# Genesis and fading away of persistent currents in a Corbino disk geometry

Yuriy Yerin  Dipartimento di Fisica e Geologia, Universitá degli Studi di
Perugia, Via Pascoli, 06123 Perugia, Italy  V.P. Gusynin  Bogolyubov Institute
for Theoretical Physics, National Academy of Science of Ukraine, 14-b
Metrologicheskaya Street, Kiev, 03143, Ukraine  S.G. Sharapov  Bogolyubov
Institute for Theoretical Physics, National Academy of Science of Ukraine,
14-b Metrologicheskaya Street, Kiev, 03143, Ukraine  Kyiv Academic University,
03142 Kyiv, Ukraine  A.A. Varlamov  CNR-SPIN, via del Fosso del Cavaliere 100,
00133, Rome, Italy

###### Abstract

The detailed analytical and numerical analysis of the electron spectrum,
persistent currents, and their densities for an annulus placed in a constant
magnetic field (Corbino disk geometry) is presented. We calculate the current
density profiles and study their dependence on the inner and outer radii of
the annular. We study evolution of the persistent currents and track their
emergence and decay for different limiting cases of such a geometry, starting
from a nanodot and ending by a macroscopic circle. Our analytical results for
the currents are confirmed by the agreement between the integration of the
corresponding current densities and the application of the Byers-Yang formula,
when it is applicable. Among other results we find the general expression for
the persistent current in a narrow annulus, which in the one channel
approximation reproduces the well-known result for quasi-one dimensional
mesoscopic metallic ring. Moreover it allows to analyze the multi-channel case
of a relatively wide annulus. Our study can be used for more accurate
treatment and interpretation of the experimental data with measurements of the
persistent currents in different doubly-connected systems.

##  I Introduction

While the Landau diamagnetism of free electron gas Landau  is often regarded
as a standard textbook knowledge, the discussion of the role of the edge
states arising in finite systems is less known, although it is only one year
younger Teller . This analysis addressed the naturally risen question, why the
Landau’s calculation of magnetization and similar treatment of the
nondissipative transport coefficients remain correct for large enough system
with finite boundaries Heuser . It is remarkably that these studies revealed
the presence of the macroscopic nondissipative persistent edge currents
flowing along the boundaries of the sample.

The persistent currents can also exist in doubly-connected systems due to the
Aharonov-Bohm effect Aharonov1959PR . For instance, it was predicted Kulik1 ;
Kulik2010  that in a hollow thin-walled normal metallic cylinder or ring with
the small enough radius $R$ threaded by a magnetic flux $\Phi$ the persistent
current can flow. Its magnitude oscillates as
$I\sim(\left|e\right|{v_{F}}/R)\sin({2\pi\Phi/{\Phi_{0}}})$, where $v_{F}$ is
the Fermi velocity and $\Phi_{0}=hc/e$ is the magnetic flux quantum. The
diamagnetic currents in the restricted geometry, including rings, were studied
in between 60-70’s e.g. in Prange ; Bogachek ; Nedorezov . It was also
demonstrated that the account for such geometrical effects can lead to the
magnetic response of the magnitude larger than the Landau diamagnetic moment
(see the reviews in Refs. Richter ; Gurevich .)

Subsequently the properties of persistent currents were studied in Ref.
Buttiker ; Gefen1 ; Gefen2  within different approaches for ballistic and
diffusive regimes of conductivity. It was revealed that the condition of their
feasibility consists of the requirement that the size of a system should be
mesoscopic, i.e. the radius of a ring has to be of the order of the electron
mean free path at zero temperature.

Another insight into the nature of these currents was obtained after the
discovery of the quantum Hall effect in 2D MOSFET structures Klitzing1980 . It
was shown in Ref. MacDonald1984PRB  that in the rectangular geometry with the
characteristic sizes much larger than the electron magnetic length the
quantized Hall current may be expressed as the difference between diamagnetic
currents flowing along the two edges (see also Ref. Mineev2007PRB  for a more
recent discussion of link between quantum Hall effect and diamagnetism). When
in a state of thermodynamic equilibrium the chemical potential of these edges
is the same, the edge currents cancel each other and the total one caused by
the applied external magnetic field is zero.

Interestingly, a more simple rectangular geometry was considered in Ref.
MacDonald1984PRB  two years after the same problem was studied in the annular
geometry Halperin1982PRB . This is not surprising because the annular geometry
of the Corbino disk represents a practical realization of the cylinder
geometry suggested by Laughlin for the gedanken experiment explaining quantum
Hall effect. The sizes of the disk in Ref. Halperin1982PRB  are assumed to be
macroscopic, i.e. its inner and outer radii along with the width of the ring
strongly exceeds the electron magnetic length. Again if the chemical potential
of the two edges is the same, the currents at the inner and outer edge flow in
the opposite directions and there is no net current around the annulus.

It could seem that the contradiction exists between the two above discussed
approaches. Summing of edge currents in annulus results in their cancellation,
while the one-dimensional treatment of the thin ring demonstrates the
existence of the persistent current in it. This imaginary controversy is
related to the reconstruction of the electron spectrum in the annulus as it
becomes of the microscopic size. Indeed, when its dimensions approach the
magnetic length, the staircase of Landau energy levels undergoes non-
negligible alteration, and the edge currents start to overlap.

Persistent currents owing to edge effects of the diamagnetic response in the
bulk disks and due to the Aharonov–Bohm effect in a mesoscopic samples were
initially predicted as the tiny effects, and they were hardly detectable
experimentally at those times. Nevertheless, enormous advances in
nanotechnology of the last two decades renewed interest to this elusive
quantum mechanical phenomenon. Recently the magnetic response of individual
gold rings with the typical radii of the order of 1 $\mu m$ and the comparable
width indeed has been measured at very low temperatures. It has been found
that the response of sufficiently small rings in applied magnetic field can be
attributed to the predicted in Ref. Buttiker ; Gefen1 ; Gefen2  persistent
currents. Their amplitudes were found in a rather good agreement to the
corresponding theory for quasi-one-dimensional rings Moler ; Glazman2009 .

It is important to note that in all Refs. Kulik1 ; Buttiker ; Gefen1 ; Gefen2
the consideration of persistent currents was carried out solely for a quasi-
one-dimensional mesoscopic ring within the assumption of its infinitesimal
small width. Contrary, in Ref. Varlamov  the current density distribution was
studied for the macroscopic Corbino disk placed in classically strong magnetic
field under the assumption that all its sizes strongly exceed the electron
magnetic length.

The authors of Ref. Avishai1993PRB  investigated the persistent current in the
same macroscopic annulus being in ballistic regime as a function of electron
density. They found the violent fluctuations of the current (in sign and in
absolute value) which is quite unusual for systems without disorder. It was
demonstrated that these fluctuations result from the overlapping of the sign-
changing currents produced by the inner and outer edge states. The same
authors in Ref. Avishai1993PA  considered numerically the case of the large
disk with the infinitesimal inner radius in the strong magnetic field.

The goal of the present paper is to establish a bridge between the listed
above different approaches for calculation of the persistent current in the
systems of various scales. In purpose to do this we perform calculations for
the annulus (Corbino disk geometry) in applied perpendicular homogeneous
magnetic field with the inner and outer radii arbitrary with respect to the
magnetic length and among themselves. We start from the microscopic solution
of the eigenvalues and eigenfunctions problem for the electron in a magnetic
field in the case of doubly connected geometry. Basing on it we succeed in the
unified way to find the emergence of the first current states in quantum dots,
genesis of the persistent currents in quasi-one-dimensional rings, current
oscillatory behavior and the current density profiles in finite size metallic
annulus. Finally, we observe fading away of these currents when the sizes of
the annulus become substantially macroscopic.

##  II Model and general relations

###  II.1 Model

We consider an annulus with the inner and the outer radii $r_{1}$ and $r_{2}$,
correspondingly, subjected to a constant magnetic field $\bf{B}$ applied
perpendicularly to its plane described by the Hamiltonian

$H=\frac{{{{\bf{p}}^{2}}}}{{2m_{e}}}=\frac{1}{{2m_{e}}}{\left({-i\hbar\bm{\nabla}+\frac{e}{c}{\bf{A}}\left({\bf{r}}\right)}\right)^{2}},$
(1)

where $m_{e}$ is the effective electron mass. Due to the axial symmetry of the
problem it is convenient to study it in the polar coordinates, where the
vector potential is written in the symmetric gauge
${\bf{A}}\left({\bf{r}}\right)=(A_{r},A_{\varphi})=\frac{1}{2}\left({0,Br}\right)$.
Consequently, the Schrödinger equation acquires the form

$\displaystyle\left[{-\frac{{{\hbar^{2}}}}{{2m_{e}}}\left({\frac{{{\partial^{2}}}}{{\partial{r^{2}}}}+\frac{1}{r}\frac{\partial}{{\partial
r}}+\frac{1}{{{r^{2}}}}\frac{{{\partial^{2}}}}{{\partial{\varphi^{2}}}}}\right)-\frac{{i\hbar{\omega_{c}}}}{2}\frac{\partial}{{\partial\varphi}}+\frac{1}{8}m_{e}\omega_{c}^{2}{r^{2}}}\right]\psi\left({r,\varphi}\right)=E\psi\left({r,\varphi}\right).$
(2)

Here $-e<0$ is the charge of the electron and ${\omega_{c}}=eB/(m_{e}c)$ is
the cyclotron frequency. The boundary conditions imposed on the wave function
$\psi\left({{r},\varphi}\right)$ correspond to the impenetrability of the disk
edges, i.e. the mandatory requirement

$\psi\left({{r_{1}},\varphi}\right)=\psi\left({{r_{2}},\varphi}\right)=0.$ (3)

Separation of the variables in Eq. (2)

$\psi\left({r,\varphi}\right)=f\left(r\right){e^{-im\varphi}}$ (4)

reduces it to the differential equation for the radial component of the wave
function

$\left[{\\!-\\!\frac{{{\hbar^{2}}}}{{2m_{e}}}\left({\frac{{{\partial^{2}}}}{{\partial{r^{2}}}}\\!+\\!\frac{1}{r}\frac{\partial}{{\partial
r}}\\!-\\!\frac{{{m^{2}}}}{{{r^{2}}}}}\right)+\frac{1}{8}m_{e}\omega_{c}^{2}{r^{2}}}\right]f\left(r\right)\\!=\\!\tilde{E}f\left(r\right),$
(5)

where the energy $\tilde{E}$ is shifted in respect to $E$ as
$\tilde{E}=E+\hbar\omega_{c}m/2$. Introducing the dimensionless energies
$\varepsilon={E}/(\hbar{\omega_{c}})$,
$\tilde{\varepsilon}={\tilde{E}}/({\hbar{\omega_{c}}})$ and the dimensionless
variable $\rho=r/l$ (where $l=\sqrt{\hbar c/(|e|B)}$ is the magnetic length)
one can simplify Eq. (5):

$\left[{\frac{{{\partial^{2}}}}{{\partial{\rho^{2}}}}+\frac{1}{\rho}\frac{\partial}{{\partial\rho}}-\frac{{{\rho^{2}}}}{4}+2\tilde{\varepsilon}-\frac{{{m^{2}}}}{{{\rho^{2}}}}}\right]f\left(\rho\right)=0.$
(6)

Its general solution can be written in terms of two Whittaker functions
${M_{\kappa,\mu}}\left(z\right)$ and ${W_{\kappa,\mu}}\left(z\right)$:

$f\left(\rho\right)=\frac{1}{\rho}\left[{{C_{1}}{M_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left(\frac{\rho^{2}}{2}\right)+{C_{2}}{W_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left(\frac{\rho^{2}}{2}\right)}\right].$
(7)

Below we will analyze the spectral properties of Eq. (6) and the asymptotic
behavior of radial functions (7).

###  II.2 Dispersive Landau levels in the Corbino geometry

Applying of the boundary conditions (3) at Eq. (7) yields a transcendental
equation for the energy levels
$\tilde{\varepsilon}_{n,m}\equiv\tilde{\varepsilon}$:

${W_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho_{1}^{2}}{2}}\right)\\!{M_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho_{2}^{2}}{2}}\right)\\!-\\!{M_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho_{1}^{2}}{2}}\right)\\!{W_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho_{2}^{2}}{2}}\right)\\!=\\!0.$
(8)

Here the two quantum numbers appear, viz. the first (principal) $n$
corresponds to the level number in Landau problem, while the second, azimuthal
one $m$, characterizes the angular momentum and the latter is an analogue of
the wave-vector component $k_{y}$ in the Landau gauge Landau . Let us recall,
that in the case of the Landau gauge the quantum number $k_{y}$ determines the
position of the potential minimum $x_{0}=l^{2}k_{y}$ in the coordinate space.
In the case under consideration its role passes to the value
$r_{m}=\sqrt{2|m|}l$, which is nothing else as the position along the radial
coordinate of the maximum in the probability of the electron state with
quantum number $m$ for the given $n$.

The same impenetrability boundary conditions allow to simplify the form of
radial part of the wave function (7)

$\displaystyle
f_{nm}\left(\rho\right)=\frac{C}{\rho}\left[W_{{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho_{1}^{2}}{2}}\right)\\!{M_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho^{2}}{2}}\right)\right.-$
$\displaystyle\left.{M_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho_{1}^{2}}{2}}\right)\\!{W_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho^{2}}{2}}\right)\right],$
(9)

leaving in it the only constant $C$. The latter is determined from the
normalization condition:

$\begin{split}{(Cl)^{-2}}=2\pi&\int\limits_{{\rho_{1}}}^{{\rho_{2}}}{\frac{d\rho}{\rho}}\left[{{W_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho_{1}^{2}}{2}}\right){M_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho^{2}}{2}}\right)-}\right.\\\
&{\left.{{M_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho_{1}^{2}}{2}}\right){W_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho^{2}}{2}}\right)}\right]^{2}}.\end{split}$
(10)

Accordingly, the eigenfunction of the one particle problem is

$\psi_{nm}\left(\mathbf{r}\right)=f_{nm}\left(r\right)e^{-im\varphi}$ (11)

(see Eqs. (4) and (9)).

Let us note that the Whittaker functions in Eq. (8), in the case when the
parameter $\tilde{\varepsilon}_{n,m}$ equals to

$\tilde{\varepsilon}_{n,m}=n+\frac{1}{2}\left({\left|m\right|+1}\right),\qquad
n=0,1,\ldots,$ (12)

turn out linearly dependent and they reduce to Laguerre polynomials. It is why
these trivial solutions, corresponding to the infinite system with the
spectrum (this problem in the symmetric gauge was addressed in Frenkel1930 )

${\varepsilon_{n,m}}=n+\frac{1}{2}\left({\left|m\right|-m+1}\right),\quad
n=0,1,\ldots,\quad m\geq-n,$ (13)

we exclude from consideration.

In Figure 1 one can see the series of the energy levels $\varepsilon_{n,m}$ as
the function of the angular quantum number $m$ for different inner and outer
radius of the disk obtained from the numerical solution of Eq. (8).

Figure 1: Energy levels as a function of angular quantum number $m$ for an
annulus with a fixed inner radius $\rho_{1}=3$ and different outer radii
$\rho_{2}=6$ (a), $\rho_{2}=9$ (b), $\rho_{2}=10$ (c) and $\rho_{2}=13$ (d).

One observes that as the outer radius $\rho_{2}$ increases the Landau levels
inside the ring flatten and approach the values (13) for the infinite system.

###  II.3 Persistent currents

####  II.3.1 General expressions

The current density operator in the representation of field operator $\psi$ is

${\bf{\hat{j}}}=\frac{{ie\hbar}}{{2m_{e}}}\left({{\psi^{*}}\bm{\nabla}\psi-\bm{\nabla}{\psi^{*}}}\psi\right)-\frac{e^{2}}{{m_{e}c}}{\bf{A}}\psi^{*}{\psi}.$
(14)

For the Fermi gas with chemical potential $\mu$ the current density
expectation value reads AGD

$\begin{split}&\mathbf{j}\left(\mathbf{r}\right)=-\frac{e}{{2m_{e}}}\left(\boldsymbol{\pi}\left(\mathbf{r}\right)+{\boldsymbol{\pi}^{*}}\left(\mathbf{r}^{\prime}\right)\right)\times\\\
&\sum_{\begin{subarray}{c}m=-\infty\\\
n=0\end{subarray}}^{\infty}{{\left.{\psi_{nm}}\left(\mathbf{r}\right)\psi_{nm}^{*}\left(\mathbf{r}^{\prime}\right)\right|}_{\mathbf{r}^{\prime}=\mathbf{r}}}n_{F}\left(E_{n,m}\right),\end{split}$
(15)

where $\boldsymbol{\pi}=-i\hbar\bm{\nabla}+\frac{e}{c}\mathbf{A}(\mathbf{r})$
is the gauge invariant momentum operator, $\psi_{nm}$ is given by Eq. (11),
and $n_{F}(E)=[\exp((E-\mu)/T)+1]^{-1}$ is the Fermi-Dirac distribution
function. In the considered limit of low temperatures ($T\ll\hbar\omega_{c}$)
is reduced to the Heaviside function $\theta(\mu-E_{n,m})$.

Generally speaking the chemical potential $\mu$ is the sophisticated function
of the electron density and applied magnetic field (see for example Mineev1 ).
The electron density is fixed in a closed system. Moreover, in the following
we will be interested in the energy level dispersion and current distribution
as the functions of the system size for a chosen value of the magnetic field.
Thus in the further consideration we assume chemical potential as the
constant.

Based on the expressions for radial and tangential components of a momentum

${\pi_{r}}=-i\hbar\frac{\partial}{{\partial
r}},\qquad{\pi_{\varphi}}=-\frac{{i\hbar}}{r}\frac{\partial}{{\partial\varphi}}+\frac{{eBr}}{{2c}},$
(16)

one can find the corresponding expressions for the current density:

$\begin{split}{j_{r}}=-\frac{ie\hbar}{2m_{e}}&\left(\frac{\partial}{\partial
r}-\frac{\partial}{\partial
r^{\prime}}\right)\sum_{\begin{subarray}{c}m=-\infty\\\
n=0\end{subarray}}^{\infty}\theta(\mu-E_{n,m})\times\\\
&\left.\psi_{nm}\left({r,\varphi}\right)\psi_{nm}^{*}\left(r^{\prime},\varphi\right)\right|_{r^{\prime}=r}\equiv
0\end{split}$ (17)

and

$\begin{split}{j_{\varphi}}(r)=\frac{{e\hbar}}{{2m_{e}{l^{2}}}}\sum_{\begin{subarray}{c}m=-\infty\\\
n=0\end{subarray}}^{\infty}{\left({\frac{{r_{m}^{2}}}{r}{\rm{sgn}}(m)-r}\right)}\times\\\
{\left|{{\psi_{nm}}\left({r,\varphi}\right)}\right|^{2}}\theta(\mu-
E_{n,m}).\end{split}$ (18)

Full current flowing in the disk is

$I=\int_{r_{1}}^{r_{2}}j_{\varphi}(r)dr=\sum_{\begin{subarray}{c}m=-\infty\\\
n=0\end{subarray}}^{\infty}I_{nm}\theta(\mu-E_{n,m})$ (19)

with the partial component $I_{nm}$ carried by the state with definite quantum
numbers $n,m$:

$\begin{split}I_{nm}&=\frac{e\hbar}{2m_{e}}{\int\limits_{{r_{1}}}^{{r_{2}}}{\left({\frac{{2m}}{r}-\frac{r}{l^{2}}}\right)}}f_{nm}^{2}\left(r\right)dr\\\
&=\frac{e\hbar}{2m_{e}}\left(2m\int\limits_{{r_{1}}}^{{r_{2}}}\frac{{dr}}{r}f_{nm}^{2}\left(r\right)-\frac{1}{2\pi
l^{2}}\right),\end{split}$ (20)

where in the second term we used the normalization condition

$\int
d^{2}r|\psi_{nm}(r,\varphi)|^{2}=2\pi\int_{r_{1}}^{r_{2}}rdrf_{nm}^{2}(r)=1.$
(21)

One can see that Eqs. (19) and (20) are in agreement to Eq. (7) in
Halperin1982PRB . Note that the first paramagnetic part in Eqs. (18) and (20)
originates from the gradient term in Eq. (14) and is related to the spacial
inhomogeneity of the current flow in the disk. The second term of the
corresponding equations $\sim B$ is diamagnetic.

####  II.3.2 Byers-Yang formula

It is worth to mention that a more complicated problem with the annulus placed
in the constant magnetic field and threaded by the flux $\Phi$ can still be
considered basing on the solution (7) in terms of the Whittaker functions.
This occurs because adding a vector potential
${\bf{A}}_{\eta}\left({\bf{r}}\right)=\left({0,\Phi_{0}\eta/(2\pi r)}\right)$
corresponding to the magnetic field
$\mathbf{B}_{\eta}(\mathbf{r})=\nabla\times{\mathbf{A}}_{\eta}=\mathbf{e}_{z}\Phi_{0}\eta\delta^{2}(\mathbf{r})$
does not alter the structure of Eq. (2). One can easily check that the
corresponding solution for the problem that involves a superposition of the
constant field and flux can be written by mere replacement $m\to m-\eta$. The
energy spectrum of the infinite system is still given by Eq. (8) with the
shifted azimuthal quantum number.

Under certain conditions the current $I_{nm}$ carried by the state with
definite quantum numbers $n,m$ can be found using the Byers-Yang formula
Byers1961PRL ; Imry.book :

$I_{nm}=-\frac{e}{h}\frac{\partial
E_{n,m}(\eta)}{\partial\eta}=\frac{e}{h}\frac{\partial E_{n,m}(\eta)}{\partial
m}.$ (22)

The latter expresses the fact that the persistent current is the thermodynamic
quantity conjugated to the flux through the ring. Although initially the
Byers-Yang formula was introduced for a system with a hole threaded by the
flux, the second equality in Eq. (22) (see Ref. Avishai1993PA  for a detailed
discussion) allows one to use it for the description of the current in the
system with hole in a constant magnetic field with $\eta=0$. Yet, considering
the spectrum for the infinite system, Eq. (8), one can see that this formula
is ill defined for $m=0$. We note that the Byers-Yang formula follows from the
Hamiltonian given by Eq. (1) if one utilizes the Hellman-Feynman theorem. We
will use this formula below to estimate the values of $I_{nm}$.

Finally we note that in what follows we use the dimensionless units for the
brevity of notations. Yet we will restore the units to underline the physics.

##  III The cases amenable to analytical solution

###  III.1 Asymptotic analysis of the eigenvalue problem for wide annulus

The solution of transcendental equation (8) admits the rich variety of
asymptotic representations. Below we analyze the approximation of a wide
annulus $\rho_{2}\gg\rho_{1}$, and, besides the first case with $m=0$, in the
following the large angular momentum limit ($m\gg 1$) will be in the focus of
our discussion. For the sake of convenience they are graphically classified in
Figure 2.

Another important parameter is the cyclotron radius of the orbits for the most
essential electrons being at the highest Landau levels (which energies are
close to the chemical potential):
$r_{c}=mv_{F}/(eB)=l\sqrt{2\mu/(\hbar\omega_{c})}$. It determines the
effective width of the region where the edge currents flow and in the
following consideration will remain arbitrary with respect to the inner and
outer radii of the annulus.

Figure 2: Graphical representation of applicability of the analytical
expressions for energy levels in a annulus under the conditions specified in
figures (see details in the text). (a) Eq. (23) is valid in the entire disk.
(b) Eq. (26) is for the bulk region, while inner and outer orange domains
correspond Eq. (24) and Eq. (25) respectively. (c) Eq. (32) corresponds for a
narrow annulus (ring). (d) Eq. (47) is applicable for the small annulus with
the infinitesimal inner radius. The proportions are not to scale. The figures
are illustrative in nature.

####  III.1.1 The states with zero angular momentum in a disk with a small
hole

Let us start from consideration of the electron spectrum in a wide disk
($\rho_{2}\gg 1$) with the small hole ($\rho_{1}\ll 1$). In this case one can
find the explicit expression for the energy levels corresponding the states
with $m=0$. Basing on the asymptotic expressions of the Whittaker functions
with $m=0$ for small arguments one can arrive to Eq. (59) (see details of the
derivation in Appendix A in parts 2 and 3). Corresponding energy levels for
not very large quantum numbers $n+1\ll 1/\rho_{1}^{2}$ are given by:

$\varepsilon_{n,0}=n+\frac{1}{2}+\frac{1}{{\ln\left({\frac{2}{{\rho_{1}^{2}}}}\right)-\psi\left({n+1}\right)-2\gamma}},$
(23)

where $\psi(z)$ is the digamma function and $\gamma=0.577...$ is the Euler-
Mascheroni constant. One can see, that the Landau equidistant “ladder”
distorts, and this distortion increases with the growth of the level number.

####  III.1.2 The states with large angular momentum close to the edges

In an annulus with the radius of the inner hole larger than magnetic length
($\rho_{1}\gtrsim 1$) the edge states are formed in the both border areas.
Near the edges when the center of the wave function $\rho_{m}=\sqrt{2|m|}$
satisfies the condition $|\rho_{m}-\rho_{i}|<1$ ($i=1,2$) the energy levels
have the following form (see Eq. (73) in the Appendix A)

$\begin{split}\varepsilon_{n,m}&=2\left(n+\frac{3}{4}\right)-\frac{\Gamma(n+1/2)}{\pi
n!}(2n+1)\frac{\rho_{m}^{2}-\rho_{1}^{2}}{\rho_{m}}\\\
&+2G(2n+2)\left(\frac{\Gamma(n+3/2)}{\pi
n!}\right)^{2}\frac{(\rho_{m}^{2}-\rho_{1}^{2})^{2}}{\rho_{m}^{2}},\end{split}$
(24)
$\begin{split}\varepsilon_{n,m}&=2\left(n+\frac{3}{4}\right)-\frac{\Gamma(n+1/2)}{\pi
n!}(2n+1)\frac{\rho_{2}^{2}-\rho_{m}^{2}}{\rho_{m}}\\\
&+2G(2n+2)\left(\frac{\Gamma(n+3/2)}{\pi
n!}\right)^{2}\frac{(\rho_{2}^{2}-\rho_{m}^{2})^{2}}{\rho_{m}^{2}},\end{split}$
(25)

respectively, with $G(2z)=\psi(z+1/2)-\psi(z)$. They correspond to the
energies of the skipping electrons subjected to the parabolic potential with
the minimum shifted with respect to the inner/outer edge. In the first term of
these expressions one can recognize the spectrum for the harmonic oscillator
with reflecting wall at the minimum of potential (see the Problem 2.12 in Ref.
Galitsky_problems ). These peculiarities of the electron energy levels near
the edges of the ribbon placed in magnetic field were anticipated in
Halperin1982PRB  (also cf. Eqs. (7) and (9) in Refs. Heuser ; Varlamov ,
respectively).

####  III.1.3 The states with large angular momentum far from the edges

In the bulk of the disk not too close to the edges, when
$\rho_{1}<\rho_{m}<\rho_{2}$ and $|\rho_{m}-\rho_{i}|\gg 1$, the electron
states are localized. The corresponding spectrum tends to the Landau one (see
details of the derivation in Appendix A in part 5):

$\begin{split}&\varepsilon^{(i)}_{n,m}=n+\frac{1}{2}+\frac{1}{\sqrt{2\pi}n!}x_{i}^{2n+1}e^{-x_{i}^{2}/2},\\\
&x_{i}^{2}=\rho_{m}^{2}(\rho_{i}^{2}-1-\ln\frac{\rho_{i}^{2}}{\rho_{m}^{2}}),\quad\rho_{m}=\frac{r_{m}}{l},\end{split}$
(26)

and corresponds to the energy spectrum of the ribbon in magnetic field Heuser
.

####  III.1.4 Currents carried by the states with large angular momentum

To find the current $I_{nm}$ carried by the state with definite quantum
numbers $n,m$ in the different regions of the annulus we apply Byers-Yang
formula (22).

First, it is easy to see from Eq. (26) that inside the disk,
$\rho_{1}<\rho_{m}<\rho_{2}$ and $|\rho_{m}-\rho_{i}|\gg 1$ the current
$I_{nm}=0$, because the Landau levels are flat up to the exponentially small
correction in the momentum $m$.

Near the edges of the disk, $|\rho_{m}-\rho_{i}|<1$, from Eqs. (24) and (25)
one obtains the currents

$I_{n}(\rho_{i})\equiv
I_{nm}(\rho_{i})=(-1)^{i}\frac{\Gamma(n+1/2)(2n+1)}{\pi^{2}n!}\frac{e\omega_{c}}{\rho_{i}},$
(27)

where we set $\rho_{m}=\rho_{i}$, so that the partial contributions
$I_{nm}(\rho_{i})$ are independent on $m$. Let us note that the currents at
the inner and outer edges of the disk do not coincide due to the difference in
curvatures.

The partial contribution to the full persistent current is

$I_{n}^{\mathrm{tot}}=-{\mathcal{J}}_{0}\frac{2\Gamma(n+1/2)(2n+1)}{\pi
n!}\frac{a_{0}^{2}}{l}\left(\frac{1}{r_{1}}-\frac{1}{r_{2}}\right),$ (28)

where in order to explicitly highlight the current dimensionality we used the
value of the current carried by an electron in hydrogen atom

${\mathcal{J}}_{0}=\frac{m_{e}e^{5}}{2\pi\hbar^{3}}\approx$1.05\text{\times}{10}^{-3}\text{\,}\mathrm{A}$$
(29)

with and
$a_{0}=\hbar^{2}/(m_{e}e^{2})\approx$0.053\text{\,}\mathrm{n}\mathrm{m}$$ as
the Bohr radius. For future comparison with experiment it worth to note that
the magnetic length $l=$26\text{\,}\mathrm{n}\mathrm{m}$/\sqrt{B[\mbox{T}]}$
and the field is measured in Tesla. One can see that in the simple rectangular
geometry when $r_{1},r_{2}\rightarrow\infty$ the current (28) turns zero
Heuser .

Let us consider the effect of discussed persistent current on the quantum Hall
effect measurements in the macroscopic
($2r_{1}=$0.9\text{\,}\mathrm{m}\mathrm{m}$$ and
$2r_{2}=$3.9\text{\,}\mathrm{m}\mathrm{m}$$) annulus Dolgopolov1992 . Assuming
that the potential difference between the disk edges is kept to be zero, we
estimate the magnitude of the current $I_{n}^{\mathrm{tot}}$ induced by the
disk curvature. Since the filling factor in Ref. Dolgopolov1992  was below 3,
we restrict our evaluation only by one channel, what gives
$|I_{n}^{\mathrm{tot}}|=$6.92\text{\times}{10}^{-13}\text{\,}\mathrm{A}$$ for
$B=$10\text{\,}\mathrm{T}$$ and $n=0$. Such small additional persistent
current cannot affect the measurements of the quantum Hall effect (for
example, in Ref. Dolgopolov1992  corresponding currents were five orders more:
$\sim${10}^{-8}\text{\,}\mathrm{A}$$).

In order to estimate the magnitude of the total persistent current in such a
ring basing on Eq. (28) it is necessary to perform in it the summation over
the quantum numbers $n,m$. The principal quantum number $n$ changes over all
filled states below the chemical potential up to its maximal value
$\mu/{(\hbar\omega_{c})}\gg 1$, while the azimuthal quantum number $m$ can be
fixed by the value of radius $r_{2}=\sqrt{2|m|}l$. Summation in Eq. (28) is
easily performed using the Stirling’s formula what results in:

$\begin{split}I^{\mathrm{tot}}&=-\frac{8{\mathcal{J}}_{0}}{3\pi}\left(\frac{\mu}{\hbar\omega_{c}}\right)^{3/2}\frac{a_{0}^{2}}{l}\left(\frac{1}{r_{1}}-\frac{1}{r_{2}}\right)\\\
&=-\frac{2\sqrt{2}{\mathcal{J}}_{0}}{3\pi}\frac{a_{0}^{2}r_{c}^{3}}{l^{4}}\left(\frac{1}{r_{1}}-\frac{1}{r_{2}}\right).\end{split}$
(30)

Bearing in mind the above use of the Byers-Yang formula one has to remember
that Eq. (30) is valid only when the cyclotron radius $r_{c}$ is much smaller
than the width of the disk, $r_{c}\ll r_{2}-r_{1}$, when the edge currents do
not overlap. One can see that the persistent current Eq. (30) is inversely
proportional to magnetic field and is determined by the difference of the edge
curvatures.

###  III.2 The spectral problem and persistent currents in a narrow annulus

####  III.2.1 Energy spectrum

For an annulus with the large inner radius as it becomes more and more narrow
($\rho_{2}\rightarrow\rho_{1}$) the energy levels tend to grow up (see Fig. 9
(right panel) in Appendix A). This fact is easy to understand, bearing in mind
that in the problem under consideration we have the interplay between the
Landau and size quantization of the energy levels. When the disk becomes more
and more narrow the first level of size quantization rises up like in the
narrow quantum well with infinitely high walls. As shown in Appendix B, Eq.
(8) for eigenenergies acquires a much simpler form that involves a combination
of the Bessel functions of the first and second kind:

$\begin{split}{J_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}{\rho_{1}}}\right){Y_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}{\rho_{2}}}\right)\\\
-{J_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}{\rho_{2}}}\right){Y_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}{\rho_{1}}}\right)=0.\end{split}$
(31)

When the magnetic field disappears, $\hbar\omega_{c}\to 0$, then
$\sqrt{2\tilde{\varepsilon}}\rho_{1,2}\to r_{1,2}\sqrt{2m_{e}E}/\hbar$, so
that Eq. (31) reduces to the well-known equation that describes, for example,
motion of a free particle on the annulus Suzuki1996NC .

The equation (31) can be solved in the high energy approximation (see Eq. (85)
in Appendix B) what results in the following spectrum

$E_{n,m}=\frac{\hbar^{2}}{2m_{e}}\left(\frac{\pi^{2}n^{2}}{d^{2}}+\frac{m^{2}-1/4}{r^{2}}\right)-\frac{\hbar\omega_{c}}{2}m$
(32)

with $n=1,2,\ldots,$ and $m=-\infty,\ldots,0,\ldots,\infty$ and
$d=r_{2}-r_{1}$ being the width of the annulus. We stress that both Eq. (31)
and Eq. (32) are valid for $d\ll\sqrt{2\pi}l\ll r$ (see Appendix B). The first
term of Eq. (32) is nothing else that the energy spectrum of the free electron
gas confined in quantum well of the width $d$. Writing Eq. (32) we also
neglected the term $\sim d^{2}$ which is present in Eq. (85) in Appendix B and
replaced $r_{2}$ by $r=r_{1}\approx r_{2}$.

We note that the spectrum (32) can be rewritten in terms of the flux $\Phi=\pi
r^{2}B$ that goes through the ring

$E_{n,m}\\!=\\!\frac{\hbar^{2}}{2m_{e}}\left\\{\frac{\pi^{2}n^{2}}{d^{2}}\\!+\\!\frac{1}{r^{2}}\left[\left(m\\!-\\!\frac{\Phi}{\Phi_{0}}\right)^{2}\\!-\\!\frac{\Phi^{2}}{\Phi_{0}^{2}}\\!-\\!\frac{1}{4}\right]\right\\}.$
(33)

When the disk becomes very narrow, $d\to 0$, it is sufficient to restrict
ourselves by considering in Eqs. (32) and (33) only the lowest, $n=1$ level.
Then one can see that the term $(m-\Phi/\Phi_{0})^{2}$ coincides with the
spectrum of the one-dimensional ring Gefen1 . The extra terms
$\Phi^{2}/\Phi_{0}^{2}+1/4$ appear because of the homogeneity of magnetic
field in our model, while the spectra in Refs. Kulik1 ; Gefen1  were obtained
for the ring threaded by the flux.

####  III.2.2 Wave function

The radial wave function corresponding to the eigenenergies given by Eq. (31)
reads

$\begin{split}f_{nm}(\rho)=C\left[{{Y_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}{\rho_{1}}}\right)J_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}{\rho}}\right)\right.\\\
\left.-{J_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}{\rho_{1}}}\right){Y_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}{\rho}}\right)\right].\end{split}$
(34)

The last expression can be further simplified in the large energy limit
supposing the validity of the conditions $r=r_{1}\approx r_{2}\gg l$. The
latter inequality imposes the restriction on a magnetic field, which cannot be
too weak: $\omega_{c}\gg\hbar/(m_{e}r^{2})$. Physically this means that the
quantization related to the magnetic field must dominate on the size
quantization of the tangential electron motion along the ring. As result, the
wave function acquires the form (86) which turns out to be more convenient for
further calculation of the partial current $I_{nm}$.

####  III.2.3 Persistent currents in a large and narrow annulus

Let us now consider the current in a large and narrow annulus under the
conditions $r=r_{1}\approx r_{2}\gg l$. The partial current $I_{nm}$ carried
by the state with definite quantum numbers $n,m$ can be written down by
substituting the integral (89) to the second line of Eq. (20)

$\begin{split}{I_{m}}\equiv{I_{nm}}&=\frac{e\hbar}{2\pi
m_{e}}\left(\frac{m}{r^{2}}-\frac{1}{2l^{2}}\right)\\\ &=\frac{e\hbar}{2\pi
m_{e}r^{2}}\left(m-\frac{\Phi}{\Phi_{0}}\right),\end{split}$ (35)

where within our approximation for the wave function Eq. (34) $I_{nm}$ turn
out to be independent on the principal quantum number $n$, thereby
${I_{nm}}\equiv{I_{m}}$ (see details in Appendix B). The second line of Eq.
(35) is identical to the corresponding expression for the partial current in
Gefen1 , in spite of the fact that the given above spectrum (33) contains the
additional terms.

One can easily check that exactly the same expression for $I_{nm}$ follows
directly from the Byers-Yang formula (22) with the derived above spectrum
(32).

The found spectrum (32) and wave function (34) in the above approximations
will allow us to obtain the value of full current flowing in the narrow
annulus vs its radius and width.

Substitution of Eq. (34) into Eq. (18) for the tangential component of the
current density and subsequent straightforward integration leads to the
cumbersome formula that can be expressed in terms of sine and cosine integral
functions. However, within our assumption about the large scale annulus
($r_{1}\approx r_{2}=\rho l$, $\rho\gg 1$) with the small width $d\lesssim
l\sqrt{2\pi}$ ($\delta=d/l\lesssim\sqrt{2\pi}$) we can simplify the expression
for the current to the form [see Eqs. (87) - (89) in Appendix B]

$\begin{split}I=\sum_{\begin{subarray}{c}n=1\end{subarray}}^{\infty}\sum_{\begin{subarray}{c}m=-\infty\end{subarray}}^{\infty}I_{m}\theta(\mu-
E_{n,m})\end{split}$ (36)

The case of very narrow ring, when only one level of dimensional quantization
occurs below the chemical potential, was studied by the authors of Gefen1 ;
Kulik1 . We can reproduce their result accounting in Eq. (36) for only the
lowest principal quantum number ($n=1$) and performing the remaining summation
over azimuthal number by means of Poisson formula:

$I=\frac{{e\hbar
k_{F}}}{{2\pi^{2}{m_{e}}{r}}}\sum\limits_{k=1}^{\infty}{\sin}\left({\frac{{2\pi
k\Phi}}{{{\Phi_{0}}}}}\right)\frac{{\cos(2\pi k{k_{F}}r)}}{{k}},$ (37)

where $k_{F}$ is the Fermi wave-vector. Here is assumed that
$\mu\gg\hbar\omega_{c}$ and $k_{F}r\gg 1$. The obtained magnitude of current
oscillations can be compared to that one revealed in experiment Moler . Taking
$v_{F}=$1.2\text{\times}{10}^{8}\text{\,}\mathrm{c}\mathrm{m}\mathrm{/}\mathrm{s}$$
for gold, $r=$670\text{\,}\mathrm{n}\mathrm{m}$$, one finds
$I_{0}\approx$10\text{\,}\mathrm{n}\mathrm{A}$$ that is one order larger the
observed values. Such overestimate is related to the fact that we do not
consider accompanied effects such as temperature, disorder and the electron-
electron interaction.

In Ref. Richter,  the problem of calculation of the persistent current in the
large thin ring was considered in the semiclassical approximation basing on
the Bohr-Sommerfeld quantization rule. This allowed the authors to avoid the
sophisticated study of the Bessel function cross-product zeros (see Eq. (83))
and get the explicit expression for the density of levels distribution in the
simple form. With its use they obtained the complete analytical expression for
the persistent current in the large ring accounting for the contributions of
all semiclassical electron trajectories and, additionally, for the finite
temperatures.

In case of the annulus of the finite width the averaging of Eq. (37) over the
varying radius $r$ results in strong cancellation of oscillations, the total
current strongly decreases. In order to get its value we can perform both
summations in Eq. (36) exactly. The presence of theta function in it implies
the finite limits in both summations:

$\begin{split}I=\frac{e\hbar}{4\pi
m_{e}l^{2}}\sum_{n=1}^{N_{max}}\sum_{m=M_{min}}^{M_{max}}{\left({\frac{2m}{\rho^{2}}-1}\right)}.\end{split}$
(38)

which are determined by the explicit expression for spectrum (32). The
condition that the argument of theta-function in Eq. (38) remains positive for
fixed value of the chemical potential yields the constraints for the azimuthal
quantum number $m$:

$\begin{gathered}{M_{\min}}=\left[{-\rho\sqrt{\frac{{2\mu}}{{\hbar{\omega_{c}}}}-\frac{{{\pi^{2}}{n^{2}}}}{{{\delta^{2}}}}+\frac{{{\rho^{2}}}}{4}}+{\rho^{2}}}\right],\hfill\\\
{M_{\max}}=\left[{\rho\sqrt{\frac{{2\mu}}{{\hbar{\omega_{c}}}}-\frac{{{\pi^{2}}{n^{2}}}}{{{\delta^{2}}}}+\frac{{{\rho^{2}}}}{4}}+{\rho^{2}}}\right].\hfill\\\
\end{gathered}$ (39)

with $[\ldots]$ denoting the integer part.

What concerns the summation over the principal quantum number its upper limit
$N_{max}$ can be determined from the condition of the positiveness of the
square root in Eq. (39):

${N_{\max}}=\left[{\frac{\delta}{\pi}\sqrt{\frac{{2\mu}}{{\hbar{\omega_{c}}}}+\frac{{{\rho^{2}}}}{4}}}\right].$
(40)

In the following we assume that $N_{max}\gg 1$, i.e. the width of the disk is
not too small and is limited by the conditions
${\pi{{\left({2\mu/(\hbar{\omega_{c}})+{\rho^{2}}/4}\right)}^{-1/2}}}\ll\delta\lesssim\sqrt{2\pi}$.
If this were not so, $N_{max}$ would become less than $1$ and because of such
a strong size quantization there would be no level left under the chemical
potential.

The summation over $m$ in Eq. (38) is trivial and results in

$I=\frac{{e\hbar}}{{4\pi{m_{e}}{l^{2}}}}\sum\limits_{n=1}^{{N_{max}}}{\left[{\frac{{2\pi\rho}}{\delta}\sqrt{N_{\max}^{2}-{n^{2}}}+1}\right]}.$
(41)

The characteristic dependence of the total current flowing in the narrow ring
versus the strength of magnetic field is shown in Fig. 3. One can see, that
due to the presence of the integer part in Eq. (40), this dependence keeps
saw-like character imposed on the general growth.

Figure 3: Total current flowing in the narrow annulus ($d=50\,nm$) versus the
strength of magnetic field. The radius $r=600\,nm$, concentration of the 2DEG
is $10^{12}cm^{-2}$. The red line illustrates a linear dependence on the
strength of a weak magnetic field in accordance with Eq. (44). The magnetic
length remains larger than the annulus width (see inset).

To analyse the various regimes and in view of $N_{max}\gg 1$ the summation
over $n$ in Eq. (41) can be performed applying the Euler–Maclaurin formula,
what leads to:

$\begin{split}I=\frac{{e\hbar\rho}}{{4{m_{e}}{\delta
l^{2}}}}&\left[{\frac{{\pi
N_{\max}^{2}}}{2}-N_{\max}^{2}\arcsin\left({\frac{1}{{{N_{\max}}}}}\right)}\right.\\\
&\left.-\sqrt{N_{\max}^{2}-1}+\frac{\delta}{\pi\rho}N_{max}\right].\end{split}$
(42)

Keeping the leading term of Eq. (42) and substituting in it Eq. (40) we obtain
the explicit expression for the current enveloping curve:

$I=\frac{{e\hbar
rd}}{{8\pi{m_{e}}{l^{4}}}}\left({\frac{{2\mu}}{{\hbar{\omega_{c}}}}+\frac{{{r^{2}}}}{{4{l^{2}}}}}\right).$
(43)

We can analyze two different regimes of Eq. (43).

  * •

$\hbar/(m_{e}r^{2})\ll\omega_{c}\ll\sqrt{\mu/(m_{e}r^{2})}$. In this limit of
weak enough fields, the magnitude of the current increases linearly with
growth of magnetic field:

$I=\frac{e^{2}rd\mu}{4\pi c\hbar^{2}}B=\frac{m_{e}e^{2}rd}{4c}n_{2D}B.$ (44)

  * •

$\sqrt{\mu/(m_{e}r^{2})}\ll\omega_{c}\ll\mu/\hbar$. In this interval the
magnetic field becomes strong enough, though still remaining below the ultra-
quantum limit (when the second term under the square root in Eq. (40) starts
to dominate on the first one); Eq. (43) reduces to

$I=\frac{{e^{4}{r^{3}}}d}{{32\pi c^{3}\hbar^{2}{m_{e}}}}B^{3},$ (45)

and the magnitude of the persistent current increases more rapidly ($I\sim
B^{3}$ instead of $I\sim B$).

Figure 4: Current density as a function of a dimensionless radial coordinate
for a narrow annulus with a fixed width $d=0.5$ and different inner and outer
radii $\rho_{1}=11$, $\rho_{2}=11.5$ (black) $\rho_{1}=11.25$,
$\rho_{2}=11.75$ (blue), $\rho_{1}=11.5$, $\rho_{2}=12$ (green) and
$\rho_{1}=11.75$, $\rho_{2}=12.25$ (red). The chemical potential is chosen to
be $\mu=20\hbar\omega_{c}$ and $T=0$. Inset shows the values of the current in
units of $\frac{{2{m_{e}}{l^{2}}}}{{\pi e\hbar}}$ in an annulus, where the
color lines correspond to an appropriate current density profile of the
system.

Now we proceed to the discussion of the current radial distribution over the
narrow annnulus under the same conditions $d\lesssim\sqrt{2\pi}l\ll r$ and
$\hbar\omega_{c}\ll\mu$ ($N_{max}\gg 1$). It can be found by means of the
numerical analysis of Eq. (18) with the wave functions and spectrum determined
by Eqs. (7) and (8). As one can see from Fig. 4 the current density changes
its sign as one moves from the inner to the outer edge. Moreover, one can
notice that the total current $I$ (the integral of the current density) also
changes its value and direction with the growth of the annulus radii (see the
values in the upper right corner of Fig. 4). The latter fact is in accordance
with the observed alterations of the current values in Fig. 3 as a function of
the magnetic field.

###  III.3 The spectral problem and persistent current in a small annulus with
the infinitesimal inner radius

####  III.3.1 Energy spectrum and wave function

In the case when the inner radius of an annulus is smaller than the magnetic
length one can forget about it and approximate the disk by the solid one
without hole in the centre at all (${\rho_{1}}=0$). The energy spectrum in
such a case is determined by the zeroes of the confluent hypergeometric
function (of the first kind or Kummer’s confluent hypergeometric function)
(see e.g. Ref. Rensink ):

$\Phi\left({\frac{1}{2}-\varepsilon_{nm}+\frac{{\left|m\right|-m}}{2},\left|m\right|+1,\frac{\rho_{2}^{2}}{2}}\right)=0.$
(46)

Being interested in the properties of a “small” disk we assume that its the
outer radius is smaller than the magnetic length: $r_{2}\lesssim l$ (weak
magnetic field approximation). Corresponding spectrum acquires the form (cf.
Eq. (32))

${E_{n,m}}=\frac{\hbar^{2}}{2m_{e}}\frac{{j_{nm}^{2}}}{{r_{2}^{2}}}-\frac{\hbar\omega_{c}}{2}m,$
(47)

where ${j_{nm}}$ is the n-th zero of ${J_{\left|m\right|}}\left(z\right)$.

The radial component of the wave function can be also evaluated (see Eqs.
(90)-(94) in Appendix C)

$f_{nm}\left(r\right)=\frac{1}{\sqrt{\pi}{r_{2}{J_{\left|m\right|+1}}\left({{j_{nm}}}\right)}}{J_{\left|m\right|}}\left({{j_{nm}}\frac{r}{{{r_{2}}}}}\right).$
(48)

As in the case of the previous subsection these analytical findings will allow
us to study the nontrivial full current genesis vs the size of such a small
annulus.

####  III.3.2 Persistent currents

Figure 5: (a) The full current as a function of a dimensionless radius
$\rho_{2}$ for a small annulus with a infinitesimal $\rho_{1}$. The chemical
potential is equal to $\mu=20\hbar\omega_{c}$ and $T=0$. Inset in (a) shows
the zoom of the plot, where the current has negative values. (b) The negative
contribution $I_{1}$ to the full current, while (c) is the positive
contribution $I_{2}$.

In the case of a disk of the outer radius $r_{2}\lesssim l$ with the hole much
smaller magnetic length ($r_{1}\to 0$) the total current can be determined by
Eq. (18) for the tangential component of the current density and corresponding
expression for radial part of the wave-function (48). Integration can be
explicitly performed in terms of the Bessel functions (see Appendix C):

$I=I_{1}+I_{2}.$ (49)

Here

${I_{1}}=-\frac{{e\hbar}}{{4\pi{m_{e}}{l^{2}}}}\sum_{\begin{subarray}{c}m=-\infty\\\
n=1\end{subarray}}^{\infty}\theta(\mu-E_{n,m})$ (50)

is the diamagnetic current and

$I_{2}=\frac{e\hbar}{2\pi m_{e}r_{2}^{2}}\sum_{\begin{subarray}{c}m=-\infty\\\
n=1\end{subarray}}^{\infty}\frac{A_{nm}\mbox{sgn}(m)}{J^{2}_{|m|+1}(j_{nm})}\theta(\mu-
E_{n,m})$ (51)

is the paramagetic current with $A_{nm}$ given by Eq. (96) and
$\mbox{sgn}(0)=0$.

####  III.3.3 Derivation using Byers-Yang formula

One can also verify that Eqs. (49) - (51) follow directly from the Byers-Yang
formula (22). Differentiating the spectrum (47) one obtains

$I_{nm}=\frac{e\hbar}{2m_{e}}\left(\frac{{\rm sign}(m)}{\pi
r_{2}^{2}}j_{nm}\frac{\partial
j_{n\nu}}{\partial\nu}\Big{|}_{\nu=|m|}-\frac{1}{2\pi l^{2}}\right),$ (52)

where it was taken into account that roots $j_{nm}$ of the equation
$J_{|m|}(z)=0$ depend on $|m|$. Substituting the derivative of these roots
$\partial j_{nm}/\partial|m|$ [see Eq. (98) in Appendix C] in Eq. (52) we
arrive at the final result

$I_{nm}=\frac{e\hbar}{2m_{e}}\left(\frac{A_{nm}{\rm sign}(m)}{\pi
r_{2}^{2}J^{2}_{|m|+1}(j_{nm})}-\frac{1}{2\pi l^{2}}\right).$ (53)

As one can easily see, the second term of Eq. (53) corresponds to $I_{1}$ and
the first term to $I_{2}$, respectively.

####  III.3.4 Emergence of the current states in the small annulus

The numerical simulation of Eq. (49) is represented in Figure 5. With the
chemical potential equal to $20\hbar\omega_{c}$ and in the vicinity of the
zero temperature two leaps of the current are clearly observed. The behaviour
of these leaps can be easily understood if we consider separately two
contributions of the currents $I_{1}$ and $I_{2}$, where $I_{1}$ given by Eq.
(50) is the negative and independent of the radius $\rho_{2}$ and $I_{2}$
represented by Eq. (51) is inversely proportional to the value of outer
radius. As long as all energy levels Eq. (47) of the small annulus are located
above the given value of the chemical potential there is no current in an
annulus (see inset in Fig. 5a). The first leap is connected with the emergence
of the quantum state with $m=0$ below the chemical potential. Due to this the
full current dependence has only one contribution from $I_{1}$ given by Eq.
(50) with zero part from $I_{2}$ and, therefore, is the constant until the
certain value of $\rho_{2}$ (Fig. 5 b), where another energy levels with the
nonzero azimuthal number $m$ give rise a new leap and a new constant.

Further increase of the radius allows to involve other energy levels with
$m\neq 0$. This leads to the activation of the second contribution $I_{2}$
given by Eq. (51). As a result together with the second leap in $I_{1}$ (Fig.
5b) similar effect occurs for $I_{2}$ (Fig. 5c).

##  IV Annulus of the arbitrary sizes: numerical analysis

In principle, the expression for the current density (18) allows to
investigate its profile for the disk of arbitrary radii (in practice the
computation for a wide enough disk is very time-consuming).

Figure 6: Current density as a function of a dimensionless radial coordinate
for an annulus with a fixed inner radius $\rho_{1}=3$ and different outer
radii $\rho_{2}=5.25$ (black) $\rho_{2}=5.5$ (brown), $\rho_{2}=6$ (blue),
$\rho_{2}=6.5$ (green) and $\rho_{2}=7$ (red) in (a) and $\rho_{2}=9$ (black),
$\rho_{2}=10$ (blue), $\rho_{2}=11$ (green), $\rho_{2}=12$ (yellow) and
$\rho_{2}=13$ (red) in (b). The chemical potential is chosen to be
$\mu=1.1\hbar\omega_{c}$ and $T=0$.

We studied the current density as a function of a radial coordinate for an
annulus with a fixed inner radius $\rho_{1}=3$ and the set of outer radii from
$\rho_{2}=5.25$ to $\rho_{2}=13.0$. The corresponding results are presented in
Figures 6 and 7.

Figure 6 shows the the current density as the function of the dimensionless
radial coordinate of an annulus $\rho$ for the fixed inner radius
$\rho_{1}=3$. Relatively narrow annulus shows almost a sinusoidal type of the
behaviour with the maximal positive value of the current density near the
inner radius and the minimal negative one near the outer edge of a system
(Fig. 6a, black line).

With the increasing of the width disk (the same with the increasing of the
outer radius) the deformation of the current density profile is begun. One can
see in Figure 6 a (red line) that in the middle part of the annulus the
current density profile starts to be flattened with the zero value. This
suppression is clearly seen in Fig. 6, where the evolution of the current
density profile for the wide disk is shown. Such a behaviour can be easily
understood from the energy spectrum for a wide disk when electrons strive to
approach Landau levels as it can be seen in Fig. 1 b and as a result make
infinitesimal contributions to the current density distribution inside the
disk. Moreover, together with the flattened zero part of $\j_{\varphi}$ the
double changing of the current density sign is observed near in the vicinity
of the inner and outer edge of the annulus.

The occurrence of such complicated current density profiles should be taken
into account in experiments with persistent currents in a ring, where as we
have shown already in Fig. 6 the local magnetic response of a system can be
changed significantly even for a ring with the relatively small width.

The numerical calculations of the full current as the function of the inverse
$\rho_{2}$ are presented in Fig. 7. They were obtained by numerical
integration of the previously obtained current densities for the fixed radius
$\rho_{1}$. The dependence of total current on $\rho_{2}^{-1}$ exhibits
unambiguously the decay of the persistent current with the increase of the
outer radius of the disk. This result is not surprising and is in agreement
with Eq. (30), obtained from Byers-Yang formula.

Figure 7: Total current as a function of a dimensionless inverse radial
coordinate $1/\rho_{2}$ for an annulus with a fixed inner radius $\rho_{1}=3$.
The chemical potential is equal to $\mu=1.1\hbar\omega_{c}$ and $T=0$. Insets
show current density profiles for $\rho_{2}=7$ and $\rho_{2}=13$.

##  V Conclusions

We have presented a comprehensive analytic and numeric study of the electron
spectra and occurrence of persistent current in the 2DEG filling the annulus
of arbitrary dimensions subjected to constant magnetic field. The results
obtained in this article can be summarized as follows.

i) The case of a nanodot. When the outer radius of the annulus is small with
respect to the magnetic length, $r_{2}\lesssim l$, while that one of the inner
hole is infinitesimaly small (see Fig. 2d) no current flows in the system. The
only available states here correspond to zero azimuthal number which do not
carry paramagnetic current (see Eq. (51) and take into account that
$A_{n0}=0$). The diamagnetic contribution is also absent, because
$E_{n,0}>\mu$. As the outer radius increases, we observe how the first current
state appears simultaneously with the emergence of the first $m\neq 0$ state
(see Fig. 5).

ii) The case of a narrow annulus. This is another case amendable for the
analytic solution: i.e. when the annulus radius $r$ is much larger than the
magnetic length, while its width is less or comparable with it
($d\lesssim\sqrt{2\pi}l\ll r$, see Fig. 2c) and the magnetic field is not
ultraquantum ($\hbar\omega_{c}\ll\mu$).

When the ring is very narrow and only one level of radial dimensional
quantization occurs below the chemical potential (i.e. $d\lesssim
k_{F}^{-1}=\hbar/\sqrt{2m_{e}\mu}$), we reproduce the persistent current
oscillations occurring in a nanoring Gefen1 ; Kulik1 . Yet, our general
expression for current Eq. (36) together with the spectrum (33) allows also to
analyze the multi-channel case of a relatively wide annulus. The obtained Eq.
(41) reproduces the vanishing saw-like oscillations at the background of the
persistent current growing with the increase of magnetic field (see Fig. 3 and
Eqs. (44) and (45)).

The analysis of Eq. (18) allows us to study the radial distribution of the
current density over the narrow annnulus. As one can see from Fig. 4 the
current density changes its sign as one moves from the inner to the outer
edge. Moreover, the total current $I$ also changes its value and direction
with the growth of the annulus radii. The numerical analysis presented in Fig.
4 allows to see the alteration of the current direction frequently observed in
experiment with nanorings (see Ref. Moler ).

iii) The case of a wide disc. We have succeeded to make a considerable
progress in study of the general case of the annulus of arbitrary dimensions
$r_{2}>r_{1}\gtrsim l$ (wide disc with a hole in the center). Detailed
analysis of the spectral problem for the edge states resulted in Eqs. (24),
(25) (Fig. 2b). Further application of the Byers-Yang formula allowed us to
reveal that the total current in a wide disk is inversely proportional to the
strength of magnetic field and is determined by the difference in curvatures
($r_{1}^{-1}-r_{2}^{-1}$) of the inner and outer edges [see Eq. (30)]. This
clearly demonstrates its fading away in the case of the standard rectangular
geometry (see Refs. Heuser ; MacDonald1984PRB ) and the annular geometry but
neglecting the curvature effects Halperin1982PRB ; Varlamov . Moreover, we
have revealed the structure of current density profiles in such a geometry.
This information can be important for the understanding of future non-invasive
experiments (by means of scanning SQUID microscope) studied the magnetic
response in a wide disc.

The obtained results allowed us to apply them for the analysis of some
experimental findings (see Ref. Moler ) and to establish very reasonable
coincidence [see the estimates after Eq. (30)]. Yet, their validity can be
restricted both by disorder and by electron-electron interaction (see e.g.
Glazman1992 .) The latter becomes noticeable for the electronic states at the
almost empty quantized levels rounding for example the abrupt teeth in Fig. 3.

Recently a large progress was achieved in the high-resolution (on the
submicrometer scale) imaging of the magnetic field and reconstructing current
density distribution in graphene ribbons Tetienne2017ScAdv . Furthermore there
is a hope that the same approach can be applied to thin film systems. We
stress that the considered homogeneous magnetic field geometry corresponds to
the real experimental conditions better than the Aharonov-Bohm flux
penetrating the ring.

Our study points out clear evidence of the geometry significance for more
precise and accurate interpretations of experiments with persistent current
density distribution in mesoscopic rings and similar systems.

###### Acknowledgements.

V.P.G. and S.G.Sh. acknowledge a support by the National Research Foundation
of Ukraine grant (2020.02/0051) ”Topological phases of matter and excitations
in Dirac materials, Josephson junctions and magnets”. Y.Y. acknowledges
support by the CarESS project. A.A.V. is grateful to Yu. Galperin, A. Kavokin,
and V.B. Shikin for valuable discussions. S.G.Sh. thanks V. Kagalovsky for
useful discussion. The authors are grateful to C. Petrillo for critical
reading of the manuscript and valuable comments.

##  Appendix A Derivation of asymptotic expressions for the eigenvalue problem

###  A.1 Alternative form of the equation for eigenenergies of the Corbino
disk

Using the relation (in notations of Bateman1 ) between Whittaker functions and
confluent hypergeometric functions of the first $\Phi(a,b,z)$ and the second
kind $\Psi(a,b,z)$, respectively, Eq. (8) can be written the following form

$\displaystyle\Psi\left(\frac{1}{2}-\epsilon+\frac{|m|-m}{2},|k|+1;\frac{\rho_{1}^{2}}{2}\right)\Phi\left(\frac{1}{2}-\epsilon+\frac{|m|-m}{2},|m|+1;\frac{\rho_{2}^{2}}{2}\right)$
$\displaystyle-\Phi\left(\frac{1}{2}-\epsilon+\frac{|m|-m}{2},|m|+1;\frac{\rho_{1}^{2}}{2}\right)\Psi\left(\frac{1}{2}-\epsilon+\frac{|m|-m}{2},|m|+1;\frac{\rho_{2}^{2}}{2}\right)=0.$
(54)

This form turns out to be useful for finding different analytical asymptotic
solutions in some limits.

###  A.2 Eigenvalue equation for the disk with large outer radius

We start investigation of the energy levels distribution from the case of the
disk with large outer radius $\rho_{2}\gg 1$ and arbitrary inner one
$\rho_{1}<\rho_{2}$. The general Eq. (8) in this case can be simplified using
the asymptotic expressions for Whittaker functions for large arguments
Abramowitz ,

${M_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho^{2}}{2}}\right)\approx{e^{\rho^{2}/4}}{\left({\frac{\rho^{2}}{2}}\right)^{-\tilde{\varepsilon}}}\left[\frac{{\Gamma\left({1+\left|m\right|}\right)}}{{\Gamma\left({\frac{1}{2}+\frac{1}{2}\left|m\right|-\tilde{\varepsilon}}\right)}}+O\left(\frac{1}{\rho^{2}}\right)\right],$
(55)

and

${W_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho^{2}}{2}}\right)\approx{e^{-\rho^{2}/4}}{\left({\frac{\rho^{2}}{2}}\right)^{\tilde{\varepsilon}}}\left[1+O\left(\frac{1}{\rho^{2}}\right)\right].$
(56)

In result one arrive to the following equation for the energy levels

${W_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{\rho_{1}^{2}}{2}}\right)=0.$
(57)

In other words, the energy dispersion relation for a Corbino disk with the
very large external radius and fixed internal radius is determined by zeros of
the Whittaker function. It is worth to mention that the roots of Eq. (57) can
be approximated by those ones of Bessel or Airy functions (see, e.g. Gabutti
). The numerical solutions of Eq. (57) are presented in Fig. 8.

Figure 8: (Left) The energy levels $\epsilon_{n,m}$ of the very large disk
(see Eq. (57)) for the inner radius $\rho_{1}=0.25$ (black lines) and
$\rho_{1}=5$ (red lines) as a function of continuous variable $m$. (Right)
Three-dimensional representation of energy levels as a function of the quantum
number $m$ considered as a continuous variable and the inner radius
$\rho_{1}$.

###  A.3 Solutions with zero angular momentum

####  A.3.1 Numerical analysis for a hole of an arbitrary size

Now we analyze the energy levels distribution basing on Eq. (8) in the
particular case of the quantum number $m=0$ and consider their dependence on
the external radius $\rho_{2}$ with fixed internal radius $\rho_{1}$. Figure 9
illustrates their characteristic deviations from the standard Landau spectrum.
One can see that for $\rho_{1}=0.25$ (left panel) the energy levels tend to
constant values that differ from the half integer Landau spectrum even for
large $\rho_{2}$. This obviously is the consequence of the inner hole
presence. It will be shown below that the dispersion relation returns to the
Landau spectrum with small corrections when $\rho_{1}\to 0$, i.e. the central
hole disappears.

For sufficiently large $\rho_{1}>1$ the absence of the low energy levels (see
the right panel of Fig. 9) reflects the fact that they sharply rise when $m=0$
(see Fig. 1d). In other words, the flattening of the spectrum and approaching
usual half-integer Landau levels takes place for sufficiently large $m$ only
as clearly see from Fig. 1 (d).

Figure 9: The dependence of of the energy levels with $m=0$ on the external
radius $\rho_{2}$ for the two values of the internal radius: $\rho_{1}=0.25$
(left panel) and $\rho_{1}=3$ (right).

####  A.3.2 The states with zero angular momentum in a disk with a small hole

The mentioned above deviation of the spectrum from the standard Landau one can
be found out analytically by considering the specific limit $\rho_{1}\ll 1$
and $\rho_{2}\to\infty$ (plane with a small hole). Using asymptotic expansions
for the Whittaker function for the case $m=0$ (i.e.
$\tilde{\varepsilon}=\varepsilon$) with a small radius $\rho_{1}$

${W_{\varepsilon,0}}\left({\frac{\rho^{2}}{2}}\right)=\frac{1}{2}\frac{{\sqrt{2}\left({-\psi\left({\frac{1}{2}-\varepsilon}\right)-2\gamma+2\ln\left({\frac{1}{\rho}}\right)+\ln
2}\right)}}{{\Gamma\left({\frac{1}{2}-\varepsilon}\right)}}\rho+O({\rho^{3}}).$
(58)

one arrives at the transcendental equation with the digamma function
$\psi(z)$:

$-\psi\left({\frac{1}{2}-\varepsilon}\right)-2\gamma+2\ln\left({\frac{1}{{{\rho_{1}}}}}\right)+\ln
2=0,$ (59)

which for $\rho_{1}\ll 1$ gives the energy levels determined by the Eq. (23)
One can see that when $\rho_{1}\to 0$ this spectrum tends to the standard
Landau one, in complete agreement with the left panel of Fig. 9.

###  A.4 Asymptotic of solutions with $m\to\infty$

Let us pass to the analysis of the energy levels with large angular momentum:
$m\to\pm\infty$. We will do this basing on the same Eq. (57), obtained in the
assumption of $\rho_{2}\gg 1$, but do not requiring any more $\rho_{1}\ll 1$,
i.e. we just fix $\rho_{1}<\rho_{2}$.

####  A.4.1 General relations

One can rewrite Eq. (57) using the relation between the Whittaker function and
the confluent hypergeometric function of the second kind $\Psi(a,b,z)$
Bateman1 :

$\Psi\left(\frac{1}{2}-\epsilon+\frac{|m|-m}{2},|m|+1;\frac{\rho_{1}^{2}}{2}\right)=0.$
(60)

The last equation also follows directly from Eq. (A.1) in the limit
$\rho_{2}\to\infty$. It is valid for the states with arbitrary angular
momentum, while for $m\geq 0$ Eq. (60) reads as

$\Psi\left(\frac{1}{2}-\epsilon,m+1;m\lambda_{m}\right)=0.$ (61)

Here we introduced the parameter $\lambda_{m}=\rho_{1}^{2}/(2m)$.

The function $\Psi(a,b;x)$ for fixed $a$ and fixed $\lambda=x/b>0$ has the
asymptotic expansion given by Eq. (13.8.5) from NIST :

$\Psi(a,b;x)\sim
b^{-\frac{a}{2}}e^{\frac{\zeta^{2}b}{4}}\left[\lambda\left(\frac{\lambda-1}{\zeta}\right)^{a-1}U(a-\frac{1}{2},\zeta\sqrt{b})-\left(\lambda\left(\frac{\lambda-1}{\zeta}\right)^{a-1}-\left(\frac{\zeta}{\lambda-1}\right)^{a}\right)\frac{U(a-\frac{3}{2},\zeta\sqrt{b})}{\zeta\sqrt{b}}\right]+O\left(\frac{1}{b}\right)$
(62)

for $b\to\infty$ uniformly in compact $\lambda$-intervals of $(0,\infty)$ and
compact real $a$-intervals. Here $\zeta=\sqrt{2(\lambda-1-\ln\lambda)}$ with
${\rm sign}(\zeta)={\rm sign}(\lambda-1)$, and the function $U(a,x)$ is
related to the parabolic cylinder function $U(a,x)=D_{-a-1/2}(x)$. The
functions $U(a,x)$ and $U(a,-x)$ at large positive $x$ behave as

$\displaystyle U(a,x)\simeq
e^{-\frac{x^{2}}{4}}x^{-a-\frac{1}{2}}\left[1-\frac{(a+1/2)(a+3/2)}{2x^{2}}+O\left(\frac{1}{x^{4}}\right)\right],\quad
x\to\infty,$ $\displaystyle
U(a,-x)\simeq\frac{\sqrt{2\pi}}{\Gamma(1/2+a)}e^{\frac{x^{2}}{4}}x^{a-\frac{1}{2}}-\sin(\pi
a)e^{-\frac{x^{2}}{4}}x^{-a-\frac{1}{2}},\quad x\to\infty,$ (63)

therefore we can neglect the second term in the expansion (62) and the
equation (61) reduces to (note that $\lambda<1$ and ${\rm sign}(\zeta)<0$)

$U(-\epsilon,-\zeta\sqrt{m})=0,\quad\zeta=\sqrt{2(\lambda-1-\ln\lambda)}>0.$
(64)

####  A.4.2 The energy spectrum of skipping electrons

Let us consider now the case when the center of wave function ($\rho_{m}$) is
located quite close to one of the edges: $|2m-\rho_{1}|<1$.

i) When $|2m-\rho_{1}|=0$ one can rewrite Eq. (60), using the asymptotic
expression for fixed $a$ and large $b$ in confluent hypergeometric function
(see Eq. (13.8.7) in NIST ):

$\Psi(a,b;b)=\sqrt{\pi}(2b)^{-\frac{a}{2}}\left[\frac{1}{\Gamma\left(\frac{a+1}{2}\right)}-\frac{(a+1)\sqrt{8/b}}{\Gamma\left(\frac{a}{2}\right)}+O\left(\frac{1}{b}\right)\right],$
(65)

that gives

$\frac{1}{\Gamma\left(\frac{3-2\epsilon}{4}\right)}-\frac{(3/2-\epsilon)\sqrt{8/m}}{\Gamma\left(\frac{1-2\epsilon}{4}\right)}=0.$
(66)

Since $m\to\infty$, the energy levels are given by the poles of
$\Gamma\left(\frac{3-2\epsilon}{4}\right)$. which are at
$\epsilon=2n+3/2,\,n=0,1,2,\dots$. Solving the last equation we obtain the
behavior of energy levels at edges of the disk:

$\varepsilon_{n,m}\simeq 2n+3/2+\frac{(3/2-2n)\Gamma(n+3/2)}{\pi
n!}\sqrt{8/m},\quad m\to\infty.$ (67)

ii) When $\rho_{m}$ is near the edge $\rho_{1}$ but $|2m-\rho_{1}|<1$ we
consider Eq.(64) for $\lambda$ close to $1$ ($\lambda\lesssim 1$) where
$\zeta(\lambda)\simeq 1-\lambda$ and

$U(-\epsilon,(\lambda-1)\sqrt{m})=0.$ (68)

Using the formulas (19.3.5) from [Abramowitz, ],

$U(a,0)=\frac{\sqrt{\pi}}{2^{\frac{a}{2}+\frac{1}{4}}\Gamma\left(\frac{3}{4}+\frac{a}{2}\right)},\quad
U^{\prime}(a,0)=-\frac{\sqrt{\pi}}{2^{\frac{a}{2}-\frac{1}{4}}\Gamma\left(\frac{1}{4}+\frac{a}{2}\right)},\quad
U^{\prime\prime}(a,0)=\frac{a\sqrt{\pi}}{2^{\frac{a}{2}+\frac{5}{4}}\Gamma\left(\frac{3}{4}+\frac{a}{2}\right)},$
(69)

and keeping the terms up to $x^{2}$ in the expansion, we get the equation

$\frac{1+ax^{2}/2}{\Gamma\left(\frac{3-2\epsilon}{4}\right)}=\frac{\sqrt{2}x}{\Gamma\left(\frac{1-2\epsilon}{4}\right)},\quad
x=(\lambda-1)\sqrt{m}.$ (70)

As $x\to 0$, the energies are given by poles of gamma function
$\Gamma\left(\frac{3-2\epsilon}{4}\right)$ which are at $\epsilon=2n+3/2$,
$n=0,1,\dots$. Near the poles, writing $\epsilon=2n+3/2+\delta$ we have

$\frac{\Gamma\left(-n-1/2-\delta/2\right)}{\Gamma\left(-n-\delta/2\right)}=\sqrt{2}x+O\left(x^{3}\right).$
(71)

Expanding in $\delta$ the equation reduces to

$\frac{\delta}{2}\left[1-\frac{\delta}{2}G(2n+2)\right]=\frac{\sqrt{2}\Gamma(n+3/2)}{\pi\Gamma(n+1)}x,$
(72)

where $G(2z)=\psi(z+1/2)-\psi(z)$ is the known function and we used the
relation for digamma function $\psi(-n-1/2)=\psi(n+3/2)$. Thus we finally find

$\varepsilon_{n,m}=2n+3/2+\frac{\Gamma(n+1/2)}{\sqrt{2}\pi
n!}(2n+1)\frac{\rho_{1}^{2}-\rho_{m}^{2}}{\sqrt{m}}+G(2n+2)\left(\frac{\Gamma(n+3/2)}{\pi
n!}\right)^{2}\frac{(\rho_{1}^{2}-\rho_{m}^{2})^{2}}{m}$ (73)

[compare with Eq. (7) in Ref. Heuser, ]. We see the doubling of the frequency
(with a shift) and the linear and quadratic corrections due to the edge.

####  A.4.3 The energy spectrum of the electrons rotating on the cyclotron
orbits far from the edge

Now let us pass to the energy spectrum of electrons rotating on the cyclotron
orbits far from the edges: $|2m-\rho_{1}|>1$. Equation (63) gives

$\displaystyle\frac{\sqrt{2\pi}}{\Gamma(1/2-\epsilon)}+\sin(\pi\epsilon)e^{-\frac{x^{2}}{4}}x^{2\epsilon}=0,\quad
x=\zeta(\lambda)\sqrt{m},\quad\zeta(\lambda)=\sqrt{2(\lambda-1-\ln\lambda)}.$
(74)

This is the same equation as for the energy spectrum of the states in the bulk
of a ribbon Heuser ; Varlamov  hence we can write

$\varepsilon_{n,m}=n+\frac{1}{2}+\frac{1}{\sqrt{2\pi}n!}x^{2n+1}e^{-x^{2}/2},\quad
x=\zeta(\lambda)\sqrt{m},\,\lambda=\frac{\rho_{1}^{2}}{\rho_{m}^{2}}<1.$ (75)

and arrive at Eq. (26).

###  A.5 Back to Corbino disk

Until now we analyzed the case of the infinite system with a hole. We checked
that a similar consideration is valid for the Corbino disk with the two edges
described by the general equation for eigenvalues (A.1). Using the asymptotic
expression (62) for $\Psi$-functions and the analogous one for

$\Phi(a,b;\lambda b)\simeq
b^{-\frac{a}{2}}e^{\frac{\zeta^{2}b}{4}}\left[\lambda\left(\frac{\lambda-1}{\zeta}\right)^{a-1}U(a-\frac{1}{2},-\zeta\sqrt{b})+\left(\lambda\left(\frac{\lambda-1}{\zeta}\right)^{a-1}-\left(\frac{\zeta}{\lambda-1}\right)^{a}\right)\frac{U(a-\frac{3}{2},-\zeta\sqrt{b})}{\zeta\sqrt{b}}\right]+O\left(\frac{1}{b}\right),$
(76)

for the fixed $\lambda_{1}=\rho_{1}^{2}/(2m)$ and
$\lambda_{2}=\rho_{2}^{2}/(2m)$ Eq. (A.1) acquires the form

$U(-\epsilon,\zeta_{1}\sqrt{m})U(-\epsilon,-\zeta_{2}\sqrt{m})-U(-\epsilon,-\zeta_{1}\sqrt{m})U(-\epsilon,\zeta_{2}\sqrt{m})=0,\quad\zeta_{i}=\zeta(\lambda_{i}){\rm
sign}(\lambda_{i}-1).$ (77)

When $\rho_{1}<\sqrt{2m}<\rho_{2}$ we have $\lambda_{1}<1$ and
$\lambda_{2}>1$, hence in this case we can write

$U(-\epsilon,-\zeta_{1}\sqrt{m})U(-\epsilon,-\zeta_{2}\sqrt{m})-U(-\epsilon,\zeta_{1}\sqrt{m})U(-\epsilon,\zeta_{2}\sqrt{m})=0,$
(78)

with positive $\zeta_{i}=\sqrt{2(\lambda_{i}-1-\ln\lambda_{i})}$. Thus, for
$m\to\infty$, the equation splits in two:

$U(-\epsilon,-\zeta_{1}\sqrt{m})=0\quad\mbox{and}\quad
U(-\epsilon,-\zeta_{2}\sqrt{m})=0.$ (79)

In the bulk, when $\rho_{1}<\sqrt{2m}<\rho_{2}$, where $\sqrt{2m}-\rho_{1}\gg
1$ and $\rho_{2}-\sqrt{2m}\gg 1$ (it is assumed that $\rho_{1}\gg 1$ and
$\rho_{2}-\rho_{1}\gg 1$), the solutions of these equations give us the
spectrum of electrons rotating along the cyclotron orbits (cp. Eq. (75)):

$\varepsilon_{n,m}=n+\frac{1}{2}+\frac{1}{\sqrt{2\pi}n!}x^{2n+1}e^{-x^{2}/2},\quad
x=\zeta(\lambda_{i})\sqrt{m},\,\lambda_{i}=\frac{\rho_{i}^{2}}{\rho_{m}^{2}}.$
(80)

The energy spectra of the electrons skipping along the edges are determined by
Eqs. (24) - (25).

##  Appendix B Large and narrow Corbino disk

As one can see from Fig. 9 (right panel) for a narrow Corbino disk with the
large inner and outer radii energy tends to be large. By means of the
asymptotic expressions of Whittaker functions for large parameter $\epsilon$
(see Eqs. (13.21.1) and (13.21.2) in NIST )

${M_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{1}{2}{\rho^{2}}}\right)\approx\frac{\rho}{{\sqrt{2}}}\Gamma\left({\left|m\right|+1}\right){\tilde{\varepsilon}^{-\frac{{\left|m\right|}}{2}}}{J_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}\rho}\right),$
(81)

and

${W_{\tilde{\varepsilon},\frac{{\left|m\right|}}{2}}}\left({\frac{1}{2}\rho^{2}}\right)\approx\frac{\rho}{{\sqrt{2}}}\Gamma\left({\tilde{\varepsilon}+\frac{1}{2}}\right)\left[{\sin\left({\pi\tilde{\varepsilon}-\frac{{\pi\left|m\right|}}{2}}\right){J_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}\rho}\right)-\cos\left({\pi\tilde{\varepsilon}-\frac{{\pi\left|m\right|}}{2}}\right){Y_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}\rho}\right)}\right],$
(82)

we arrive at the new equation (31) for energy levels, which is much simpler
than Eq. (8) or its equivalent Eq. (A.1). The asymptotics (81) and (82) are
valid when the parameter $\rho^{2}/(8\tilde{\epsilon})<1$ (see the expansion
(90) below and the relation between Whittaker and confluent hypergeometric
functions).

Denoting in Eq. (31) $x=\sqrt{2\tilde{\epsilon}}\rho_{1}$ and
$\lambda=\rho_{2}/\rho_{1}$ one can rewrite it in the form coinciding with Eq.
(10.21.45) in NIST

$J_{|m|}(x)Y_{|m|}(\lambda x)-J_{|m|}(\lambda x)Y_{|m|}(x)=0.$ (83)

Then for $\lambda>1$ and $\lambda-1\ll 1$ one can take the first two terms of
the asymptotic expansion for $n$th positive zeros of the Bessel functions
cross-product (see Eq. (10.21.50) in NIST )

$x_{n}=\frac{\pi n}{\lambda-1}+\frac{4m^{2}-1}{8\pi
n}\frac{\lambda-1}{\lambda},\qquad n=1,2,\ldots.$ (84)

Accordingly, the solutions of Eq. (83) are

$\tilde{\epsilon}=\frac{1}{2}\left[\frac{\pi n}{\delta}+\frac{m^{2}-1/4}{2\pi
n}\frac{\delta}{\rho_{1}\rho_{2}}\right]^{2},\qquad n=1,2,\ldots,$ (85)

where $\delta=\rho_{2}-\rho_{1}$. Restoring units and neglecting the term
$\sim\delta^{2}$ ($d^{2}/r_{1}r_{2}\ll 1$) one arrives at the final expression
(32) for the spectrum of the narrow disk. The given above inequality
$\rho^{2}/(8\tilde{\epsilon})<1$ results in the condition $\delta\rho<2\pi$
which for $\rho>\delta$ implies that $\delta<\sqrt{2\pi}$.

Using the large argument asymptotic of Bessel functions (see Eqs. (9.2.1) and
(9.2.2) in Abramowitz ) one obtains from Eq. (34) the following expression for
the wave function

$f_{nm}(\rho)=\frac{\sin(\sqrt{2\tilde{\epsilon}}(\rho-\rho_{1}))}{l\sqrt{\pi\delta\rho}},$
(86)

where the normalization constant $C$ is determined by the condition (21).

The formula (20) for the current $I_{nm}$ carried by the state with definite
quantum numbers $n,m$ contains the following integral
$\int_{\rho_{1}}^{\rho_{2}}d\rho f_{nm}^{2}(\rho)/\rho$. For large narrow
Corbino disk we can use the found above asymptotic of the wave function (86)
and obtain

$\begin{split}\int\limits_{\rho_{1}}^{\rho_{2}}\frac{d\rho}{\rho}f_{nm}^{2}(\rho)&=\frac{\sqrt{2\tilde{\epsilon}}}{\pi
l^{2}\delta}\left[\frac{-1+\cos(2\sqrt{2\tilde{\epsilon}}\delta)}{2\sqrt{2\tilde{\epsilon}}\rho_{2}}+\sin(2\sqrt{2\tilde{\epsilon}}\rho_{1})\left(\mbox{Ci}(2\sqrt{2\tilde{\epsilon}}\rho_{1})-\mbox{Ci}(2\sqrt{2\tilde{\epsilon}}\rho_{2}\right)\right.\\\
&-\left.\cos(2\sqrt{2\tilde{\epsilon}}\rho_{1})\left(\mbox{Si}(2\sqrt{2\tilde{\epsilon}}\rho_{1})-\mbox{Si}(2\sqrt{2\tilde{\epsilon}}\rho_{2})\right)\right],\end{split}$
(87)

where $\mbox{Si}(z)$ and $\mbox{Ci}(z)$ are sine integral and cosine integral
functions, respectively. Using their asymptotic at large argument,

$\mbox{Si}(z)\approx\frac{\pi}{2}-\frac{\cos
z}{z},\qquad\mbox{Ci}(z)\approx\frac{\sin z}{z},\qquad z\gg 1,$ (88)

and that in the leading approximation from Eq. (85) follows that
$\sqrt{2\tilde{\epsilon}}\delta=\pi n$, we arrive at the following simple
result

$\int\limits_{\rho_{1}}^{\rho_{2}}\frac{d\rho}{\rho}f_{nm}^{2}(r)=\frac{1}{2\pi
l^{2}\rho_{1}\rho_{2}}.$ (89)

##  Appendix C Small Corbino disk with infinitesimal inner radius

In this case one can approximate a Corbino disk as a solid disk without a hole
in the centre with the conditions ${\rho_{2}}<1$ and ${\rho_{1}}=0$. The
solution for energy spectrum and is given by zeroes of the confluent
hypergeometric function (46) (see e.g. Rensink ). We rewrite this expression
by means of the formula that represents the confluent hypergeometric function
as the series of the Bessel functions of the first kind (see Eq. (13.3.7) in
Abramowitz )

$\Phi\left({\frac{1}{2}-\varepsilon+\frac{{\left|m\right|-m}}{2},\left|m\right|+1,\frac{1}{2}\rho_{2}^{2}}\right)=\Gamma(|m|+1){e^{\frac{1}{4}\rho_{2}^{2}}}{\tilde{\varepsilon}^{-\frac{1}{2}\left|m\right|}}\mathop{\sum}\limits_{p=0}^{\infty}{A_{p}}{\left(\frac{\rho_{2}^{2}}{8\tilde{\varepsilon}}\right)^{\frac{p}{2}}}{J_{\left|m\right|+p}}\left({\sqrt{2\tilde{\varepsilon}}{\rho_{2}}}\right),$
(90)

where coefficients satisfy the recurrence relation

$\left({n+1}\right){A_{n+1}}=\left({n+\left|m\right|}\right){A_{n-1}}-2\tilde{\varepsilon}{A_{n-2}},$
(91)

and ${A_{0}}=1$, ${A_{1}}=0$, ${A_{2}}=\frac{1}{2}\left|m\right|+\frac{1}{2}$.

If the expansion parameter $\rho_{2}/(2\sqrt{2\tilde{\varepsilon}})$ is small
then we can keep the first term in Eq. (90). This leads to the equation for
eigenvalues

${J_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}{\rho_{2}}}\right)=0,$
(92)

which has the solutions $\sqrt{2\tilde{\varepsilon}}\rho_{2}=j_{nm}$ with
${j_{nm}}$ being the n-th root of the equation
${J_{\left|m\right|}}\left(z\right)=0$. The corresponding energy spectrum
$E_{n,m}$ is given by Eq. (47) in the main text. The expansion parameter
$\rho_{2}/(2\sqrt{2\tilde{\varepsilon}})<1$ in Eq. (90) becomes
$\rho_{2}^{2}/(2j_{nm})<\rho_{2}^{2}/(2j_{10})$. Thus, using the value of the
lowest root $j_{10}\approx 2.4$, we estimate that $r_{2}<2.2l$ which
determines the range of validity of the considered approximation. Note that
Eq. (92) also follows directly from Eq. (31) by taking there $\rho_{1}=0$.

In turn, the radial component of the wave function has the form

$f_{nm}\left(r\right)=C{J_{\left|m\right|}}\left({\sqrt{2\tilde{\varepsilon}}\rho}\right)=CJ_{|m|}\left(j_{nm}\frac{r}{r_{2}}\right).$
(93)

The constant $C$ can be found from the normalization condition (21) (see Eq.
(6.521.1) in Gradstein ):

${C^{2}}=\frac{1}{\pi{r_{2}^{2}J_{\left|m\right|+1}^{2}\left({{j_{nm}}}\right)}}.$
(94)

Therefore, the radial component of the wave function is given by Eq. (48) in
the main text. In this case the full current given by Eqs. (19) and (20)
acquires the following form

$I=\frac{{e\hbar}}{2\pi{{m_{e}}r_{2}^{2}}}\sum_{\begin{subarray}{c}m=-\infty\\\
n=1\end{subarray}}^{\infty}\frac{\theta(\mu-
E_{n,m})}{J_{\left|m\right|+1}^{2}\left({{j_{nm}}}\right)}{\int\limits_{0}^{{r_{2}}}dr{\left({\frac{{2m}}{r}-\frac{r}{l^{2}}}\right)}}J_{\left|m\right|}^{2}\left({{j_{nm}}\frac{r}{{{r_{2}}}}}\right).$
(95)

The integration of the first term in the bracket can be done using the
following formula (Eq. (1.8.3.17) in Prudnikov2 )

$A_{nm}\equiv
2|m|\int\limits_{0}^{1}\frac{dr}{r}J^{2}_{|m|}(j_{nm}r)=1+J_{0}^{2}(j_{nm})-2\sum\limits_{k=0}^{|m|-1}J^{2}_{k}(j_{nm}),\qquad|m|\geq
1,$ (96)

while the second term is integrated using the normalization condition (21).
Thus we arrive at Eq. (49) in the main text.

To verify that Eqs. (49) - (51) follow directly from the Byers-Yang formula
(22) one needs to calculate explicitly the derivative $\partial
j_{n\nu}/\partial\nu$ of the roots $j_{n\nu}$ of the equation
$J_{\nu}(j_{n\nu})=0$ with $\nu=|m|$. It can be expressed as follows [see Ref.
Watson-Bessels-book,  Section 15.6, Eq. (2)]:

$\frac{\partial
j_{n\nu}}{\partial\nu}=\frac{2\nu}{j_{n\nu}J^{2}_{\nu+1}(j_{n\nu})}\int\limits_{0}^{1}\frac{dx}{x}J^{2}_{\nu}(j_{n\nu}x).$
(97)

Accordingly, one obtains

$\frac{\partial
j_{nm}}{\partial|m|}=\frac{2|m|}{j_{nm}J^{2}_{|m|+1}(j_{nm})}\int\limits_{0}^{1}\frac{dz}{z}J^{2}_{|m|}(j_{n\nu}z)=\frac{A_{nm}}{j_{nm}J^{2}_{|m|+1}(j_{nm})},$
(98)

where in the last identity the definition (96) for $A_{nm}$ is taken into
account.

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