/is-MC-bad.

##  2 Polynomials and least-squares polynomial approximation

In this section, we describe multivariate orthogonal polynomials and
polynomial approximation via LS. In order to keep the technical level
reasonable, we consider the finite-dimensional case only. The infinite-
dimensional case is introduced in §LABEL:s:polyapp-inf-dim.

###  2.1 Univariate notation

Let $\varrho$ be a probability measure on $[-1,1]$ and write
$L^{2}_{\varrho}([-1,1])$ for the corresponding Lebesgue space of square-
integrable functions $f:[-1,1]\rightarrow\mathbb{C}$. We assume that $\varrho$
generates a unique sequence of orthonormal polynomials
$\\{\psi_{\nu}\\}_{\nu\in\mathbb{N}_{0}}\subset L^{2}_{\varrho}([-1,1])$. In
other words,

$\langle\psi_{\nu},\psi_{\nu^{\prime}}\rangle=\delta_{\nu,\nu^{\prime}},\quad\forall\nu,\nu^{\prime}\in\mathbb{N}_{0},\quad\text{and}\quad\mathrm{span}\\{\psi_{0},\ldots,\psi_{n}\\}=\mathbb{P}_{n},\quad\forall
n\in\mathbb{N}_{0},$

where $\mathbb{P}_{n}$ is the space of polynomials of degree at most $n$. Note
that this is a mild assumption (see, e.g., [narayan2018computation, §2.1]).
Two particular cases we focus on in this paper are the uniform and Chebyshev
(arcsine) measures

$\,\mathrm{d}\varrho(y)=2^{-1}\,\mathrm{d}y,\qquad\,\mathrm{d}\varrho(y)=(\pi\sqrt{1-y^{2}})^{-1}\,\mathrm{d}y,$

which generate the Legendre and (first kind) Chebyshev polynomials,
respectively.

###  2.2 Multivariate polynomial approximation

Let $d\in\mathbb{N}$ and consider the symmetric hypercube
$\mathcal{U}=[-1,1]^{d}$. We define a probability measure over $\mathcal{U}$
via tensor products. Abusing notation, we write
$\varrho=\varrho\times\cdots\times\varrho$ for this measure, where on the
right-hand side $\varrho$ denotes the probability measure on $[-1,1]$.

Let $L^{2}_{\varrho}(\mathcal{U})$ be the Lebesgue space of square-integrable
functions $f:\mathcal{U}\rightarrow\mathbb{C}$. We construct an orthonormal
polynomial basis for this space via tensor products. Writing
$\bm{\nu}=(\nu_{1},\ldots,\nu_{d})\in\mathbb{N}^{d}_{0}$ for an arbitrary
multi-index, we define

$\Psi_{\bm{\nu}}(\bm{y})=\psi_{\nu_{1}}(y_{1})\cdots\psi_{\nu_{d}}(y_{d}),\qquad\bm{y}=(y_{i})^{d}_{i=1}\in\mathcal{U}.$

The set $\\{\Psi_{\bm{\nu}}\\}_{\bm{\nu}\in\mathbb{N}^{d}_{0}}\subset
L^{2}_{\varrho}(\mathcal{U})$ forms an orthonormal basis. Hence any function
$f\in L^{2}_{\varrho}(\mathcal{U})$ has the convergent expansion

(7)
$f=\sum_{\bm{\nu}\in\mathbb{N}^{d}_{0}}c_{\bm{\nu}}\Psi_{\bm{\nu}},\qquad\text{where
}c_{\bm{\nu}}=\langle
f,\Psi_{\bm{\nu}}\rangle_{L^{2}_{\varrho}(\mathcal{U})}=\int_{\mathcal{U}}f(\bm{y})\overline{\Psi_{\bm{\nu}}(\bm{y})}\,\mathrm{d}\varrho(\bm{y}).$

We are interested in $n$-term approximations to such functions. An $n$-term
approximation to $f$ based on an multi-index set $S\subset\mathbb{N}^{d}_{0}$,
$|S|=n$, has the form

(8) $f_{S}=\sum_{\bm{\nu}\in S}c_{\bm{\nu}}\Psi_{\bm{\nu}}.$

Due to Parseval’s identity, the error of such an approximation is determined
by the size of the coefficients $c_{\bm{\nu}}$ not included in $S$.
Specifically,

(9) ${\|f-f_{S}\|}^{2}_{L^{2}_{\varrho}(\mathcal{U})}=\sum_{\bm{\nu}\notin
S}|c_{\bm{\nu}}|^{2}.$

###  2.3 Choosing $S$

In standard multivariate polynomial approximation it is common to make an _a
priori_ choice of $S$ with some prescribed structure (see, e.g.,
[guo2020constructing]). Given $\ell\in\mathbb{N}_{0}$, classical examples are
the _tensor-product_ index set of order $\ell$,

$S^{\mathsf{TP}}_{\ell}=\left\\{\bm{\nu}=(\nu_{k})_{k=1}^{d}\in\mathbb{N}_{0}^{d}:\max\\{\nu_{1},\ldots,\nu_{d}\\}\leq\ell\right\\},$

or the _total degree_ index set of order $\ell$,

$S^{\mathsf{TD}}_{\ell}=\left\\{\bm{\nu}=(\nu_{k})_{k=1}^{d}\in\mathbb{N}_{0}^{d}:\nu_{1}+\cdots+\nu_{d}\leq\ell\right\\}.$

However, the cardinality of these index sets scales exponentially with the
dimension $d$, making them poorly suited in all but low-dimensional problems.
Specifically, $|S^{\mathsf{TP}}_{\ell}|=(\ell+1)^{d}$ and
$|S^{\mathsf{TD}}_{\ell}|={\ell+d\choose d}$ for any $\ell\in\mathbb{N}_{0}$
and $d\in\mathbb{N}$. Applying Stirling’s formula, one deduces that
$|S^{\mathsf{TD}}_{\ell}|\sim\ell^{d}/d!$ as $\ell\to\infty$ for fixed $d$.
This effect can be partially mitigated by working with _hyperbolic cross_
index sets (see, e.g., [guo2020constructing]). Yet, the cardinalities of these
sets also grow exponentially with dimension.

Alternatively, one can look to replace such isotropic sets with anisotropic
variants. High-dimensional functions may be highly anisotropic, i.e., they may
depend more strongly on some variables than others. The aforementioned index
sets fail to capture this behaviour, thus potentially leading to poor
approximations. However, selecting a good anisotropic set a priori requires
detailed knowledge about the function $f$, which is typically unavailable in
practice.

A theoretical alternative is provided by the concept of best $n$-term
approximation. Here, one does away with the classical notion of polynomial
degree (as described by the parameter $\ell$ in $S^{\mathsf{TP}}_{\ell}$ and
$S^{\mathsf{TD}}_{\ell}$) and instead seeks an optimal index set in a
function-dependent manner by minimizing the error (9) over all possible sets.
A best $n$-term approximation $f_{n}$ of $f$ is thus defined as

(10) $f_{n}=f_{S^{*}},\qquad\text{where
}S^{*}\in{\underset{}{\operatorname{argmin}}}\\{{\|f-f_{S}\|}_{L^{2}_{\varrho}(\mathcal{U})}:S\subset\mathbb{N}^{d}_{0},\
|S|=n\\}.$

It follows straightforwardly from (9) that $S^{*}$ consists of those multi-
indices corresponding to the $n$ largest coefficients $c_{\bm{\nu}}$ of $f$ in
absolute value; that is to say,
$S^{*}=\\{\bm{\nu}_{1},\bm{\nu}_{2},\ldots,\bm{\nu}_{n}\\}$, where
$\bm{\nu}_{1},\bm{\nu}_{2},\ldots$ are such that
$|c_{\bm{\nu}_{1}}|\geq|c_{\bm{\nu}_{2}}|\geq\cdots$.

Unfortunately, the best $n$-term approximation is a theoretical benchmark. It
is generally impossible to construct, since doing so would generically involve
computing and sorting infinitely-many coefficients – something that clearly
cannot be done from the finite data (1). A more practical approach is
therefore to construct a set $S$ – or more precisely, a nested sequence of set
$S^{(1)}\subseteq S^{(2)}\subseteq\cdots$ – in a function dependent, adaptive
manner. This is typically done via a greedy procedure, with multi-indices
being added according to some importance criterion. The aforementioned ALS
method is a procedure of this type. We discuss it further in §LABEL:s:ALS and
§LABEL:ss:adaptive-LS.

Regardless of how the index set (or sets) is constructed, however, it is
generally useful to restrict one’s attention sets with certain structure. A
lower set (also known as monotone or downward closed – see, e.g.,
[cohen2015approximation]) is a set $S\subseteq\mathbb{N}^{d}_{0}$ for which

$(\bm{\nu}\in S\text{ and }\bm{\mu}\leq\bm{\nu})\ \Rightarrow\ \bm{\mu}\in S,$

(here the inequality $\bm{\mu}\leq\bm{\nu}$ is understood componentwise).
Lower sets are ubiquitous in multivariate polynomial approximation, with
(isotropic or anisotropic) tensor-product, total degree and hyperbolic cross
index sets all being examples. Lower sets are also commonly employed in
adaptive strategies such as ALS (see §LABEL:ss:adaptive-LS), so as to make the
greedy selection procedure tractable. Indeed, given a lower set $S^{(l)}$
there are only finitely many multi-indices
$\bm{\nu}\in\mathbb{N}^{d}_{0}\backslash S^{(l)}$ for which
$S^{(l)}\cup\\{\bm{\nu}\\}$ is also lower.

###  2.4 Weighted least-squares polynomial approximation

Fix a set multi-indices $S\subset\mathbb{N}^{d}_{0}$, $|S|=n$, and consider
$m\geq n$ noisy samples (1) of a function $f\in L^{2}_{\varrho}(\mathcal{U})$
at sample points $\bm{y}_{1},\ldots,\bm{y}_{m}\in\mathcal{U}$. We wish to
compute an approximation to $f$ from the polynomial space defined by $S$:
namely, the subspace

$\mathcal{P}_{S}=\mathrm{span}\\{\Psi_{\bm{\nu}}:\bm{\nu}\in S\\}\subset
L^{2}_{\varrho}(\mathcal{U}).$

Given a positive and almost everywhere finite weight function
$w:\mathcal{U}\rightarrow\mathbb{R}$, we define a weighted least-squares
approximation to $f$ as

(11)
$\hat{f}\in{\underset{p\in\mathcal{P}_{S}}{\operatorname{argmin}}}{\frac{1}{m}}\sum^{m}_{i=1}w(\bm{y}_{i})|f(\bm{y}_{i})+e_{i}-p(\bm{y}_{i})|^{2}.$

This is readily computed by solving an algebraic LS problem for the
coefficients of $\hat{f}$. Indeed, let $\bm{\nu}_{1},\ldots,\bm{\nu}_{n}$ be
an enumeration of the indices in $S$. Then

$\hat{f}=\sum^{n}_{i=1}\hat{c}_{i}\Psi_{\bm{\nu}_{i}},\quad\text{where
}\hat{\bm{c}}=(\hat{c}_{i})^{n}_{i=1}\in{\underset{\bm{z}\in\mathbb{C}^{n}}{\operatorname{argmin}}}{\left\|\bm{A}\bm{z}-\bm{f}\right\|}_{2}^{2},$

and the LS matrix $\bm{A}\in\mathbb{C}^{m\times n}$ and vector
$\bm{f}\in\mathbb{C}^{m}$ are given by

(12)
$\bm{A}=\left(\sqrt{w(\bm{y}_{i})/m}\Psi_{\bm{\nu}_{j}}(\bm{y}_{i})\right)^{m,n}_{i,j=1},\qquad\bm{f}=\left(\sqrt{w(\bm{y}_{i})/m}(f(\bm{y}_{i})+e_{i})\right)^{m}_{i=1}.$

##  3 Theory of (weighted) least-squares approximation

In this section, we present some elementary theory for weighted LS
approximation. For the sake of generality, in the majority of this section we
consider an arbitrary $n$-dimensional subspace $\mathcal{P}\subset
L^{2}_{\varrho}(\mathcal{U})$ (i.e., not necessarily a polynomial subspace of
the form $\mathcal{P}=\mathcal{P}_{S}$). We write $\\{\Psi_{i}\\}^{n}_{i=1}$
for an orthonormal basis for $\mathcal{P}$. However, for convenience, we make
the mild assumption that the constant function $p(\bm{y})=1$,
$\forall\bm{y}\in\mathcal{U}$, is an element of $\mathcal{P}$. Note that this
always holds when $\mathcal{P}=\mathcal{P}_{S}$ and $S$ is a lower set.

###  3.1 Accuracy and stability

Given sample points $\bm{y}_{1},\ldots,\bm{y}_{m}\in\mathcal{U}$ and a weight
function $w:\mathcal{U}\rightarrow\mathbb{R}$, define the discrete semi-norm

${\|f\|}^{2}_{\mathsf{disc},w}=\frac{1}{m}\sum^{m}_{i=1}w(\bm{y}_{i})|f(\bm{y}_{i})|^{2},\quad\forall
f\in L^{2}_{\varrho}(\mathcal{U})\cap C(\mathcal{U}),$

(here $C(\mathcal{U})$ is the set of continuous functions on $\mathcal{U}$)
and the discrete stability constants

$\displaystyle\alpha_{w}$
$\displaystyle=\inf\left\\{{\|p\|}_{\mathsf{disc},w}:p\in\mathcal{P},\
{\left\|p\right\|}_{L^{2}_{\varrho}(\mathcal{U})}=1\right\\},$
$\displaystyle\beta_{w}$
$\displaystyle=\sup\left\\{{\|p\|}_{\mathsf{disc},w}:p\in\mathcal{P},\
{\left\|p\right\|}_{L^{2}_{\varrho}(\mathcal{U})}=1\right\\},$

Notice that $\alpha_{w}=\sigma_{\min}(\bm{A})$ and
$\beta_{w}=\sigma_{\max}(\bm{A})$, where is the LS matrix (12).

######  Lemma 3.1 (Accuracy and stability of weighted LS).

Let $\mathcal{P}\subset L^{2}_{\varrho}(\mathcal{U})\cap C(\mathcal{U})$ with
$1\in\mathcal{P}$, $f\in L^{2}_{\varrho}(\mathcal{U})\cap C(\mathcal{U})$,
$\bm{e}\in\mathbb{C}^{m}$, $\bm{y}_{1},\ldots,\bm{y}_{m}\in\mathcal{U}$ and
$w:\mathcal{U}\rightarrow\mathbb{R}$ be such that $w(\bm{y}_{i})>0$, $\forall
i\in[m]:=\\{1,\ldots,m\\}$. If $\alpha_{w}>0$ then the problem

(13)
$\min_{p\in\mathcal{P}}{\frac{1}{m}}\sum^{m}_{i=1}w(\bm{y}_{i})|f(\bm{y}_{i})+e_{i}-p(\bm{y}_{i})|^{2}$

has a unique solution $\hat{f}$. This solution satisfies

(14)
${\|f-\hat{f}\|}_{L^{2}_{\varrho}(\mathcal{U})}\leq\inf_{p\in\mathcal{P}}\left\\{{\|f-p\|}_{L^{2}_{\varrho}(\mathcal{U})}+\frac{1}{\alpha_{w}}{\|f-p\|}_{\mathsf{disc},w}\right\\}+\frac{\beta_{w}}{\alpha_{w}}{\left\|\bm{e}\right\|}_{\infty}.$

Also, the ($2$-norm) condition number of the LS matrix satisfies
$\mathrm{cond}(\bm{A})=\beta_{w}/\alpha_{w}$.

This result is standard (see, e.g., [migliorati2014analysis, Prop. 1]). We
include a short proof in §LABEL:s:proofs for completeness. Note that the last
statement is immediate, since
$\mathrm{cond}(\bm{A})=\sigma_{\max}(\bm{A})/\sigma_{\min}(\bm{A})$ by
definition. The main takeaway from this lemma is that the accuracy and
stability of $\hat{f}$ are determined by the size of the constants
$\alpha_{w}$ and $\beta_{w}$. In the next subsection, we discuss how to
control these constants in the case of random sampling. In doing so, we also
obtain an estimate for the sample complexity.

###  3.2 Sample complexity

We now consider the sample complexity of weighted LS approximation in the case
of random sampling. Specifically, we now assume that
$\bm{y}_{1},\ldots,\bm{y}_{m}\sim_{\mathrm{i.i.d.}}\mu$, where $\mu$ is some
probability measure with support in $\mathcal{U}$.

The analysis of this type of sampling strategy involves the (reciprocal)
Christoffel function of the subspace $\mathcal{P}$ [cohen2013stability]:

$\mathcal{K}(\mathcal{P})(\bm{y})=\sup\left\\{|p(\bm{y})|^{2}:p\in\mathcal{P},\
{\left\|p\right\|}_{L^{2}_{\varrho}(\mathcal{U})}=1\right\\},\quad\forall\bm{y}\in\mathcal{U}.$

Observe that $\mathcal{K}(\mathcal{P})(\bm{y})\geq 1$,
$\forall\bm{y}\in\mathcal{U}$, since, by assumption, the function
$1\in\mathcal{P}$. It is also a short argument to show that
$\mathcal{K}(\mathcal{P})$ has the equivalent expression

(15)
$\mathcal{K}(\mathcal{P})(\bm{y})=\sum^{n}_{i=1}|\Psi_{i}(\bm{y})|^{2},\quad\forall\bm{y}\in\mathcal{U},$

where $\\{\Psi_{i}\\}^{n}_{i=1}$ is any orthonormal basis for $\mathcal{P}$.
Given $\mathcal{K}$ and a weight function
$w:\mathcal{U}\rightarrow(0,\infty)$, we now also define

(16)
$\kappa(\mathcal{P};w)=\sup_{\bm{y}\in\mathcal{U}}w(\bm{y})\mathcal{K}(\mathcal{P})(\bm{y}).$

######  Theorem 1.

[Sample complexity of weighted LS with random sampling] Let
$\mathcal{P}\subset L^{2}_{\varrho}(\mathcal{U})\cap C(\mathcal{U})$ with
$\dim(\mathcal{P})=n$ and $1\in\mathcal{P}$, $0<\epsilon<1$ and $\mu$ be a
probability measure on $\mathcal{U}$ such that

(17) $\,\mathrm{d}\mu(\bm{y})=(w(\bm{y}))^{-1}\,\mathrm{d}\varrho(\bm{y})$

for some strictly positive and finite almost everywhere weight function
$w:\mathcal{U}\rightarrow\mathbb{R}$. Let
$\bm{y}_{1},\ldots,\bm{y}_{m}\sim_{\mathrm{i.i.d.}}\mu$, where $m$ satisfies

(18) $m\geq 7\cdot\kappa(\mathcal{P};w)\cdot\log(2n/\epsilon).$

Then the following holds with probability at least $1-\epsilon$. For any $f\in
L^{2}_{\varrho}(\mathcal{U})\cap C(\mathcal{U})$, the solution $\hat{f}$ of
the weighted LS problem (13) is unique and satisfies

(19)
${\|f-\hat{f}\|}_{L^{2}_{\varrho}(\mathcal{U})}\leq\inf_{p\in\mathcal{P}}\left\\{{\|f-p\|}_{L^{2}_{\varrho}(\mathcal{U})}+2{\|f-p\|}_{\mathsf{disc},w}\right\\}+2{\|\bm{e}\|}_{\infty}.$

Moreover, the condition number satisfies $\mathrm{cond}(\bm{A})\leq 2$.

This result is based on standard ideas, in particular, the use of the matrix
Chernoff inequality (see, e.g., [cohen2017optimal, Thm. 2.1]). We include a
short proof in §LABEL:s:proofs for completeness. Note that the number $7$ in
(18) is somewhat arbitrary. As can be seen from the proof, one can replace it
with a smaller constant at the expense of larger constants in the error and
condition number bounds.

The most important aspect of this result is the sample complexity bound (18).
Since $\mu$ is a probability measure, (17) implies that

(20) $\int_{\mathcal{U}}(w(\bm{y}))^{-1}\,\mathrm{d}\varrho(\bm{y})=1,$

and orthonormality and the alternative expression (15) for
$\mathcal{K}(\mathcal{P})$ imply that

(21)
$\int_{\mathcal{U}}\mathcal{K}(\mathcal{P})(\bm{y})\,\mathrm{d}\varrho(\bm{y})=n$

and

$\kappa(\mathcal{P};w)=\int_{\mathcal{U}}\kappa(\mathcal{P};w)/w(\bm{y})\,\mathrm{d}\varrho(\bm{y})\geq\int_{\mathcal{U}}\mathcal{K}(\mathcal{P})(\bm{y})\,\mathrm{d}\varrho(\bm{y})=n.$

Hence $\kappa(\mathcal{P};w)\geq n$ for any $\mathcal{P}$ and $w$. As a
result, the sample complexity bound (18) is always at least log-linear in $n$.
In the next subsection, we show that it is generally superlinear in $n$.
However, in §LABEL:ss:opt-samp-scheme we show that it is always possible to
choose the sampling measure $\mu$ so as to achieve the optimal value of the
right-hand side of (18).

###  3.3 The case of Monte Carlo sampling

MC sampling corresponds to the choice $\mu=\varrho$, in which case (17) holds
with $w=1$. The approximation (13) is correspondingly an unweighted LS
approximation. In this case, the sample complexity estimate (18) takes the
form

(22) $m\geq c\cdot\kappa(\mathcal{P})\cdot\log(n/\epsilon),\quad\text{where
}\kappa(\mathcal{P})=\kappa(\mathcal{P};1)={\left\|\mathcal{K}(\mathcal{P})\right\|}_{L^{\infty}(\mathcal{U})}.$

Recall that $\kappa(\mathcal{P})\geq n=\dim(\mathcal{P})$. Unfortunately,
$\kappa(\mathcal{P})$ can be arbitrarily large, even in cases such as
Chebyshev and Legendre polynomial approximation. We now state two standard
results, which are based on [chkifa2015discrete, chkifa2014high,
chkifa2018polynomial] (see also [adcock2022sparse, Props. 5.13 & 5.17]).


# Zero-energy states in graphene quantum dot with wedge disclination

Ahmed Bouhlal  Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb
Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Ahmed Jellal
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  Nurisya Mohd Shah
Laboratory of Computational Sciences and Mathematical Physics, Institute for
Mathematical Research (INSPEM), Universiti Putra Malaysia, 43400 UPM Serdang,
Selangor, Malaysia  Department of Physics, Faculty of Science, Universiti
Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

###### Abstract

We investigate the effects of wedge disclination on charge carriers in
circular graphene quantum dots subjected to a magnetic flux. Using the
asymptotic solutions of the energy spectrum for large arguments, we
approximate the scattering matrix elements, and then study the density of
states. It is found that the density of states shows several resonance peaks
under various conditions. In particular, it is shown that the wedge
disclination is able to change the amplitude, width, and positions of
resonance peaks.

###### pacs:

81.05.ue; 73.63.-b; 73.23.-b; 73.22.Pr
Keywords: Graphene, quantum dot, electrosatic potential, magnetic flux, wedge
disclination, density of states.

##  I Introduction

Graphene is the focus of much theoretical and experimental research due to its
electronic and optical properties [1, 2, 3, 4]. Graphene has become a primary
material for future nanoelectronic devices [5]. Most of the research conducted
so far is based on the mobility of electrons, which is determined by the
diffusion of charged impurities [6, 7]. The two graphene conduction and
valence bands touch each other at two asymmetric points noted $(K,K^{\prime})$
are called Dirac points. In the vicinity of these Dirac points, the energy
dispersion is linear, which is the origin of graphene’s unique electronic
properties like high electrical conductivity [8, 5]. At normal incidence, the
electrons transmit completely, which is due to a Klein tunneling effect, and
it is therefore difficult to control them with external fields in the graphene
[9, 10, 11]. However, it has been shown theoretically that it is still
possible to create quasi-bound states (QBS) using external electric fields
such as the magnetic field [12, 13, 14].

The notion of quantum dots is used in modern physics to study some physical
phenomena. It is essential to the progress of quantum computing and
spintronics. The reasons for choosing the application of graphene are twofold:
on the one hand, because graphene is considered a two-dimensional zero energy
gap system [15] and on the other hand, because the quantum dots in graphene
have the potential to be used as spin qubits [16]. The study of the
possibility of electron confinement in graphene quantum dots is particularly
important in this context [17, 18]. The experimental activity will be based on
the confinement of Dirac fermions in graphene quantum dots [19]. Due to the
absence of a band gap, the confinement of charge carriers under the effect of
an electrostatic potential using metallic gates is impossible in graphene. The
use of a magnetic flux in graphene quantum dots allows for better control and
confinement of fermions in the presence of an electrostatic potential [20]. It
is well demonstrated that the magnetic flux is responsible for the kinematic
angular momentum taking integer values, allowing the appearance of states that
cannot be confined by the electrostatic potential alone [21]. The disk-shaped
quantum dot has a circular symmetry from which the angle of incidence is a
characteristic of the movement of electrons, knowing that the latter at their
non-zero angular momentum are confined within the quantum dot [14, 22]. Like
graphene, it is well demonstrated that it is an ideal two-dimensional system
where the energy dispersion is linear and has a density of states that cancels
at zero energy [5]. In this work, we study electrostatic confinement in the
presence of magnetic flux with wedge disclination.

Insulators and superconductors react significantly to topological defects and
exhibit quasi-zero energy states around the defect [23]. The effect of a
Coulombian electrostatic potential in the presence of a uniform magnetic field
on the energy spectrum of graphene has been studied recently [24]. Prior to
the year 2014, two-terminal conductance was the only technique used to study
the confinement of Dirac fermions in a graphene quantum dot [14, 25, 26]. In
2014, Martin Schneider developed a technique based on density of states (DOS)
analysis to study the possibility of confinement in a small graphene region
[27]. These studies are extended by examining how the information of the
confinement signature has been deduced by the analysis of the density of
states, and in parallel, the effect of the gap energy on the information of
the system has also been studied [28]. We wish to consider in this context the
simple disinclination via the construction of Volterra [29].Our goal in this
paper is to investigate the properties of a graphene monolayer in which the
crystal symmetry is locally modified by the conical geometry mentioned above.

We consider a circular graphene quantum dot of radius $R$ surrounded by a
circular ring of undoped graphene of external radius $L$ bound by a metallic
contact in the presence of a magnetic field with a wedge disclination $n=0,\pm
1,\pm 2$. By restricting ourselves to the asymptotic behavior of Hankel
functions for large arguments, we determine the scattering matrix. DOS is
calculated as a function of magnetic flux $\Phi$, applied electrostatic
potential $V_{0}$, and wedge disclination $n$. We numerically analyze the DOS
under various conditions of the physical parameters. This allows us to show
different oscillatory behaviors and to find resonances. We show that the DOS
peaks strongly depend on the momentum quantum number $m$. During the scanning
of the gate voltage, the zero energy bound states cause resonant peaks that
become narrow if the size $L$ of the undoped graphene sheet containing the
quantum dot increases, see Fig. 1. Subsequently, we discover that $n$ acts by
changing the resonance peaks. Indeed, this behavior contrasts strongly with
that of $n=2$ (square defect), which moves to the right from its initial
position where $n=0$ (defect-free), and then the resonance becomes more and
more acute. In the opposite case, where $n=-2$ (octagon defect), we observe
the creation of a new peak that corresponds to $m=5/2$.

The paper is organized as follows. In section II, we establish a theoretical
model describing electrostatically confined charge carriers in geometry
obtained by Volterra construction. We calculate the scattering matrix by
restricting to the asymptotic behavior of Hankel functions in section III,
which are obtained in the solution of the energy spectrum. Using the
scattering matrix, we explicitly determine the associated DOS in section IV.
In section V, we numerically analyze the DOS and show it exhibits a set of
resonance peaks. Under various conditions, the effects of magnetic flux and
the wedge $n$ on these peaks will be investigated. Finally, we conclude our
results.

##  II Theoretical model

The configuration we study consists of a graphene quantum dot (QD) connected
to a ring-shaped metal contact, and the two are separated by an undoped
intrinsic graphene layer.

Figure 1: (color online) Graphene quantum dots of radius $R$ surrounded by an
undoped intrinsic graphene sheet in the form of a ring of outer radius $L$ and
coupled to source and drain reservoirs with a corner of angle $\theta$
removed. $R(\theta)$ is the curve along which the spinor is rotated.

The Hamiltonian of a graphene monolayer that describes the motion of a single
electron in the presence of an axially symmetric external potential $V(r)$ at
low energy is

$H=v_{F}\left({\tau_{z}}\sigma_{x}p_{x}+\sigma_{y}p_{y}\right)+V(r)\mathbb{I}$
(1)

where $p=(p_{x},p_{y})$ is the momentum operator, $v_{F}=10^{6}$ m/s is the
Fermi velocity, $\sigma_{i},\tau_{i}$ ($i=x,y,z$) are Pauli matrices denoting
the sublattice and valley degrees of freedom in the basis of the two
sublattices of $A$ and $B$ atoms, respectively. The potential barrier $V(r)$
is defined as follows

$V(r)=\left\\{\begin{array}[]{lll}0,&{R}<r<{L}\\\ -\hbar v_{F}V_{0},&r<{R}\\\
-\hbar v_{F}V_{\infty},&r>{L}\end{array}\right.$ (2)

with the parameters $V_{0}$ and $V_{\infty}$ are chosen to be positive because
the two regions outside the ring of radius $R$ and $L$ are doped with
electrons. The highly doped lead graphene is modeled by taking the limit
$V_{\infty}\to\infty$. The Hamiltonian (1) acts on the spinor with four
components

$\Psi(\vec{r})=\left(\psi_{A+}(\vec{r})\psi_{B+}(\vec{r})\psi_{A-}(\vec{r})\psi_{B-}(\vec{r})\right)^{T}$
(3)

where the valley indices $+(-)$ refer to the two Dirac points $K(K^{\prime})$
in the Brillouin zone. For a two dimensional graphene, the angular boundary
condition for a Dirac spinor when it follows a closed path is given by

$\Psi(r,\theta=2\pi)=e^{i\pi{\tau_{z}}\sigma_{z}}\Psi(r,\theta=0)$ (4)

and we will see that the presence of the wedge disclination modifies the
angular boundary conditions in (4). Indeed, it is known that the deformation
in a honeycomb lattice falls in the description at the continuum boundary
undergoes a pseudo-gauge field coupled to the electron pulse by a vector
potential [30, 31, 32]. As shown in Fig. 1, the Volterra construction can be
used to introduce the wedge disclination by removing a sector of
$\delta=n\pi/3$. The integer $n$ represents the type of disclination in the
honeycomb lattice, with $n=-2$ representing an isolated octagon defect, $n=-1$
representing an isolated heptagon defect, $n=0$ representing an isolated
defect in the no defect case, $n=1$ representing an isolated pentagon defect,
and $n=2$ representing an isolated square defect. As a result, the wave
function (4) becomes

$\Psi(r,\theta=2\pi)=-e^{i2\pi[1-(n/6)]{\tau_{z}}\sigma_{z}/2}\Psi(r,\theta=0).$
(5)

Exchanging the $K$ and $K^{\prime}$ blocks in the Hamiltonian using the
unitary transformation $e^{i\pi\sigma_{y}\tau_{y}/2}$ yields the angular
boundary condition for any number $n$ [33, 34]

$\Psi(r,\theta=2\pi)=-e^{i2\pi\left[-\frac{n}{4}\sigma_{y}\tau_{y}+(1-(n/6))\frac{1}{2}{\tau_{z}}\sigma_{z}\right]}\Psi(r,\theta=0)$
(6)

For a local gauge transformation, (6) can be written in two singular gauge
transformations

$\Psi(r,\theta=2\pi)=F(\phi)G_{n}(\theta)\Psi(r,\theta=0)$ (7)

where $F(\phi)=e^{i\frac{\phi}{2}{\tau_{z}}\sigma_{z}}$ and
$G_{n}(\theta)=e^{i\frac{n\theta}{4}\tau_{y}\sigma_{y}}$, as shown in Fig. 1
with $\theta=\frac{\phi}{1-\frac{n}{6}}\in[0,2\pi]$.

Let us now consider an external magnetic flux $\Phi$ applied to the quantum
dot (QD)

$\vec{A}=\frac{\hbar\Phi}{\Omega_{n}e\Phi_{0}r}\vec{e}_{\theta}$ (8)

where $\Phi_{0}=h/e$ is the flux quantum and $\vec{e}_{\theta}$ is the unit
vector along the azimuthal direction and the wedge disclination is described
by $\Omega_{n}=1-\frac{n}{6}$ [35] with $n=0,\pm 1,\pm 2$. For this, we
replace the momentum $\vec{p}$ by the conical momentum $\vec{p}+e\vec{A}$ in
the Hamiltonian (1) and use the transformations of $F(\phi)$ and
$G_{n}(\theta)$. This process yields

$\tilde{H}(r,\theta)=F^{\dagger}G_{n}^{\dagger}HG_{n}F$ (9)

such that the Hamiltonian (1) is now given by

$\displaystyle\tilde{H}(r,\theta)=$ $\displaystyle\hbar
v_{F}\left(k_{r}-\frac{i}{2r}\right)\sigma_{x}$ (10) $\displaystyle+\hbar
v_{F}\left(k_{\theta}+\frac{\Phi_{i}}{\Omega_{n}r}+\frac{n}{4\Omega_{n}r}\tau_{z}\right)\sigma_{y}+V(r)\mathbb{I}$

where $k_{r}=-i\frac{\partial}{\partial r}$,
$k_{\theta}=-\frac{i}{r\Omega_{n}}\frac{\partial}{\partial\theta}$ and
$\Phi_{i}=\frac{\Phi}{\Phi_{0}}$ is the dimensionless flux. It is easy to
verify that the total angular momentum $J_{z}=L_{z}+\frac{\hbar}{2}\sigma_{z}$
commutes with the Hamiltonian (1). As a result, the eigenfunctions can be
separated as

$\Psi(r,\theta)=e^{im\theta}\left(\begin{array}[]{c}\chi_{A}(r)\\\
\chi_{B}(r)\\\ \end{array}\right)$ (11)

with $m=\pm 1/2,\pm 3/2\cdots$ are eigenvalues of $J_{z}$.

We solve the eigenvalue equation $H\Psi_{\tau}=E\Psi_{\tau}$ in the three
regions: $0<r<R$, $R<r<L$ and $r>L$. After some algebras, we obtain two
coupled radial equations

$\displaystyle\left[\frac{\partial}{\partial
r}+\frac{1}{2r}+\frac{1}{r\Omega_{n}}\left(m+\Phi_{i}{+\frac{n\tau}{4}}\right)\right]\chi_{B}={(\epsilon-
V_{i})\chi_{A}}$ (12) $\displaystyle\left[-\frac{\partial}{\partial
r}-\frac{1}{2r}+\frac{1}{r\Omega_{n}}\left(m+\Phi_{i}{+\frac{n\tau}{4}}\right)\right]\chi_{A}={(\epsilon-
V_{i})\chi_{B}}$ (13)

where dimensionless parameters $\epsilon=\frac{E}{\hbar v_{F}}$ and
$V_{i}=\frac{V(r)}{\hbar v_{F}}$ have been defined. We look for a second order
equation for one component and reusing one of (12-13) to find the other
component of the wave function. Now, injecting (12) into (13) to end up with

$\left[\rho^{2}\frac{\partial^{2}}{\partial\rho^{2}}+\rho\frac{\partial}{\partial\rho}+\rho^{2}-\left(\mu-\frac{1}{2}\right)^{2}\right]\chi_{A}(\rho)=0$
(14)

where we have set
$\mu=\frac{1}{\Omega_{n}}\left(m+\Phi_{i}{+\frac{n\tau}{4}}\right)$ and
$\rho=kr$ with $k=\epsilon-V_{i}$ is the wave number. (14) can be reformulated
as the Bessel differential equation. In general, the solution is a linear
combination of the Hankel function of first $H^{+}_{n}\left(\rho\right)$ and
of second $H^{-}_{n}\left(\rho\right)$ kinds. Then we combine them to get the
proper eigenspinors

$\psi_{k,\mu}^{\pm}(r)=e^{i\left(\mu-\frac{1}{2}\right)\theta}\sqrt{\frac{k}{4\pi}}\begin{pmatrix}H^{\pm}_{\mu-1/2}(kr)\\\
i~{}\mathrm{sign}(\mu)e^{i\theta}H^{\pm}_{\mu+1/2}(kr)\end{pmatrix}$ (15)

which describe incoming (-) (propagating from $r=0$) or outgoing (+)
(propagating from $r=\infty$) for circular waves. Note that, $H_{n}^{(\pm)}$
are linear combinations of Bessel $J_{n}$ and Neumann $Y_{n}$, namely
$H_{n}^{(\pm)}=J_{n}\pm iY_{n}$. We specify the value of $k$ which differs
from one region to another where the solutions (15) apply. For $r<R$: $k\equiv
k_{0}=\epsilon+V_{0}$, for $R<r<L$: $k=\epsilon$ and for $r>L$: $k\equiv
k_{\infty}=\epsilon+V_{\infty}$. Recall that the half-integer Bessel function
are valid, i.e., $Y_{1/2}(x)=-J_{-1/2}(x)=-\sqrt{\frac{2}{\pi x}}\cos x$,
$Y_{-1/2}(x)=J_{1/2}(x)=\sqrt{\frac{2}{\pi x}}\sin x$. It is clearly seen that
the function $\frac{\cos(kr)}{\sqrt{r}}$ is divergent at the origin. Therefore
for $\mu=0$, we have

$\psi_{\kappa,0}^{\pm}(r)=\frac{e^{\pm i\kappa
r}}{\sqrt{8\pi^{2}r}}\begin{pmatrix}\pm e^{-i\theta}\\\ 1\end{pmatrix}.$ (16)

At the center of the quantum dot ($R\to 0$) and for the wave function to be
regular at $r=0$, we have the solution inside the quantum dot ($r<R$)

$\psi_{k,\mu}(r)=e^{i\left(\mu-\frac{1}{2}\right)\theta}\sqrt{\frac{k_{0}}{4\pi}}\begin{pmatrix}J_{\mu-1/2}(kr)\\\
i~{}\mathrm{sign}(\mu)e^{i\theta}J_{\mu+1/2}(kr).\end{pmatrix}$ (17)

For the region $R<r<L$, the eigenspinors can be written as a linear
combination of the two solutions of (15)

$\displaystyle\psi_{2}(r)$ $\displaystyle=$ $\displaystyle
a_{\mu}\psi_{k,\mu}^{-}(r)+b_{\mu}\psi_{k,\mu}^{+}(r)$ (18)

with $a_{\mu}$, $b_{\mu}$ being arbitrary constants. Inside the circular ring
${R}<r<{L}$ and at zero energy ($\epsilon=0$), the radial components have the
forms

$\chi_{A}(r)=a_{+}r^{\mu-\frac{1}{2}},\qquad\chi_{B}(r)=a_{-}r^{-\mu-\frac{1}{2}}.$
(19)

To avoid divergence, we require the constraints $a_{+}=0$ for $\mu>0$ and
$a_{-}=0$ for $\mu<0$.

We use the asymptotic behavior of Hankel functions in the limit of a heavily
doped lead where $k_{{\infty}}L\gg 1$. That is

$H^{(\pm)}_{n}(x)\approx(2/\pi x)^{1/2}e^{\pm
i(x-n\frac{\pi}{2}-\frac{\pi}{4})}.$ (20)

As a results the eigenspinors $\psi_{1}(r)$ inside the quantum dot ($r<R$) and
$\psi_{3}(r)$ outside the quantum ring ($r>L$) can be simplified to

$\displaystyle\psi_{1}(r)$ $\displaystyle=$
$\displaystyle\sqrt{k_{0}}\begin{pmatrix}J_{\mu-1/2}(k_{0}r)\\\
J_{\mu+1/2}(k_{0}r)\end{pmatrix}$ (21) $\displaystyle\psi_{3}(r)$
$\displaystyle=$ $\displaystyle
c_{\mu}\frac{e^{-ik_{\infty}r}}{\sqrt{r}}\begin{pmatrix}1\\\
-1\end{pmatrix}+d_{\mu}\frac{e^{ik_{\infty}r}}{\sqrt{r}}\begin{pmatrix}1\\\
1\end{pmatrix}$ (22)

##  III Scattering matrix

The coefficients $a_{\mu}(\epsilon)$, $b_{\mu}(\epsilon)$, $c_{\mu}(\epsilon)$
and $d_{\mu}(\epsilon)$ can be determined using the continuity of the wave
function at $r=R$ and $r=L$ and the regularity at $r=0$. The first two
coefficients are linked via the scattering matrix $S_{\mu}(\epsilon)$

$\displaystyle{d}_{\mu}(\epsilon)=\mathcal{S}_{\mu}(\epsilon){c}_{\mu}(\epsilon)$
(23) $\displaystyle S_{\mu}(\epsilon)=-\frac{\det D^{(-)}}{\det D^{(+)}}$ (24)

where the matrices $D^{(\pm)}$ are given by

$D^{(\pm)}=\begin{pmatrix}0&\sqrt{k}H^{(+)}_{\mu-\frac{1}{2}}(kR)&\sqrt{k}H^{(-)}_{\mu-\frac{1}{2}}(kR)&\sqrt{k_{0}}J_{\mu-\frac{1}{2}}(k_{0}R)\\\
0&\sqrt{k}H^{(+)}_{\mu+\frac{1}{2}}(kR)&\sqrt{k}H^{(-)}_{\mu+\frac{1}{2}}(kR)&\sqrt{k_{0}}J_{\mu+\frac{1}{2}}(k_{0}R)\\\
\frac{e^{\mp
ik_{\infty}L}}{\sqrt{L}}&-\sqrt{k}H^{(+)}_{\mu-\frac{1}{2}}(kL)&-\sqrt{k}H^{(-)}_{\mu-\frac{1}{2}}(kL)&0\\\
\mp\frac{e^{\mp
ik_{\infty}L}}{\sqrt{L}}&-i\sqrt{k}H^{(+)}_{\mu+\frac{1}{2}}(kL)&-i\sqrt{k}H^{(-)}_{\mu+\frac{1}{2}}(kL)&0\end{pmatrix}.$
(25)

For small distances, we have

$\displaystyle J_{n}(x)\sim\frac{1}{n!}\left(\frac{x}{2}\right)^{n}$ (26)
$\displaystyle
Y_{n}(x)\sim\left\\{\begin{array}[]{c}-\frac{\Gamma(n)}{\pi}\left(\frac{2}{x}\right)^{n},~{}~{}n>0\\\
\frac{2}{\pi}\ln\left(\gamma_{E}\frac{x}{2}\right),~{}~{}n=0\end{array}\right.$
(29)

with $\ln\left(\gamma_{E}\right)=0.577\cdots$ is defined as an Euler constant.
Indeed, for small $\epsilon$, we can develop the scattering matrix as a
function of $k$ in the internal region ($R<r<L$). Then, we write

$\displaystyle\mathcal{S}_{\mu}(\epsilon)=e^{-2ik_{\infty}L+i|\mu|\pi}\left[\mathcal{S}_{\mu}^{(0)}+kL\mathcal{S}_{\mu}^{(1)}+{\cal
O}(\epsilon^{2})\right]$ (30)

such that

$\mathcal{S}_{\mu}^{(0)}=\frac{1+i\mathcal{J}_{\mu}R_{L}^{2|\mu|}}{1-i\mathcal{J}_{\mu}R_{L}^{2|\mu|}}$
(31)

and we used the abbreviation

${\cal
J}_{\mu}=\frac{J_{|\mu|+\frac{1}{2}}\left(k_{0}{R}\right)}{J_{|\mu|-\frac{1}{2}}\left(k_{0}R\right)}$
(32)

where $R_{L}=\frac{R}{L}$. We distinguish two cases for $S_{\mu}^{1}$. Indeed,
in the absence of magnetic flux and in the defect-free case ($\phi=0,n=0$), we
have

$\displaystyle\mathcal{S}^{(1)}_{\mu}=$
$\displaystyle\displaystyle-\frac{2i}{2|\mu|-1}{\cal S}^{(0)}_{\mu}$ (33)
$\displaystyle+\frac{8i|\mu|+2i[(2|\mu|+1){\cal
J}_{\mu}^{2}-(2|j|-1)]R_{L}^{2|\mu|+1}}{(4|\mu|^{2}-1)(1-i{\cal
J}_{j}R_{L}^{2|\mu|})^{2}}$

as well as for $|\mu|=\frac{1}{2}$

$\mathcal{S}^{(1)}_{\pm 1/2}=\frac{i(1-R_{L}^{2})-2i{\cal
J}_{\frac{1}{2}}^{2}R_{L}^{2}\ln(R_{L})}{(1-i{\cal
J}_{\frac{1}{2}}R_{L})^{2}}.$ (34)

In the presence of magnetic flux and wedge disclination, we find that the
results of (33) remain valid for $\mu\neq 0$. We compute the scattering matrix
$S_{0}(\epsilon)$ for the case $\mu=0$. We show that it is constant and
independent of $V_{0}$, such as

$S_{0}=e^{-2i(k_{\infty}-k_{0})R}e^{-2i(k-k_{\infty})(R-L)}.$ (35)

##  IV Density of states

Using the scattering matrix for low energy (30), we will calculate the density
of states at zero energy in terms of the potential $V(r)$. The local density
of states (LDOS) $\rho(r,\epsilon)$ is given by the derivative of the
scattering matrix $S(\epsilon,V(r))$ [36, 37, 38]

$\rho(r,\epsilon)=-\frac{1}{2\pi
i}\mbox{Tr}\,{S_{\mu}(\epsilon,V(r))}^{\dagger}\left(\frac{\delta{S_{\mu}(\epsilon,V(r))}}{\delta
V(r)}\right).$ (36)

We calculate the total density of states (DOS) of the circular quantum dot by
integrating $\rho(r,\epsilon)$ in region $r<L$ [27]. This yields to

$\rho_{dot}(\epsilon)=-\frac{1}{2\pi
i}\int_{r<L}\mbox{Tr}{S_{\mu}(\epsilon,V(r))}^{\dagger}\left(\frac{\delta{S_{\mu}(\epsilon,V(r))}}{\delta
V(r)}\right)~{}dr.$ (37)

We can now calculate $\rho_{\rm dot}$ at zero energy using (37), which gives

$\rho_{dot}=\frac{1}{2\pi i\hbar
v_{F}}\sum_{\mu}~{}\mathcal{S}^{(0)*}_{\mu}\left[\frac{\partial\mathcal{S}^{(0)}_{\mu}}{\partial
V_{0}}+L\mathcal{S}^{(1)}_{\mu}\right].$ (38)

It is worth noting that DOS is related to the point conductance [39, 27] and
is dependent on the Wigner-Smith delay [40].

Taking into account the continuity of the eigenspinors (17) and (19) at $r=R$
for zero energy, the resonance condition is well established

$J_{|\mu|-1/2}(V^{\prime}_{0}{R})=0$ (39)

where $V^{\prime}_{0}=V_{0}$. At resonance, one can write

${\cal J}_{\mu}\approx\frac{1}{R(V^{\prime}_{0}-V_{0})}.$ (40)

For zero energy, we show that the DOS has a Lorentzian variation on $V_{0}$.
When there is no magnetic flux ($\phi=0$) and $\mu|\neq 1/2$, DOS takes the
form at zero energy

$\rho_{dot}=\frac{4R|\mu|}{\pi\hbar
v_{F}(2|\mu|-1)}\frac{\Gamma}{4R^{2}(V_{0}-V^{\prime}_{0})^{2}+\Gamma^{2}}$
(41)

while for $|\mu|=1/2$, we have

$\rho_{dot}=\frac{2R}{\pi\hbar v_{F}}\left(1-\ln
R_{L}\right)\frac{\Gamma}{4R^{2}(V_{0}-V^{\prime}_{0})^{2}+\Gamma^{2}}$ (42)

where the resonance width is given by

$\Gamma=2\left(R_{L}\right)^{2|\mu|}$ (43)

It is worth noting that (35) demonstrates that there is a constant that is
independent of the gate voltage $V_{0}$ as well as a non-resonant contribution
to DOS $\rho_{dot}$. In order to study the possibility of electrostatic
confinement in the graphene quantum dot reflected in DOS, we numerically
analyze the analytical results obtained above. The next paragraph presents the
studies on the effect of wedge disclination on the zero energy states in
quantum-circular graphene.

##  V Numerical results and discussions

Figure 2: (color online) DOS as a function of gate voltage $RV_{0}$ at
$\epsilon=0$ with $R/L=0.2$, $\Phi_{i}=0$ for $(n,\tau)$ such that (a): 0,
(b): $(1,1)$, (d): $(1,-1)$, (e): $(2,1)$, (f): $(2,-1)$. (g): $(-1,1)$. (h):
$(-1,-1)$, (i): $(-2,1)$, (j): $(-2,-1)$. Here (c) is a zoom of (b) for
$5.6<V_{0}R<6.6$ and resonances are labeled by angular momentum
$|m|=1/2,\cdots,13/2$.

We start at this point with the numerical analysis of the results obtained
analytically. First, we investigate the effect of ratio $R_{L}=\frac{R}{L}$,
magnetic flux $\Phi_{i}$, and wedge disclination index $n$ values on the
density of states of a quantum dot in graphene surrounded by an undoped hands-
on graphene region with zero energy $\epsilon=0$ and coupled to source and
drain reservoirs. We analyze the possibility of confining electrons in the
quantum-circular of monolayer graphene by evaluating the DOS of such a
circular quantum dot defined by a gate in the presence of two conductors.
Indeed, because the parameters of our theory are
$\mu=\frac{1}{\Omega_{n}}\left(m+\Phi_{i}{+\frac{n\tau}{4}}\right)$, then we
choose to numerically analyze the DOS versus the gate voltage $RV_{0}$ under
suitable conditions of the physical parameters characterizing our system.

DOS as a function of $V_{0}R$ is shown in Fig. 2 by varying the values of the
wedge disclination index $n=0,\pm 1,\pm 2$, and valley $\tau=\pm 1$. Recall
that each value of $n$ represents a different type of default, such as $n=0$
for a defect isolated in the absence of a defect (a), $n=1$ for a defect
isolated in a pentagon (b), $n=2$ for a defect isolated in a square (c),
$n=-1$ for a defect isolated in a heptagon (d), and $n=-2$ for a defect
isolated in an octagon (e). We notice that the DOS presents resonance peaks
with oscillatory behavior that are marked by their angular momentum $m$. For
$n=0$, we notice that the position of the resonances, as well as the scale of
the amplitude and width, are in good agreement with the results obtained in
the literature [14, 41, 27, 28]. We find that the bound states of the electron
in graphene quantum dots with a wedge disclination using density of states
analysis are strongly dependent on the value of $n$. When changing the value
of n, the positions as well as the amplitude and width size change. Positive
values of $n$ and $\tau$ shift the peaks to the right. This is due to peak
suppression. For $n=2$ and $\tau=1$, we can see that only four peaks remain,
with increasing amplitudes and decreasing width sizes (see Fig. 2e). In the
case where $n$ takes a negative value and $\tau=1$, we observe that the
resonance peaks change their initial positions and move to the left, which
explains the appearance of three peaks corresponding to $m=5/2$, $m=11/2$ and
$m=13/2$ when $n=-2$ and $\tau=1$. Their amplitudes decrease as the size of
the width increases. In the opposite case where $\tau=-1$, the effect of wedge
disclination $n$ is reversed, such that for negative values of $n$ with
$\tau=-1$, the resonance peaks move to the right with an increase in amplitude
and a decrease in width. The condition (39) shows that there are three
resonance peaks that are labeled according to their angular momentum $m=1/2$.
However, they are missing for two cases: $(n=2,\tau=-1)$ and $(n=-2,\tau=1)$,
as shown in Fig. 2f,i.

The characteristics of the resonance are found to be dependent on the value of
$n$ and the valley $\tau$. The resonance positions shift to the right if $\tau
n>0$ and become very straight and very important for $n=2$ and $\tau=1$,
whereas they shift to the left for $\tau n<0$. When $n>0$, the influence of
this magnitude where $n>0$ on the behavior of the resonance peaks is reversed.
This result becomes very clear when the DOS is plotted in Fig. 3 for the first
or the second resonance by varying the values of $n$ and $\tau$. In the case
where $\tau=-1$, the parameter $n$ has no effect on the position or
characteristics of the second resonance peak, as shown in Fig. 3d. We note
that the peaks of the second DOS resonance are very narrow compared to those
of the first resonance. As $n$ increases in Fig. 3c, the resonance peaks
become very narrow, such that their amplitude increases but their width
decreases.

Figure 3: (color online) DOS as a function of gate voltage $V_{0}R$ at
$\epsilon=0$ with $R/L=0.2$, $\Phi_{i}=0$ for $n=0$ (blue line), $n=1$ (green
line), $n=2$ (magenta line), $n=-1$ (purple line), $n=-2$ (orange line). (a,
b): $m=1/2$ and (c, d): $m=3/2$ with (a, c): $\tau=1$ and (b, d): $\tau=-1$.

Figure 4: (color online) DOS as a function of gate voltage $V_{0}R$ at
$\epsilon=0$ with $\Phi_{i}=0$ for $m=1/2$ by changing the ratio $R/L=0.08$
(blue line), $R/L=0.1$ (green line), $R/L=0.13$ (magenta line), $R/L=0.16$
(purple line), $R/L=0.2$ (orange line). The pair $(n,\tau)$ is (a): $0$, (b):
$(1,1)$, (c): $(1,-1)$, (d): $(2,1)$, (e): $(\pm 2,\mp 1)$, (f): $(-1,1)$,
(g): $(-1,-1)$, (h): $(-2,-1)$.

In Fig. 4, we only consider the first resonance and investigate its influence
on the magnitude of the point-lead coupling $R/L$, where $L$ denotes the
distance between the source and the drain. We observe that the resonance peaks
have Lorentzian shapes whose properties depend on several parameters,
including the index $n$ and the ratio $R/L$. The resonance peaks become
narrower when the quantum dot is very far from the source, i.e., when $R/L=0$,
as well as when the defect is chosen to be of square type, $n=2$, and for the
valley $K$ ($\tau=1$). As a result, the DOS is a series of peaks influenced by
the gate voltage $V_{0}R$, $n$, $R/L$, $m$, and $\tau$. The width of the
resonances is independent of the coupling to the source and drain, which is
determined by $R/L$, and the height increases when $R/L$ is very low, $n=2$,
and $\tau=1$.

Figure 5: (color online) The width of resonance peak as a function of wedge
disclination $n$ at $\epsilon=0$ with $R/L=0.2$, $\Phi_{i}=0$ for (a):
$m=1/2$, (b): $m=3/2$, (c): $m=5/2$, and (d): $7=3/2$, $\tau=1$ (blue line)
and $\tau=-1$ (green line).

The width of the resonance peak as a function of wedge disclination $n$ is
shown in Fig. 5 for $m=1/2,3/2,5/2,7/2$, and $\tau=\pm 1$. For higher values
of $m$ with $n=2$ and $\tau=1$, one sees that the resonances become very
sharp, which is consistent with the width scale (39). The results presented in
both Fig. 5b and Fig. 3d show that there is no influence of $n$ on the
characteristics and position of the second resonance peak for $\tau=-1$. On
the other hand, the variation of the amplitude of resonance peaks is inversely
related to the variation of their width. Consequently, these numerical values
confirm the results obtained above.

Figure 6: (color online) DOS as a function of gate voltage $V_{0}R$ at
$\epsilon=0$ with $R/L=0.2$, $\Phi_{i}=\frac{1}{2}$ for $(n,\tau)$ such that
(a): $0$, (c): $(1,1)$, (d): $(1,-1)$, (e): $(2,1)$, (f): $(2,-1)$, (g):
$(-1,1)$, (h): $(-1,-1)$, (i): $(-2,1)$, (j): $(-2,-1)$. Here (b) is a zoom of
(d) for $5.6<V_{0}R<6.6$ and resonances are labeled by $|\mu|=1,\cdots,6$.

We now introduce a magnetic flux carried by a half magnetic flux of
$\Phi_{i}=1/2$, yielding an integer total angular momentum denoted by $\mu$,
which labels the resonance peaks. In Fig. 6, the DOS is shown as a function of
the gate voltage $V_{0}R$, with $R/L=0.2$ for $n=0,\pm 1,\pm 2$, and $\tau=\pm
1$. The electrons circling the flux tube collect an Aharonov-Bohm phase $\pi$,
canceling out the effect of the Berry phase, and the electron wave function
collects via pseudo-rotation while moving in a circular motion. We find that
when $\Phi_{i}=1/2$, the system enters a state with zero angular momentum,
$\mu=0$, which cannot be confined by an electrostatic potential, even in the
presence of a magnetic field. We obtain results that are consistent with
previous simulations [14, 41, 27, 28] in terms of the position and scale of
the width by changing the resonances $R/L$ when $n=0$ in Fig. 6. Furthermore,
one can observe extremely small resonance peaks in the presence of a magnetic
flux.

For $\tau=\pm 1$, the DOS of the first and second resonances $\mu=1,2$ is
represented in Fig. 7 by varying the value of the wedge disclination $n=0$
(blue line), $n=1$ (green line), $n=2$ (magenta line), $n=-1$ (purple line),
and $n=-2$ (orange line).

Figure 7: (color online) DOS as a function of gate voltage $V_{0}R$ at
$\epsilon=0$ with $R/L=0.2$, ${\Phi_{i}}=1/2$ for various wedge of
disclination values: $n=0$ (blue line), $n=1$ (green line), $n=2$ (magenta
line), $n=-1$ (purple line), $n=-2$ (orange line). (a, b): $\mu=1$ and (c, d):
$\mu=2$ with (a, c): $\tau=1$ and (b, d): $\tau=-1$.

Figure 8: (color online) The width of the resonance peak as a function of
wedge disclination $n$ at $\epsilon=0$ with $R/L=0.2$, $\Phi_{i}=\frac{1}{2}$
for (a): $\mu=1$, (b): $\mu=2$, (c): $\mu=3$ and (d): $\mu=4$ such that
$\tau=1$ (blue line) and $\tau=-1$ (green line).

We notice that the effect of a wedge disclination becomes very important, as
shown in Fig. 6e for $n=2$ and $\tau=1$. We find very narrow resonance peaks
well shifted to the right, which leads to a suppression of the four peaks. For
$(n=-2,\tau=1$ and under the influence of a magnetic flux, however, we notice
a decrease in the amplitude of the peaks as well as the appearance of three
peaks, indicating that the number of bound states increases, see Fig. 6.

Figure 9: (color online) DOS as a function of gate voltage $V_{0}R$ at
$\epsilon=0$ with $\Phi_{i}=1/2$ and for the first resonance $\mu=1$ by
changing the ratio $R/L=0.08$ (blue line), $R/L=0.1$ (green line), $R/L=0.13$
(magenta line), $R/L=0.16$ (purple line), $R/L=0.2$ (orange line). With the
pair $(n,\tau)$ such that (a): 0, (b): $(1,1)$, (c): $(1,-1)$, (d): $(2,1)$,
(e): $(2,-1)$, (f): $(-1,1)$, (g): $(-1,-1)$, (h): $(-2,1)$, (i): $(-2,-1)$.

Figure 10: (color online) Amplitude of the resonance peak as function of the
magnetic flux $\Phi_{i}$ at $\epsilon=0$ with $R/L=0.2$ for $n=1$ (blue line),
$n=2$ (green line) and $V_{0}^{\prime}R=V_{0}R$.

We investigate the effect of the ratio $R/L$ on DOS for the first resonance
$\mu=1$ in the presence of a magnetic flux $\Phi_{i}=1/2$ with $\tau=\pm 1$ by
varying the value of a wedge disclination $n$. By comparing the results of the
two Figs. 4 and 9, we show that the analysis can be made quantitative by
taking a specific peak and studying the effect of the ratio $R/L$. We observe
that DOS becomes very saturated when the limit $R/L\to 0$, which corresponds
to a weak coupling between the quantum dot and a ring-shaped metallic.
Finally, the behavior of the resonance peaks is affected by three physical
quantities, which are the wedge disclination value $n$, the magnetic flux
$\Phi_{i}=1/2$, and the ratio $R/L$. To give a better presentation of the
result, we draw another curve that represents the width of the resonance peaks
in the presence of magnetic flux $\Phi_{i}=1/2$.

The width of the resonance peak for $\mu=1,2,3,4$ is plotted in Fig. 8 for two
values of $\tau=+1(-1)$. Comparing the results of the two Fig. 5 and Fig. 8,
we see that the wedge disclination effect becomes very important for the
valley $K(\tau=1)$. Also, we notice that its effect enjoys the presence of a
magnetic flux and for $\tau=1$, because it is better.

The amplitude of a DOS resonance peak as a function of magnetic flux
$\Phi_{i}$ for the angular momentum $m=1/2$ and wedge disclination $n=1,2$ is
shown in Fig. 10. We see that the amplitude of a resonance peak grows
exponentially as a function of the magnetic flux and the variation becomes
very important when the value of $n$ is fixed at the value $2$ and for
$\tau=1$. This result is very similar to the previous results obtained in [28,
42]. We conclude that the wedge disclination $n$ is a physical quantity that
can alter the behavior of the resonance peaks that make up the DOS.
Furthermore, its effect is amplified in the presence of a magnetic flux. As a
result, we can say that it can be used as a tunable physical parameter for the
control of the transport properties of our system.

##  VI Conclusion

We considered a model of graphene quantum dot in the presence of magnetic flux
$\Phi_{i}$ undergoing a default, which can be understood from Volterra’s
constructions. To treat the problem properly, we started by solving the
corresponding Dirac equation and analytically determining the energy spectra,
taking into account the cylindrical symmetry. We found that the solutions are
expressed in terms of Hankel functions. Then, we obtained an approximate
formula for the density of states (DOS) as a function of physical parameters
such as magnetic flux, angular momentum, wedge disclination, and applied
electrostatic potential.

The analysis of the results shows that even if graphene does not present a gap
in its electronic energy spectrum, we can have the confinement of electrons
under the effect of an external electrostatic excitation in a quantum dot
surrounded by a large sheet of undoped graphene. We have shown that the
density of states is an alternative technique that can be used to study the
phenomenon of fermions confinement in a graphene quantum dot. We have
performed an analysis that considers the integrated density of states on the
dot and an undoped graphene sheet. The formalism we have followed in this
paper can easily be extended to this type of measurement configuration.
However, with respect to the qualitative analysis of the width, amplitude, and
position of the peaks as well as the suppression and creation of new resonance
peaks.

We presented our numerical results in terms of the $R/L$ ratio, $\Phi_{i}$
magnetic flux, and wedge disclination $n$. We have shown that the presence of
wedge disclination modifies the behavior of the resonance peaks and also
changes their positions. We found that for negative values of $n$, they can
create new resonance peaks and minimize the resonance phenomenon by decreasing
the amplitude of the peaks and increasing their widths, thus increasing the
number of bound electron states. On the other hand, we have shown that the
result of adding a positive $n$ term that represents the defect is a narrow
peak that becomes very narrow in the presence of magnetic flux as well as the
suppression of some peaks. Finally, we discovered that the density of states
is very divergent in the presence of a very strong magnetic field and large
values of $n=2$ (square defect). These results could have potential
applications in nanoelectronics.

## References

  * Novoselov  _et al._ [2004] K. S. Novoselov, A. K. Geim, S. V. Morozov, D.-e. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films, science 306, 666 (2004).
  * Chen  _et al._ [2008] J.-H. Chen, C. Jang, S. Xiao, M. Ishigami, and M. S. Fuhrer, Intrinsic and extrinsic performance limits of graphene devices on sio2, Nature nanotechnology 3, 206 (2008).
  * Geim and Novoselov [2010] A. K. Geim and K. S. Novoselov, The rise of graphene, in  _Nanoscience and technology: a collection of reviews from nature journals_ (World Scientific, 2010) pp. 11–19.
  * Schedin  _et al._ [2007] F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson, and K. S. Novoselov, Detection of individual gas molecules adsorbed on graphene, Nature materials 6, 652 (2007).
  * Neto  _et al._ [2009] A. C. Neto, F. Guinea, N. M. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Reviews of modern physics 81, 109 (2009).
  * Martin  _et al._ [2008] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. v. Smet, K. Von Klitzing, and A. Yacoby, Observation of electron–hole puddles in graphene using a scanning single-electron transistor, Nature physics 4, 144 (2008).
  * Das  _et al._ [2008] A. Das, S. Pisana, B. Chakraborty, S. Piscanec, S. K. Saha, U. V. Waghmare, K. S. Novoselov, H. R. Krishnamurthy, A. K. Geim, A. C. Ferrari, _et al._ , Monitoring dopants by raman scattering in an electrochemically top-gated graphene transistor, Nature nanotechnology 3, 210 (2008).
  * Novoselov  _et al._ [2007] K. Novoselov, S. Morozov, T. Mohinddin, L. Ponomarenko, D. Elias, R. Yang, I. Barbolina, P. Blake, T. Booth, D. Jiang, _et al._ , Electronic properties of graphene, physica status solidi (b) 244, 4106 (2007).
  * Young and Kim [2009] A. F. Young and P. Kim, Quantum interference and klein tunnelling in graphene heterojunctions, Nature Physics 5, 222 (2009).
  * Klein [1929] O. Klein, Die reflexion von elektronen an einem potentialsprung nach der relativistischen dynamik von dirac, Zeitschrift für Physik 53, 157 (1929).
  * Katsnelson  _et al._ [2006] M. Katsnelson, K. Novoselov, and A. Geim, Chiral tunnelling and the klein paradox in graphene, Nature physics 2, 620 (2006).
  * Pereira Jr  _et al._ [2006] J. M. Pereira Jr, V. Mlinar, F. Peeters, and P. Vasilopoulos, Confined states and direction-dependent transmission in graphene quantum wells, Physical Review B 74, 045424 (2006).
  * Matulis and Peeters [2008] A. Matulis and F. M. Peeters, Quasibound states of quantum dots in single and bilayer graphene, Physical Review B 77, 115423 (2008).
  * Bardarson  _et al._ [2009] J. H. Bardarson, M. Titov, and P. Brouwer, Electrostatic confinement of electrons in an integrable graphene quantum dot, Physical review letters 102, 226803 (2009).
  * Dóra  _et al._ [2009] B. Dóra, M. Gulácsi, and P. Sodano, Majorana zero modes in graphene with trigonal warping, physica status solidi (RRL)–Rapid Research Letters 3, 169 (2009).
  * Recher and Trauzettel [2010] P. Recher and B. Trauzettel, Quantum dots and spin qubits in graphene, Nanotechnology 21, 302001 (2010).
  * Ponomarenko  _et al._ [2008] L. A. Ponomarenko, F. Schedin, M. I. Katsnelson, R. Yang, E. W. Hill, K. S. Novoselov, and A. K. Geim, Chaotic dirac billiard in graphene quantum dots, Science 320, 356 (2008).
  * Jacobsen  _et al._ [2012] A. Jacobsen, P. Simonet, K. Ensslin, and T. Ihn, Transport in a three-terminal graphene quantum dot in the multi-level regime, New Journal of Physics 14, 023052 (2012).
  * Hämäläinen  _et al._ [2011] S. K. Hämäläinen, Z. Sun, M. P. Boneschanscher, A. Uppstu, M. Ijäs, A. Harju, D. Vanmaekelbergh, and P. Liljeroth, Quantum-confined electronic states in atomically well-defined graphene nanostructures, Physical Review Letters 107, 236803 (2011).
  * De Martino  _et al._ [2007] A. De Martino, L. Dell’Anna, and R. Egger, Magnetic confinement of massless dirac fermions in graphene, Physical review letters 98, 066802 (2007).
  * Pal  _et al._ [2011] G. Pal, W. Apel, and L. Schweitzer, Electric transport through circular graphene quantum dots: Presence of disorder, Physical Review B 84, 075446 (2011).
  * Calvo [2011] M. Calvo, Electrostatic quantum dots in a suspended graphene monolayer, Physical Review B 84, 235413 (2011).
  * De Souza  _et al._ [2014] J. De Souza, C. de Lima Ribeiro, and C. Furtado, Bound states in disclinated graphene with coulomb impurities in the presence of a uniform magnetic field, Physics Letters A 378, 2317 (2014).
  * Choudhari and Deo [2014] T. Choudhari and N. Deo, Graphene with wedge disclination in the presence of intrinsic and rashba spin orbit couplings, EPL (Europhysics Letters) 108, 57006 (2014).
  * Heinl  _et al._ [2013] J. Heinl, M. Schneider, and P. W. Brouwer, Interplay of aharonov-bohm and berry phases in gate-defined graphene quantum dots, Physical Review B 87, 245426 (2013).
  * Schneider and Brouwer [2011] M. Schneider and P. W. Brouwer, Resonant scattering in graphene with a gate-defined chaotic quantum dot, Physical Review B 84, 115440 (2011).
  * Schneider and Brouwer [2014] M. Schneider and P. W. Brouwer, Density of states as a probe of electrostatic confinement in graphene, Physical Review B 89, 205437 (2014).
  * Bouhlal  _et al._ [2021] A. Bouhlal, A. Jellal, and H. Bahlouli, Density of states analysis of electrostatic confinement in gapped graphene, Solid State Communications 333, 114335 (2021).
  * Furtado  _et al._ [1994] C. Furtado, B. G. da Cunha, F. Moraes, E. B. de Mello, and V. Bezzerra, Landau levels in the presence of disclinations, Physics Letters A 195, 90 (1994).
  * Vozmediano  _et al._ [2010] M. A. Vozmediano, M. Katsnelson, and F. Guinea, Gauge fields in graphene, Physics Reports 496, 109 (2010).
  * Guinea  _et al._ [2010] F. Guinea, M. Katsnelson, and A. Geim, Energy gaps and a zero-field quantum hall effect in graphene by strain engineering, Nature Physics 6, 30 (2010).
  * Ghaemi  _et al._ [2012] P. Ghaemi, J. Cayssol, D. N. Sheng, and A. Vishwanath, Fractional topological phases and broken time-reversal symmetry in strained graphene, Physical Review Letters 108, 266801 (2012).
  * Lammert and Crespi [2000] P. E. Lammert and V. H. Crespi, Topological phases in graphitic cones, Physical review letters 85, 5190 (2000).
  * Rüegg and Lin [2013] A. Rüegg and C. Lin, Bound states of conical singularities in graphene-based topological insulators, Physical Review Letters 110, 046401 (2013).
  * Nazarov  _et al._ [2009] Y. V. Nazarov, Y. Nazarov, and Y. M. Blanter,  _Quantum transport: introduction to nanoscience_ (Cambridge university press, 2009).
  * Langer and Ambegaokar [1961] J. Langer and V. Ambegaokar, Friedel sum rule for a system of interacting electrons, Physical Review 121, 1090 (1961).
  * Buttiker [1993] M. Buttiker, Capacitance, admittance, and rectification properties of small conductors, Journal of Physics: Condensed Matter 5, 9361 (1993).
  * Büttiker  _et al._ [1994] M. Büttiker, H. Thomas, and A. Prêtre, Current partition in multiprobe conductors in the presence of slowly oscillating external potentials, Zeitschrift für Physik B Condensed Matter 94, 133 (1994).
  * Büttiker [2000] M. Büttiker, Time-dependent transport in mesoscopic structures, Journal of Low Temperature Physics 118, 519 (2000).
  * Smith [1960] F. T. Smith, Lifetime matrix in collision theory, Physical Review 118, 349 (1960).
  * Titov  _et al._ [2010] M. Titov, P. Ostrovsky, I. Gornyi, A. Schuessler, and A. Mirlin, Charge transport in graphene with resonant scatterers, Physical review letters 104, 076802 (2010).
  * Gutiérrez-Rubio and Stauber [2015] A. Gutiérrez-Rubio and T. Stauber, Mass-profile quantum dots in graphene and artificial periodic structures, Physical Review B 91, 165415 (2015).