# Circulating current in 1D Hubbard rings with long-range hopping: Comparison
between exact diagonalization method and mean-field approach

Madhumita Saha<EMAIL_ADDRESS>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

The interplay between Hubbard interaction, long-range hopping and disorder on
persistent current in a mesoscopic one-dimensional conducting ring threaded by
a magnetic flux $\phi$ is analyzed in detail. Two different methods, exact
numerical diagonalization and Hartree-Fock mean field theory, are used to
obtain numerical results from the many-body Hamiltonian. The current in a
disordered ring gets enhanced as a result of electronic correlation and it
becomes more significant when contributions from higher order hoppings, even
if they are too small compared to nearest-neighbor hopping, are taken into
account. Certainly this can be an interesting observation in the era of long-
standing controversy between theoretical and experimental results of
persistent current amplitudes. Along with these we also find half-flux quantum
periodic current for some typical electron fillings and kink-like structures
at different magnetic fluxes apart from $\phi=0$ and $\pm\phi_{0}/2$. The
scaling behavior of current is also discussed for the sake of completeness of
our present analysis.

###### pacs:

71.27.+a, 73.23.Ra, 73.23.-b

##  I Introduction

The phenomenon of persistent current was first established during $1980$’s
when Büttiker and his group have shown butt1  that a small conducting ring
carries a net circular current in presence of an Aharonov-Bohm (AB) flux
$\phi$, and it does not vanish even in presence of disorder. This is a pure
quantum mechanical phase coherence phenomenon and a direct consequence of AB
effect imry1 .

The experimental verification of it came after few years when Levy and co-
workers did the pioneering experiment levy  considering an ensemble of
$10^{7}$ independent Cu rings. Later several other experiments jari ; bir ;
chand ; mailly1 ; mailly ; blu  were carried out to confirm the existence of
persistent current in different conducting loops. At the same time a
substantial amount of theoretical work gefen ; yang ; cheu2 ; mont ; bouz ;
giam ; san1 ; san2 ; madhu ; ambe ; schm1 ; schm2 ; bary  has also revealed
several unique features along this direction. From the single ring measurement
levy  a period of $\phi_{0}$ ($=ch/e$, the elementary flux-quantum) current
has been observed, while some other experiments jari ; ram  have shown both
$\phi_{0}/2$ and $\phi_{0}$ periodic currents. This half flux-quantum
($\phi_{0}/2$) periodic persistent current is not well explained from
theoretical calculations and certainly it demands proper analysis. The another
confliction between theoretical and experimental results is associated with
the prediction of low-field currents, viz, slope of the current with flux
$\phi$ in the limit $\phi\rightarrow 0$. Some experiments levy ; jari  have
shown diamagnetic (negative slope) nature, while paramagnetic (positive slope)
response has been noticed from other experiments chand . These observations do
not really match well with theoretical observations. Many attempts have been
made along this line and it has been observed that cheu2  the slope strongly
depends on electronic filling, randomness, impurities, etc.

Finally, the most important controversy has been raised during the prediction
of current amplitude. Only for nearly ballistic rings comparable currents are
obtained, while for diffusive rings the calculated current is much smaller and
in some cases it is even less than two orders of magnitude compared to the
experimental observations. In order to resolve this long-standing anomaly in
diffusive rings, several attempts have been made considering the effects of
different parameters. It has been verified that on-site Coulomb correlation
plays a dominant role mont ; bouz ; giam  to enhance current amplitude in a
disordered ring, though the enhancement is not sufficient to make it
comparable to experimental observations. Later few theoretical models have
been given considering the effect of higher order hopping integrals san3 ;
san4  in addition to the usual nearest-neighbor hopping (NNH) in favor of
current enhancement upto a certain level. So naturally the question comes how
the current becomes affected if one includes the effect of both on-site
Coulomb interaction and long-range hopping. This part has not been discussed
so far, to the best of our knowledge.

In the present work we actually focus towards this direction. Here we consider
interacting rings with higher-order hopping integrals and determine energy
eigenvalues by exact numerical diagonalization of many-body tight-binding
Hamiltonian. In this method we restrict ourselves to rings with few number of
electrons, because of our computational limitations, as dimension of the
matrix raises rapidly with ring size $N$ and total number of electrons
$N_{e}$. Different combinations of up and down spin electrons are taken into
account and from our results we observe some similarities in current-flux
characteristics for odd $N_{e}$ as well as even $N_{e}$, which certainly help
us to generalize the results for larger ring size with higher electrons.
However, it is always good to have some definite results of large interacting
rings to understand the interplay between long-range hopping, disorder and e-e
interaction. We thus require an appropriate technique to work out such a many-
body problem. Here we use Hartree-Fock (HF) mean-field theory kato ; kam ; sil
; san5  to find energy eigenvalues and hence persistent currents, and, compare
these currents with exact exact numerical diagonalization analysis.
Scrutinizing the results we show that persistent current behaves almost
identical in these two methods, but one can miss some important signatures of
current at some typical fluxes, particularly in the limit $\phi\rightarrow 0$.
This is an artifact of the mean-field calculations.

Several interesting features are obtained from our calculations. The current
gets enhanced in a disordered ring as a result of Coulomb repulsion and it
becomes more effective when higher order hopping integrals are included. Under
this situation current becomes quite comparable to that of a perfect ring. In
addition we also discuss the appearance of half flux-quantum periodicity for
some particular electron fillings and some kink-like structures at different
fluxes apart from $\phi=0$ and $\pm\phi_{0}/2$. Finally, scaling behavior of
current is also discussed, for the sake of completeness.

The rest of the work is arranged as follows. In Sec. II we describe the ring
model and theoretical procedure to construct the Hamiltonian matrix and
calculation of persistent current as a function of flux $\phi$. Later, in Sec.
III we discuss numerical results, and finally, a brief summary is given in
Sec. IV.

##  II Model and Theoretical Formulation

###  II.1 Model and Tight-Binding Hamiltonian

The schematic diagram of our model is illustrated in Fig. 1, where a
mesoscopic ring is subjected to on-site Hubbard interaction and long-range
hoppings, viz, second-neighbor hopping (SNH) and third-neighbor hopping (TNH)
including nearest-neighbor hopping. Under the application of an AB flux
$\phi$, a net circulating current is established in the ring.

To describe this model we employ a tight-binding (TB) framework. For a
$N$-site ring the TB Hamiltonian looks like,

H $\displaystyle=$
$\displaystyle\sum_{j,\sigma}\epsilon_{j}c_{j,\sigma}^{\dagger}c_{j,\sigma}+t\sum_{j,\sigma}\left[e^{i\theta_{1}}c_{j,\sigma}^{\dagger}c_{j+1,\sigma}\right.$
(1)
$\displaystyle\left.+\,e^{-i\theta_{1}}c_{j+1,\sigma}^{\dagger}c_{j,\sigma}\right]$
$\displaystyle+\,t_{1}\sum_{j,\sigma}\left[e^{i\theta_{2}}c_{j,\sigma}^{\dagger}c_{j+2,\sigma}+e^{-i\theta_{2}}c_{j+2,\sigma}^{\dagger}c_{j,\sigma}\right]$
$\displaystyle+\,t_{2}\sum_{j,\sigma}\left[e^{i\theta_{3}}c_{j,\sigma}^{\dagger}c_{j+3,\sigma}+e^{-i\theta_{3}}c_{j+3,\sigma}^{\dagger}c_{j,\sigma}\right]$
$\displaystyle+\,U\sum_{j}c_{j,\uparrow}^{\dagger}c_{j,\uparrow}c_{j,\downarrow}^{\dagger}c_{j,\downarrow}$

where $\epsilon_{j}$ gives the site energy for $j$th site and
$c_{j,\sigma}^{\dagger}$ ($c_{j,\sigma}$) represents the creation
(annihilation) operator. $t$ describes the nearest-neighbor hopping strength
and due to this hopping a phase factor $\theta_{1}$ ($=2\pi\phi/\phi_{0}$) is
introduced into the Hamiltonian. Similarly, $t_{1}$ and $t_{2}$ denote the SNH
and TNH integrals, respectively, with the

Figure 1: (Color online). Schematic view of a mesoscopic ring threaded by an
Aharonov-Bohm flux $\phi$. A net circulating current $I$ is established in the
ring.

associated phase factors $\theta_{2}=2\theta_{1}$ and
$\theta_{3}=3\theta_{1}$. The parameter $U$ measures the Hubbard interaction
strength. Setting $\epsilon_{j}=$ constant $\forall j$ we get a perfect ring
and without loss of generality we can fix it to zero, while for a disordered
ring site energies are chosen randomly from a Box distribution function of
width $W$.

###  II.2 Method I: Exact Numerical Diagonalization

Using the TB Hamiltonian, Eq. 1, we construct matrix elements through the
prescription $\textbf{H}_{mn}=\langle\psi_{m}|\textbf{H}|\psi_{n}\rangle$,
where $|\psi_{m}\rangle$ and $|\psi_{n}\rangle$ are basis states associated
with total number of up ($N_{up}$) and down ($N_{dn}$) spin electrons in a
ring. For example, for a ring with two opposite spin electrons (viz,
$N_{up}=N_{dn}=1$) the state vectors are like:
$|\psi_{m}\rangle=c_{p\uparrow}^{\dagger}c_{q\downarrow}^{\dagger}|0\rangle$
and
$|\psi_{n}\rangle=c_{k\uparrow}^{\dagger}c_{l\downarrow}^{\dagger}|0\rangle$,
where $|0\rangle$ is the vacuum state. In a similar way, we can choose the
basis states for any ring with different $N_{up}$ and $N_{dn}$ and construct
the required many-body Hamiltonian matrix. Diagonalizing this matrix, we find
energy eigenvalues and eventually determine persistent current at absolute
zero temperature from the relation

$\displaystyle I(\phi)$ $\displaystyle=$ $\displaystyle-\frac{\partial
E_{0}(\phi)}{\partial\phi},$ (2)

where $E_{0}(\phi)$ is the ground state energy.

###  II.3 Method II: Hartree-Fock Mean-Field Approach

In this method the interacting Hamiltonian H is decoupled into two non-
interacting parts sil ; san5 , say, $\textbf{H}_{\uparrow}$ and
$\textbf{H}_{\downarrow}$, associated with up and down spin electrons.
Diagonalizing these non-interacting Hamiltonians ($\textbf{H}_{\uparrow}$ and
$\textbf{H}_{\downarrow}$), we find energy eigenvalues, and eventually,
calculate the ground state energy at absolute zero temperature for a system
containing $N_{up}$ and $N_{dn}$ spin electrons from the relation

$E_{0}=\sum_{n=1}^{N_{up}}E_{n\uparrow}+\sum_{n=1}^{N_{dn}}E_{n\downarrow}-\sum_{i=1}^{N}U\langle
n_{i\uparrow}\rangle\langle n_{i\downarrow}\rangle$ (3)

where $E_{n\sigma}$’s are the energy eigenvalues and $\langle
n_{i\sigma}\rangle=\langle c_{i\sigma}^{\dagger}c_{i\sigma}\rangle$.

Finally, persistent current is determined from the relation given in Eq. 2.

##  III Results and Discussion

We present our results in two different sub-sections. In the first part (sub-
section A) we determine persistent current by evaluating ground state energy
from exact numerical diagonalization of many-body Hamiltonian. Starting from a
ring with two opposite spin electrons we consider upto six-electron system.
For a fixed $N_{e}$ ($=N_{up}+N_{dn}$), different combinations of $N_{up}$ and
$N_{dn}$ are taken into account to explore all the basic features of
persistent current in interacting rings. While, in the other part (sub-section
B), persistent current is found out by determining ground state energy from HF
mean-field approach.

Throughout the analysis we set $t=-1\,$eV ($t_{1}$ and $t_{2}$ are variable
and they are mentioned in appropriate places) and all the energies are scaled
with respect to it. The current is measured in unit of $et/h$.

###  III.1 Exact numerical diagonalization analysis

####  III.1.1 Impurity free rings

Let us first concentrate on interacting rings without any impurity i.e.,
$\epsilon_{j}=0$ $\forall\;j$.

Case 1 - Ring with two electrons: To have an interacting ring with two
electrons we consider one up and one down spin electron i.e.,
$N_{up}=N_{dn}=1$. For this configuration we select basis states as
$|\psi_{m}\rangle=c_{p\uparrow}^{\dagger}c_{q\downarrow}^{\dagger}|0\rangle$
and
$|\psi_{n}\rangle=c_{k\uparrow}^{\dagger}c_{l\downarrow}^{\dagger}|0\rangle$
and the dimension of matrix becomes $N^{2}\times N^{2}$ for a $N$-site ring.

Figure 2 displays the variation of ground state energy and corresponding
persistent current with $\phi$ for a $10$-site ring having two opposite spin
electrons for different values of correlation strength $U$. In the $1$st row
the results are shown when the ring is described with only NNH integral, while
in the $2$nd row we add the effect of SNH integral and finally in the last row
both SNH and TNH integrals are taken into account. The results are noteworthy.
(i) The ground state energy exhibits $U$-independent energy region across
$\phi=\pm\phi_{0}/2$ and the width of this region gets increased with
increasing $U$, while it decreases with the inclusion of higher order hopping
integrals, as clearly seen by comparing the $E_{0}$-$\phi$ spectra given in
Figs. 2(a), (c) and (e). This behavior is nicely reflected in current-flux
characteristics yielding a kink-like structure. A sudden change in direction
of current takes place as a result of this kink, and interestingly we see that
inside a kink the slope of the current remains unchanged irrespective of $U$.
(ii) For a fixed correlation strength $U$, the slope of $E_{0}$-$\phi$ curve
gets increased with higher order hopping which results a larger current. Here
it is important to note that though TNH strength is much weaker than

Figure 2: (Color online). Dependence of ground state energy with flux $\phi$
and associated current for a $10$-site ring with two opposite spin electrons
for different values of $U$, where the $1$st, $2$nd and $3$rd rows correspond
to the ring with only NNH; NNH and SNH; and NNH, SNH and TNH integrals,
respectively. Here we set $t_{1}=-0.1$ and $t_{2}=-0.05$.

SNH, it contributes significantly to raise the current which is visible from
the spectra given in Figs. 2(d) and (f). Certainly, more current is expected
in presence of additional higher order hopping, beyond TNH, though their
strengths are vanishingly small. Physically this is easily understandable
since the incorporation of additional hopping allows electrons to hop further
and thereby increases electronic current. (iii) A reduction of current is
observed with increasing $U$ when all the other physical parameters describing
the system remain unchanged. This is solely due to the repulsive nature of
electron-electron (e-e) interaction. It suggests that lesser and lesser
current is expected with increasing $U$, though it eventually reaches to a
finite value (not shown here) and never drops to zero since $N_{e}<N$, which
we verify through our detailed numerical analysis.

To have a deeper insight about the role of SNH and TNH integrals in Fig. 3 we
present the variation of ground state energy along with persistent current as
a function of $\phi$ considering $U=1$ for the same ring size as taken in Fig.
2 for some typical values of $t_{1}$ and $t_{2}$. From the spectra given in
$1$st row we see that with increasing $t_{1}$ current gets enhanced

Figure 3: (Color online). Variation of ground state energy and corresponding
current as function of $\phi$ for a $10$-site ring with two opposite spin
electrons considering $U=1$ for different values of $t_{1}$ and $t_{2}$. For
the $1$st row the ring is described with NNH and SNH, while in the $2$nd row
the effect of TNH is also included where $t_{1}$ is fixed at $-0.2$.

following energy-flux diagram but it ($t_{1}$) reduces the kink size. Further
enhancement of current can be also made possible with the inclusion of TNH
(see Fig. 3(d)), and like $t_{1}$, $t_{2}$ is also responsible for the
reduction of kink size, though these kinks do not vanish even for larger
$t_{1}$ and $t_{2}$ as long as e-e interaction is there. For this two-spin
system we get only $\phi_{0}$ periodic current.

Case 2 - Ring with three electrons: Now we consider a ring with one down spin
and two up spin electrons to have a three-electron ring system, and here we
choose
$|\psi_{1}\rangle=c_{p\uparrow}^{\dagger}c_{q\uparrow}^{\dagger}c_{r\downarrow}^{\dagger}|0\rangle$
and
$|\psi_{2}\rangle=c_{k\uparrow}^{\dagger}c_{l\uparrow}^{\dagger}c_{m\downarrow}^{\dagger}|0\rangle$.
With this configuration the dimension of the matrix for a $N$-site ring
becomes: $N^{2}(N-1)/2\times N^{2}(N-1)/2$. Two different cases are analyzed
depending on the filling factor those are as follows.

$\bullet$ Half-filled ring: In Fig. 4 we present the variation of ground state
energy $E_{0}$ and its corresponding persistent

Figure 4: (Color online). $E_{0}$-$\phi$ and corresponding $I$-$\phi$
characteristics of a $3$-electron ($N_{up}=2$ and $N_{dn}=1$) system in the
half-filled limit (viz, $N=3$). Here SNH and TNH are not applicable.

current as a function of AB flux $\phi$ in the half-filled limit for three
typical values of $U$. Two important features are observed. (i) Ground state
energy exhibits half-flux-quantum ($\phi_{0}/2$) periodicity with $\phi$, and
it is nicely reflected in the current-flux characteristics. To illustrate this
fact we can set the interaction strength $U$ to zero, since it does not affect
$\phi_{0}/2$ periodicity (which is shown from the $E_{0}$-$\phi$ and
$I$-$\phi$ spectra). Due to this simplification we can

Figure 5: (Color online). $E_{0}$-$\phi$ and $I$-$\phi$ curves for a $8$-site
ring with three electrons ($N_{up}=2$, $N_{dn}=1$), where the first, second
and third rows correspond to the ring with only NNH; NNH and SNH; NNH, SNH and
TNH integrals, respectively. The other parameters are: $t_{1}=-0.1$ and
$t_{2}=-0.05$.

handle the problem very easily as it becomes now a non-interacting one. For
$N=3$, three energy eigenstates are obtained where each of these states can
have maximum two opposite spin electrons. Therefore, ground state energy is
obtained by placing two opposite spin electrons in the lowest non-interacting
energy level and the third electron (up spin) in the first excited level.
Quite interestingly, we observe that all such distinct energy levels have
$\phi_{0}$ flux-quantum periodicity. But as long as the highest accessible
energy level (here it is $1$st excited state) contains a single electron,
instead of two, the net ground state energy of the system gets strange
$\phi_{0}/2$ periodicity. This peculiar $\phi_{0}/2$ periodicity also remains
unchanged even in presence of $U$, which we claim from our results. (ii) No
kink-like structure appears in $I$-$\phi$ curves which implies that at half-
filling no $U$-independent state contributes to the lowest energy for any flux
window.

For such filling, current decreases rapidly with the correlation strength $U$,
as there is no empty site.

$\bullet$ Non-half-filled ring: The results of a three-electron non-half-
filled ring ($N=8$) are shown in Fig. 5 which exhibit some interesting
patterns unlike the half-filled case. (i) From the spectra it is shown that
the $\phi_{0}/2$ periodicity disappears and the current, associated with
ground state energy, regains its usual $\phi_{0}$ periodicity. (ii) The kink-
like structure, associated with $U$-independent states, appears in persistent
current across $\phi=0$ at sufficiently large value of $U$, and the width of
such kink decreases with the inclusion of higher order hopping integrals like
SNH and TNH. In this context it is important to note that for two-electron
case, kink appears as long as the interaction is turned on. The other features
viz, the reduction of current with $U$ and its enhancement with higher order
hopping integrals remain same as discussed in a two-electron ring system.

Case 3 - Ring with four electrons: For four-electron system we consider two
distinct configurations depending on $N_{up}$ and $N_{dn}$. In one
configuration we set $N_{up}=3$ and $N_{dn}=1$, while for the other we choose
$N_{up}=N_{dn}=2$.

$\blacksquare$ Configuration I – Ring with $N_{up}=3$ and $N_{dn}=1$:
The dimension of the Hamiltonian matrix for this setup becomes
$(N^{2}/6)(N-1)(N-2)\times(N^{2}/6)(N-1)(N-2)$, where the basis states are:
$|\psi_{1}\rangle=c_{p\uparrow}^{\dagger}c_{q\uparrow}^{\dagger}c_{r\uparrow}^{\dagger}c_{s\downarrow}^{\dagger}|0\rangle$
and
$|\psi_{2}\rangle=c_{k\uparrow}^{\dagger}c_{l\uparrow}^{\dagger}c_{m\uparrow}^{\dagger}c_{n\downarrow}^{\dagger}|0\rangle$.

The characteristics features of ground state energy and corresponding current
for a half-filled ring with four electrons ($N_{up}=3$ and $N_{dn}=1$) are
shown in Fig. 6. At $U=0$, a sharp transition

Figure 6: (Color online). $E_{0}$-$\phi$ and $I$-$\phi$ curves of a four-site
ring ($N=4$) with $N_{e}=4$ ($N_{up}=3$ and $N_{dn}=1$) for different values
of $U$ considering NNH only.

occurs in persistent current around $\phi=\pm\phi_{0}/2$, while for finite $U$
the current gets reduced and varies continuously with $\phi$. This continuous-
like variation is not quite trivial as obtained in traditional disordered
case. While, the reduction of current with $U$ is anticipated easily for this
half-filled case.

Away from half-filling some kink-like structures are visible in $I$-$\phi$
spectra reflecting the $E_{0}$-$\phi$ curves, and the most crucial point is
that these kinks are generated as a result of $U$-dependent states, not like
$U$-independent states as described earlier. The results are shown in Fig. 7
for a $8$-site ring considering the effects of higher order hopping (SNH and
TNH) integrals

Figure 7: (Color online). Energy and current spectra for a $8$-site ring with
$N_{up}=3$ and $N_{dn}=1$ for some typical values of $U$, where the first,
second and third rows correspond to the ring with only NNH; NNH and SNH; NNH,
SNH and TNH integrals, respectively. Here we choose $t_{1}=-0.1$ and
$t_{2}=-0.05$.

for different values of $U$. The kink appears for a sufficiently large value
of $U$ and its width gradually decreases as a result of higher order hopping.

$\blacksquare$ Configuration II – Ring with $N_{up}=2$ and $N_{dn}=2$:
For this configuration we choose many-body basis states as:
$|\psi_{1}\rangle=c_{p\uparrow}^{\dagger}c_{q\uparrow}^{\dagger}c_{r\downarrow}^{\dagger}c_{s\downarrow}^{\dagger}|0\rangle$
and
$|\psi_{2}\rangle=c_{k\uparrow}^{\dagger}c_{l\uparrow}^{\dagger}c_{m\downarrow}^{\dagger}c_{n\downarrow}^{\dagger}|0\rangle$.
Here the dimension of the Hamiltonian matrix becomes
$(N^{2}/4)(N-1)^{2}\times(N^{2}/4)(N-1)^{2}$, and, it is higher compared to
the Configuration I, i.e., ring with three up and one down spin electrons.

The results for a half-filled ring are displayed in Fig. 8. It is

Figure 8: (Color online). Variation of ground state energy and corresponding
current for a four-site ring with $N_{up}=2$ and $N_{dn}=2$ for different
strengths of $U$ considering NNH only.

shown that in absence of electronic correlation ($U=0$), current exhibits a
sharp transition similar to that as obtained in the previous $4$-electron
half-filled ring, but the transition takes place at $\phi=0$. All the other
features, viz, the continuous like variation of current with $\phi$ and its
reduction as a result of electronic correlation remain almost similar as
presented in Fig. 6. But there is a crucial difference between these two sets
of electronic configurations. Comparing the current spectra given in Figs.
6(b) and 8(b) it is seen that the suppression of current due to e-e
interaction becomes more significant for the ring with $N_{up}=3$ and
$N_{dn}=1$ compared to the other case. The reason is that in the first case
where $N_{up}>N_{dn}$, the hopping probability of up spin electrons is less
compared to the ring with $N_{up}=N_{dn}$ since in the later case pairing of
up and down spin electrons in a single site is possible as a result of hopping
term for

Figure 9: (Color online). $E_{0}$-$\phi$ and $I$-$\phi$ curves for a $7$-site
ring with $N_{up}=2$ and $N_{dn}=2$ for some typical values of $U$ considering
$t_{1}=-0.1$ and $t_{2}=-0.05$, where three different rows correspond to the
identical meaning as given in Fig. 7.

low $U$, yielding a contribution to the current. For large $U$ in both these
two cases current gets reduced.

Focusing on the spectra given in Fig. 9 apparently it seems that the features
are almost identical with the results shown in Fig. 7, but there is a striking
difference between the two. For this configuration ($N_{up}=N_{dn}=2$) kink,
associated with $U$-dependent states, appears for any non-zero correlation
strength $U$, unlike the previous case ($N_{up}=3$ and $N_{dn}=1$) where a
critical $U$ (say $U_{c}$) is required. Apart from this, other qualitative
features are almost identical. For all these four-electron ring systems we get
only $\phi_{0}$ periodic persistent current.

Case 4 - Ring with five electrons: Now we move to the five-electron ring
system and here we also consider two distinct configurations depending on
$N_{up}$ and $N_{dn}$. In one configuration we set $N_{up}=3$ and $N_{dn}=2$,
while for the other we fix $N_{up}=4$ and $N_{dn}=1$.

$\blacksquare$ Configuration I – Ring with $N_{up}=3$ and $N_{dn}=2$:
For this configuration the many-body basis states get the forms:
$|\psi_{1}\rangle=c_{p\uparrow}^{\dagger}c_{q\uparrow}^{\dagger}c_{r\uparrow}^{\dagger}c_{s\downarrow}^{\dagger}c_{t\downarrow}^{\dagger}|0\rangle$
and
$|\psi_{2}\rangle=c_{k\uparrow}^{\dagger}c_{l\uparrow}^{\dagger}c_{m\uparrow}^{\dagger}c_{n\downarrow}^{\dagger}c_{o\downarrow}^{\dagger}|0\rangle$,
and the dimension of the Hamiltonian for a $N$-site ring becomes
$(N^{2}/12)(N-1)^{2}(N-2)\times(N^{2}/12)(N-1)^{2}(N-2)$.

The characteristic features of ground state energy along with persistent
current for a half-filled five-electron ($N_{up}=3$ and $N_{dn}=2$) ring

Figure 10: (Color online). $E_{0}$-$\phi$ and $I$-$\phi$ characteristics for a
five-electron ($N_{up}=3$ and $N_{dn}=2$) half-filled ring, where the first
row corresponds to the ring with only NNH integral and the second row
describes the ring with NNH and SNH ($t_{1}=-0.1$) integrals. Here TNH does
not make any sense.

system are shown in Fig. 10. In the first row, results are given when the ring
is described with only NNH, while in the other row we consider the effect of
SNH integral too. For this five-site ring inclusion of TNH does not make any
sense, since long-range hopping in shortest path is only the physically
accessible hopping parameter. Several notable features are obtained. (i) For
the ring described with only NNH, current exhibits half-flux-quantum
($\phi_{0}/2$) periodicity and it is not at all affected by the electronic
correlation. This is exactly the same what we get in a three-electron half-
filled ring (see Fig. 4). (ii) But surprisingly we see that the $\phi_{0}/2$
periodicity disappears when the effect of higher order hopping is included
(Fig. 10(d)). From these results we can generalize that $\phi_{0}/2$
periodicity is specific to the odd-half-filled rings described with only NNH
integral. Other properties like the reduction of current with $U$ and the role
of higher order hopping on current enhancement remain same as discussed for
other cases.

The results for a five-electron ($N_{up}=3$ and $N_{dn}=2$) non-half-filled
ring are placed in Fig. 11 considering some typical values of $U$. All the
basic features of ground state energy and corresponding current shown in this
spectra (Fig. 11) are almost similar to those given in Fig. 5 for a three-
electron non-half-filled ring. For this

Figure 11: (Color online). Variation of ground state energy and corresponding
current for a five-electron ($N_{up}=3$ and $N_{dn}=2$) non-half-filled ring
($N=7$) for some typical values of $U$, where the $1$st, $2$nd and $3$rd rows
represent the ring with only NNH; NNH and SNH ($t_{1}=-0.4$); NNH, SNH and TNH
($t_{2}=-0.2$) integrals, respectively.

five-electron system we can also observe kink-like structure in persistent
current above a critical value of $U$, like the three-electron interacting
ring. The critical value $U_{c}$ is very large for this $7$-site ring (which
is not shown here in the figure) and it ($U_{c}$) becomes smaller for larger
ring which we confirm through our detailed numerical calculations. The role
played by higher order hopping integrals remains same as discussed in other
non-half-filled rings and in all these cases current gives $\phi_{0}$
periodicity.

$\blacksquare$ Configuration II – Ring with $N_{up}=4$ and $N_{dn}=1$:
For this set-up the basis states read as
$|\psi_{1}\rangle=c_{p\uparrow}^{\dagger}c_{q\uparrow}^{\dagger}c_{r\uparrow}^{\dagger}c_{s\uparrow}^{\dagger}c_{t\downarrow}^{\dagger}|0\rangle$
and
$|\psi_{2}\rangle=c_{k\uparrow}^{\dagger}c_{l\uparrow}^{\dagger}c_{m\uparrow}^{\dagger}c_{n\uparrow}^{\dagger}c_{o\downarrow}^{\dagger}|0\rangle$.
Here the dimension of the Hamiltonian matrix is:
$(N^{2}/24)(N-1)(N-2)(N-3)\times(N^{2}/24)(N-1)(N-2)(N-3)$ and it is less
compared to the dimension of a matrix for a $N$-site ring having three up and
two down spin electrons.

The half-flux-quantum periodicity is still preserved for a half-filled

Figure 12: (Color online). Ground state energy and corresponding current as a
function of $\phi$ for a half-filled five-electron ring with $N_{up}=4$ and
$N_{dn}=1$, where the top row represents the ring with NNH only, and, the
bottom row describes the ring with NNH and SNH ($t_{1}=-0.1$) integrals. TNH
integral is redundant here.

Figure 13: (Color online). $E_{0}$-$\phi$ and $I$-$\phi$ curves for a five-
electron ($N_{up}=4$ and $N_{dn}=1$) non-half-filled ring ($N=7$) for some
particular values of $U$, where three different rows correspond to the
identical meaning as given in Fig. 11.

five-electron ring even when we set $N_{up}=4$ and $N_{dn}=1$. But the
requirement is that the ring should be described with only NNH hopping, and
the electronic correlation does not have any explicit dependence on it. But as
long as the effect of higher order hopping is incorporated, $\phi_{0}/2$
periodicity gets lost and the current regains its conventional $\phi_{0}$
periodicity. These features are clearly presented in Fig. 12 and consistent
with the results shown in the spectra Fig. 10.

In the non-half-filled case some anomalous features appear in persistent
current associated with ground state energy in presence of e-e interaction.
The results for a seven-site ring are displayed in Fig. 13 considering
$N_{up}=4$ and $N_{dn}=1$. For $U=0$, sharp transitions are available, while
sudden phase change in current takes place at some typical fluxes in presence
of $U$ due to $U$-dependent states as seen from the $I$-$\phi$ spectra. Here
only $\phi_{0}$ periodic current is obtained.

Case 5 - Ring with six electrons: Finally, we consider six-electron system to
study the characteristic properties of persistent current. Depending on up and
down spin electrons we focus on two distinct electronic configurations. For
one configuration we set $N_{up}=N_{dn}=3$ and for the other configuration we
fix $N_{up}=4$ and $N_{dn}=2$.

$\blacksquare$ Configuration I – Ring with $N_{up}=3$ and $N_{dn}=3$:
The many-body basis states for this configuration look like
$|\psi_{1}\rangle=c_{p\uparrow}^{\dagger}c_{q\uparrow}^{\dagger}c_{r\uparrow}^{\dagger}c_{s\downarrow}^{\dagger}c_{t\downarrow}^{\dagger}c_{u\downarrow}^{\dagger}|0\rangle$
and
$|\psi_{2}\rangle=c_{j\uparrow}^{\dagger}c_{k\uparrow}^{\dagger}c_{l\uparrow}^{\dagger}c_{m\downarrow}^{\dagger}c_{n\downarrow}^{\dagger}c_{o\downarrow}^{\dagger}|0\rangle$,
and the dimension of the Hamiltonian matrix for a $N$-site ring becomes
$(N^{2}/36)(N-1)^{2}(N-2)^{2}\times(N^{2}/36)(N-1)^{2}(N-2)^{2}$.

In Fig. 14 we present the dependence of ground state energy along with
persistent current for a six-site ring considering three

Figure 14: (Color online). Ground state energy and current spectra for a six-
site ring with three up and three down spin electrons for three distinct
values of $U$, where the first row corresponds to the ring with only NNH and
the second row represents the ring with NNH and SNH integrals. Here we set
$t_{1}=-0.8$.

up and three down spin electrons for some typical values of $U$. At $U=0$ the
current provides a sharp jump across $\phi=\pm\phi_{0}/2$, while it varies
continuously with reduced amplitude in presence of finite e-e interaction.
Exactly identical behavior is obtained for the four-electron half-filled ring.
Thus it can be emphasized that there exists a striking similarity between all
half-filled rings with even number of electrons under NNH approximation. With
the addition of higher order hopping (here it is SNH), some anomalous
oscillations appear as a result of phase reversal of ground state energy.

Away from half-filling kink-like structure appears following the variation of
ground stage energy and the width of this kink decreases

Figure 15: (Color online). $E_{0}$-$\phi$ and $I$-$\phi$ curves for a seven-
site ring with three up and three down spin electrons, where the $1$st, $2$nd
and $3$rd rows correspond to the ring with only NNH; NNH and SNH
($t_{1}=-0.6$); NNH, SNH and TNH ($t_{2}=-0.5$) integrals, respectively.

with the incorporation of additional hopping integrals (see Fig. 15), similar
to the four-electron ring system. In addition, current exhibits more complex
structure in presence of TNH for large $U$.

$\blacksquare$ Configuration II – Ring with $N_{up}=4$ and $N_{dn}=2$:
The basis states for this choice of up and down spin electrons are:
$|\psi_{1}\rangle=c_{p\uparrow}^{\dagger}c_{q\uparrow}^{\dagger}c_{r\uparrow}^{\dagger}c_{s\uparrow}^{\dagger}c_{t\downarrow}^{\dagger}c_{u\downarrow}^{\dagger}|0\rangle$
and
$|\psi_{2}\rangle=c_{j\uparrow}^{\dagger}c_{k\uparrow}^{\dagger}c_{l\uparrow}^{\dagger}c_{m\uparrow}^{\dagger}c_{n\downarrow}^{\dagger}c_{o\downarrow}^{\dagger}|0\rangle$.
This configuration sets the dimension of the matrix for a $N$-site ring as
$(N^{2}/48)(N-1)^{2}(N-2)(N-3)\times(N^{2}/48)(N-1)^{2}(N-2)(N-3)$ and it is
lower than the dimension of a identical size ring containing three up and
three down spin electrons.

The characteristics features of ground state energy together with persistent
current for a half-filled six-electron ring with $N_{up}=4$ and $N_{dn}=2$ are
shown in Fig. 16 for some typical values of $U$. Comparing the spectra given
in Figs. 14 and 16 it is noticed that almost identical variation of current is
obtained apart from a shifting along the flux line. In one case the sharp
transition for $U=0$ takes place at $\phi=\pm\phi_{0}/2$ (Fig. 14), while in
the other case it occurs at $\phi=0$ (Fig. 16) for the identical strength of
$U$. These features, in fact, are quite similar to the results discussed for
two different configurations in the half-filled four-electron ring system.
Certainly, these observations demand that all half-filled rings with even
number of electrons exhibit qualitatively identical patterns.

Quite similar argument can also be applicable for the non-half-filled case.

Figure 16: (Color online). $E_{0}$-$\phi$ and $I$-$\phi$ spectra for a six-
site ring with $N_{up}=4$ and $N_{dn}=2$, where the upper and lower rows
represent the similar meaning as described in Fig. 14 with identical parameter
values.

Figure 17: (Color online). Dependence of ground state energy and associated
current with flux $\phi$ for a seven-site ring considering $N_{up}=4$ and
$N_{dn}=2$, where different rows correspond to the identical meaning as stated
in Fig. 15. Here we set $t_{1}=-0.5$ and $t_{2}=-0.4$.

Apart from certain kinks at different fluxes, current exhibit more complex
structures in presence of higher order hopping as clearly seen from the
spectra given in Fig. 17, like the non-half-filled ring with three up and
three down spin electrons (see Fig. 15). Due to smaller size of the ring
(where $N_{e}$ is very close to $N$) current gets suppressed significantly
with increasing the correlation strength and this effect dominates for the
maximum unequal distribution among up and down spin electrons keeping the
total number of electrons constant. For all these six-electron ring systems we
get only $\phi_{0}$ periodic persistent current, like two-electron and four-
electron cases.

####  III.1.2 Effect of impurity

Now we investigate the role of impurities on persistent current and the
interplay between Hubbard interaction and long-range hopping on it. To
introduce impurities in a ring we choose on-site potentials ($\epsilon_{j}$)
randomly from a ‘Box’ distribution function of width $W$ setting the window
region $-W/2$ to $W/2$. Since the response strongly depends on disordered
configurations, we compute all the results taking average over a large number
of distinct such configurations and below we present some of these essential
results.

Case I - Ring with two electrons: To investigate the critical role played by
disorder in Fig. 18 we display the dependence of ground state energy together
with persistent current as a function of $\phi$ for a twenty-site ring
considering two opposite spin electrons setting the disorder strength $W=1$.
In presence of disorder current becomes vanishingly small (red curve of Fig.
18(b)) for the non-interacting ring ($U=0$) described with NNH integral. This
is solely due to electronic localization as electrons get pinned at different
sites associated with the on-site potentials, and, it becomes even smaller for
higher $W$ (not shown here in the figure to save space). Now under this
situation current gets enhanced if we include the effect of electronic
correlation(green line of Fig. 18(b)). This correlation tries to homogenize
the system providing enhanced current.

Figure 18: (Color online). $E_{0}$-$\phi$ and $I$-$\phi$ characteristics for a
$20$-site ring ($N=20$) with two-opposite spin electrons in presence of
disorder ($W=1$) for some typical values $U$ considering the effect of higher
order hopping integrals.

But the enhancement becomes more significant when the contributions from
higher order hopping integrals are taken into account. This is clearly seen
from the cyan and purple curves of Fig. 18(b). Thus it can be emphasized that
the effect of localization as a result of disorder in NNH model can be
suppressed significantly by introducing higher order hopping integrals in
presence of e-e interaction. The available current for the disordered ring in
presence of higher order hopping integrals is quite comparable to the current
obtained for a perfect non-interacting ring described with only NNH integral
which, on the other hand, is very close to the experimental observations.
Therefore, only NNH model is not sufficient to verify experimental results
where the rings invariably contain disorder and certainly we have to go beyond
NNH model.

Apart from this we also find that sudden phase reversal of current takes place
around $\phi=\pm\phi_{0}/2$ due to $U$-independent states, like the two-
electron ordered ring, but for this disordered case the kink appears beyond a
critical value of $U$ ($=U_{c}$, say) and it depends on ring size as well as
disorder strength. Whereas, any non-zero value of $U$ yields kink-like
structure in the ordered ring. For all these cases current exhibits $\phi_{0}$
flux-quantum periodicity.

Case II - Ring with three electrons: Finally, let us focus on the response of
a three-electron disordered ring.

In Fig. 19 we present the variation of ground state energy together with
persistent current for a half-filled ($N=3$) disordered ring, described with
NNH integral, considering $N_{up}=2$ and $N_{dn}=1$. To reveal the precise
role of disorder on periodicity of ground state energy as well as current with
$\phi$, here we superimpose the result of an ordered interacting ring (green
curve). From the spectra it is seen

Figure 19: (Color online). $E_{0}$-$\phi$ and $I$-$\phi$ characteristics for a
three-electron ($N_{up}=2$ and $N_{dn}=1$) ring in the half-filled case in
presence of disorder. For the sake of comparison, the result of an ordered
ring (green line) is also given.

that $\phi_{0}/2$ periodicity no longer exists in presence of disorder, and
both the energy and current get usual $\phi_{0}$ periodicity.

For this three-site ring it is quite hard to view the competition between

Figure 20: (Color online). Ground state energy and corresponding current for a
three-electron ($N_{up}=2$ and $N_{dn}=1$) ring in the non-half-filled case
($N=10$) in presence of disorder.

$U$ and $W$, and thus, we focus on the spectra given in Fig. 20 where the
results are shown for a ten-site ring. A significant enhancement of current
takes place in presence of electronic correlation and higher order hopping
integrals, and, sometimes this enhancement becomes two orders of magnitude
larger compared to the NNH model.

Before we finish this sub-section we would like to point out that the
characteristic features of ground state energies and corresponding persistent
currents for other disordered rings with four, five and even six electrons are
reviewed thoroughly and from our analysis we find exactly similar kind of
enhancement and identical periodic nature of current in the presence of e-e
interaction and long-range hopping.

Scaling behavior

The results analyzed so far are worked out for some typical interacting rings
taking different combinations of up and down spin electrons, and, these
results exhibit certain similarities among the rings with even number (two,
four and six) of electrons as well as rings with odd number (three and five)
of electrons. To make the analysis a self-contained one, now we discuss the
dependence of current with ring size $N$ and from it we establish its scaling
behavior.

Figure 21 displays the dependence of current ($I^{\mbox{\tiny max}}$) as a
function of $N$ considering $U=1$ and $N_{e}=2$ ($N_{up}=N_{dn}=1$), where

Figure 21: (Color online). Dependence of persistent current with ring size $N$
considering $U=1$ and $N_{e}=2$ ($N_{up}=N_{dn}=1$). The dotted points,
corresponding to the currents, are evaluated following the exact
diagonalization method. Using these dots a scaling relation between the
current and ring size is established which generates the continuous curve. Two
cases are presented depending on $W$ where (a) and (b) correspond to $W=0$ and
$1$, respectively.

$I^{\mbox{\tiny max}}$ is obtained by taking the maximum absolute value of
current computed within the flux range $0$ to $\phi_{0}$. Two different cases,
one is for $W=0$ and other is for $W=1$, are described those are given in (a)
and (b), respectively. The dotted points, corresponding to current for
different $N$, are computed following the exact diagonalization method, and,
using these dots we construct a scaling behavior of current which generates
the continuous line. The scaling relation gets the form: $I^{\mbox{\tiny
max}}=CN^{-\xi}$, where the constant $C$ becomes $30$ for both ordered and
disordered cases, while the exponent $\xi$ becomes different in these two
cases. It is $1.65$ for $W=0$ and $1.75$ for $W=1$. Usually, for a particular
filling this constant ($C$) depends on the disorder strength, but for this
typical $N_{e}$ we find its

Figure 22: (Color online). Same as Fig. 21 with $N_{e}=3$, where $N_{up}=2$
and $N_{dn}=1$.

identical value both for the ordered and disordered rings which we confirm
through elaborate numerical calculations.

In Fig. 22 we present the variation of current with ring size $N$ for
$N_{e}=3$ (considering $N_{up}=2$ and $N_{dn}=1$), keeping all other physical
parameters the same as taken in Fig. 22. Like two-electron system, here we
also get similar kind of scaling relation, viz, $I^{\mbox{\tiny
max}}=CN^{-\xi}$, where the constant $C$ becomes $22$ and $16$, and, the
exponent $\xi$ reaches to $1.45$ and $1.55$ for the ordered and disordered
cases, respectively.

In the same way we can also check scaling behavior of persistent current in
higher-electron systems (i.e., rings with four, five, six and more electrons),
and from our analysis it can be emphasized that in each case current obeys the
identical scaling relation, viz, $I\propto N^{-\xi}$. This exponent $\xi$
gradually decreases, both for ordered and disordered cases, with $N_{e}$ and
for $N_{e}\geq 5$ it almost reaches to the limiting value $1.33$ at $U=1$,
which is consistent with the previous analysis done by Gendiar et al gen
using bosonization techniques. Our analysis clearly suggests that the current
decreases algebraically with ring size $N$.

###  III.2 Hartree-Fock mean field analysis

In this sub-section we present numerical results computed from HF mean-field
approach and compare these results with those obtained from the other method
i.e., exact numerical diagonalization.

Figure 23 displays the dependence of ground state energy together with
persistent current as a function of flux $\phi$ for some typical ordered rings
(even $N$) described with only NNH integral in the limit of half-filling
considering different values of electronic correlation strength $U$. Two cases
are analyzed depending on the ring size $N$. In the first row we present the
results for $N=6$ and $N_{up}=N_{dn}=3$, to compare these spectra with our
previous data (top row of Fig. 14) obtained from

Figure 23: (Color online). Dependence of ground state energy and corresponding
persistent current as a function of $\phi$ for some typical ordered rings,
described with only NNH integral, in the half-filled band case considering
different values of $U$. In the first row we choose $N=6$ and
$N_{up}=N_{dn}=3$, while in the second row these parameters are: $N=50$ and
$N_{up}=N_{dn}=25$.

Figure 24: (Color online). $E_{0}$-$\phi$ and $I$-$\phi$ spectra for some
typical ordered rings, described with only NNH integral, in the half-filled
band case considering different values of $U$. In the first row we choose
$N=5$, $N_{up}=3$ and $N_{dn}=2$, while in the second row these parameters
are: $N=41$, $N_{up}=21$ and $N_{dn}=20$.

exact numerical diagonalization method. Comparing the spectra Figs. 23(b) and
14(b) we see that MF results are highly consistent with our data obtained from
exact numerical diagonalization of the many-body Hamiltonian. Therefore, we
can certainly rely on MF solutions and extend our analysis to larger lings. As
illustrative example, in the second row of Fig. 23 we present the MF results
for a $50$-site ring, and they exhibit almost identical features, like a
$6$-site ring, which yield the invariant nature of energy- and current-flux
characteristics for different ring sizes when the filling factor remains
constant.

In a similar way we now focus our attention on the rings with odd $N$. The
results are shown in Fig. 24 where we present the behavior of ground state
energy and corresponding persistent current, obtained from MF technique, for
two different ring sizes. The spectra for $N=5$ (first row of Fig. 24) can be
directly compared to the results given in the first row of Fig. 10 since all
the physical parameters are same for these two cases. Quite interestingly we
see that almost all the features, viz, suppression of current with $U$ and
half flux-quantum periodicity as a function of $\phi$ remain unchanged. But,
for narrow regions of $\phi$ the slope of the currents for non-zero $U$ shows
opposite signature in these two different numerical methods, while it (slope)
remains

Figure 25: (Color online). Variation of persistent current for a $60$-site
disordered ($W=1$) ring, where (a) $N_{up}=N_{dn}=30$ and (b)
$N_{up}=N_{dn}=15$. The red line corresponds to the disordered non-interacting
($U=0$) ring with only NNH integral, while introducing the effect of e-e
interaction ($U=0.5$) in this ring we get cyan curve. The green curve
represents the ring with disorder, e-e interaction ($U=0.5$) and SNH integral
($t_{1}=-0.2$), whereas the purple line corresponds to the ring with disorder,
e-e interaction ($U=0.5$), SNH ($t_{1}=-0.2$) and TNH ($t_{2}=-0.05$)
integrals.

same for other flux regions. This is the peculiarity of MF calculations and
can be noticed for any other ordered rings with odd $N$ in the limit of half-
filling. To substantiate it in the second row of Fig. 24 we present the
results for a $41$-site ring considering $N_{up}=21$ and $N_{dn}=20$, and see
that all the properties exhibited by a $5$-site ring are nicely followed in
this bigger ring.

Finally, let us focus on the MF results given in Fig. 25 where we present the
variation of persistent current for a $60$-site ring as a function of $\phi$
considering two different band fillings those are placed in (a) and (b),
respectively. Four distinct cases are analyzed depending on the physical
parameters of the system. The red curve corresponds to the current for the
non-interacting ($U=0$) disordered ($W=1$) ring described with only NNH
integral. While introducing the effect of e-e interaction in this ring we get
cyan curve. The green line represents the current for the ring with disorder,
e-e interaction and SNH integral, whereas incorporating further the effect of
TNH integral in this ring we get the current shown by the purple curve. At
half-filling all the currents are quite comparable with each other (Fig.
25(a)) and higher order hopping integrals do not actually play any such
significant role in enhancing the current since it is counterbalanced by the
repulsive Coulomb interaction. On the other hand, at less than half-filling an
enhancement of current takes place as a result of $U$ and it becomes more
significant in presence of higher order hopping integrals (Fig. 25(b)) as
there exists empty sites where electrons can hop quite easily yielding more
current. These mean-field results are fully consistent with our exact
numerical diagonalization analysis (see Figs. 19 and 20).

##  IV Closing Remarks

To conclude, in this work we have investigated the behavior of ground state
energy and persistent current in interacting mesoscopic rings subjected to AB
flux $\phi$ in presence of long-range hopping and disorder. Two different
methods, exact numerical diagonalization and HF mean-field theory, have been
used to compute numerical results of the many-body systems, and, found that
the results obtained from these two methods are highly consistent with each
other. The interplay between Hubbard interaction and long-range hopping
integrals yields a significant enhancement of persistent current in disordered
ring which in some cases becomes comparable to that of an ordered ring and
reaches very close to the experimental observations. This phenomenon suggests
that only e-e interaction is not sufficient to explain the enhancement of
persistent current in a disordered ring described with NNH integral. We have
to incorporate the effect of higher order hopping integrals. From our results
we have shown that even for too small values of $t_{1}$ and $t_{2}$ compared
to $t$, current gets enhanced significantly. In addition, we have also
discussed the appearance of $\phi_{0}/2$ periodic current for some typical
electron fillings and its sensitivity on higher order hopping as well as
randomness. Finally, we have analyzed the scaling behavior of current which
suggests an asymptotic decay with ring size $N$.

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