text:
\section{INTRODUCTION}

Electronic properties of fractal lattices have been extensively investigated
in the past~\cite{domany}-\cite{meyer}, and a wealth of knowledge has been
accumulated. Such deterministic lattices provide excellent examples of
systems in between perfect periodic order and complete randomness and exhibit
electronic properties drastically different from a crystal and a disordered
material. The variable environment around any site of a
fractal structure makes such systems quite similar to the random networks. This
has motivated a series of experiments to examine the fractal behavior, magneto-resistance
and the superconductor-normal phase boundaries on Sierpinski gasket wire
networks~\cite{gordon1,gordon2,gordon3,korshu,meyer}.
The interface of theory and experiments has further been strengthened by
the recent development of non-dendritic,
perfectly self-similar fractal macromolecules of hexagonal Sierpinski
gaskets~\cite{new}.

Fractals, in general, exhibit a Cantor set like energy spectrum
~\cite{domany} that is highly degenerate, and the density of states displays
a wide variety of singularities. The spectrum may however, broaden up in the
presence of a magnetic field~\cite{banavar}. The electronic conductance
typically exhibits scaling with a multifractal distribution of the exponents
~\cite{schwalm2}. One useful tool in investigating
regular, finitely ramified fractal lattices has been the real space renormalization
group (RSRG) methods.
The RSRG approach has been successfully used over the years to unravel,
for example,
the spectral features of Koch fractals
with long range interactions~\cite{maritan}, localization aspects and the length
scaling of corner-to-corner propagation in fractal glass and several other interesting
fractal networks~\cite{schwalm1,andrade3}.

Interestingly, RSRG studies on fractal lattices reveal another remarkable
property of such systems, viz, the existence of an infinite number of
isolated {\it extended} eigenstates that coexist with an otherwise Cantor set
spectrum. This is non trivial, as the fractals as such do not have any long range
translational order. Such extended states sometimes are traced back to the
existence of multiple cycle fixed points of the Hamiltonian,
or have been shown to arise out of a local positional correlation, giving rise to
a resonant tunneling effect in some local atomic clusters~\cite{wang2,arun1,arun2,arun3}. However,
no conclusive evidence of the possible formation of a continuum of states has been
obtained until very recently, when Schwalm and Moritz~\cite{schwalm4}
have presented exhaustive numerical results to show that a family of fractals
does possess a continuum of such states. Incidentally, such continua were also
`suspected' in some of the previous studies~\cite{arun1,arun2,arun3}, where an
external magnetic field was shown to play a role.

This, to our mind, keeps the problem alive, and we look back into the
aspect of observing continuous distribution of eigenstates in an anisotropic
Sierpinski gasket (SPG) fractal. The magnetic flux, as discussed in the earlier works
\cite{arun1,arun2,arun3} through
each elementary plaquette of an isotropic SPG breaks the time reversal symmetry of the
electron hopping between nearest neighbors.
This in turn, already generates a kind of anisotropy in the hopping of
an electron when it travels along an {\it angular} bond, compared to
when it hops along a {\it horizontal} bond (Fig.~\ref{gasket}). Is
anisotropy the factor responsible for any continuum then ?
This question is the central motivation behind our present work.
Such an issue
had been addressed earlier by Hood and Southern~\cite{hood} using a
generating function approach coupled with RSRG to come to the conclusion that
in an anisotropic SPG only certain {\it hierarchical} states persist.

A Second motivation is of course the fact that, grafting a deterministic fractal
geometry on a given substrate
is quite feasible using the present day advanced nanotechnology and
lithographic methods. Therefore, any spectral peculiarity that might come out
as a consequence of the anisotropy, and is likely to get reflected in the
transport behavior, would be quite possible to observe. This may throw some
light into the unaddressed issues related to such deterministic fractal lattices, and
will definitely enrich the experimental aspects.

In this communication We examine the transmission spectrum of
finite but arbitrarily large anisotropic SPG lattices, with and
without any magnetic flux threading the system, using an RSRG decimation
method~\cite{arun1}. In addition to this, we also calculate the persistent current in
such a network, and look for any peculiarities in the behavior of the
current as well as in the Aharonov-Bohm (AB) oscillations in the transmission
that might come out as a result of the anisotropy. To the
best of our knowledge,
no results are available for persistent currents in an SPG network,
specially for the anisotropic case, and the impact of the scale invariant,
multiple loop geometry of an SPG on such properties
are either very little addressed, or unaddressed at all.
Therefore, a thorough study of the problem, together with the literature that already
exists will possibly help in obtaining some conclusive result about the spectra
of regular fractal lattices under various conditions.
\begin{figure}[ht]
{\centering\resizebox*{8cm}{5cm}{\includegraphics{fractal1.eps}} \par}
\caption{A Sierpinski gasket and its renormalization to a `dimer'}
\label{gasket}
\end{figure}

We find interesting results. Anisotropy indeed signals the
formation of {\it bands} of eigenstates with high transmittivity.
In general, it is seen that an anisotropic SPG is more conducting compared to its
isotropic counterpart. We have carefully carried out an RSRG analysis of the
recursion relations of the parameters of the Hamiltonian, and observe certain
unusual flow pattern. The transmission coefficient for arbitrarily large
lattices has been worked out. In addition to this, we present an in-depth
study of the persistent current observed in a SPG network, and the influence of the
degree of anisotropy on the current. The behavior of the persistent current is
seen to be very sensitive to the degree of anisotropy.

In what follows, we present the results of our investigation. In section II, we present
the model and the RSRG method. Section II contains the discussion on the transmission
spectrum, while, in section IV we present the results of the calculation of the persistent
current.
\vskip .3in
\section{THE MODEL AND THE METHOD}

We begin by referring to Fig.~\ref{gasket}. The Hamiltonian of the
network is given by,
\begin{equation}
{\mathbf H}=\sum_n \epsilon_n c_n^{\dagger} c_n + t e^{i\theta} \sum_{<nm>} c_n^{\dagger}
c_{m} + h.c.
\label{equ1}
\end{equation}
where,
$c_{n}$ ($c_{n^{\dagger}}$) is the annihilation (creation)
operator at the $n$th site of the gasket, the nearest neighbor hopping
integral $t = t_x$ along the `horizontal' bond and it is equal to $t_y$ along any
`angular' bond. $\theta = 2 \pi \phi/\phi_0$ is the phase acquired by the
electron in hopping along a side of the elementary triangle, as shown in Fig.~\ref{gasket}.
The phase factor $\theta$ is positive if the electron hops around an elementary
triangle in the clockwise sense, and negative otherwise.
We have chosen two kinds of `on-site' potentials depending on the local environment of any
given site. They are symbolized as $\epsilon$ and $\mu$, and the `edge' sites have
potentials $\epsilon_A$ and $\epsilon_B$ corresponding to the vertices $A$ and $B$.
For calculating the end-to-end transmission we have fixed two
semi-infinite ordered chains as leads (Fig.1) at the end sites $B$. The on-site
potential at the lead-sites has been taken to be constant and equal to $\epsilon_0$.
The nearest neighbor hopping integral in the leads
is chosen as $t_0$.

To obtain the transmission coefficient, as well as to judge the nature of the
eigenfunctions we renormalize the SPG using a decimation
technique~\cite{arun1}. We first explicitly discuss the anisotropic SPG
{\it without any magnetic field}.
The recursion relations for the on-site potentials and
the hopping integrals ( in the absence of any flux threading the gasket)
are given by,
\begin{widetext}
\begin{eqnarray}
\epsilon_{n+1} & = & \epsilon_n + \frac{p_n + 2q_n + r_n}{\alpha_n \beta_n -
\gamma_n} + s_n \nonumber \\
\mu_{n+1} & = & \mu_n + u_n + v_n \nonumber \\
\epsilon_{A,n+1} & = & \epsilon_{A,n} + \frac{(E - \epsilon_n) [ \beta_n t_{y,n}^2
+ \alpha_n t_{x,n}^2 + 2 t_{x,n}^2 t_{y,n}^2 (E - \epsilon_n + t_{x,n}) ]}
{\alpha_n \beta_n - \gamma_n} \nonumber \\
\epsilon_{B,n+1} & = & \epsilon_{B,n} + \frac{\beta_n t_{y,n} (E - \epsilon_n + t_{x,n}) +
\gamma_n t_{y,n}}{\alpha_n \beta_n - \gamma_n} \nonumber \\
t_{x,n+1} & = & \frac{[ (E - \epsilon_n) t_{x,n} + t_{y,n}^2]^2 + t_{x,n} t_{y,n}^2 (E - \mu_n + 2t_{x,n})
- t_{x,n}^4}{( E - \epsilon_n + t_{x,n} ) [ \beta_n - t_{y,n}^2 - t_{x,n} ( E - \mu_n) ]} \nonumber \\
t_{y,n+1} & = & \frac{t_{y,n}^2 ( E - \mu_n + 2 t_{x,n} )}{\beta_n - t_{y,n}^2 - t_{x,n} ( E - \mu_n)}
\end{eqnarray}
\end{widetext}
Here, we have defined,
\begin{eqnarray}
\alpha_n & = & (E - \epsilon_n)^2 - t_{x,n}^2 \nonumber \\
\beta_n & = & (E - \epsilon_n) (E - \mu_n) - t_{y,n}^2 \nonumber \\
\gamma_n & = & t_{y,n}^2 (E - \epsilon_n + t_{x,n})^2 \nonumber \\
p_n & = & t_{y,n}^2 \beta_n (E - \epsilon_n) \nonumber \\
q_n & = & t_{x,n} t_{y,n}^2 (E - \epsilon_n) (E - \epsilon_n + t_{x,n}) \nonumber \\
r_n & = & t_{x,n}^2 \alpha_n (E - \epsilon_n) \nonumber \\
s_n & = & \frac{2t_{y,n}^2 (E - \epsilon_n)}{\beta_n - t_{y,n}^2 - t_{x,n}(E - \mu_n)} \nonumber \\
u_n & = & \frac{2t_{x,n}^2 (E - \epsilon_n + t_{x,n}) + t_{y,n}^2 (E - \mu_n)}{(E - \mu_n)
(E - \epsilon_n + t_{x,n})} \nonumber \\
v_n & = &\frac{t_{y,n}^2 (E - \mu_n + 2t_{x,n})^2}{(E - \mu_n)[\beta_n - t_{y,n}^2 - t_{x,n}(E - \mu_n)]}
\end{eqnarray}

Using the above set of equations, we renormalize an $n$-th generation SPG $(n-1)$- times
to reduce it to a simple triangle. The top vertex is then decimated to generate an
effective diatomic molecule $B-B$ clamped between the leads, as shown in Fig.1(b).
The transmission coefficient is then obtained by the standard formula~\cite{stone},
\begin{widetext}
\begin{equation}
T = \frac{4 \sin^2(qa)}{[M_{1,2} - M_{2,1} + ( M_{11} - M_{22} ) \cos (qa)]^2 +
(M_{11} + M_{22})^2 \sin^2(qa)}
\end{equation}
\end{widetext}
where, the matrix elements corresponding to the `diatomic molecule' are given by,
$M_{11} = (E - \epsilon_{B,n})^2/(t_0 t_{x,n}) - t_{x,n}/t_0$,
$M_{12} = -M_{21} = -(E - \epsilon_{B,n})/t_{x,n}$, and $M_{22} = -t_0/t_{x,n}$.
Before ending this section we just point out that the same decimation technique has been
used to obtain the recursion relations in the presence of a magnetic field.
The recursion relations in the latter case are much more complicated compared to the field-free
case,
and we decide not to present them here to save space. The role of the magnetic field will  however,
be discussed in appropriate places.
The results are now presented below.

\begin{figure}[ht]
{\centering\resizebox*{15cm}{12cm}{\includegraphics[angle=-90] {spfig2.eps}} \par}
\caption{Transmission across a sixth generation
isotropic gasket for (a) $\phi = 0$,
and for (b) $\phi = \phi_0/4$. We have chosen $\epsilon_j = 0$ for all
the vertices of the SPG as well as for the sites in the leads, and
$t_x = t_y = t_0 = 1$ throughout.}
\label{trans1}
\end{figure}

\section{RESULTS AND DISCUSSION}
\subsection{The transmission spectrum and the electron states}

The transmission coefficients for the zero-field case and for
flux $\phi = \phi_0/4$ threading an elementary triangle have been shown in
Fig.~\ref{trans1} for a sixth generation SPG in the isotropic limit
for the sake of completeness. As is already
known~\cite{arun1}, the magnetic field makes an isotropic gasket
more conducting with the signature of {\it continuous} zones of transmission.
When a slight anisotropy is introduced, and the magnetic field is taken off,
it is interesting to observe that, first of all the SPG becomes much more
conducting than its isotropic counterpart, and second, once again patches of
continuum appear in the transmission spectrum. One such example is presented
in Fig.~\ref{trans2} with $t_x = 0.9$ and $t_y = 1$.
With increasing degree of anisotropy
the continua become much more pronounced at places. We have not been able to
provide an analytical proof as to whether real continua appear even in such
non-translationally invariant systems, but have carried out extensive numerical
investigations involving very fine scan of the `apparently' continuous bands
found in the transmission spectrum. In every case we find the fine structure
in these `special' zones free from any real gaps.
Even, the apparently sharp peaks in the transmission spectrum show up
finite width when the energy is scanned minutely in their neighborhood.
Thus, it is tempting to
claim that anisotropy introduces {\it continuous bands} of high transmission
in fractals such as an SPG. In a recent work
Schwalm and Moritz~\cite{schwalm4} have discussed precisely this issue in the
case of a different class of hierarchical lattices. This latest observation gives us
confidence in regards of the existence of bands in fractal lattices which can
coexist with even a fragmented Cantor like spectrum.
\begin{figure}[ht]
{\centering\resizebox*{15cm}{12cm}{\includegraphics[angle=-90]{spfig3.eps}} \par}
\caption{(a) Transmission across a sixth generation
anisotropic gasket for $\phi = 0$ and with $t_x = 0.9$ and $t_y = 1$.
(b) Fine scan of the energy interval
$[-1.92,-1.8]$ with $\delta E = 10^-7$. Other parameters are the same as in
Fig.2, and energy is measured in unit of the hopping in the leads.
}
\label{trans2}
\end{figure}

Are these conducting states extended ?
To understand this, we have critically examined the flow of the hopping
integrals $t_x$ and $t_y$ under successive RSRG steps. We find two different
types of behavior. In general, for an arbitrarily chosen energy both
$t_x$ and $t_y$ flow to zero quickly under iteration. This is understandable, as
the spectrum of such a fractal lattice is fragmented in general. So, any
arbitrarily chosen energy, in all probability, would either fall in a gap, or
would correspond to a localized eigenstate.
However, Hood and Southern~\cite{hood} have eliminated the possibility of
exponentially localized states in an anisotropic SPG.
The flow of the hopping integrals under RSRG changes its pattern when
we choose an energy from what apparently looks like a continuum in the transmission
spectrum in Fig.~\ref{trans2}(a). For any energy within the {\it continuous} portion
of the spectrum $t_x$ remains non-zero for arbitrary number of
RSRG loops while the hopping $t_y$ ultimately flows to zero.
This feature is always true no matter whether the initial
value of $t_x$ is less or greater than the initial value of $t_y$, and persists
for any degree of anisotropy. That is, the anisotropy somehow develops a `preference'
for $t_x$, and breaks the symmetry at all scales of length.
Thus, in the infinite lattice limit the entire SPG
fractal consists of isolated {\it dimers} coupled by the hopping integral
$t_{x,n \rightarrow \infty}$. For a finite (but arbitrarily large) SPG we have
coupled the semi-infinite leads to the two base atoms ($B-B$). Thus, as $t_y$ becomes
(practically) zero under RSRG operation, we are left with a perfectly ordered $1$-d
chain of atoms with an effective, energy dependent  potential connected to its
nearest neighbor by a hopping amplitude $t_{x,n \rightarrow \infty}$ (Fig.~\ref{rsrg}).
So, any energy $E$ that will map the original
SPG to such a configuration will definitely correspond to a conducting state provided,
the energy falls within the allowed band of the leads. What is peculiar about such energy
values is that, they apparently form a completely continuous band. We have scanned the
spectrum very carefully.
One such result of scanning is exhibited in Fig.~\ref{trans2}(b), where
we have chosen, quite arbitrarily, a certain portion of the spectrum shown in Fig.2a, and
scanned the selected zone in an energy interval of $10^{-7}$. It is obvious that the
spectrum retains its continuous character.

Before we end this sub-section, it is worth mentioning that, getting a non-zero transmission
is definitely connected to the fact that we have attached the leads at the two base atoms
of the SPG. Talking about an infinite SPG, we should appreciate that we can build an infinite
SPG using a {\it top down} approach, when one takes a triangle and keeps on
piercing it in the SPG design an infinite number of times. We can keep the leads attached to
the base pair of $B$ atoms from the beginning. So, the effective ordered chain of atoms
which the SPG boils down to, when we pick up an energy from the continuum, will always
be found clamped between the leads. The corresponding state will be extended in the sense that
a finite number of sites of the parent SPG will have non-zero amplitudes of the wave function.
In the reverse approach, when one builds an infinite SPG by placing SPG's of previous generations
on top of each other, following the growth rule, it is of course not very meaningful to talk
about the base atoms such as the $B-B$ pair, or even fixing the leads at the ends. So, at most the
infinite system can break up into diatomic clusters as depicted in Fig.~\ref{rsrg}.
\begin{figure}[ht]
{\centering\resizebox*{8cm}{8cm}{\includegraphics {fractal2.eps}} \par}
\caption{Separation of  a finite Sierpinski gasket under RSRG into
detached diatomic clusters and linear ordered chain clamped between the leads.}
\label{rsrg}
\end{figure}

\subsection{The Aharonov-Bohm oscillations}

The anisotropic SPG displays a wide variety of AB-oscillations which are
sensitive to the energy chosen, as well as on the degree of anisotropy. The
period of oscillations has been found to be equal to $\phi_0/2$ if we choose,
say, $E = 0$. The AB-oscillations are displayed in Fig.~\ref{ab} for the
cases $(t_x = t_y =1)$, $(t_x = 1, t_y = 2)$ and $(t_x = 1, t_y = 10)$.
The simple oscillation profile in the isotropic case develops into a profile
with multiple peaks and valleys (Fig.~\ref{ab}(b)) as the anisotropy grows
larger and finally, in the case of very large degree of anisotropy, the AB-oscillation
profile consists of delta-like peaks at special values of the magnetic flux, indicating
that, the anisotropic gasket triggers ballistic transmission for $E = 0$ only at certain
special values of the magnetic flux.
\begin{figure}[ht]
{\centering\resizebox*{15cm}{12cm}{\includegraphics [angle=-90]{aharonov.eps}} \par}
\caption{(a) Aharonov-Bohm oscillations in a $4$th generation SPG for $E = 0$.
(a) Isotropic SPG with $t_x = 1$ and $t_y = 1$, (b) anisotropic SPG with
$t_x = 1$, and $t_y = 2$, and (c) anisotropic SPG with $t_x = 1$, and $t_y = 10$.
The potential and other parameters are the same as in the previous figures.}
\label{ab}
\end{figure}

\subsection{The persistent current}

It is well known that a simple Aharonov-Bohm flux threading a metallic or
semiconducting ring generates an equilibrium persistent current~\cite{gefen}-
\cite{jin}. With reference to our SPG fractal, where the same flux penetrates
each elementary triangular plaquette, the eigenvalues and the eigenstates are
flux periodic with period $\phi_0$,and hence, as is well known, the equilibrium
persistent current carried by the energy level $E_n$ is given by,
\begin{equation}
I_n = -c \frac{\partial E_n}{\partial \phi}
\end{equation}
and, the total current in the ring is, $I (\phi) = \sum_{n=1}^{N_e} I_n$ where,
$N_e$ is the total number of electrons in the system. It is clear that the nature
of the energy spectrum plays a major role in dictating the persistent current in a
system. We therefore first have a look at the
flux dependence of the eigenvalue spectrum of a $4$-th generation
SPG fractal (Fig.~\ref{spectrum}). For the sake of comparison, we present the
spectrum of the SPG in the isotropic limit (Fig.~\ref{spectrum}, top panel), which
shows multiple band crossings, as observed by Rammal and Toulose~\cite{rammal}
earlier. The band crossings, related to the symmetry and degeneracy of the eigenvalues
at various values of $\phi/\phi_0$ are already discussed in details in reference~\cite{rammal}.
\begin{figure}[ht]
{\centering\resizebox*{7cm}{14cm}{\includegraphics{spectrum.eps}} \par}
\caption{Flux dependence of the eigenvalue spectrum of a $4$th generation
Sierpinski gasket. (a) The isotropic gasket with $t_x = t_y =1$,
(b) the anisotropic gasket with $t_x =1$, $t_y = 2$, and (c) with
$t_x =1$ and $t_y = 10$. In each cast the on-site potential is taken
to be zero throughout.}
\label{spectrum}
\end{figure}
\begin{figure}[ht]
{\centering\resizebox*{15cm}{14cm}{\includegraphics{current.eps}} \par}
\caption{Persistent current in a $4$th generation
Sierpinski gasket. (a) The isotropic gasket with $t_x = t_y =1$. The solid
and the dashed curves correspond to $N_e = 4$ and $15$ respectively.
(b) The anisotropic gasket with $t_x =1$, $t_y = 2$ (solid line), and
$t_x = 1$, $t_y = 10$ (dashed line) with $N_e = 21$, and (c) Anisotropic
gasket with $N_e = 25$ when
$t_x =1$, $t_y = 2$ (solid line) and $t_x = 1$, $t_y = 10$ (dashed line).
In each cast the on-site potential is taken
to be zero throughout.}
\label{current}
\end{figure}

With the introduction of anisotropy in the hopping integrals ( $t_x \ne t_y$ ) gaps start
to open up in the spectrum lifting the degeneracy in places. As the degree of anisotropy is
increased, the spectrum shows clear signature of having groups of states lying close in energy
and separated by wider global gaps, that bring a flavor of sub-band structures in the spectrum.
The middle and the bottom panels of Fig.~\ref{spectrum} represent the anisotropic situation
with $t_x = 1$, $t_y = 2$  and  $t_x = 1$, $t_y = 10$ respectively, the last case may be taken
as a case of large anisotropy. The flattening of the bands in this case is noteworthy, which
finally brings in non trivial changes in the persistent current.

We have examined in details the variation of the persistent current for both the isotropic
and anisotropic SPG's. for the isotropic case with $t_x = t_y = 1$, the current does not
exhibit serious deviations (in its qualitative features) from that in a single loop ~\cite{gefen}
as long as $N_e$ is small. With increasing values of $N_e$, a slight rounding off of the
$I(\phi)$ is observed. The overall current keeps on increasing as $N_e$ increases. The first
variation shows up for $N_e = 10$, when additional kinks appear at $\phi = \phi_0/2$.
With increasing $N_e$ the SPG starts displaying features different from what one finds in
simple loop structures. The features persist as we approach the half filled case which, in
this study is $N_e = 21$, when a precise $\phi_0/2$ periodicity in $I(\phi)$ is observed.
In Fig.~\ref{current}(a) we show the variation of the persistent current when $N_e=4$ (solid line)
and $N_e=15$ (dashed line).

Introduction of anisotropy in the values of the hopping integral brings in different
features in the $I(\phi)$-$\phi$ curves. Though several basic features such as the
$\phi_0$ periodicity, or the special case of $\phi_0/2$ period at $N_e=21$ persist, the
degree of anisotropy, i.e. the relative values of $t_x$ and $t_y$ strongly influence the
current. Needless to say, the filling factor also plays a crucial role in determining the
current profiles. For example, we have found that with $(t_x,t_y) = (1,10)$, the current drops
compared to the case when $(t_x,t_y) = (1,2)$ when we take $N_e = 21$. This has been tested
with several values of the hopping and we come to the conclusion that at half filling, the
effect of anisotropy is to reduce the overall current. The current profile becomes smooth
with the discontinuous features of isotropic cases for low $N_e$ as we introduce anisotropy.
This is mainly due to the gaps which open up in the spectrum as the anisotropy is introduced.
In this aspect, anisotropy plays a role equivalent to the role that disorder plays in one
dimensional rings.

\begin{center}
{\bf Acknowledgment}
\end{center}

The authors are thankful to Santanu K. Maiti for helping out with the graphics, and
useful comments on the results of this paper.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

The recent discovery of iron oxypnictide superconductors\cite{Kamihara,Ren,xChen,gChen,Kito} with transition temperatures of up to $T_c \sim 55K$ has stimulated much interest in the relationship between the mechanism of the superconductivity and the orbital degrees of freedom. First-principles calculations have predicted the band structure with hole Fermi pockets around the $\Gamma$ point and  electron Fermi pockets around the $M$ point\cite{Singh,Ishibashi,Nakamura}. By using the weak-coupling approaches based on  multiorbital models, the spin-singlet $s$-wave pairing is predicted, where the order parameter of this pairing changes its sign between  hole and  electron Fermi pockets (sign-reversing $s$-wave pairing)\cite{Kuroki,Mazin,Wang,Nomura,Yanagi}.
This unconventional $s$-wave pairing is expected to emerge owing to the effect of antiferromagnetic spin fluctuations. Since the strong correlation between electrons is considered to play an important role in the superconductivity of iron oxypnictides as well as in that of  high-$T_c$ cuprates,
 nonperturbative and reliable approaches  are required.

As a nonperturbative approach, the exact diagonalization (ED) method has been extensively applied in the Hubbard,  $d$-$p$, and  $t$-$J$ models\cite{newref,tJ-Ladder}. Although these models are much simplified and mostly limited to one dimension, it has elucidated some important effects of a strong correlation on  superconductivity.  Using the ED method, we have studied the one-dimensional (1D) two-orbital Hubbard model in the presence of the band splitting $\Delta$. It is  found that the superconducting (SC) phase appears in the vicinity of the partially polarized ferromagnetism when the exchange (Hund's rule) coupling $J$ is larger than its critical value on the order of $\Delta$\cite{Sanohund}. The result suggests that  spin triplet pairing emerges owing to the effect of  ferromagnetic spin fluctuation. In the case of $\Delta=0$, spin triplet superconductivity has also been discussed on the basis of bosonization\cite{Shelton,Shen,Lee} and  numerical\cite{Sakamoto,Shirakawa,Sano-ICM} approaches. Previous works, however, were restricted to the case of a single Fermi surface, and the effects of  electron and hole Fermi pockets on  superconductivity have not been discussed therein.

In this study,  we investigate the 1D two-orbital Hubbard model with  electron and hole Fermi pockets, where the Fermi level crosses both the upper and lower bands in the presence of a finite band splitting $\Delta$.   Using the ED method, the Luttinger liquid parameter $K_{\rho}$ is calculated  to obtain the SC phase diagram as a function of on-site Coulomb interactions in a wide parameter region including both weak- and strong-correlation regimes.
It would clarify the effects of a strong correlation on  superconductivity in  iron oxypnictides. We also calculate  various pairing correlation functions and discuss a possible pairing symmetry. Although our model is much simplified and limited to one dimension, we expect that the essence of the  superconducting mechanism of iron oxypnictides can be discussed.

\section{Model and Formulation}
We consider the one-dimensional two-orbital Hubbard  model given by the following Hamiltonian:
\begin{eqnarray}
 H&=&t\sum_{i,m,\sigma}(c_{i,m,\sigma}^{\dagger} c_{i+1,m,\sigma}+h.c.)
      \nonumber \\
  &+&\frac{\Delta}{2}\sum_{i,\sigma}(n_{i,u,\sigma}-n_{i,l,\sigma})
     +U\sum_{i,m}n_{i,m,\uparrow}n_{i,m,\downarrow}
      \nonumber \\
  &+&U'\sum_{i,\sigma}n_{i,u,\sigma}n_{i,l,-\sigma}
    +(U'-J)\sum_{i,\sigma}n_{i,u,\sigma}n_{i,l,\sigma}
      \nonumber \\
  &-&J\sum_{i} (c_{i,u,\uparrow}^{\dagger} c_{i,u,\downarrow}
          c_{i,l,\downarrow}^{\dagger} c_{i,l,\uparrow}+h.c.)
      \nonumber \\
     &-&J'\sum_{i} (c_{i,u,\uparrow}^{\dagger} c_{i,u,\downarrow}^{\dagger}
        c_{i,l,\uparrow} c_{i,l,\downarrow}+h.c.),
\label{hund-Hamil}
 \end{eqnarray}
where $c^{\dagger}_{i,m,\sigma}$ stands for the creation operator of an
 electron with a spin $\sigma \ (= \uparrow, \downarrow)$ and an orbital $m \ ( =u,l)$ at site $i$  and
   $n_{i,m,\sigma}=c_{i,m,\sigma}^{\dagger}c_{i,m,\sigma}$.
Here, $t$ represents the hopping integral between the same orbitals and
we set $t=1$ in this study.
The interaction parameters $U$, $U'$, $J$, and $J'$ stand for the intra- and inter-orbital direct Coulomb interactions, exchange  (Hund's rule) coupling, and  pair-transfer, respectively.
 $\Delta$ denotes the energy difference between the two atomic orbitals.
For simplicity, we impose the relation $J=J'$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=6.5cm]{63893Fig1.eps}
\end{center}
 \caption{Schematic diagrams of (a) the model Hamiltonian, (b) the band  structure in the noninteracting case, and (c) a corresponding two-dimensional Fermi surface  related to our 1D model. }
 \label{bandmodel}
\end{figure}

The model in eq. (\ref{hund-Hamil})  is schematically shown in  Fig. \ref{bandmodel}(a).
In a noninteracting case ($U=U'=J=0$), the Hamiltonian eq. (\ref{hund-Hamil}) yields dispersion relations representing the upper and lower band energies:
$ \epsilon^{u}(k)=2t \cos(k) + \frac{\Delta}{2}$ and
$ \epsilon^{l}(k)=2t \cos(k) - \frac{\Delta}{2}$,
where $k$ is the wave vector. This  band structure  is schematically shown in  Fig. \ref{bandmodel}(b). When the Fermi level $E_{k_{\rm F}}$ crosses  both the upper  and lower bands, the system is  metallic with electron and hole Fermi pockets  corresponding to a characteristic band structure of the FeAs plane in iron oxypnictides, as shown in Fig. \ref{bandmodel}(c).

We numerically diagonalize the  model Hamiltonian up to 6 sites (12 orbitals) and estimate the Luttinger liquid parameter $K_{\rho}$ from the  ground-state energy of finite-size systems  using the standard Lanczos algorithm.\cite{newref}
To reduce the finite size effect, we impose the boundary condition (periodic or  antiperiodic)  on upper and lower orbitals independently  and chose  both boundary conditions to minimize $|K_{\rho}^0-1|$, where  $K_{\rho}^0$ represents the $K_{\rho}$ of the finite-size system in a noninteracting case. The typical deviation of $K_{\rho}$ from unity becomes about $\sim 0.1$ for a six-site system.
For simplicity, we will redefine $K_{\rho}$ as a renormalize value calculated using $K_{\rho}/K_{\rho}^0$,  hereafter.

On the basis of the Tomonaga-Luttinger liquid theory \cite{Solyom,Voit,Emery1,Balentz,Fabrizio}, various types of correlation functions are determined by the single parameter $K_\rho$ in the model which is isotropic in spin space.
For a single-band model with two Fermi points, $\pm k_{F}$, the SC correlation function decays as $\sim r^{-(1+\frac{1}{K_{\rho}})}$, while the CDW and SDW correlation functions decay as $\sim \cos({2k_F r}) r^{-(1+K_{\rho})}$.
Thus, the SC correlation  is dominant for $K_{\rho}>1$, while the CDW or SDW correlation is dominant for $K_{\rho}<1$.
On the other hand, for a two-band model with four Fermi points, i.e., $\pm k_{F_1}$ and $\pm k_{F_2}$,  low-energy excitations are given by a single gapless charge mode with a gapped spin mode\cite{Balentz,Fabrizio,Emery}. In this case, the SC  and  CDW correlations decay as $\sim r^{-\frac{1}{2K_{\rho}}}$ and $\sim \cos[{2(k_{F_2}-k_{F_1}) r}] r^{-2K_{\rho}}$, respectively, while the SDW correlation decays exponentially. Hence, the SC correlation is dominant for  $K_\rho >0.5$, while the CDW correlation is dominant for $K_\rho <0.5$.
In either case, the SC correlation increases with the exponent $K_\rho $,
 and then  $K_{\rho}$ is regarded as a good  indicator of  superconductivity.\cite{Ladder}
As the noninteracting $K_{\rho}$ is always unity, we assume that the condition of $K_{\rho}>1$ for our model corresponds to   the superconducting state  realized in oxypnictide superconductors.

\section{Phase Diagram}
\begin{figure}[t]
\begin{center}
\includegraphics[width=7.0cm]{63893Fig2.eps}
\end{center}
\caption{$K_{\rho}$ as a function of $U'(=4J)$ for $n=5/3$ (10 electrons/6 sites)  at $\Delta=0.8, 1.6, 1.9, 2.3$, and 2.8.
The singlet ground state  changes into a partially  polarized ferromagnetic ($S$=1) state   at $U'\simeq$ 2.5, 3.2, and 4.1  for $\Delta=$ 1.9, 2.3, and 2.8, respectively.
The inset shows  the energy difference $E_0(\phi)-E_0(0)$  as a function of the external flux $\phi$ for $n=2/3$ (6 electrons/9 sites) at $\Delta=1.2$.
}
\label{kr-all}
\end{figure}
Figure \ref{kr-all} shows   $K_{\rho}$ as a function of  $U'$ for several values of $\Delta$ at an electron density  $n=5/3$ (10 electrons/6 sites), where we set  $J=U'/4$ with  $U=U'+2J$.
When $U'$ increases, $K_{\rho}$  decreases for a small $U'$, while it increases  for a large $U'$ in the case of $\Delta \ge 1.9$, and then  becomes larger than unity for $U'>2.3$ in the case of $\Delta= 1.9$.
When $J(=U'/4)$ is larger than a certain critical value, the ground state changes into the partially polarized ferromagnetic state with a total spin $S=1$ from the singlet  state with $S=0$. We find that  superconductivity is  most enhanced in the vicinity of the partially polarized ferromagnetic state. To confirm  superconductivity, we calculate  the energy difference of the ground state, $E_0(\phi)-E_0(0)$,  as a function of the external flux $\phi$. As shown in the inset of Fig.  \ref{kr-all},  anomalous flux  quantization is clearly observed for $\Delta=1.9$ while not for $\Delta=0.8$.

\begin{figure}[t]
\begin{center}
\includegraphics[width=6.6cm]{63893Fig3.eps}
\end{center}
\caption{Phase diagram of the ground state with  $K_{\rho}$ on the  $U'-J$  parameter plane with $U=U'+2J$ for $n=5/3$ (10 electrons/6 sites)  at $\Delta=1.9$.
  }
\label{U-Jsouz}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=6.6cm]{63893Fig4.eps}
\end{center}
\caption{Phase diagram of the ground state with  $K_{\rho}$ on the  $U-U'$  parameter plane with  $J=U'/4$ for $n=5/3$ (10 electrons/6 sites)  at $\Delta=1.9$.
  }
\label{U-Upsouz}
\end{figure}

In Fig. \ref{U-Jsouz}, we show the  phase diagram of the ground state  on the $U'-J$ parameter plane  under  the condition of $U=U'+2J$ for $n=5/3$ (10 electrons/6 sites)  at $\Delta=1.9$.
It contains the  singlet state with  $S=0$  together with partially polarized ferromagnetic states with $S=1$ and  $S=2$.
The singlet state with $K_\rho>1$, where we call it the $SC$ phase,  appears near the partially polarized  ferromagnetic  region at $J \stackrel{>}{_\sim} U'$. It extends from the attractive  region ($U'<0$) to  the realistic parameter region with $J \sim U'/4>0$, which is expected to correspond to that in the case of iron oxypnictides\cite{Nakamura}. We have confirmed  that  similar phase diagrams are also obtained  for $\Delta=2.3$ and 2.6.

Figure \ref{U-Upsouz} shows the phase diagram of the ground state on the  $U-U'$ plane under  the condition of $J=U'/4$ for $n=5/3$ (10 electrons/6 sites)  at $\Delta=1.9$.
We observe two types of SC phases with $K_\rho>1$, the SC \Roman{u'-large} for $U>U'$ and the SC \Roman{u-large} for $U<U'$, in the wide parameter region including both weak- and strong-correlation regimes. Note that the SC  \Roman{u'-large}  corresponds to the SC phase shown in  Fig. \ref{U-Jsouz} and belongs to the realistic parameter region mentioned before.

\section{Pairing Correlation}

\begin{figure}[t]
\begin{center}
\includegraphics[width=6.5cm]{63893Fig5.eps}
\end{center}
\caption{Schematic diagrams of various types of  superconducting pairing symmetries, $S^{\rm l}_{\rm on}$, $S^{\rm u}_{\rm on}$, $S^{\rm l}_{\rm nn}$, $S^{\rm u}_{\rm nn}$, and $S^{\rm lu}_{\rm on}$  with spin singlet pairings  and  $T^{\rm l}_{\rm nn}$,  $T^{\rm u}_{\rm nn}$, and $T^{\rm lu}_{\rm on}$ with spin triplet pairings.
  }
\label{hub-pair}
\end{figure}

To examine the nature of these SC phases, we calculate  SC pairing correlation  functions for the various types of pairing symmetries  schematically shown in Fig. \ref{hub-pair}. Explicit forms of the SC pairing correlation  functions $C(r)$  are given by
\begin{eqnarray}
  S^{\rm l}_{\rm on}(r)&=&\frac1{N}\sum_{i} \langle c^\dagger_{i,l,\uparrow}
 c^\dagger_{i,l,\downarrow} c_{i+r,l,\downarrow} c_{i+r,l,\uparrow}\rangle,

\nonumber \\
    S^{\rm u}_{\rm on}(r)&=&\frac1{N}\sum_{i} \langle c^\dagger_{i,u,\uparrow} c^\dagger_{i,u,\downarrow}  c_{i+r,u,\downarrow} c_{i+r,u,\uparrow}\rangle,
\nonumber  \\
 S^{\rm l}_{\rm nn}(r)&=&\frac1{2N}\sum_{i}  \langle (c^\dagger_{i,l,\uparrow} c^\dagger_{i+1,l,\downarrow}-c^\dagger_{i,l,\downarrow}c^\dagger_{i+1,l,\uparrow})
  \nonumber  \\ &\times&
(c_{i+r+1\downarrow}c_{i+r,l,\uparrow}-c_{i+r+1,l,\uparrow} c_{i+r,l,\downarrow})\rangle,
\nonumber \\
      S^{\rm u}_{\rm nn}(r)&=&\frac1{2N}\sum_{i}   \langle (c^\dagger_{i,u,\uparrow} c^\dagger_{i+1,u,\downarrow}-  c^\dagger_{i,u,\downarrow}c^\dagger_{i+1,u,\uparrow})   \nonumber
\nonumber \\
   &\times& (c_{i+r+1,u,\downarrow}c_{i+r,u,\uparrow}-c_{i+r+1,u,\uparrow} c_{i+r,u,\downarrow})\rangle,
\nonumber \\
 S^{\rm lu}_{\rm on}(r)&=&\frac1{2N_u}\sum_{i}  \langle (c^\dagger_{i,l,\uparrow} c^\dagger_{i,u,\downarrow}-  c^\dagger_{i,l,\downarrow}c^\dagger_{i,u,\uparrow})   \nonumber
\nonumber \\
 &\times &(c_{i+r,u,\downarrow}c_{i+r,l,\uparrow}-c_{i+r,u,\uparrow}c_{i+r,l,\downarrow})\rangle,
\nonumber \\
  T^{\rm l}_{\rm nn}(r)&=&\frac1{2N_u}\sum_{i}  \langle (c^\dagger_{i,l,\uparrow} c^\dagger_{i+1,l,\downarrow}+ c^\dagger_{i,l,\downarrow}c^\dagger_{i+1,l,\uparrow})   \nonumber
\nonumber \\
 &\times &(c_{i+r+1\downarrow}c_{i+r,l,\uparrow}+c_{i+r+1,l,\uparrow}c_{i+r,l,\downarrow})\rangle,
\nonumber \\
  T^{\rm u}_{\rm nn}(r)&=&\frac1{2N_u}\sum_{i}  \langle (c^\dagger_{i,u,\uparrow} c^\dagger_{i+1,u,\downarrow}+ c^\dagger_{i,u,\downarrow}c^\dagger_{i+1,u,\uparrow})   \nonumber       \\
 &\times &(c_{i+r+1,u,\downarrow}c_{i+r,u,\uparrow}+c_{i+r+1,u,\uparrow} c_{i+r,u,\downarrow})\rangle,
\nonumber \\
 T^{\rm lu}_{\rm on}(r)&=&\frac1{2N_u}\sum_{i} \langle (c^\dagger_{i,l,\uparrow} c^\dagger_{i,u,\downarrow}+c^\dagger_{i,l,\downarrow}c^\dagger_{i,u,\uparrow})   \nonumber       \\
 &\times &(c_{i+r,u,\downarrow}c_{i+r,l,\uparrow}+c_{i+r,u,\uparrow} c_{i+r,l,\downarrow})\rangle,
\nonumber
\end{eqnarray}
where   $S^{\rm l}_{\rm on}(r)$, $S^{\rm u}_{\rm on}(r)$, $S^{\rm l}_{\rm nn}(r)$, $S^{\rm u}_{\rm nn}(r)$, and  $S^{\rm lu}_{\rm on}(r)$   denote  the singlet pairing  correlation functions  on the same site in the lower orbital,  on the same site in the upper orbital,  between  nearest-neighbor  sites in the lower orbital,     between  nearest-neighbor sites in the upper orbital, and between the lower and upper orbitals on the same site, respectively.
Furthermore, $T^{\rm l}_{\rm nn}(r)$,  $T^{\rm u}_{\rm nn}(r)$,  and $T^{\rm lu}_{\rm on}(r)$ are  the triplet pairing   correlation functions between  nearest-neighbor  sites in the lower orbital, between nearest-neighbor  sites in the upper orbital, and  between the lower and upper orbitals on the same site,  respectively.

In Fig. \ref{sc-cor1}, we show the absolute values of various types of  SC pairing correlation  functions  $|C(r)|$  for $n=5/3$ (10 electrons/6 sites) at $\Delta=1.9$,  $U'=4J=1.0$,  and $U=-0.4$. Here, the electronic state of the system belongs to the SC \Roman{u-large}  phase, although the phase diagram for $U<0$ is not explicitly shown in Fig. \ref{U-Upsouz}.  Note that $|T^{\rm u}_{\rm nn}(r)|<10^{-4}$ and $S^{\rm lu}_{\rm on}(r=3)=T^{\rm lu}_{\rm on}(r=3)=0$, which are not shown in Fig. \ref{sc-cor1}.
We find that $S^{\rm u}_{\rm on}(r)$ and $S^{\rm u}_{\rm nn}(r)$  decay very slowly  as functions of $r$, and  $|S^{\rm u}_{\rm on}(r=3)|$  is the largest among the various $|C(r=3)|$ values.
Therefore, a relevant  pairing symmetry for the SC \Roman{u-large} phase seems to be the spin singlet pairing in the upper orbital band,  mainly consist of 'on-site' pairing. It is considered that such a pairing in attractive region with $U<0$ is due to the intra-orbital attraction $U$. On the other hand, in repulsive region with $U'>U>0$, the pairing may be due to  charge fluctuation, which is enhanced by a large inter-orbital repulsion $U'$  similarly to that in the case of the $d$-$p$ model in the presence of the inter-orbital repulsion $U_{pd}$\cite{Sano-Udp-all}.

\begin{figure}[h]
\begin{center}
\includegraphics[width=7.0cm]{63893Fig6.eps}
\end{center}
\caption{Absolute values of various types of  SC pairing correlation  functions  $|C(r)|$  as functions of $r$  for $n$=5/3 (10 electrons/6 sites) at  $\Delta=1.9$, $U'(=4J)=1.0$, and $U=-0.4$, corresponding to the SC \Roman{u-large} phase.}
\label{sc-cor1}
\end{figure}

Next, we discuss the superconductivity in the SC \Roman{u'-large} phase including the realistic parameter region  mentioned before.
Figure. \ref{sc-cor2} shows  the absolute values of various types of  SC pairing correlation  functions  $|C(r)|$  for $n=5/3$ (10 electrons/6 sites) at $\Delta=1.9$,  $U'=4J=1.0$,  and $U=2.4$, where the system belongs to the SC \Roman{u'-large}  phase, as shown in Fig. \ref{U-Upsouz}.
Here,  $|T^{\rm u}_{\rm nn}(r)|$,  $|S^{\rm lu}_{\rm on}(r=3)|$, and $|T^{\rm lu}_{\rm on}(r=3)|$  are not shown,  because these correlation functions are very small or zero.
We find that $|S^{\rm u}_{\rm on}(r)|$ is considerably suppressed  compared with $|S^{\rm u}_{\rm nn}(r)|$ in contrast to that in the case of the SC \Roman{u-large} phase. Furthermore,  $|S^{\rm l}_{\rm nn}(r)|$  increases with increasing $r$ except at $r=2$. Therefore, the relevant  pairing symmetry for the SC \Roman{u'-large} phase seems to be an extended spin singlet pairing, and mainly consist of nearest-neighbor site pairing.

\begin{figure}[h]
\begin{center}
\includegraphics[width=7.0cm]{63893Fig7.eps}
\end{center}
\caption{Absolute values of various types of  SC pairing correlation  functions  $|C(r)|$  as functions of $r$  for $n$=5/3 (10 electrons/6 sites) at  $\Delta=1.9$, $U'(=4J)=1.0$, and $U=2.4$, corresponding to the SC \Roman{u'-large} phase.}
\label{sc-cor2}
\end{figure}

Recently, weak-coupling approaches such as RPA and perturbation expansions have shown that the sign-reversing  $s$-wave ($s_{\pm}$-wave) pairing is realized in  iron oxypnictide superconductors\cite{Kuroki,Mazin,Wang,Nomura,Yanagi}.
 The order parameter of such a pairing is considered to change its sign between  hole and the electron Fermi pockets.
To compare our result with the result obtained by weak-coupling approaches, we  examine  the SC  pairing correlation function between  the $lower$ and $upper$ orbitals, such as
\[
  S^{\rm l-u}_{\rm nn}(r)=\frac1{2N}\sum_{i}  <{\Delta^{l}_{\rm nn}(i)}^\dagger \Delta^{u}_{\rm nn}(i+r)>
\]
with
\[
{\Delta^{m}_{\rm nn}(i)}^\dagger= c^\dagger_{i,m,\uparrow} c^\dagger_{i+1,m,\downarrow}-  c^\dagger_{i,m,\downarrow}c^\dagger_{i+1,m,\uparrow}
\ \   (m=l,u).
\]
We also define $T^{\rm l-u}_{\rm nn}(r)$ as well as $S^{\rm l-u}_{\rm nn}(r)$ in the above equation.

When the $s_{\pm}$-wave pairing is dominant, the values of the interorbital SC pairing  correlation function  are expected to be negative, since the Fermi surface of the lower (upper) orbital band in our model corresponds to a hole (electron) Fermi pocket, as shown in Fig. \ref{bandmodel}(b).
 In Fig. \ref{dd-pp}, we show the interorbital pairing  correlation functions  $S^{\rm l-u}_{\rm nn}(r)$  and  $T^{\rm l-u}_{\rm nn}(r)$(see also inset) for the same parameters in Fig. \ref{U-Upsouz} corresponding to the SC \Roman{u'-large}  phase.
We see that the values of $T^{\rm l-u}_{\rm nn}(r)$  are positive and very small, while those of $S^{\rm l-u}_{\rm nn}(r)$  are negative except at $r=3$ and not so small.
This result suggests that the relevant pairing symmetry of the SC \Roman{u'-large}  phase is the spin-singlet $s_{\pm}$-wave pairing, which agrees with the result of the weak coupling approaches.
Therefore, we expect that the $s_{\pm}$-wave pairing proposed on the basis of the result of weak-coupling approaches is realized in the wide parameter region including both weak- and strong-correlation regimes.

\begin{figure}[h]
\begin{center}
\includegraphics[width=7.0cm]{63893Fig8.eps}
\end{center}
\caption{Pairing correlation functions  $S^{\rm l-u}_{\rm nn}(r)$ and  $T^{\rm l-u}_{\rm nn}(r)$.
Here, we show  the absolute value of the correlation functions at  $U=2.4$, $\Delta=1.9$, and $U'(=4J)=1.0$ for $n$=5/3 (10 electrons/6 sites).
The inset shows  a schematic diagram of the pairing symmetries, $S^{\rm l-u}_{\rm nn}(r)$  and  $T^{\rm l-u}_{\rm nn}(r)$.
}
\label{dd-pp}
\end{figure}

Finally, we discuss the mechanism of superconductivity in the SC \Roman{u'-large} phase.
The $s_{\pm}$-wave pairing  is considered to be mediated by the antiferromagnetic fluctuation due to the nesting effect between  electron and hole pockets \cite{Kuroki,Mazin,Wang,Nomura,Yanagi}. At first glance, the SC \Roman{u'-large} phase is located adjacent to  the partial ferromagnetic phase (S=1), and then, the ferromagnetic fluctuation seems to be related to superconductivity.
To examine the relationship between  spin fluctuation and superconductivity, we also calculate the spin correlation function for  finite-size systems, where a short-range spin correlation is considered to be crucial for the superconductivity.
We obtain the ferromagnetic and antiferromagnetic components of the spin correlation as a function of $U'(=4J)$ for a fixed $U=1.5$ (see also Fig. \ref{U-Upsouz}) and find that  antiferromagnetic (ferromagnetic) correlation increases (decreases) with decreasing $U'$ together with increasing $K_\rho$ (not shown).
Therefore, we conclude that the antiferromagnetic spin fluctuation is responsible for the $s_{\pm}$-wave pairing in the SC \Roman{u'-large} phase.

 \section{Summary and Discussion}
We have investigated the superconductivity  of the  one-dimensional two-orbital Hubbard model in the case of electron and hole Fermi pockets  corresponding to a characteristic band structure of iron oxypnictide superconductors.
To obtain reliable results including those in the strong-correlation regime, we  used  the exact diagonalization method and  calculated the critical exponent $K_{\rho}$ on the basis of  the Luttinger liquid theory. It has been found that the system shows two types of SC phases, the SC \Roman{u'-large} for $U>U'$ and the SC \Roman{u-large} for $U<U'$, in the wide parameter region including both weak- and strong-correlation regimes.

We have also calculated various types of SC pairing correlation functions in the realistic parameter region of the iron oxypnictides. The result indicates that the most dominant pairing for the SC \Roman{u'-large} phase is the intersite intraorbital spin-singlet with sign reversal of the order parameters between  two Fermi pockets.  The result is consistent with the sign-reversing $s_{\pm}$-wave pairing that has recently been proposed on the basis of the result obtained by  weak-coupling approaches for iron oxypnictide superconductors. This indicates that the $s_{\pm}$-wave pairing  is  realized not only in a weak-correlation regime but also in a strong-correlation regime.
We have also calculated the spin correlation function and found that antiferromagnetic spin fluctuation is responsible for the $s_{\pm}$-wave pairing in the SC \Roman{u'-large} phase.

As for the SC \Roman{u-large} phase, the most dominant pairing is found to be the on-site intraorbital spin-singlet pairing, which is consistent with the ordinary $s$-wave pairing of BCS superconductors.
However, the superconducting mechanism of this phase is due to the charge fluctuation enhanced by the interorbital Coulomb interaction and  is different from the conventional BCS superconductivity due to the electron-phonon interaction.
Although the SC \Roman{u-large} phase seems to be realized only for the unrealistic parameter region in our model, it might be realized  for a realistic parameter region  in  the $d$-$p$ model, which is  closer to  iron oxypnictides\cite{Yanagi,Sano-Udp-all}.
We will address such a problem by applying the present method in the $d$-$p$ model in the future.


text:
\section{Introduction}
Colouring a graph is the problem of finding the minimal number of colours required to assign each vertex a colour such that no adjacent vertices have the same colour. It is a classic problem in computer science and one of Karp's~21 NP-complete problems \cite{Karp72}. Colouring remains NP-complete if bounded to three colours \cite{Stockmeyer1973}. Also, if a 3-colourable graph is given, it remains NP-hard to find a colouring, even if four colours are allowed \cite{4col3colNPhard}.

Dually chordal graphs are closely related to chordal graphs, hypertrees and $\alpha$-acyclic hypergraphs, which also explains the name of this graph class (see \cite{duallyChordal} for more). Section~\ref{sec:DuallyChordal} will show, that Colouring is NP-complete for dually chordal graphs if bounded to four colours. So the aim of this paper is to find a linear time algorithm for the 3-Colouring problem.

Next to the classic vertex colouring problem, there are the problems of colouring only edges or edges and vertices of a graph. For dually chordal graphs they have been investigated by de~Figueiredo et~al.~\cite{deFigueiredo1999147}.

\section{Basic Notions}\label{sec:BasicNotions}
Let $G=(V,E)$ be a graph with the vertex set $V$, the edge set $E$. For this paper, any graph is finite, undirected, connected  and without loops or multiple edges. For a set $U \subseteq V$, $G[U]$ denotes the induced subgraph of $G$ with the vertex set $U$. Let $\overline{G}=(V,\overline{E})$ be the complement of $G$, such that $\overline{E} = \{ uv \mid u,v \in V; u \neq v; uv \notin E \}$.

A set of vertices $S \subseteq V$ is a \emph{clique} if for each pair $u,v \in S$ ($u \neq v$), $u$ and $v$ are adjacent. A clique of size $i$ is denoted as $K_i$. The number of vertices in a maximum clique in $G$ is the \emph{clique number $\omega(G)$} of $G$. Let a \emph{diamond} be a graph with four vertices and five edges. The edge connecting the vertices with degree 3 is called \emph{mid-edge}. A \emph{chordless cycle} $C_k$ has $k$ vertices $v_1, \ldots, v_k$ and the edges $v_iv_{i+1}$ (index arithmetic modulo $k$). If a chordless cycle has at least five vertices, it is also called $hole$. A \emph{wheel} $W_k$ is a $C_k$ plus a vertex $v_w$ and the edges $v_wv_i$ ($1 \leq i \leq k$). A chordless cycle $C_k$ or wheel $W_k$ is \emph{odd} if $k$ is odd.

Let $N(v) := \{ u \in V \mid uv \in E \}$ denote the \emph{open} and $N[v] := N(v) \cup \{v\}$ the \emph{closed neighbourhood} of the vertex $v$. A graph is \emph{locally connected} if for all vertices $v$ the open neighbourhood $N(v)$ is connected.

A vertex $u \in N[v]$ is a \emph{maximum neighbour} of $v$, if for all $w \in N[v]$, $N[w] \subseteq N[u]$ holds. Note that $u = v$ is not excluded. A vertex ordering $(v_1, \ldots, v_n)$ is a \emph{maximum neighbourhood ordering} if every $v_i$ ($1 \leq i \leq n$) has a maximum neighbour in $G[\{v_i, \ldots, v_n \}]$.

A vertex $v$ is an articulation point of $G$ if $G[V \setminus \{v\}]$ is not connected. If a graph has no articulation point it is \emph{biconnected} (or \emph{2-connected}). Maximal biconnected subgraphs are called \emph{blocks}. Any connected graph can decomposed into a block tree. If two blocks intersect, they have exactly one common vertex which is an articulation point.

\bigskip

An \emph{independent set} is a vertex set $I \subseteq V$ such that for all vertices $u,v \in I$, $uv \notin E$ holds. A graph is \emph{$k$-colourable} if $V$ can be partitioned into $k$ independent sets $V_1, \ldots, V_k$ with $V = V_1 \cup \ldots \cup V_k$ and $V_i \cap V_j = \emptyset$ ($i \neq j$). The \emph{chromatic number} $\chi(G)$ of a graph $G$ is the lowest $k$ such that $G$ is $k$-colourable. The \emph{$k$-Colourability} problem asks if a graph is $k$-colourable. Accordingly the \emph{Colourability} problem asks for the chromatic number of a graph. It is easy to see, that 2-Colourability can be solved in linear time for each graph.

A graph $G$ is \emph{perfect} if for each induced subgraph $G^*$ of $G$ the chromatic and the clique number are equal ($\omega(G^*) = \chi(G^*)$). Chudnovsky et~al.~\cite{StrongPerfectGraph} have shown that a graph is perfect if and only if it is $\left( C_{2n+5}, \overline{C_{2n+5}} \right)$-free for all $n \geq 0$. This is known as the Strong Perfect Graph Theorem.  Perfect graphs can be recognised and coloured in polynomial time \cite{RecogBerge}\cite{ColorPerfect}.

For a graph $G$, $K(G)$ denotes the \emph{clique graph} of $G$, where each vertex in $K(G)$ represents a maximal clique of $G$ and two vertices are connected if the corresponding cliques have a common vertex, i.e. $K(G)=(V, E)$ with $V = \{ k \mid \text{$k$ is a maximal clique of $G$}\}$ and $E= \{ k_1k_2 \mid k_1 \cap k_2 \neq \emptyset \}$. A graph $G$ is \textit{chordal} if it is $C_k$-free ($k \geq 4$). A graph is \emph{clique-chordal} if its clique graph is chordal.

\section{Dually Chordal Graphs}\label{sec:DuallyChordal}
Dually chordal graphs are originally defined by a maximum neighbourhood ordering.

\begin{definition}
    A graph is \emph{dually chordal} if and only if it has a maximum neighbourhood ordering.
\end{definition}

Brandstädt et~al.~\cite{duallyChordal} give an overview of characterisations for dually chordal graphs. One of it allows to recognise graphs of this class in linear time (with an algorithm by Tarjan and Yannakakis~\cite{RecogAlgor}). Another characterisation was developed in~\cite{LeitertDiploma2012}:

\begin{lemma}[\cite{duallyChordal}, \cite{LeitertDiploma2012}]\label{theo:dcDef}
    Let $P_T(u,v)$ the set of vertices on the path from $u$ to $v$ in the tree $T$, with $u,v \notin P_T(u,v)$. For a graph $G=(V,E)$ the following conditions are equivalent:
    \begin{enumerate}
        \item  $G$ is dually chordal.
        \item \label{item:dcDefCliqueTree} There is a spanning tree $T$ of $G$ such that every maximal clique of $G$ induces a subtree in $T$.
        \item \label{item:dcDefPathTree} There is a spanning tree $T$ of $G$ such that for every edge $uv \in E$ the following holds: $\forall\,w \in P_T(u,v): uw, vw \in E$
    \end{enumerate}
\end{lemma}

\begin{lemma}[\cite{LeitertDiploma2012}]\label{lem:CliqueTreeEqualPathTree}
	 The spanning trees in conditions~\ref{item:dcDefCliqueTree} and \ref{item:dcDefPathTree} of Lemma~\ref{theo:dcDef} are equal.
\end{lemma}

Since $P_T(u,v)$ does not include the vertices $u$ and $v$, (according to the notation for the open and closed neighbourhood) let $P_T[u,v] := P_T(u,v) \cup \{u,v\}$.

\begin{corollary}
    If $uv \in E$ then $P_T[u,v]$ is a clique.
\end{corollary}

Is is easy to see that an arbitrary graph becomes dually chordal if a vertex adjacent to all vertices is added. This leads to a reduction for several NP-complete problems, including Colourability. The idea for this method was already given by Brandstädt et~al.~\cite{Brandstaedt199843}.

\begin{theorem}\label{theo:4colDC}
	4-Colourability is NP-complete for dually chordal graphs.
\end{theorem}

\begin{proof}
	Let $G=(V,E)$ be an arbitrary graph. Thus, $G_{dc}=(V \cup \{ u \}, E \cup \{uv \mid v \in V \})$ with $u \notin V$ is dually chordal. It follows that $G$ is 3-colourable if and only if $G_{dc}$ is 4-colourable. Because 3-Colourability is NP-complete in general~\cite{Stockmeyer1973}, 4-Colourability is NP-complete for dually chordal graphs.
	\qed
\end{proof}

\section{$K_4$-free Dually Chordal Graphs}\label{sec:K4Free}
Obviously a graph has to be $K_4$-free to be 3-colourable. So this section investigates $K_4$-free dually chordal graphs.

For this section let $G=(V,E)$ be a $K_4$-free dually chordal graph. The spanning tree $T=(V,E_t)$ of $G$ and the set $P_T[u,v]$ are defined as in condition~\ref{item:dcDefPathTree} of Theorem~\ref{theo:dcDef} and the paragraph after Lemma~\ref{lem:CliqueTreeEqualPathTree}.

\begin{corollary}\label{cor:K4FreePEdges}
    If $uv \in E$ then $|P_T[u,v]| \leq 3$, i.e. $uv \in E \land uv \notin E_t \Rightarrow \exists!\, w: uw,wv \in E_t$.
\end{corollary}

For the next lemma the notion of a \emph{route} is defined:

\begin{definition}
    A \emph{route} on $G$ is a list of vertices $(v_1,\ldots,v_k)$ such that $v_iv_{i+1} \in E$ or $v_i=v_{i+1}$ for all $i < k$.
\end{definition}

\begin{lemma}\label{lem:CircleWheel}
    Let $\mathcal{C}$ be a $C_k$ in $G$ with $k \geq 4$. There is a vertex $w$ such that $\mathcal{C}$  and $w$ form a wheel. Also no edge of $\mathcal{C}$ is in $T$.
\end{lemma}

\begin{proof}
    Let $\mathcal{C}$ be a chordless cycle with the vertices $\{c_1, \ldots, c_k\}$ ($k \geq 4$) and the edges $c_ic_{i+1}$ (index arithmetic modulo $k$). Based on Corollary~\ref{cor:K4FreePEdges} for every edge there is the set of vertices $P_T[c_i, c_{i+1}] = \{ c_i, w_i, c_{i+1} \}$ with $c_i = w_i$ if and only if $c_ic_{i+1} \in E_t$. Thus, it is possible to build a route $\rho=(c_1,w_1,c_2,\ldots,w_k,c_1)$ on $T$ (see Figure~\ref{fig:circleWheel}).

\begin{figure}[htb]
    \centering
    \includegraphics{fig_circleWheel}
    \caption{The circle $\mathcal{C}$ and the route $\rho$ (waved).}
    \label{fig:circleWheel}
\end{figure}

Because all the edges of $\rho$ are in $T$, $\rho$ also induces a tree. Therefore $\rho$ can be eliminated by removing leaves. Let $(\ldots, v_{i-1}, v_i, v_{i+1}, \ldots)$ be a subroute of $\rho$ and $v_i$ a leaf. Thus, if $v_{i-1} \neq v_i$ and $v_i \neq v_{i+1}$ then $v_{i-1} = v_{i+1}$, so $v_{i-1}$ or $v_{i+1}$ can also removed from $\rho$ with $v_i$.

Now assume that no vertex of $\mathcal{C}$ is a leaf. In this case a vertex $w_i$ is a leaf. So it follows, that $c_i = c_{i+1}$. Therefore there must be a leaf $c_i$ and also $w_{i-1}=w_i$ (otherwise $c_{i-1}c_{i+1} \in E$). This allows to remove $c_i$ and $w_i$ from $\rho$.

Continuing this procedure, it follows that $w_1=\ldots=w_k$ and $c_i \neq w_i$ for all $i$. Thus, no edge $c_ic_{i+1}$ is an edge of $T$ and there is a vertex $w$, such that $\mathcal{C}$ and $w$ form a wheel.
\qed
\end{proof}

\begin{theorem}\label{theo:3colIffPerfect}
    A dually chordal graph $G$ is 3-colourable if and only if $G$ is perfect and $K_4$-free.
\end{theorem}

\begin{proof}
$\Leftarrow:$ By definition of perfect graphs.

$\Rightarrow:$ $G$ is 3-colourable, so it is $K_4$-free. Let $k \geq 7$. It is easy to see, that no $\overline{C_k}$ is 3-colourable. Therefore, $G$ has no $\overline{C_k}$. From the 3-colourability it also follows that there is no odd wheel in $G$ and by Lemma~\ref{lem:CircleWheel} no odd hole\footnote{also no $\overline{C_6}$}. Thus, by the Strong Perfect Graph Theorem, $G$ is perfect (recall, $C_5 = \overline{C_5}$).
\qed
\end{proof}

Theorem~\ref{theo:3colIffPerfect} now leads to a polynomial time algorithm for 3-Colourability of dually chordal graphs.

\section{Linear Time Algorithm}\label{sec:LinTimeAlgo}
After showing that there is a polynomial time algorithm, this section presents a linear time algorithm for 3-Colourability of dually chordal graphs based on the following theorem:

\begin{theorem}\label{theo:locConConstruction}
    Each connected locally connected graph can be constructed by starting with an arbitrary edge and then add only vertices that has at least two adjacent neighbours in the already constructed graph.
\end{theorem}

\begin{proof}
Let $(u, v, v_1, \ldots, v_k)$ be the ordering the vertices are added to construct a locally connected graph $G=(V,E)$ starting with an edge $uv$. Also let $V_i := \{u, v, v_1, \ldots, v_i \}$ with $V_0 := \{ u, v \}$ and $0 \leq i \leq k$.

Now let $G_i := G[V_i]$ ($0 \leq i < k$) be a locally connected subgraph of $G$. Assume, there is no vertex~$w$ adjacent to two vertices in $G_i$. Then $G$ is not connected or $w$ has only one neighbour~$v_n$ in $G_i$. In second case, because $G$ is locally connected, there is a vertex set $S \subseteq N(v_n) \setminus V_i$ connecting $w$ with the neighbours of $v_n$ that are already in $G_i$. Thus, there is a vertex $s \in S$ adjacent to $v_n$ and a neighbour of $v_n$ in $G_i$. Therefore, set $v_{i+1} := s$.
\qed
\end{proof}

Theorem~\ref{theo:locConConstruction} allows to colour graphs whose blocks are locally connected by using the following strategy: Select an uncoloured vertex with a minimal number of available colours and give it an available one. Repeat this until all vertices are coloured or there is no available colour for a vertex. Algorithm~\ref{algo:3colLocCon} describes this more detailed:

\begin{algo}\label{algo:3colLocCon}
\ \\
\textbf{Input:} A graph $G=(V,E)$ whose blocks are locally connected.\\
\textbf{Output:} A 3-colouring for $G$ if and only if $G$ is 3-colourable.

\begin{enumerate}
    \item Give every vertex $v$ a set of available colours $c(v) := \{1,2,3\}$. Also every vertex gets the possibility to get marked as coloured. At the beginning no vertex is marked.

    \item \textbf{While} There are uncoloured vertices

    \begin{enumerate}
        \item Select an uncoloured vertex~$v$ for which $|c(v)|$ is minimal.
        \item If $c(v) = \emptyset$, \textbf{STOP:} $G$ is not 3-colourable.
        \item Disable all but one colour in $c(v)$ and mark $v$ as coloured.
        \item Disable $c(v)$ for all neighbours of $v$ ($\forall\,u \in N(v): c(u) := c(u) \setminus c(v)$).
    \end{enumerate}
\end{enumerate}
\end{algo}

\begin{theorem}\label{theo:3colLocConAlog}
    Algorithm~\ref{algo:3colLocCon} works correctly and can be implemented in linear time.
\end{theorem}

\begin{proof}
\ \par \emph{Correctness}.
After selecting the first two vertices, the algorithm only selects vertices with two adjacent neighbours, i.e. with $|c(v)| \leq 1$, until all vertices of a locally connected block are coloured (Theorem~\ref{theo:locConConstruction}). Therefore, the colouring is unique for a locally connected block.

Also, if a vertex~$v$ with $|c(v)| = 2$ and the coloured neighbour~$u$ is selected, $u$ is an articulation point. This allows to treat $u$ and $v$ as the first coloured vertices of a locally connected graph. Therefore, each available colour for $v$ leads to a correct colouring.

\medskip

\emph{Complexity}.
It is easy to see, that line~1 runs in linear time and line~2(b) in constant time such as line~2(c). Line~2(d) is bounded by the number of neighbours. Because $\sum_{v \in V} |N(v)| = 2|E|$, the full time needed (overall iterations) for line~2(d) is in ${\cal O}(|E|)$.

Line~2(a) can be implemented by using three doubly linked lists as queues. The lists include the uncoloured vertices. One is for vertices with $|c(v)| = 3$, one for $|c(v)| = 2$ and one for $|c(v)| \leq 1$. If there is also a pointer from each vertex to its entry in the list, adding and removing a vertex can be done in constant time. Thus, line~2(a) runs in constant time.
\qed
\end{proof}

Another linear time algorithm for 3-Colourability of locally connected graphs was presented by Kochol~\cite{LinearThreeColoring}, but is more complicated.

\bigskip

Dually chordal graphs are a subclass of clique-chordal graphs \cite{duallyChordal}. The next lemma will show, that a 3-colouring for clique-chordal graphs can be be found with Algorithm~\ref{algo:3colLocCon}.

\begin{lemma}
    Each block of a clique-chordal graphs is locally connected.
\end{lemma}

\begin{proof}
	Let $G=(V, E)$ be a clique-chordal graph and $K=(V_k, E_k)$ its clique graph. Assume that $G$ has a block which is not locally connected. Then there is a vertex~$v$ such that its neighbourhood is not connected and it is no articulation point. Therefore, there is a chordless cycle ${\cal C}$ fulfilling the conditions that there is no path in $G[N(v)]$ connecting the two neighbours of $v$ in ${\cal C}$.

	Let ${\cal C}_G$ be the smallest ${\cal C}$ and using the following notation: $u$, $v$, $w$ and $a$, $b$, $c$ are vertices of ${\cal C}_G$ with $\{ uv, vw, ab, cd \} \subseteq E$, $u \notin \{ b, c \}$, $v \notin \{a, b, c \}$, $w \notin \{ a, b \}$, $a \notin \{ v, w \}$, $b \notin \{ u, v, w \}$ and $c \notin \{ u, v \}$. If ${\cal C}_G$ is a $C_4$, then $u=a$ and $w=c$. Figure~\ref{fig:cycleProofCliqueChordal} illustrates this notation.

\begin{figure}[htb]
    \centering
    \includegraphics{fig_cycleProofCliqueChordal}
    \caption{The cycle ${\cal C}_G$ and the vertex~$n$.}
    \label{fig:cycleProofCliqueChordal}
\end{figure}

    There is no clique in $G$ that contains two edges of ${\cal C}_G$, otherwise ${\cal C}_G$ would not be chordless. Thus, each edge represents another clique of $G$. Let $k_{uv}$, $k_{vw}$, $k_{ab}$ and $k_{bc}$ denote these cliques. Because the cliques have a common vertex if the corresponding edges have one, they are forming a cycle ${\cal C}_K$ in $K$.

    $K$ is chordal. Therefore, ${\cal C}_K$ has at least one chord. Without loss of generality, let $k_{uv}k_{bc}$ be such a chord. Thus, $k_{uv}$ and $k_{bc}$ have a common vertex~$n$ that is adjacent to $u$, $v$, $b$ and $c$.

    If $w = c$, $n$ is adjacent to $w$. If $w \neq c$, the sequence $(v, n, c, \ldots, w)$ induces a chordless cycle smaller than ${\cal C}_G$. In both cases ${\cal C}_G$ is not the smallest cycle~${\cal C}$ such that there is no path in $G[N(v)]$ connecting the two neighbours of $v$ in~${\cal C}$.

    Therefore, $G$ is nearly locally connected. \qed
\end{proof}

It follows:

\begin{corollary}
	The 3-Colourability problem can be solved in linear time for dually chordal and clique chordal graphs.
\end{corollary}

\section{Conclusion}
After introducing dually chordal graphs and investigating the connection between $K_4$-free dually chordal and perfect graphs, this paper presented a linear time algorithm to find a 3-colouring by using the structure of locally connected graphs. This algorithm also computes
a correct 3-colouring for each graph whose blocks are locally connected.

\subsubsection{Acknowledgement.}
The author is grateful to H.\,N. de~Ridder for stimulating discussions.