# On the way to meet the experimental observation of persistent current in a
mesoscopic cylinder: A mean field study

Santanu K. Maiti<EMAIL_ADDRESS>Theoretical Condensed Matter
Physics Division, Saha Institute of Nuclear Physics, Sector-I, Block-AF,
Bidhannagar, Kolkata-700 064, India  Department of Physics, Narasinha Dutt
College, 129 Belilious Road, Howrah-711 101, India

###### Abstract

The behavior of persistent current in a mesoscopic cylinder threaded by an
Aharonov-Bohm flux $\phi$ is carefully investigated within a Hartree-Fock mean
field approach. We examine the combined effect of second-neighbor hopping
integral and Hubbard correlation on the enhancement of persistent current in
presence of disorder. A significant change in current amplitude is observed
compared to the traditional nearest-neighbor hopping model and the current
amplitude becomes quite comparable to experimental realizations. Our analysis
is found to exhibit several interesting results which have so far remained
unaddressed.

###### pacs:

73.23.-b, 73.23.Ra.

##  I Introduction

Over the last many years appearance of persistent current in metallic single-
channel rings and multi-channel cylinders has drawn much attention in
theoretical as well as in experimental research. In mesoscopic regime where
dimensions of a system are comparable to mean free path of an electron the
phase coherence of electronic states is of fundamental importance and the
existence of dissipationless current in a mesoscopic conducting ring threaded
by an Aharonov-Bohm (AB) flux $\phi$ is a direct consequence of quantum phase
coherence. In this new quantum regime, two important aspects appear at low
temperatures and they are as follows.
$\bullet$ The phase coherence length $L_{\phi}$ i.e., the length scale for
which an electron maintains its phase memory, increases significantly with the
lowering of temperature and becomes comparable to the system size $L$.
$\bullet$ The energy levels of these small finite size systems are discrete.
These two are the most essential criteria for the existence of persistent
current in a small metallic ring/cylinder due to the application of an
external magnetic flux $\phi$. In the pioneering work of Büttiker, Imry and
Landauer butt , the appearance of persistent current in metallic rings has
been explored. Later, many excellent experiments levy ; chand ; mailly ; jari
; deb ; reu  have been carried out in several ring and cylindrical geometries
to reveal the actual mechanisms of persistent current. Though much efforts
have been paid to study persistent current both theoretically cheu1 ; cheu2 ;
peeters1 ; peeters2 ; peeters3 ; mont ; mont1 ; alts ; von ; schm ; ambe ;
abra ; bouz ; giam ; yu ; belu ; ore ; xiao1 ; xiao2 ; san1 ; san2 ; san3  as
well as experimentally levy ; chand ; mailly ; jari ; deb ; reu , yet several
drawbacks still exist between the theory and experiment, and the full
knowledge about it in this scale is not well established even today.

The results of the single loop experiments are significantly different from
those for the ensemble of isolated loops. Persistent currents with expected
$\phi_{0}$ periodicity have been observed in isolated single Au rings chand
and in a GaAs-AlGaAs ring mailly . Levy et al. levy  found oscillations with
period $\phi_{0}/2$ rather than $\phi_{0}$ in an ensemble of $10^{7}$
independent Cu rings. Similar $\phi_{0}/2$ oscillations were also reported for
an ensemble of disconnected $10^{5}$ Ag rings deb  as well as for an array of
$10^{5}$ isolated GaAs-AlGaAs rings reu . In a recent experiment, Jariwala et
al. jari  obtained both $\phi_{0}$ and $\phi_{0}/2$ periodic persistent
currents in an array of thirty diffusive mesoscopic Au rings. Except for the
case of the nearly ballistic GaAs-AlGaAs ring mailly , all the measured
currents are in general one or two orders of magnitude larger than those
expected from the theory.

Free electron theory predicts that at $T=0$, an ordered one-dimensional
metallic ring threaded by magnetic flux $\phi$ supports persistent current
with maximum amplitude $I_{0}=ev_{F}/L$, where $v_{F}$ is the Fermi velocity
and $L$ is the circumference of the ring. Metals are intrinsically disordered
which tends to decrease the persistent current, and the calculations show that
the disorder-averaged current $\langle I\rangle$ crucially depends on the
choice of the ensemble cheu2 ; mont ; mont1 . The magnitude of the current
$\langle I^{2}\rangle^{1/2}$ is however insensitive to the averaging issues,
and is of the order of $I_{0}l/L$, $l$ being the elastic mean free path of the
electrons. This expression remains valid even if one takes into account the
finite width of the ring by adding contributions from the transverse channels,
since disorder leads to a compensation between the channels cheu2 ; mont .
However, the measurements on an ensemble of $10^{7}$ Cu rings levy  reported a
diamagnetic persistent current of average amplitude $3\times 10^{-3}$
$ev_{F}/L$ with half a flux-quantum periodicity. Such $\phi_{0}/2$
oscillations with diamagnetic response were also found in other persistent
current experiments consisting of ensemble of isolated rings deb ; reu .

Measurements on single isolated mesoscopic rings on the other hand detected
$\phi_{0}$-periodic persistent currents with amplitudes of the order of
$I_{0}\sim ev_{F}/L$, (closed to the value for an ordered ring). Theory and
experiment mailly  seem to agree only when disorder is weak. In another recent
nice experiment Bluhm et al. blu  have measured the magnetic response of $33$
individual cold mesoscopic gold rings, one ring at a time, using a scanning
SQUID technique. They have measured $h/e$ component and predicted that the
measured current amplitude agrees quite well with theory cheu1  in a single
ballistic ring mailly  and an ensemble of $16$ nearly ballistic rings raba .
However, the amplitudes of the currents in single-isolated-diffusive gold
rings chand  were two orders of magnitude larger than the theoretical
estimates. This discrepancy initiated intense theoretical activity, and it is
generally believed that the electron-electron correlation plays an important
role in the disordered diffusive rings abra ; bouz ; giam . An explanation
based on the perturbative calculation in presence of interaction and disorder
has been proposed and it seems to give a quantitative estimate closer to the
experimental results, but still it is less than the measured currents by an
order of magnitude, and the interaction parameter used in the theory is not
well understood physically. Most of these theoretical results have been
obtained based on a tight-binding framework within the nearest-neighbor
hopping (NNH) approximation. This is an important approximation and it has
been shown that within the NNH model electronic correlation provides a small
enhancement of current amplitude in disordered materials i.e., a weak
delocalizing effect is observed in presence of electron-electron (e-e)
interaction. As an attempt in the present work we modify the traditional NNH
model by incorporating the effects of higher order hopping integrals, at least
second-neighbor hopping (SNH), in addition to the NNH integral. It is also
quite physical since electrons have some finite probabilities to hop from one
site to other sites apart from nearest-neighbor with reduced strengths. We
will show that the inclusion of higher order hopping integrals gives
significant enhancement of current amplitude and it reaches quite closer to
the current amplitude of ordered systems.

The other important controversy comes for the determination of the sign of
low-field currents and still it is an unresolved issue between theoretical and
experimental results. In an experiment on persistent current Levy et al. levy
have shown diamagnetic nature for the measured currents at low-field limit.
While, in other experiment Chandrasekhar et al. chand  have obtained
paramagnetic response near zero field limit. Jariwala et al. jari  have
predicted diamagnetic persistent current in their experiment and similar
diamagnetic response in the vicinity of zero field limit were also supported
in an experiment done by Deblock deb  et al. on Ag rings. Yu and Fowler yu
have shown both diamagnetic and paramagnetic responses in mesoscopic Hubbard
rings. Though in a theoretical work Cheung et al. cheu2  have predicted that
the direction of current is random depending on the total number of electrons
in the system and the specific realization of the random potentials. Hence,
prediction of the sign of low-field currents is still an open challenge and
further studies on persistent current in mesoscopic systems are needed to
remove the existing controversies.

In the present paper we address the behavior of persistent current in an
interacting mesoscopic ring with finite width threaded by an Aharonov-Bohm
flux $\phi$. A simple tight-binding Hamiltonian is used to illustrate the
system and all the calculations are performed within a mean field approach.
Using a generalized Hartree-Fock (HF) approximation kato ; kam , we compute
numerically persistent current ($I$) as functions of AB flux $\phi$, total
number of electrons $N_{e}$, e-e interaction strength $U$, second-neighbor
hopping integral, disorder strength $W$ and system size $N$. The main
motivation of the present work is to illustrate the effects of higher order
hopping integrals on the enhancement of persistent current in disordered
mesoscopic cylinders. Our results can be utilized to study magnetic response
in any interacting mesoscopic system.

In what follows, we present the results. In Section II, we describe the
geometric model and generalized Hartree-Fock theory to study magnetic response
in the model quantum system. Section III contains the numerical results.
Finally, in Section IV we draw our conclusions.

##  II Model and synopsis of the theoretical formulation

Let us start by referring to Fig. 1, where a small metallic cylinder is
threaded by a magnetic flux $\phi$. The filled black circles correspond to the
positions of the atomic sites in the cylinder. To predict

Figure 1: (Color online). Schematic view of a $1$D mesoscopic cylinder
penetrated by a magnetic flux $\phi$. The red dashed line corresponds to the
second-neighbor hopping integral and the filled black circles represent the
positions of the atomic sites. A persistent current $I$ is established in the
cylinder.

the size of a cylinder we use two parameters $N$ and $M$, where the $1$st one
($N$) represents total number of atomic sites in each circular ring and the
other one ($M$) gives total number of identical circular rings. For the
description of our model quantum system we use a tight-binding (T-B) framework
and in order to incorporate the effect of higher order hopping integrals to
the Hamiltonian here we consider second-neighbor hopping (SNH) (shown by the
red dashed line in Fig. 1) in addition to the nearest-neighbor hopping (NNH)
of electrons. Considering both NNH and SNH integrals the T-B Hamiltonian for
the cylindrical system in Wannier basis looks in the form,

$\displaystyle H_{\mbox{c}}$ $\displaystyle=$
$\displaystyle\sum_{i,j,\sigma}\epsilon_{i,j,\sigma}c_{i,j,\sigma}^{\dagger}c_{i,j,\sigma}+\sum_{i,j,\sigma}t_{l}\left[e^{i\theta_{l}}c_{i,j,\sigma}^{\dagger}c_{i,j+1,\sigma}\right.$
(1) $\displaystyle+$
$\displaystyle\left.h.c.\right]+\sum_{i,j,\sigma}t_{d}\left[e^{i\theta_{d}}c_{i,j,\sigma}^{\dagger}c_{i+1,j+1,\sigma}+h.c.\right]$
$\displaystyle+$
$\displaystyle\sum_{ij}Uc_{i,j,\uparrow}^{\dagger}c_{i,j,\uparrow}c_{i,j,\downarrow}^{\dagger}c_{i,j,\downarrow}$

where, ($i,j$) represent the co-ordinate of a lattice site. The index $i$ runs
from $1$ to $M$, while the integer $j$ goes from $1$ to $N$.
$\epsilon_{i,j,\sigma}$ is the on-site energy of an electron at the site
($i,j$) of spin $\sigma$ ($\uparrow,\downarrow$). $t_{l}$ and $t_{d}$ are the
NNH and SNH integrals, respectively. Due to the presence of magnetic flux
$\phi$ (measured in unit of the elementary flux quantum $\phi_{0}=ch/e$), a
phase factor $\theta_{l}=2\pi\phi/N$ appears in the Hamiltonian when an
electron hops longitudinally from one site to its neighboring site, and
accordingly, a negative sign comes when the electron hops in the reverse
direction. $\theta_{d}$ is the associated phase factor for the diagonal motion
of an electron between two neighboring concentric rings. No phase factor
appears when an electron moves along the vertical direction which is set by
proper choice of the gauge for the vector potential $\vec{A}$ associated with
the magnetic field $\vec{B}$, and this choice makes the phase factors
($\theta_{l}$, $\theta_{d}$) identical to each other for the longitudinal and
diagonal motions. Since the magnetic field corresponding to the AB flux $\phi$
does not penetrate anywhere of the surface of the cylinder, we ignore Zeeman
term in the above tight-binding Hamiltonian (Eq. 1).
$c_{i,j,\sigma}^{\dagger}$ and $c_{i,j,\sigma}$ are the creation and
annihilation operators, respectively, of an electron at the site ($i,j$) with
spin $\sigma$. $U$ is the on-site Hubbard interaction term.

###  II.1 Decoupling of the interacting Hamiltonian

To get the energy eigenvalues of the interacting model quantum system
described by the above tight-binding Hamiltonian given in Eq. 1, first we
decouple the interacting Hamiltonian using generalized Hartree-Fock approach,
the so-called mean field approximation. In this procedure, the full
Hamiltonian is completely decoupled into two parts. One is associated with the
up-spin electrons, while the other is related to the down-spin electrons with
their modified site energies. For up and down spin Hamiltonians, the modified
site energies are expressed in the form,

$\epsilon_{i,j,\uparrow}^{\prime}=\epsilon_{i,j,\uparrow}+U\langle
n_{i,j,\downarrow}\rangle$ (2)
$\epsilon_{i,j,\downarrow}^{\prime}=\epsilon_{i,j,\downarrow}+U\langle
n_{i,j,\uparrow}\rangle$ (3)

where, $n_{i,j,\sigma}=c_{i,j,\sigma}^{\dagger}c_{i,j,\sigma}$ is the number
operator. With these site energies, the full Hamiltonian (Eq. 1) can be
written in the decoupled form as,

$\displaystyle H_{\mbox{c}}$ $\displaystyle=$
$\displaystyle\sum_{i,j}\epsilon_{i,j,\uparrow}^{\prime}n_{i,j,\uparrow}+\sum_{i,j}t_{l}\left[e^{i\theta_{l}}c_{i,j,\uparrow}^{\dagger}c_{i,j+1,\uparrow}+h.c.\right]$
(4) $\displaystyle+$
$\displaystyle\sum_{i,j}t_{d}\left[e^{i\theta_{d}}c_{i,j,\uparrow}^{\dagger}c_{i+1,j+1,\uparrow}+h.c.\right]$
$\displaystyle+$
$\displaystyle\sum_{i,j}\epsilon_{i,j,\downarrow}^{\prime}n_{i,j,\downarrow}+\sum_{i,j}t_{l}\left[e^{i\theta_{l}}c_{i,j,\downarrow}^{\dagger}c_{i,j+1,\downarrow}+h.c.\right]$
$\displaystyle+$
$\displaystyle\sum_{i,j}t_{d}\left[e^{i\theta_{d}}c_{i,j,\downarrow}^{\dagger}c_{i+1,j+1,\downarrow}+h.c.\right]$
$\displaystyle-$ $\displaystyle\sum_{i,j}U\langle
n_{i,j,\uparrow}\rangle\langle n_{i,j,\downarrow}\rangle$ $\displaystyle=$
$\displaystyle H_{\uparrow}+H_{\downarrow}-\sum_{i,j}U\langle
n_{i,j,\uparrow}\rangle\langle n_{i,j,\downarrow}\rangle$

where, $H_{\uparrow}$ and $H_{\downarrow}$ correspond to the effective tight-
binding Hamiltonians for the up and down spin electrons, respectively. The
last term is a constant term which provides an energy shift in the total
energy.

###  II.2 Self consistent procedure

With these decoupled Hamiltonians ($H_{\uparrow}$ and $H_{\downarrow}$) of up
and down spin electrons, now we start our self consistent procedure
considering initial guess values of $\langle n_{i,j,\uparrow}\rangle$ and
$\langle n_{i,j,\downarrow}\rangle$. For these initial set of values of
$\langle n_{i,j,\uparrow}\rangle$ and $\langle n_{i,j,\downarrow}\rangle$, we
numerically diagonalize the up and down spin Hamiltonians. Then we calculate a
new set of values of $\langle n_{i,j,\uparrow}\rangle$ and $\langle
n_{i,j,\downarrow}\rangle$. These steps are repeated until a self consistent
solution is achieved.

###  II.3 Calculation of ground state energy

After achieving the self consistent solution, the ground state energy $E_{0}$
for a particular filling at absolute zero temperature ($T=0$K) can be
determined by taking the sum of individual states up to Fermi energy ($E_{F}$)
for both up and down spins. Thus, we can write the final form of ground state
energy as,

$E_{0}=\sum_{p}E_{p,\uparrow}+\sum_{p}E_{p,\downarrow}-\sum_{i,j}U\langle
n_{i,j,\uparrow}\rangle\langle n_{i,j,\downarrow}\rangle$ (5)

where, the index $p$ runs for the states up to the Fermi level.
$E_{p,\uparrow}$ ($E_{p,\downarrow}$) is the single particle energy eigenvalue
for $p$-th eigenstate obtained by diagonalizing the Hamiltonian $H_{\uparrow}$
($H_{\downarrow}$).

###  II.4 Calculation of persistent current

At absolute zero temperature, total persistent current of the system is
obtained from the expression,

$I(\phi)=-c\frac{\partial E_{0}(\phi)}{\partial\phi}$ (6)

where, $E_{0}(\phi)$ is the ground state energy for a particular filling.

In the present work we perform all the essential features of persistent
current at absolute zero temperature and use the units where $c=h=e=1$.
Throughout our numerical calculations we set the nearest-neighbor hopping
strength $t_{l}=-1$ and fix $M=2$ i.e., cylinders with two identical rings.
Energy scale is measured in unit of $t_{l}$.

##  III Numerical results and discussion

Following the above theoretical prescription now we start to analyze our
numerical results. We describe the results in three different parts. In the
first part, we consider perfect cylinders with only nearest-neighbor hopping
integral. In the second part, disordered cylinders described with only NNH
integral are considered. Finally, in the third part we discuss the effect of
second-neighbor hopping (SNH) integral on the enhancement of persistent
current in disordered cylinders.

###  III.1 Perfect cylinders with NNH integral

For perfect cylinders we choose
$\epsilon_{i,j,\uparrow}=\epsilon_{i,j,\downarrow}=0$ for all ($i,j$). Since
here we consider the cylinders described with NNH integral only, the second-
neighbor hopping strength $t_{d}$ is fixed to zero.

####  III.1.1 Energy-flux characteristics

As illustrative examples, in Fig. 2 we show the variation of ground state
energy levels as a function of magnetic flux $\phi$ for some typical
mesoscopic cylinders where $N$ is fixed at $5$ (odd $N$). In (a) the results
are given for the quarterly-filled ($N_{e}=5$) cylinders, while in (b) the
curves correspond to the results for the half-filled ($N_{e}=10$) cylinders.
The red, green and blue lines represent the ground state energy levels for
$U=0$, $0.5$ and $1$, respectively. It is observed that the ground state
energy shows oscillatory behavior as a function of $\phi$ and the energy
increases as the electronic correlation strength $U$ gets increased. Most
significantly we see that the ground state energy levels provide two

Figure 2: (Color online). Ground state energy levels as a function of flux
$\phi$ for some perfect cylinders with $N=5$ and $M=2$. The red, green and
blue curves correspond to $U=0$, $0.5$ and $1$, respectively. (a) Quarter-
filled case and (b) Half-filled case.

Figure 3: (Color online). Ground state energy levels as a function of flux
$\phi$ for some perfect cylinders considering $N=8$ and $M=2$. The red, green
and blue curves correspond to $U=0$, $0.5$ and $1$, respectively. (a) Quarter-
filled case and (b) Half-filled case.

different types of periodicities depending on the electron filling. At
quarter-filling, ground state energy level gives $\phi_{0}$ ($=1$, since
$c=e=h=1$ in our chosen unit system) flux-quantum periodicity. On the other
hand, at half-filling it shows $\phi_{0}/2$ flux-quantum periodicity. The
situation becomes quite different when the total number of atomic sites $N$ in
individual rings is even. For our illustrative purposes in Fig. 3 we plot the
lowest energy levels as a function of $\phi$ for some typical mesoscopic
cylinders considering $N=8$ (even $N$). The curves of different colors
correspond to the identical meaning as in Fig. 2. From the spectra given in
Figs. 3(a) (quarter-filled case) and (b) (half-filled case) it is clearly
observed that the ground state energy levels vary periodically with AB flux
$\phi$ exhibiting only $\phi_{0}$ flux-quantum periodicity. Thus it can be
emphasized that the appearance of half flux-quantum periodicity strongly
depends on the electron filling as well as on the oddness and evenness of the
total number of atomic sites $N$ in individual rings. Only for the half-filled
cylinders with odd $N$, the lowest energy level gets $\phi_{0}/2$ periodicity
with flux $\phi$. Now it is important to note that this half flux-quantum
periodicity does not depend on the width ($M$) of the cylinder and also it is
independent of the Hubbard correlation strength $U$. Hence, depending on the
system size and filling of electrons variable periodicities are observed in
the variation of lowest energy level. It may provide an important signature in
studying magnetic response in nano-scale loop geometries.

####  III.1.2 Current-flux characteristics

In Fig. 4 we display the current-flux characteristics for some impurity free
mesoscopic cylinders considering $M=2$. In (a) the

Figure 4: (Color online). Persistent current as a function of flux $\phi$ for
some ordered mesoscopic cylinders considering $M=2$. (a) Half-filled case with
$N=15$. The red, green and blue curves correspond to $U=0$, $1.5$ and $2$,
respectively. (b) Quarter-filled case with $N=20$. The red, green and blue
curves correspond to $U=0$, $2$ and $3$, respectively.

results are given for the half-filled case where we set $N=15$. The red line
corresponds to the current for the non-interacting ($U=0$) case, while the
green and blue lines represent the currents when $U=1.5$ and $2$,
respectively. From the curves we notice that the current amplitude gradually
decreases with the increase of electronic correlation strength $U$. The reason
is that at half-filling each site is occupied by at least one electron of up
spin or down spin, and the placing of a second electron of opposite spin needs
more energy due to the repulsive effect of $U$. Thus conduction becomes
difficult as it requires more energy when an electron hops from its own site
and situates at the neighboring site. Now both for the non-interacting and
interacting cases, current shows half flux-quantum periodicity as a function a
$\phi$ obeying the energy-flux characteristics since here we choose odd $N$
($N=15$). The behavior of the persistent currents for even $N$ is shown in (b)
where we set $N=20$. The currents are drawn for the quarter-filled case i.e.,
$N_{e}=20$, where the red, green and blue curves correspond to $U=0$, $2$ and
$3$, respectively. The reduction of current amplitude with the increase of
Hubbard interaction strength is also observed for this quarter-filled case,
similar to the case of half-filled as described earlier. But the point is that
at quarter-filling, the reduction of current amplitude is much smaller
compared to the half-filled situation. This is quite obvious in the sense that
at less than half-filling ‘empty’ lattice sites are available where electrons
can hop easily without any cost of extra energy and the conduction becomes
much easier than the half-filled situation. In this quarter-filled case,
persistent currents provide only $\phi_{0}$ flux-quantum periodicity following
the $E$-$\phi$ diagram. From these current-flux characteristics it can be
concluded that for ‘ordered’ cylinders current amplitude always decreases with
the enhancement in Hubbard correlation strength $U$.

###  III.2 Disordered cylinders with NNH integral

In order to describe the effect of impurities on electron transport now we
focus our attention on the results of some typical disordered cylinders
described with NNH integral. Here we consider the diagonal disordered
cylinders i.e., impurities are introduced only at the site energies without
disturbing the hopping integrals. The site energies in each concentric ring
are chosen from a correlated distribution function which looks in the form,

$\epsilon_{j,\uparrow}=\epsilon_{j,\downarrow}=W\cos\left(j\lambda\pi\right)$
(7)

where, $W$ is the impurity strength. $\lambda$ is an irrational number and we
choose $\lambda=(1+\sqrt{5})/2$, for the sake of our illustration. Setting
$\lambda=0$, we get back the pure system with uniform site energy $W$. Now,
instead of considering site energies from a correlated distribution function,
as mentioned above in Eq. 7, we can also take them randomly from a “Box”
distribution function of width $W$. But in the later case we have to take the
average over a large number of disordered configurations (from the stand point
of statistical average) and since it is really a difficult task in the aspect
of numerical computation we select the other option. Not only that in the
averaging process several mesoscopic phenomena may disappear. Therefore, the
averaging process is an important issue in low-dimensional systems.

In presence of disorder, energy levels get modified significantly. For our
illustrative purposes in Fig. 5 we plot ground state energy levels as a
function of magnetic flux $\phi$ for some disordered mesoscopic cylinders when
they are half-filled. The Hubbard interaction strength $U$ is set at $1$ and
the impurity strength $W$ is fixed to $2$. In (a) the ground state energy
level is shown for a cylinder with $N=5$ (odd), while in (b) it is presented
for a cylinder taking $N=8$ (even). Quite interestingly we see that for the
cylinder with odd $N$, the half flux-quantum periodicity of the lowest energy
level disappears in the presence of impurity and it provides conventional
$\phi_{0}$ periodicity. Hence, for cylinders with odd $N$, $\phi_{0}/2$ flux-
quantum periodicity will be observed only when they are free from any
impurity. For the disordered cylinder with even $N$ ($N=8$), the lowest energy
level as usual provides $\phi_{0}$ periodicity similar to the impurity free
cylinders containing even $N$. Apart from this periodic nature, impurities
play another significant role in the determination of the slope of the

Figure 5: (Color online). Ground state energy level as a function of flux
$\phi$ for half-filled disordered mesoscopic cylinders ($M=2$) considering
$U=1$ and $W=2$. (a) $N=5$ and (b) $N=8$.

energy levels. The slope of the lowest energy level decreases significantly
compared to the perfect case, and therefore, a prominent change in current
amplitude also takes place.

To justify the above facts, in Fig. 6 we present the variations of persistent
currents with AB flux $\phi$ for a half-filled mesoscopic cylinder, described
in the framework of NNH model, considering $N=15$ and $M=2$. The red curve
represents the current for the ordered ($W=0$) non-interacting ($U=0$)
cylinder. It shows saw-tooth like nature with flux $\phi$ providing
$\phi_{0}/2$ flux-quantum periodicity. The situation becomes completely
different when impurities are introduced in the cylinder as seen by the other
two curves. The green curve represents the current for the case only when
impurities are considered but the effect of electronic correlation is not
taken into account. It shows a continuous like nature with $\phi_{0}$ flux-
quantum periodicity. The most important observation is that the current
amplitude gets reduced enormously, even an order of magnitude, compared to the
perfect cylinder. This is due to the localization of the energy eigenstates in
the presence of impurity, which is the so-called Anderson localization. Hence,
a large difference exists in the current amplitudes of an ordered and
disordered non-interacting cylinders and it was the main controversial issue
among the theoretical and experimental predictions. Experimental verifications
suggest that the measured current amplitude is quite comparable to the
theoretical current amplitude obtained in a perfect system. To remove this
controversy, as a first attempt, we include the effect of e-e correlation in
the disordered cylinder described by the NNH model. The result is shown by the
blue curve where $U$ is fixed at $1.5$. It is observed that the current
amplitude gets increased compared to the non-interacting disordered cylinder,
though the increment is too small. Not only that the enhancement can take
place only for small values of $U$, while for large enough $U$ the current
amplitude rather decreases. This phenomenon can be implemented as follows.

Figure 6: (Color online). Persistent current as a function of flux $\phi$ for
a half-filled mesoscopic cylinder considering $N=15$ and $M=2$. The red line
corresponds to the ordered case when $U=0$, whereas the green and blue lines
correspond to the disordered case ($W=2$) when $U=0$ and $1.5$, respectively.

For the non-interacting disordered cylinder the probability of getting two
opposite spin electrons becomes higher at the atomic sites where the site
energies are lower than the other sites since the electrons get pinned at the
lower site energies to minimize the ground state energy, and this pinning of
electrons becomes increased with the rise of impurity strength $W$. As a
result the mobility of electrons and hence the current amplitude gets reduced
with the increase of impurity strength $W$. Now, if we introduce electronic
correlation in the system then it tries to depin two opposite spin electrons
those are situated together due to the Coulomb repulsion. Therefore, the
electronic mobility is enhanced which provides larger current amplitude. But,
for large enough interaction strength, no electron can able to hop from one
site to other at the half-filling since then each site is occupied either by
an up or down spin electron which does not allow other electron of opposite
spin due to the repulsive term $U$. Accordingly, the current amplitude
gradually decreases with $U$. On the other hand, at less than half-filling
though there is some finite probability to hop an electron from one site to
the other available ‘empty’ site but still it is very small. So, in brief, we
can say that within the nearest-neighbor hopping (NNH) approximation electron-
electron interaction does not provide any significant contribution to enhance
the current amplitude, and hence the controversy regarding the current
amplitude still persists.

###  III.3 Disordered cylinders with NNH and SNH integrals

To overcome the existing situation regarding the current amplitude, in this
sub-section, finally we make an attempt by incorporating the effect of second-
neighbor hopping (SNH) integral in addition to the nearest-neighbor hopping
(NNH) integral.

A significant change in current amplitude takes place when we include the
contribution of second-neighbor hopping (SNH) integral in addition to the NNH
integral. As representative examples, in Fig. 7 we plot the current-flux
characteristics for a half-filled mesoscopic cylinder considering $N=15$ and
$M=2$. The black,

Figure 7: (Color online). Persistent current as a function of flux $\phi$ for
a half-filled mesoscopic cylinder taking $N=15$ and $M=2$ in the presence of
NNH and SNH integrals. The black line corresponds to the ordered case when
$U=0$, whereas the magenta and gold lines correspond to the disordered case
($W=2$) when $U=0$ and $1.5$, respectively. Here SNH integral is fixed at
$-0.6$. The currents shown by the red, green and blue lines for the ring
described with NNH model (identical to Fig. 6) are re-plotted to judge the
effect of SNH integral over NNH model much clearly.

magenta and gold lines correspond to the results in the presence of SNH
integral, while the other three colored curves (red, green and blue) represent
the currents in the absence of SNH integral. Here we choose $t_{d}=-0.6$. The
black curve refers to the persistent current for the perfect ($W=0$) non-
interacting ($U=0$) cylinder and it achieves much higher amplitude compared to
the NNH model (red curve). This additional contribution comes from the SNH
integral since it allows electrons to hop further. In addition it is also
noticed that the current varies periodically with $\phi$ providing $\phi_{0}$
flux-quantum periodicity, instead of $\phi_{0}/2$ as in the case of NNH
integral model (red curve). Thus, it can be emphasized that $\phi_{0}/2$
periodicity will be observed only when the cylinder is (a) free from impurity,
(b) half-filled, (c) made with odd $N$, and (d) described by the nearest-
neighbor hopping model. The main focus of this sub-section is to interpret the
combined effect of SNH integral and electron-electron correlation on the
enhancement of persistent current amplitude in disordered cylinder. To do this
first we narrate the effect of SNH integral in disordered non-interacting
cylinder. The nature of the current for this particular case is shown by the
magenta curve of Fig. 7. It shows that the current amplitude gets reduced
compared to the perfect case (black line), which is expected, but the
reduction of the current amplitude is quite small than the NNH integral model.
This is due the fact that the SNH integral tries to delocalize the electronic

Figure 8: (Color online). Persistent current as a function of flux $\phi$ for
a half-filled mesoscopic cylinder taking $N=15$ and $M=2$ in the presence of
NNH and SNH integrals. The black line corresponds to the ordered case when
$U=0$, whereas the magenta and gold lines correspond to the disordered case
($W=2$) when $U=0$ and $1.5$, respectively. Here SNH integral is fixed at
$-0.8$. The currents shown by the red, green and blue lines for the ring
described with NNH model (identical to Fig. 6) are re-plotted to judge the
effect of SNH integral over NNH model much clearly.

states, and therefore, the mobility of the electrons is enriched. The
situation becomes more interesting when we include the effect of Hubbard
interaction. The behavior of the current in the presence of interaction is
plotted by the gold curve of Fig. 7 where we fix $U=1.5$. Very interestingly
we see that the current amplitude is enhanced significantly and quite
comparable to that of the perfect cylinder.

For better clarity of the results discussed above, in Fig. 8 we also present
the similar feature of persistent current for other hopping strength of SNH
integral. Here we set $t_{d}=-0.8$. From these curves we see that the current
amplitude gets enhanced more as we increase the SNH strength.

Thus, it can be predicted that the presence of SNH integral and Hubbard
interaction can provide a persistent current which may be comparable to the
measured current amplitudes. In this presentation we consider the effect of
only SNH integral as a higher order hopping integral in addition to the NNH
model, and, illustrate how such a higher order hopping integral leads an
important role on the enhancement of current amplitude in presence of Hubbard
correlation for disordered cylinders. Instead of considering only the SNH
integral we can also take the contributions from all possible higher order
hopping integrals with reduced hopping strengths. Since the strengths of other
higher order hopping integrals are too small, the contributions from these
factors are reasonably small and they will not provide any significant change
in the current amplitude. Finally, we can say that further studies are needed
by incorporating all these factors.

##  IV Closing remarks

To summarize, in the present work we have addressed the behavior of persistent
current in an interacting mesoscopic cylinder threaded by an Aharonov-Bohm
flux $\phi$. We have adopted a tight-binding Hamiltonian to describe the model
quantum system and all the numerical calculations have been done within a mean
field approximation. Using the generalized Hartree-Fock (HF) approximation, we
have computed persistent current as functions of SNH integral, impurity
strength $W$, AB flux $\phi$, electron filling $N_{e}/N$ and system size $N$.
Our numerical results have provided several interesting features and the
present study may be helpful in understanding magnetic response in nano-scale
loop geometries.

The essential features observed from our analysis are as follows.
(i) In the determination of the lowest energy level we see that the energy
level varies periodically with $\phi$ exhibiting both $\phi_{0}/2$ and
$\phi_{0}$ flux-quantum periodicities depending on the choices of the
parameters describing the tight-binding Hamiltonian.
(ii) In the NNH model, current amplitude gets significantly reduced when
impurities are introduced in the system. With the inclusion of Hubbard
interaction ($U$), the current amplitude can be enhanced, though the
enhancement becomes too small compared to the experimental verifications.
(iii) A significant change in the current amplitude takes place when the
effect of SNH integral is taken into account. The combined effect of SNH and
Hubbard interaction can provide the current which is quite comparable to the
experimental realizations. So in short we can say that the conventional NNH
model can be modified by incorporating the higher order hopping integrals.

Throughout the analysis we have kept the width of the cylinders at a fixed
value ($M=2$), for the sake of our illustration. All these results are also
valid for cylinders of larger widths. Here we have considered several
important approximations by ignoring the effects of temperature, electron-
phonon interaction, etc. Due to these factors, any scattering process that
appears in the cylinder would have influence on electronic phases. At the end,
we would like to say that we need further study in such systems by
incorporating all these effects.

ACKNOWLEDGMENTS

I acknowledge with deep sense of gratitude the illuminating comments and
suggestions I have received from Prof. S. N. Karmakar during the calculations.

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