# Magnetic response of interacting electrons in a spatially non-uniform
disordered multi-channel system: Exact and mean-field results

Arpita Koley  Physics and Applied Mathematics Unit, Indian Statistical
Institute, 203 Barrackpore Trunk Road, Kolkata-700 108, India  Santanu K.
Maiti<EMAIL_ADDRESS>Physics and Applied Mathematics Unit, Indian
Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700 108, India

###### Abstract

In this work we explore magnetic response of interacting electrons in a
spatially non-uniform disordered system, where impurities are introduced in
one sector of the geometry keeping the other one free. The interaction among
the electrons are taken in the well known Hubbard form which leads to several
anomalous features in energy spectra and flux driven circular current,
depending on the concentration of up and down spin electrons. For smaller
systems with less number of electrons we present exact results which always
give a clear picture to understand the basic mechanisms, while for large
systems having higher number of electrons mean-field results are given. The
effect of disorder is very interesting. Beyond a critical disorder, completely
contrasting signature is obtained compared to uniform disordered systems, and
the phenomenon becomes more promising when the Hubbard interaction is
included. Along with these features, we find unusual half flux-quantum
periodic current at some typical cases and a suitable explanation of it is
provided. Our detailed analysis may be utilized to study magnetic response of
interacting electrons in other similar kind of non-uniform disordered systems.

##  I Introduction

Persistent current is a well known phenomenon in the field of mesoscopic
physics. When an isolated conducting loop is threaded by a magnetic flux,
commonly referred as Aharonov-Bohm (AB) ab1 ; ab2  flux, a net current
appears. Once the current is established, it never dissipates to zero even in
presence of impurities. This fact was first successfully proposed by Büttiker,
Imry and Landauer in their seminal work pc1  during the early 80’s and soon
after this proposition interest in this area has rapidly grown up with lots of
theoretical and experimental works pc2 ; pc3 ; pc4 ; pc5 ; pc6 ; pc7 ; pc8 ;
pc9 ; pc10 ; pc11 ; pc12 ; pc13 ; pc14 ; pc15 ; pc16 ; pc17 ; liang ; chen .

Till date an enormous amount of work has been done exploring several
interesting features, though some open questions still persist even today. For
example, the interplay between electron-electron (e-e) interaction and random
disorder in a ‘spatially non-uniform’ disordered material has not been
addressed, that might be an interesting one. Traditionally impurities are
included uniformly pc18 ; pc19 ; pc20 ; pc21  throughout the sample and under
this situation the role of impurities are well studied both for the non-
interacting pc2 ; pc11  and interacting electron pc22 ; pc23 ; pc24  cases.
But no attempt has been made considering an ordered-disordered separated (ODS)
system where one part is completely free and the other part is subjected to
impurities pc25 ; pc26 ; pc27 . The recent breakthroughs in nanoscience and
nanotechnology allow us to get tailor made ODS systems in different
geometrical shapes like nanotubes, thick rings, 2D surfaces and to name a few.
Apparently it seems that the role of disorder in such systems will be quite
analogous to what we observe in the case of fully disordered ones i.e., the
Anderson type localization pc18 ; pc19 ; anlc1 ; anlc2 ; anlc3 ; anlc4 .
Though the situation is not like that, as pointed out by Zhong and Stocks pc25
in their work. Considering a shell-doped nanowire they have shown that beyond
a critical disorder, electrons get delocalized and it becomes more effective
as we increase the disorder strength. A sharp localization-to-delocalization
transition takes place in such an ODS system, and importantly, the mobility
edge (ME) skmprl  that appears is highly robust and it does not vanish even in
the limit of strong disorder. This is in complete contrast to the 3D random
disordered lattices where the ME vanishes at large disorder. Thus, in the
context

Figure 1: (Color online). ODS nanotube threaded by an AB flux $\phi$, where
the filled colored balls represent the atomic sites. Impurities are introduced
in the sites shown by the light red balls, while the sites denoted by the dark
gray balls are free from impurities. A net circular current appears in the
tube due to the flux $\phi$.

of electronic localization, an ODS system is a very good example that may
provide some non-trivial features, especially when we include the effect of
electron-electron correlation.

The other open question which is associated with the phenomenon of persistent
current, to the best of our concern, is the lack of suitable explanation of
getting half flux-quantum periodicity $\phi_{0}/2$ ($\phi_{0}=h/e$, the
elementary flux-quantum where $e$ and $h$ are the fundamental constants) with
AB flux $\phi$. Usually, the current provides $\phi_{0}$ periodicity and this
issue is well understood. But, for some typical cases depending on the filling
factor, the current exhibits $\phi_{0}/2$ periodicity, and its specific reason
is not fully clear. Thus, further probing is definitely required for the sake
of completeness. Though few attempts have been made pc6 ; pc28 ; new1 ; new2 ,
here we intend to explain the phenomenon of half flux-quantum periodicity in
other way. For the ordered nanotube, we substantiate it analytically. An
analytical treatment is always helpful to understand any phenomenon quite
easily. For the disordered cases, numerical results are given.

The present work mainly focuses on two issues. (i) The interplay between e-e
interaction and impurities in an ODS nanotube (see Fig. 1), where the e-e
correlation is included in the form of well known Hubbard model pc29 ; pc30 ;
pc31 . (ii) The possible reason of getting half flux-quantum periodic
persistent current. Now, working with a many-body system has always been a
challenging task as the main difficulty arises in finding the electronic
energy eigenvalues. In our work, we proceed in two ways. For small size
systems, we compute the energy eigenvalues by diagonalizing the full many-body
Hamiltonian. Whereas, this prescription is no longer suitable for bigger
systems, and thus, Hartree-Fock (HF) mean-field (MF) prescription pc30 ; pc31
; pc32 ; pc33 ; pc34  is employed. The interplay between disorder and e-e
interaction is really interesting which we examine critically. Along with
this, we give a suitable mathematical description of getting half flux-quantum
periodicity in persistent current for the ordered case.

Following the above introduction, in Sec. II, we describe briefly the nanotube
and its Hamiltonian. Section III contains the theoretical prescription for
calculating the results. All the numerical results are thoroughly discussed in
Sec. IV. A brief outline about the experimental feasibility of nano rings is
given in Sec. V, for the sake of completeness. Finally, we conclude in Sec. V.

##  II Nanotube and the Hamiltonian

The sketch of the nanotube is shown in Fig. 1 in which $N_{y}$ number of rings
are stacked vertically, where each of these rings contains $N_{x}$ number of
lattice sites. We introduce random impurities in the upper $N_{y}/2$ ($N_{y}$
to be considered as even) rings, keeping the other half free, to make the tube
as an ordered-disordered separated one. The tube encloses an AB flux $\phi$,
measured in units of $\phi_{0}$, which produces a net circular current in the
system.

We use a tight-binding (TB) framework to describe the system. Under nearest-
neighbor hopping (NNH) approximation, the TB Hamiltonian of the system in
presence of Hubbard interaction reads as pc29 ; pc30

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{j,k,\sigma}\epsilon_{j,k,\sigma}c_{j,k,\sigma}^{\dagger}c_{j,k,\sigma}+t_{y}\sum_{j,k,\sigma}c_{j,k,\sigma}^{\dagger}c_{j+1,k,\sigma}$
(1) $\displaystyle+$ $\displaystyle
t_{x}\sum_{j,k,\sigma}e^{i\theta}c_{j,k,\sigma}^{\dagger}c_{j,k+1,\sigma}+h.c.$
$\displaystyle+$ $\displaystyle
U\sum_{j,k}c_{j,k,\uparrow}^{\dagger}c_{j,k,\uparrow}c_{j,k,\downarrow}^{\dagger}c_{j,k,\downarrow}$

where the integers $j$ (runs from $1$ to $N_{y}$) and $k$ (varies from $1$ to
$N_{x}$) are used to refer the ring index and the lattice sites in each of
these rings, respectively. $\epsilon_{j,k,\sigma}$ denotes the on-site energy
at a particular site ($j,k$) of an electron having spin $\sigma$
($\uparrow,\downarrow$), and $U$ gives the on-site Hubbard interaction
strength. $t_{x}$ and $t_{y}$ correspond to the intra-ring and inter-ring NNH
integrals, respectively, and $i=\sqrt{-1}$. Due to the AB flux $\phi$, a phase
factor $\theta$ is introduced pc22 ; deo ; kus ; kulik ; sai  whenever an
electron hops from one site of a ring to its neighboring site, and it is
expressed as pc22 ; deo ; kus ; kulik ; sai  $\theta=2\pi\phi/N_{x}\phi_{0}$.
For our system we assume that the flux $\phi$ threads the tube axially in such
a way that the magnetic field becomes zero at the ring circumference.
Therefore, the electrons always move in the field free space, and under this
condition the electronic states depend only on the net flux (viz, $\phi$)
penetrated by individual rings in the tube pc2 . In traversing through a
particular ring path even though the magnetic field is zero, the electronic
wave function picks up a phase, and for one complete rotation the phase factor
is $2\pi\phi/\phi_{0}$. This is the well known Aharonov-Bohm effect pc2 ; deo
. As there are $N_{x}$ bonds in individual rings, the phase factor associated
in each hopping becomes $\theta=2\pi\phi/N_{x}\phi_{0}$. No such phase factor
appears during the motion along the vertical direction, since for this motion
no flux is enclosed. For the ordered sites, the site energies are uniform and
we fix them to zero, without loss of generality. While they are chosen
‘randomly’ pc18 ; pc19 ; pc22 ; kus  from a ‘Box’ distribution function of
width $W$ within the range $-W/2$ to $W/2$. $c_{j,k,\sigma}$ and
$c_{j,k,\sigma}^{\dagger}$ are the usual fermionic operators. In our work we
assume that the hopping integrals do not depend of the spin index ($\sigma$),
and also the Hubbard interaction is uniform throughout sample. Accordingly, we
drop spin label $\sigma$ from these parameters.

##  III Theoretical framework

For the above TB Hamiltonian Eq. 1 we investigate the characteristic features
of interacting electrons. To explore the behavior of flux-driven circular
current, first we need to determine the energy eigenvalues, which we compute
in two ways. For the case of exact diagonalization of full many-body
Hamiltonian matrix, within our computational capacity, we can reach the
maximum limit of six electrons with three up and three down spins, and the
total number of lattice sites that can be considered is ten. For larger
systems with higher number of electrons we use HF MF scheme. Below we discuss
in detail both the two prescriptions for the benefit of the readers.

###  III.1 Exact diagonalization of the full many-body Hamiltonian

The construction of the full many-body Hamiltonian is one of the key parts for
theoretical calculations, and once it is done we find the energy eigenvalues
by diagonalizing the Hamiltonian. Here we give an example of forming such a
Hamiltonian considering a simple system, which is a $N_{x}$-site single
channel ring ($N_{y}=1$) with two opposite spin electrons, viz,
$N_{\uparrow}=N_{\downarrow}=1$ ($N_{\uparrow}$ and $N_{\downarrow}$ are used
to refer the total number of up and down spin electrons in a system). It can
easily be generalized for multi-ring systems with higher number of electrons.
For the single ring, we can skip the strand index $j$ in order to write the TB
Hamiltonian in a simplified form and it is expressed as

$\displaystyle H_{R}$ $\displaystyle=$
$\displaystyle\sum_{k,\sigma}\epsilon_{k,\sigma}c_{k,\sigma}^{\dagger}c_{k,\sigma}+t_{x}\left(\sum_{k,\sigma}e^{i\theta}c_{k,\sigma}^{\dagger}c_{k+1,\sigma}+h.c.\right)$
(2) $\displaystyle+$ $\displaystyle
U\sum_{k}c_{k,\uparrow}^{\dagger}c_{k,\uparrow}c_{k,\downarrow}^{\dagger}c_{k,\downarrow}$
$\displaystyle=$ $\displaystyle H_{1}+H_{2}+H_{3}$

where the sub-Hamiltonians $H_{1}$, $H_{2}$ and $H_{3}$ are associated with
the site energy, NNH and the Hubbard interaction terms, respectively. Our aim
is to find out the matrix elements $\langle\eta|H|\zeta\rangle$, where
$|\eta\rangle$ and $|\zeta\rangle$ are the many-particle Wannier states. The
general forms of these states are:

$|\eta\rangle=c_{p,\uparrow}^{\dagger}c_{q,\downarrow}^{\dagger}|0\rangle~{}~{}\mbox{and}~{}~{}|\zeta\rangle=c_{r,\uparrow}^{\dagger}c_{s,\downarrow}^{\dagger}|0\rangle$

where $|0\rangle$ represents the null state. $p$, $q$, $r$ and $s$ are the
integers and they run from $1$ to $N$. Here it is important to note that, for
higher number of up and/or down spin electrons we need to consider the values
of these integers selectively satisfying the Pauli’s exclusion principle. Now,
plugging the Hamiltonian $H$ between the states $|\eta\rangle$ and
$|\zeta\rangle$, and doing some mathematical steps we evaluate all the terms.
For the three sub-Hamiltonians they are as follows.

$\displaystyle\langle\eta|H_{1}|\zeta\rangle$ $\displaystyle=$
$\displaystyle\epsilon_{p}\delta_{p,r}\delta_{q,s}+\epsilon_{q}\delta_{p,r}\delta_{q,s},$
$\displaystyle\langle\eta|H_{2}|\zeta\rangle$ $\displaystyle=$ $\displaystyle
t_{x}e^{i\theta}\left(\delta_{p,r+1}\delta_{q,s}+\delta_{q,s+1}\delta_{p,r}\right)$
$\displaystyle+$ $\displaystyle
t_{x}e^{-i\theta}\left(\delta_{p+1,r}\delta_{q,s}+\delta_{q+1,s}\delta_{p,r}\right),$
$\displaystyle\langle\eta|H_{3}|\zeta\rangle$ $\displaystyle=$ $\displaystyle
U\delta_{p,q,r,s}.$

Using these factors we can easily compute all the matrix elements with the
help of a simple algorithm to get the full many-body Hamiltonian matrix.
Following the above steps of 1D ring with two opposite spin electrons, we can
now construct the Hamiltonian of any multi-ring system with higher number of
up and down spin electrons. Finally, we find the many-body energy eigenvalues
by diagonalizing the matrix. As already mentioned, one important limitation
arises in this step which is the diagonalization of the full matrix since the
dimension of the matrix gets enhanced so rapidly with increasing the system
size and the number of electrons. As illustrative example, for a $N_{x}$-site
ring with two opposite spin electrons, the dimension is $N_{x}^{2}\times
N_{x}^{2}$. Thus, for a nanotube having (say) $N_{x}=N_{y}=20$, the dimension
of the matrix when it contains only these two opposite spin electrons becomes
$400^{2}\times 400^{2}$, which is impossible to handle (specifically within
our computational facilities). Thus, working with the full many-body
Hamiltonian, we need to restrict ourselves within small size systems with few
number of electrons.

###  III.2 Hartree-Fock Mean-Field theory

In the MF scheme pc30 ; pc31 ; pc32 ; pc33 ; pc34  we can manage to find the
energy eigenvalues of large systems. Here the fluctuations are neglected
assuming they are small enough around the mean densities. This is the key
concept of mean-field approximation (MFA). The MFA allows us to decompose the
interacting Hamiltonian into two non-interacting parts. One is associated with
the up spin electrons, while the other is involved with the down spin ones.
Under MFA, the TB Hamiltonian Eq. 1 is expressed as

$\displaystyle H_{MF}$ $\displaystyle=$ $\displaystyle
H_{\uparrow}+H_{\downarrow}-U\sum_{j,k}\langle
c_{j,k,\uparrow}^{\dagger}c_{j,k,\uparrow}\rangle\langle
c_{j,k,\downarrow}^{\dagger}c_{j,k,\downarrow}\rangle$ (3) $\displaystyle=$
$\displaystyle H_{\uparrow}+H_{\downarrow}-U\sum_{j,k}\langle
n_{j,k,\uparrow}\rangle\langle n_{j,k,\downarrow}\rangle$

where $n_{j,k,\sigma}$ ($=c_{j,k,\sigma}^{\dagger}c_{j,k,\sigma}$) corresponds
to the number operator. $\langle n_{j,k,\sigma}\rangle$ is the expectation
value of the number operator, which measures the mean occupation of electrons
i.e., the electron density at the site ($j,k$) with spin $\sigma$. The
Hamiltonians $H_{\uparrow}$ and $H_{\downarrow}$ read as

$\displaystyle H_{\uparrow}$ $\displaystyle=$ $\displaystyle
t_{y}\sum_{j,k}c_{j,k,\uparrow}^{\dagger}c_{j+1,k,\uparrow}+t_{x}\sum_{j,k}e^{i\theta}c_{j,k,\uparrow}^{\dagger}c_{j,k+1,\uparrow}+h.c.$
(4) $\displaystyle+$
$\displaystyle\sum_{j,k}\left(\epsilon_{j,k,\uparrow}+U\langle
n_{j,k,\downarrow}\rangle\right)c_{j,k,\uparrow}^{\dagger}c_{j,k,\uparrow}$

and

$\displaystyle H_{\downarrow}$ $\displaystyle=$ $\displaystyle
t_{y}\sum_{j,k}c_{j,k,\downarrow}^{\dagger}c_{j+1,k,\downarrow}+t_{x}\sum_{j,k}e^{i\theta}c_{j,k,\downarrow}^{\dagger}c_{j,k+1,\downarrow}+h.c.$
(5) $\displaystyle+$
$\displaystyle\sum_{j,k}\left(\epsilon_{j,k,\downarrow}+U\langle
n_{j,k,\uparrow}\rangle\right)c_{j,k,\downarrow}^{\dagger}c_{j,k,\downarrow}.$

As $H_{\uparrow}$ and $H_{\downarrow}$ are single particle Hamiltonians, we
can easily handle them. The last term in Eq. 3 is simply a number, as it is
the product of the averages of two number operators.

The final solution i.e., the energy eigenvalues are obtained in a self-
consistent procedure. We start with a set of initial guess values of $\langle
n_{j,k,\uparrow}\rangle$ and $\langle n_{j,k,\downarrow}\rangle$, and
determine the energy eigenvalues and eigenvectors. Using the eigenvectors a
new set of $\langle n_{j,k,\uparrow}\rangle$ and $\langle
n_{j,k,\downarrow}\rangle$ is formed, and then we compute the energy
eigenvalues from $H_{\uparrow}$ and $H_{\downarrow}$. This process continues
till the converged solution is reached.

###  III.3 Calculation of ground state energy

To calculate flux driven circular current we need to know the ground state
energy of the system. It is obtained in two ways, for the two different
prescriptions i.e., diagonalization of the full many-body Hamiltonian and the
HF MF scheme. For the previous one, since we deal with the many-body states,
the ground state energy $E_{g}$ at absolute zero temperature becomes the
lowest energy eigenvalue of the many-body Hamiltonian. While, in the other
prescription, $E_{g}$ is determined at zero temperature from the relation pc30
; gup1

$E_{g}=\sum_{m}E_{m,\uparrow}+\sum_{m}E_{m,\downarrow}-U\sum_{j,k}\langle
n_{j,k,\uparrow}\rangle\langle n_{j,k,\downarrow}\rangle$ (6)

where, $E_{m,\uparrow}$ and $E_{m,\downarrow}$ are the eigenenergies of the
single particle Hamiltonians $H_{\uparrow}$ and $H_{\downarrow}$ (Eqs. 4 and
5), respectively. Depending on the filling factors associated with up and down
spin electrons ($N_{\uparrow}$ and $N_{\downarrow}$) we take the sum of the
lowest $N_{\uparrow}$ and $N_{\downarrow}$ energy levels to get the ground
state energy $E_{g}$.

###  III.4 Calculation of persistent current

Finally, we evaluate the current in the nanotube, that appears due to the AB
flux $\phi$, from the relation pc2

$I=-\frac{\partial E_{g}}{\partial\phi}.$ (7)

It clearly suggests that the nature of flux driven current solely depends on
the slope of the ground state energy with respect to the flux $\phi$.
Vanishing slope yields zero circular current, otherwise a finite current
appears.

##  IV Numerical Results and Discussion

In this section we present our numerical results those are computed by using
the above theoretical framework. As the sites are non-magnetic in nature, we
have $\epsilon_{j,k,\uparrow}=\epsilon_{j,k,\downarrow}$. For the ordered case
we set them to zero without loss of any generality, while they are chosen
‘randomly’ pc18 ; pc19 ; pc22 ; kus  from a ‘Box’ distribution function of
width $W$. For the entire numerical calculations we select
$t_{x}=t_{y}=1\,$eV, and measure the other energies in unit of $t_{x}$. All
the circular currents are scaled by the factor $40\,\mu$A. This typical scale
factor is associated with the term $e/h$ ($\sim 40\,\mu$A/eV) that appears in
the evaluation of current.

Figure 2: Ground state energy and corresponding current as a function of flux
$\phi$ for ordered multi-ring systems with different combinations of up and
down spin electrons. In each spectrum, the results are shown for three
different values of $U$, where the solid, dotted and dashed curves correspond
to $U=0$, $2$ and $4$, respectively. The number of lattice sites in each ring
and the combination of up and down spin electrons in different cases are as
follows: (a) and (f) $N_{x}=5$, $N_{\uparrow}=N_{\downarrow}=1$; (b) and (g)
$N_{x}=5$, $N_{\uparrow}=2$, $N_{\downarrow}=1$; (c) and (h) $N_{x}=4$,
$N_{\uparrow}=N_{\downarrow}=2$; (d) and (i) $N_{x}=4$, $N_{\uparrow}=3$,
$N_{\downarrow}=2$; (e) and (j) $N_{x}=3$, $N_{\uparrow}=N_{\downarrow}=3$. In
all these cases we set $N_{y}=2$.

###  IV.1 Fully ordered multi-ring system

Before focusing on the central parts of our analysis, we start with the
energy-flux and current-flux characteristics for the completely ordered system
to understand the precise role of e-e correlation on these quantities. The
results are presented in Fig. 2, where everything is done by considering the
full many-body Hamiltonian. Here it is relevant to note that for a ‘multi-ring
system’, no analysis has been made so far using the full Hamiltonian. Five
different cases are taken into account depending on the number of up and down
spin electrons those are: ($\uparrow,\downarrow$),
($\uparrow,\uparrow,\downarrow$), ($\uparrow,\uparrow,\downarrow,\downarrow$),
($\uparrow,\uparrow,\uparrow,\downarrow,\downarrow$) and
($\uparrow,\uparrow,\uparrow,\downarrow,\downarrow,\downarrow$). In the left
column ground state energies are shown (Figs. 2(a)-(e)) for the three
different values of $U$, and the corresponding currents are presented in the
right column (Figs. 2(f)-(j)). Several atypical features are observed those
are thoroughly discussed one by one as follows.

Case I. Two-electron system: For the system with two electrons
($\uparrow,\downarrow$) a sharp change in slope in $E_{g}$ takes place across
$\phi=0$ for $U=0$ (solid line of Fig. 2(a)) which yields a saw-tooth like
behavior in the current (solid line of Fig. 2(f)). This is a well established
fact pc2 . But as we include $U$, a dip-like structure appears in the
$E_{g}$-$\phi$ curve around $\phi=0$ (dotted line of Fig. 2(a)), and most
importantly, it arises for any non-zero value of $U$. The arc length of the
dip increases with $U$ (dashed line of Fig. 2(a)). Because of this finite arc
in $E_{g}$-$\phi$ curve, kink appears in the current-flux characteristics as
clearly seen from the dotted and dashed curves of Fig. 2(f). The origin of the
kink like structure in current followed by the dip in ground state energy is
clearly described below. Now, comparing the spectra given in Fig. 2(f) we see
that the current amplitude gets reduced with $U$. This is solely associated
with the mutual repulsion of up and down spin electrons which prohibits their
motions. Due to the less number of electrons the reduction of current is not
so prominent, but it can be more properly seen from the other systems having
higher number of electrons. Both the energy and current show $\phi_{0}$
periodicity, which is commonly observed in almost all the earlier studies pc1
; pc2  involving circular current in AB loops.

Origin of the dip in $E_{g}$, and hence, the kink in $I$ due to $U$:
Appearance of a dip in the ground state energy is an interesting phenomenon
which yields a kink-like structure in the $I$-$\phi$ curve since the current
is directly proportional to the first order derivative of $E_{g}$ with respect
to $\phi$. For the system with two opposite spin electrons we can write the
full many-body Hamiltonian matrix in a ‘block diagonal’ form by selectively
choosing the basis states from two different sub-spaces associated with net
spin $S$. Here we have two possibilities of $S$: $S=0$ and $S=1$. For
instance, a basis state like
$(1/\sqrt{2})(|\uparrow,\downarrow,0,0,...\rangle-|\downarrow,\uparrow,0,0,...\rangle)$
belongs to the spin sub-space with $S=0$, while a state like
$(1/\sqrt{2})(|\uparrow,\downarrow,0,0,...\rangle+|\downarrow,\uparrow,0,0,...\rangle)$
lies in the other spin sub-space i.e., $S=1$. The eigenstates of the block
matrix of the spin sub-space $S=1$ are no longer affected by the Hubbard
correlation $U$ which we confirm through our exhaustive numerical
calculations. Only the eigenstates of the other block matrix for the spin sub-
space $S=0$ are influenced by the electronic correlation. When $U=0$, the
$U$-independent and $U$-dependent energy levels exactly touch each other at
$\phi=0$. On the other hand, for finite $U$ the $U$-independent energy level
remain unchanged whereas the energies of the other levels get increased due to
repulsive interaction. Because of this, the $U$-independent energy level
becomes the lowest energy level within a small window across $\phi=0$ which
provides a dip in $E_{g}$. As the energies of the $U$-dependent levels
increase, we get more higher dip with $U$. The dip in $E_{g}$-$\phi$ curve
results a kink in the current spectrum.

Case II. Three-electron system: For the system with three electrons
($\uparrow,\uparrow,\downarrow$), the slope of the energy and the current are
very less sensitive even at moderate $U$ (Figs. 2(b) and (g)). Unlike the two-
electron system, here a large kink appears around $\phi=0$ even when $U=0$.
The appearance of the kink is solely related to the variation of $E_{g}$ which
on the other hand depends on the number of electrons in a system. The less
sensitivity of current on $U$ for this typical three-electron case can be
physically explained as follows. As the system contains two up and one down
spin electrons, mutual repulsion between one up and one down spin electrons
occurs while there is always another up spin electron that can move in the
geometry without facing the repulsive effect. Thus, for the low or even
moderate $U$, the effect of $U$ in the current spectrum is not properly seen.
For large enough $U$, minor change is obtained which we confirm through our
detailed numerical calculations. The current provides the usual $\phi_{0}$
periodicity. Here it is worthy to note that, instead of considering a two-ring
system with $10$ lattice sites (i.e., what is taken in our present case), if
we would take a single ring ($N_{y}=1$) with $N_{x}=3$ then the current would
exhibit $\phi_{0}/2$ periodicity. More precisely we can say that, the current
shows $\phi_{0}/2$ periodicity for the odd half-filled case (not shown here in
Fig. 2(g)). Its origin will be understood from our forthcoming analysis.

Case III. Four-electron system: In the absence of electronic correlation
($U=0$), current shows a sharp transition at $\phi=0$ (solid line of Fig.
2(h)) following the $E_{g}$-$\phi$ curve (solid curve of Fig. 2(c)), like what
is observed for the two-electron system. This phenomenon disappears when the
Hubbard correlation is included. The sharp discontinuity in $I$-$\phi$ curve
is directly related to the sudden change in slope of $E_{g}$ with $\phi$. The
later one is again involved with the crossing of the neighboring energy
levels. For $U=0$, we find a crossing of the lowest energy level with the
neighboring levels across $\phi=0$ which gives rise to a sudden change in
slope. This is a general feature i.e., whenever there is an intersection among
the two or more energy levels, a sudden change in slope occurs. On the other
hand, for finite $U$ the energy levels are well separated without having any
intersection. Accordingly we get a smooth variation. The usual reduction of
current with $U$ is due to the repulsive interaction between the opposite
electrons. The reduction becomes more effective when the unoccupied lattice
sites are less, and, eventually a situation can be made when all the sites are
occupied by at least one electron (viz, the half-filled band case), then no
electron can move from one site to the other. Under this condition, the system
reaches to the insulating phase. Here also we get $\phi_{0}$ periodic circular
currents.

Case IV. Five-electron system: The results of the five-electron system
($\uparrow,\uparrow,\uparrow,\downarrow,\downarrow$) are shown in Figs. 2(d)
and (i). The kink-like structure at different $\phi$ values is clearly
understood from the variation of $E_{g}$. For this system the reduction of
current with $U$ is quite prominent since the number of electrons is quite
higher compared to the previous cases. Among the five electrons, there are two
pairs of up and down spin electrons, and hence the repulsive interaction
dominates over the motion of other single electron that can get unoccupied
sites to hop. For this five-electron system, the energy and the current
exhibit conventional $\phi_{0}$ periodicity.

Case V. Six-electron system: Finally, we focus on the spectra Figs. 2(e) and
(j) where the results of the six-electron
($\uparrow,\uparrow,\uparrow,\downarrow,\downarrow,\downarrow$) system are
presented. For this case the system is half filled as six electrons are
accommodated among the six sites. Both at $\phi=0$ and $\phi=\pm 0.5$, kinks
appear in the current following the energy-flux characteristics, as long as
the e-e interaction is included. This is similar to what we get in the other
electron systems studied above. But, the notable observation is that, unlike
$\phi_{0}$ periodicity, here we get $\phi_{0}/2$ periodicity. This is quite
atypical compared to the reported pc6 ; pc28  $\phi_{0}/2$ periodicity for the
‘ensemble averaged’ current, as in our case no such averaging is done. The
appearance of $\phi_{0}/2$ periodicity is one of the key features of our
analysis, and below we make an attempt to explain this fact with a suitable
mathematical description. Maybe more comprehensive and simpler explanations
will be available in near future.

Possible origin of getting $\phi_{0}/2$ periodic current – An analytical
treatment: To understand the basic mechanism, let us start with a simple
system, as a typical example, which is a one-dimensional five-site ($N_{x}=5$)
non-interacting ($U=0$) AB ring with five electrons
($\uparrow,\uparrow,\uparrow,\downarrow,\downarrow$). For this perfect ring,
we set the site energy to zero, without loss of any generality. Since the
periodicity in current is directly linked with the variation of ground state
energy with $\phi$, we concentrate on the ground state energy $E_{g}(\phi)$.

Suppose at any arbitrary flux $\phi_{1}$, the five eigenvalues of the ring
(arranged in higher order) are: $-a$, $-b$, $-c$, $d$ and $e$, and they should
obey the relation $-a-b-c+d+e=0$ i.e.,

$d+e=a+b+c$ (8)

since the sum of the eigenvalues is equal to the trace of the Hamiltonian
matrix of the ring. Now, we consider another magnetic flux $\phi_{2}$
($\phi_{1}\neq\phi_{2}$) and imagine that for $\phi=\phi_{2}$ all the above
mentioned eigenvalues get flipped with a negative sign i.e., they are $-e$,
$-d$, $c$, $b$ and $a$. Thus, they also satisfy the relation

$a+b+c=d+e.$ (9)

Hence, for the half-filled band case the ground state energy at $\phi_{1}$ is
$E_{g}(\phi_{1})=-2a-2b+c$, and at $\phi_{2}$ it becomes
$E_{g}(\phi_{2})=-2e-2d+c$. Using Eq. 8, we see that

$E_{g}(\phi_{1})=E_{g}(\phi_{2}).$ (10)

For even $N_{x}$, reaching to the above condition is quite obvious if similar
swapping of energy levels takes place at these two distinct fluxes.

Thus, swapping the energy eigenvalues is the necessary condition to get
identical ground state energy at two different AB fluxes. Now, the question is
that, if we set $\phi_{1}$ at any particular value within one flux-quantum
window, then can we find any other flux $\phi_{2}$ within this window such
that $E_{g}(\phi_{1})=E_{g}(\phi_{2})$.

It can be answered with the help of the energy eigenvalue equation for a
$N_{x}$-site AB ring which is

$E_{n}(\phi)=2t_{x}\cos\left[\frac{2\pi}{N_{x}}\left(n+\frac{\phi}{\phi_{0}}\right)\right]$
(11)

where $n$ is the energy level index. Thus, for any two energy levels, (say)
$n_{1}$ and $n_{2}$, the eigenvalues at $\phi_{1}$ and $\phi_{2}$ are:

$\displaystyle E_{n_{1}}(\phi_{1})$ $\displaystyle=$ $\displaystyle
2t_{x}\cos\left[\frac{2\pi}{N_{x}}\left({n_{1}}+\frac{\phi_{1}}{\phi_{0}}\right)\right],$
(12) $\displaystyle E_{n_{2}}(\phi_{2})$ $\displaystyle=$ $\displaystyle
2t_{x}\cos\left[\frac{2\pi}{N_{x}}\left({n_{2}}+\frac{\phi_{2}}{\phi_{0}}\right)\right].$
(13)

Now, to have $E_{n_{1}}(\phi_{1})=-E_{n_{2}}(\phi_{2})$, we get the condition
from Eq. 12 and Eq. 13 as

$\frac{2\pi}{N_{x}}\left({n_{2}}+\frac{\phi_{2}}{\phi_{0}}\right)-\frac{2\pi}{N_{x}}\left({n_{1}}+\frac{\phi_{1}}{\phi_{0}}\right)=\pi$
(14)

which simplifies to

$(n_{2}-n_{1})+\frac{(\phi_{2}-\phi_{1})}{\phi_{0}}=\frac{N_{x}}{2}.$ (15)

From Eq. 15, we can now clearly explain the origin of $\phi_{0}/2$
periodicity. As ($n_{2}-n_{1}$) is also an integer, $\phi_{2}-\phi_{1}$ can
achieve the value $\pm\phi_{0}/2$ only when $N_{x}$ is ‘odd’. Thus, if we set
$\phi_{1}=0$ then perfect swapping takes place at $\phi_{2}=\pm\phi_{0}/2$
which results the half flux-quantum periodicity. On the other hand, for even
$N_{x}$, $\phi_{2}-\phi_{1}$ becomes identical to $\phi_{0}$, which yields the
conventional $\phi_{0}$ flux-quantum periodicity.

The above mathematical analysis can easily be extended in multi-ring systems
to justify the occurrence of $\phi_{0}/2$ periodicity. From the physical point
of view it is inferred that, the central mechanism of getting such half flux-
quantum periodicity for the odd half-filled case relies on the particle-hole
symmetry of the system.

###  IV.2 Interplay between e-e interaction and disorder: Fully disordered and
ODS nanotubes

The situation becomes more interesting and complicated as well, when we
include impurities along with the Hubbard correlation. Two different cases are
considered depending on the distribution of impurities in the system, and they
are categorized as (i) full disordered (FD) and (ii) ordered-disordered
separated (ODS) ones. For a FD nanotube, impurities are added at all the
lattice sites as conventionally used, whereas for the ODS nanotube, impurities
are given in one half of the tube keeping the other part free. In Fig. 3 we
present the variations of current amplitudes, computed at a typical magnetic
flux $\phi=0.2$, by changing the impurity strength $W$ in a wide range. The
results of $U=0$ are also superimposed in the current spectra to have a clear
comparison between the interacting and non-interacting ($U\neq 0$) cases.

Figure 3: Absolute current, computed at a typical magnetic flux $\phi=0.2$, as
a function of the impurity strength $W$ for the FD (shown in (a) and (c)) and
ODS (given in (b) and (d)) systems. Here we take $N_{y}=2$ and $N_{x}=4$. The
left and the right columns represent the systems with four
($\uparrow,\uparrow,\downarrow,\downarrow$) and five
($\uparrow,\uparrow,\uparrow,\downarrow,\downarrow$) electrons, respectively.
The results are worked out for both the non-interacting and interacting cases.

A sharp contrasting behavior is obtained for these two kinds of disordered
multi-ring systems. In the presence of uniform disorder, current amplitude
gradually decreases with $W$ and eventually drops to zero, irrespective of the
e-e interaction strength (Figs. 3(a) and (c)). This reduction is fully
associated with the impurity induced electronic localization which is known as
Anderson localization pc18 ; pc19 ; anlc1 ; anlc2 ; anlc3 ; anlc4 . More
disorder yields more localization, and in the limit of large $W$ current
amplitude practically vanishes, as expected. Comparing the current spectra, it
is clearly seen that the rate of fall of current with $W$ becomes slower for
the interacting system than the non-interacting one. This is because of the
repulsive nature of electrons which does not favor to sit two opposite spin
electrons in one site. For the disordered system, the repulsive Coulomb
interaction eventually leads to a delocalization and thus enhanced current is
obtained compared to the interacting free one.

The effect of $W$ is reasonably different for the ODS system, where a gradual
reduction of current is not seen over the full window, rather a completely
opposite signature is obtained beyond a critical value of $W$ (Figs. 3(b) and
(d)). In the limit of weak $W$, current gets reduced with $W$ like what we get
in the FD system. Reaching to a minimum at moderate $W$, the current starts to
enhance with the impurity strength and it achieves a sufficiently large value
when $W$ is too high. The underlying physical mechanism for this atypical
nature is described as follows. The ODS system can be considered as a coupled
system where the ordered region is directly coupled to the disordered one. The
coupling between these two regions gradually reduces with $W$, and eventually
drops almost to zero when $W$ is too large. Thus, for the weak $W$ as the
disordered region is coupled to the ordered part, the energy eigenstates gets
affected by the impurities which results a reduction of current with
increasing the impurity strength. At the critical limit the scattering effect
becomes maximum resulting a minimum current amplitude. Beyond that limit, the
states start to get less influenced by the impurities since then the
decoupling between the two regions becomes more prominent. Finally, when $W$
is too large, the ordered and disordered parts are practically decoupled, and
then the contribution towards current comes from the ordered region.

Figure 4: Energy eigenvalues as a function of the impurity strength $W$ for
the two-ring ($N_{y}=2$) non-interacting ($U=0$) system, where each of these
rings contains $4$ lattice sites. Here (a) and (b) correspond to the FD and
ODS systems, respectively. For the ODS case, the decoupling between the
ordered and disordered regions in the limit of high $W$ can be easily
followed.

The delocalization process gets enhanced further with the inclusion of Hubbard
correlation, and accordingly, the critical point of current minimum gets
shifted to a larger $W$. Both for the zero and non-zero values of $U$, the
signature of the current with impurity strength remains same, as clearly seen
from the spectra. In the weak disorder regime the many-body ground state
becomes the ‘coherent’ linear combination of the localized states, emerged
from the disordered zone, and the delocalized states that are generated from
the ordered counterpart. On the other hand, in the strong disorder regime, the
ground state becomes an ‘incoherent’ product of these states. During the
transition from weak-to-strong disorder region, current provides a soft
minimum. Now, to visualize the decoupling between the two different regions
more clearly, in Fig. 4 we show the dependence of energy eigenvalues with $W$,
by varying it in a wide range. For the sake of simplification, here we set
$U=0$ so that single particle energy eigenvalues are obtained quite easily by
diagonalizing the TB Hamiltonian. For the FD system, all the eigenvalues are
affected whether $W$ is small or large (see Fig. 4(a)). On the other hand,
among the eight energy levels of the ODS system, four are affected and the
remaining four levels are almost undisturbed (Fig. 4(b)), and it becomes more
prominent when $W$ is quite large. It clearly suggests that the ordered region
gets separated from its disordered counter part.

From the results of the ODS system, presented in Figs. 3(b) and (d), a
question naturally comes that how does $W_{c}$ (critical $W$ for which the
current achieves a minimum) vary with the interaction strength $U$. The
results are shown in Fig. 5 and they

Figure 5: Dependence of $W_{c}$ as a function of $U$ for the ODS system. The
lines with filled black boxes and blue circles are associated with the four
and five-electron cases as taken in Fig. 3, respectively. The system size and
the other physical parameters remain also same as considered in Fig. 3. The
results are worked out by averaging over $100$ distinct disordered
configurations.

are computed for the identical electron systems as taken in Fig. 3. For each
$U$, the critical $W$ is obtained by scanning the current amplitude over a
wide range of $W$. Both for the four- and five-electron cases, $W_{c}$ shows
an increasing behavior providing almost a linear dependence with $U$. For
other set of electrons, we also get a similar kind of variation which we
confirm through our exhaustive numerical calculations. This enhancement of
$W_{c}$ is solely associated with the electronic delocalization in the
presence of e-e correlation, as already stated above. From this $W_{c}$-$U$
diagram we can estimate the high and low conducting natures depending on the
other physical parameters of the system.

###  IV.3 Mean-field results

All the characteristic features analyzed so far are worked out by
diagonalizing the full many-body Hamiltonian, and therefore the results are
restricted in small size systems. To examine whether the features studied
above still persist even for larger ring sizes as well, we now focus on the
mean-field results those are computed for much larger systems than the
previous ones. Before that, first we need to check the

Figure 6: Comparison between the results obtained from the exact
diagonalization of the full many-body Hamiltonian and the HF MF approach.
Currents are calculated for the ODS two-ring system having five electrons
($\uparrow,\uparrow,\uparrow,\downarrow,\downarrow$), where each ring contains
four sites. The other physical parameters are: $U=0.5$ and $W=2$. Averaging
over same number of distinct disordered configurations is taken into account,
as considered in Fig. 5.

accuracy and reliability of the MF results, particularly in the presence of
random impurities. For that, in Fig. 6 we make a comparison between the
currents obtained in the two methods used in our work. As the result of the
full TB many-body Hamiltonian is shown along with the MF result, here we
choose a small size system (eight lattice sites) with five electrons (three up
and two down). The currents evaluated in these two separate prescriptions
almost overlap with each other for the entire flux-quantum window. Thus, we
can safely use MF results for further analysis. Here it is relevant to note
that finding an optimized converged solution in the MF scheme is the most
important issue. In the absence of impurities, one can get the converged
solution quite easily even for large value of $U$. But as long as impurities
are given, it takes much longer time to reach into the final solution, and the
situation becomes more tedious for higher values of $U$. In that case we need
to check the results with different initial guess values $\langle
n_{j,k,\uparrow}\rangle$ and $\langle n_{j,k,\downarrow}\rangle$, such that no
spurious result appears.

Considering a much bigger ODS nanotube, in Fig. 7 we show the variation of
absolute current as a function of the disorder strength $W$. The nature of the
current spectrum remains unchanged like what we get for smaller rings (Figs.
3(b) and (d)). In the limit of weak $W$, current gets reduced with $W$
providing a localizing behavior, while the delocalization process gets started
when $W$ crosses its critical value. This is the generic feature of an ODS
nanotube and can be exploited in different ways. In addition, the

Figure 7: Absolute current, computed at a typical magnetic flux
$\phi=\phi_{0}/4$, as a function of $W$ for a much bigger ODS nanotube having
$N_{x}=10$ and $N_{y}=8$. Here we set $U=0.5$ and choose
$N_{\uparrow}=N_{\downarrow}=35$.

other notable aspect is that once the system goes to the delocalized phase, it
never comes back to the localized phase even at too large $W$ since for such a
system mobility edge (ME) always persists. This behavior is completely
different compared to the uniform random disordered 3D lattices, where ME
vanishes at large disorder. The ME separates the localized states from the
conducting ones skmprl , which thus yields a localization-to-delocalization
transition. In analyzing the behavior of $I$-$W$ characteristics of the ODS
system (Figs. 3(b) and (d)), it is explained that the coupling between the
ordered and disordered sectors gradually decreases with $W$, and for large
enough $W$ these two regions are practically decoupled (the phenomenon can
also be understood clearly from Fig. 4(b)). Under this situation, localized
states are emerged from the disordered part, while extended states are
generated from the other region. Therefore, a mixture of both the localized
and extended states are obtained for the ODS system which leads to the
existence of a ME. On the other hand, for the traditional ‘uniform’

Figure 8: Current-flux characteristics for ‘odd half-filled’ nanotubes
considering $N_{x}=7$ and $N_{y}=5$, where (a) and (b) correspond to the fully
ordered (FO), (c) and (d) denote ODS, and (e) and (f) represent FD nanotubes,
respectively. Here we set $W=2$ and choose $N_{\uparrow}=18$ and
$N_{\downarrow}=17$. The dashed and solid curves are associated with $U=0$ and
$U=0.02$, respectively. For the ODS case, impurities are included in the upper
two rings of the tube, keeping the other three rings free. In presence of
disorder, averaging over $2000$ distinct random configurations are taken into
account for each case.

random disordered 3D lattices ‘all the states’ become localized for large $W$,
as in this case there is no ordered region that can be decoupled from the
disordered one. Hence, for a uniform disordered lattice ME does not exist, and
one cannot get the conducting phase at large $W$, unlike an ODS system.

Finally, we focus on the current spectra shown in Fig. 8, where our main
intentions are to check whether (i) we can get half flux-quantum periodicity
in the odd half-filled band case in a multi-channel system i.e., in a nanotube
and (ii) this behavior still persists even in the presence of disorder. To
substantiate these facts, we present the results of three different kinds of
nanotubes viz, fully ordered (FO), ordered-disordered separated (ODS) and
fully disordered (FD), and in each nanotube we choose $N_{x}=7$ and $N_{y}=5$,
with $N_{\uparrow}=18$ and $N_{\downarrow}=17$ (half-filled band case). In the
disordered systems (ODS and FO), we compute the currents taking the average
over a large number ($2000$) of distinct random configurations such that more
accurate results are obtained. For the ordered nanotube, current exhibits half
flux-quantum periodicity both in the absence and presence of e-e correlation
(Figs. 8(a) and (b)), and this feature is expected from our previous analysis.
But the interesting fact is that, even for the disordered nanotube, be it
fully or partly disordered, the phenomenon of half flux-quantum periodicity
still persists which is clearly reflected from the spectra given in Figs.
8(c)-(f). From our detailed numerics we find that, the $\phi_{0}/2$
periodicity is always obtained, independent of the choice of the number of
rings in the nanotube, as long as $N_{x}$ is ‘odd’ and the system is half-
filled. This is a direct consequence of particle-hole symmetry. Once we set
even $N_{x}$, no such phenomenon is observed and we get back to the
conventional $\phi_{0}$ periodicity.

##  V Experimental Feasibility

Nowadays the designing of ring conductors at nanoscale level is not at all
complicated due to the availability of advanced nanotechnologies. Several
approaches are there like silicon etching process fb1 , electron beam
lithography pc13 , nanosphere lithography fb3 ; fb4 , and to name a few. With
the help of these techniques different sized thin and thick rings, nanotubes
and other tailor made geometries can be fabricated. Very recently Pham et al.
have established a new approach of devising a ring like structure on a
semiconducting surface using the atom manipulation technique pham . With this
prescription atomically precise quantum ring can be designed. In another work
Keyser et al. performed an experiment almost two decades ago to explore the
influence of Coulomb correlation on Aharonov-Bohm oscillations considering a
nano ring with less than ten electrons keys . Usually Coulomb interaction
becomes significant in smaller rings with few number of electrons since in
this case the interaction is not screened so much. All the physical parameters
chosen in our theoretical work are consistent with realistic systems, and
thus, we strongly believe that the results studied here can be tested in a
suitable laboratory.

##  VI Closing Remarks

To conclude, in the present work we investigate theoretically magnetic
response of interacting electrons in a ‘spatially non-uniform disordered
system’. The response is measured by the flux driven circular current in a
multi-channel nanotube whose one segment is subjected to disorder keeping the
other part free. A sharp contrasting behavior is obtained compared to the
conventional uniform disordered material. The e-e correlation is included in
the well known Hubbard form. Describing the quantum system using a tight-
binding framework, we compute the results in two ways: (i) exactly
diagonalizing the full many-body Hamiltonian and (ii) Hartree-Fock mean-field
scheme. The latter is used for bigger systems having higher number of
electrons. Several important features emerge from our analysis those are
summarized as follows.
$\bullet$ Because of the e-e interaction kink-like structure appears in the
current following a sudden change in slope around $\phi=0$ and/or
$\phi=\pm\phi_{0}/2$ in the ground state energy. These kinks are specifically
associated with the correlation among the up and down spin electrons.
$\bullet$ For the odd half-filled band case, current always exhibits
$\phi_{0}/2$ periodicity irrespective of the number of rings in the nanotube
and disorderness.
$\bullet$ The results are worked out for different sizes of the nanotubes, and
all the physical pictures are equally valid in all these cases as long as we
restrict ourselves in the meso-scale region. In the asymptotic limit
($N\rightarrow\infty$), no current will appear since it is a mesoscopic
phenomenon.
$\bullet$ Before discussing MF results, we test the accuracy of MF
calculations and see that almost for the entire flux window the current
obtained in the MF scheme matches very well with the other method where many-
body energy levels are determined.
$\bullet$ For the FD system, current amplitude decreases and eventually drops
to zero with $W$ where all the states are localized because of the disorder.
On the other hand, a completely different scenario is obtained for the ODS
nanotube, where delocalizing behavior is obtained beyond a critical $W$. This
is a robust phenomenon and does not vanish even in the large limit of $W$.
$\bullet$ Finally, the interplay between e-e correlation and disorder is also
critically examined. Depending on the winning factor current gets reduced or
enhanced.

Before an end, we would like to note that working with many-body systems has
always been a challenging task. Here we attempt to make an in-depth analysis
to discuss some important phenomena of the interacting electrons in a
unconventional disordered system, that might be useful to analyze electronic
and magnetic properties of several such other systems.

## ACKNOWLEDGMENT

SKM acknowledges the financial support of DST-SERB, Government of India, under
the Project Grant Number EMR/2017/000504 for carrying out his research.

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