Res. Rep. Fac. Eng. Mie. Univ., Vol.32,pp. 1-23(2007)
Original Paper

Ferromagnetism and Superconductivity in Two-band Hubbard
ModelsKazuhiro Sano and Yoshiaki Ōno∗(Department of Physics Engineering)(Received July 18, 2007)
Abstract
We investigate ferromagnetism and superconductivity in two types of two-band Hubbard models: the multi-orbital Hubbard model and the d-p model, with particularly paying attention toeffect of the interplay between the electron correlation and the band splitting. As strong correlation plays a crucial role for these phenomena, we employ two complementary approaches:the numerical diagonalization method for one-dimensional models and the dynamical mean-field
theory for the infinite-dimensional model. These approaches give us reliable results beyond theperturbative approximation or the usual mean-field like approximation. For the one-dimensionalmodels, we calculate the critical exponent Kρ based on the Tomonaga-Luttinger liquid theory
and determine phase diagram of superconducting(SC) region by the condition Kρ > 1. The SC
region appears near partially polarized ferromagnetic region in the multi-orbital Hubbard modelwith finite band splitting. Analysis of paring correlation functions for the ground state indicatesthat the triplet paring is relevant to the superconductivity. In the one-dimensional d-p model, we
find enhancements of the singlet and triplet paring correlations at the SC region. Although only
the singlet paring is relevant to superconductivity near half-filling, the triplet paring increases
with electron filling and is relevant to the superconductivity as well as the singlet paring near
full-filling. In the infinite-dimensional d-p model, we calculate the local green’s function on thebasis of the dynamical mean-field theory, where the effective impurity Anderson model is solved
using the numerical diagonalization method. It shows the phase diagrams of the metal-insulatortransition on the ground state at half-filling and quarter-filling as functions of the Coulomb interaction and the band splitting. We obtain the transition temperatures for the ferromagnetism andthe superconductivity as functions of the Coulomb interaction, the band splitting and the electron
filling. It indicates that the transition temperature of the triplet superconductivity is higher thanthat of the singlet superconductivity near the ferromagnetic state. These results suggest a close
relationship between the ferromagnetism and the superconductivity as a common feature of thetwo-band Hubbard models.

1

Introduction

The strongly correlated electron systems with multi-band (multi-orbital) have attracted muchinterest due to various interesting phenomena such as colossal magnetoresistance in manganites
La1−x Srx MnO3 [1], triplet-pairing superconductivity in the ruthenate Sr2 RuO4 [2], metal-insulatortransition in alkali-doped fullerides Ax C60 [3] and one-dimensional superconductivity of CuO2 double
chain in Pr2 Ba4 Cu7 O15−δ [4]. Among them, spin-state transition of cobalt oxides in La1−x Srx CoO3 [5,6, 7] give a good example displaying the effect of multi-orbital. Since the Hund’s rule coupling andthe crystal-field splitting between the t2g and eg orbitals are close to each other, the spin state of
the cobalt ion depends on temperature, doping concentration and crystal structure. With increasing temperature, the low-spin state (t62g e0g , S = 0) of the Co3+ (3d6 ) ion gradually changes into anintermediate-spin state (t52g e1g , S = 1) and/or a high-spin (HS) state (t42g e2g , S = 2)[6, 7, 8, 9]. With Sr
doping, La1−x Srx CoO3 shows a spin-glass state for x < 0.18 and a ferromagnetism for x > 0.18[10, 11].
Layered cobalt oxides such as Nax CoO2 also show a a variety of interesting properties as multiband systems. For example, large thermoelectric power is discovered in Na0.5 CoO2 [12]. Weak ferromagnetism has been observed in Na0.75 CoO2 [13]. The discovery of the superconductivity[14] in
Nax CoO2 · yH2 O with Tc ≈ 5K for x ≈ 0.35 and y ≈ 1.3 has stimulated considerable attention onthese materials. It has been suggest that large degeneracy of electronic states due to a competition∗ Department of Physics, Niigata University－1－1Kazuhiro Sano and Yoshiaki Ōno

between the Hund’s rule coupling and the crystal-field splitting plays a key role in the electronic statesof Nax CoO2 as well as of La1−x Srx CoO3 [15].
With the new findings of these interesting materials, theoretical studies on the interplay between
Coulomb interactions including the Hund’s rule coupling and band splitting by crystal-field are highly
desirable. The orbitally degenerate Hubbard model, which can be regarded as a kind of multi-band
system, has been extensively investigated to clarify the effect of orbital degrees of freedom in thepresence of intra-atomic Coulomb interaction. Many authors[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]
have studied the ferromagnetism of this model and revealed that Hund’s rule coupling plays a crucialrole in ferromagnetism.
On the other hand, multi-band Hubbard model, which includes the effect of band splitting explicitly, has also been extensively studied for a long time. In particular, there has been much theoreticalinterest in the d-p model as a model of the copper oxide high-temperature superconductor. The model
contains hopping tpd between the Cu (d-orbital) site and the O(p-orbital) site and strong repulsive
interactions at the d and p site (Ud and Up , respectively). In addition, it can contain the nearestneighbor d-p interaction Upd and/or hopping tpp between the nearest-neighbor p sites. It is widely
accepted as a basic model describing the electronic structure of the Cu-O network. Many theoreticalstudies have been performed on this model to explain the superconductivity and/or the antiferromagnetism of the copper oxides. A few work has been discussed the possibility of so called flat bandferromagnetism in a type of d-p model[27].
In the present work, we address the multi-orbital Hubbard model in one-dimension and the d-p
model in one- and infinite-dimension, while particularly paying attention to the effect of the interplay
between the Coulomb interactions and the band splitting ∆. At the point of view of ferromagnetismand superconductivity in itinerant electron systems, we discuss these two models in detail. As thestrong correlation effect plays crucial roles in these interesting phenomena, a nonperturbative andreliable approach is required. We employ the numerical diagonalization method for the multi-orbital
Hubbard model and the d-p model with finite system sizes in one-dimension, and the dynamicalmean field theory for the infinite-dimensional d-p model. These approaches are complementary foreach other in dimensionality and expected to give us reliable results beyond the perturbative or themean-field like approximations.
Numerical diagonalization approach has already been applied for the multi-orbital Hubbard modelat ∆ = 0 case[17, 18]. Although the available system size is fairly small, the results are in good agreement with the strong coupling analysis[18] and the results from the density-matrix renormalizationgroup method[23]. To examine the superconductivity, we calculate the critical exponent of the correlation functions Kρ based on the Luttinger liquid theory[28, 29, 30, 31]. The reliability of this approachhas been extensively tested for various one-dimensional models such as the Hubbard model[32], thet-J model[33] and the U -V model[34]. We can thus expect that this approach is reliable for themulti-orbital Hubbard model and the d-p model as well.
In the limit of infinite dimensions, the self-energy becomes purely site-diagonal and the dynamicalmean field theory (DMFT) becomes exact. The local Green’s function is given by the impurity Green’s
function of an effective single impurity Anderson model. This effective one-site problem is calculatedby the numerical diagonalization of the finite size clusters. By using this method, Momoi et al. studied
the multi-orbital Hubbard model at ∆ = 0[17]. They show the phase diagram of the ferromagnetism,
but superconductivity was not discussed there. By calculating the magnetization and the pairing
susceptibility at finite temperature, we show the transition temperatures for the ferromagnetism andthe superconductivity as functions of the Coulomb interaction, the band splitting and the electron
filling.

2

Luttinger Liquid Relation

At first, we briefly discuss a general argument for 1D-electron systems based on the Luttinger
liquid theory. In the Luttinger liquid theory[28, 29, 30, 31], an effective Hamiltonian of 1D models in
the Tomonaga-Luttinger regime is generally given by
H=

vσ
2π

Z L
0

£
¤ vρ
dx Kσ (∂x θσ )2 + Kσ−1 (∂x φσ )2 +
2π

Z L
0

£
¤
dx Kρ (∂x θρ )2 + Kρ−1 (∂x φρ )2 ,(1)

where vσ , vρ , Kσ and Kρ are the velocities and coupling parameters of spin and charge parts, respectively. According to the Luttinger liquid theory, some relations have been established as universal－2－2Ferromagnetism and Superconductivity in Two-band Hubbard Models

relations in one-dimensional single band models[30, 31]. In the model which is isotropic in spin space,
the coupling constant Kσ is renormalized to unity in the low energy limit and the critical exponents
of various types of correlation functions are determined by a single parameter Kρ .
For single band model, it is predicted that SC correlation is dominant for Kρ > 1 (the correlation−(1+ 1 )− 1

Kρfunction decays as ∼ rin the Tomonaga-Luttinger (TL) regime and as ∼ r Kρ in the LutherEmery (LE) regime), whereas the CDW or SDW correlations are dominant for Kρ < 1 (the correlation
functions decay as ∼ r−(1+Kρ ) in the TL regime and as ∼ r−Kρ in the LE regime). Here, the LE regime
is characterized by a gap in the spin excitation spectrum, while in the TL regime, the excitation isgapless[28, 31]. In the case of non-interacting fermion systems, the exponent Kρ is always unity. Thus,
the effective interaction between quasi-particles is attractive for Kρ > 1 whereas that is repulsive forKρ < 1.
The critical exponent Kρ is related to the charge susceptibility χc and the Drude weight D byKρ =1(πχc D)1/2 ,
2

(2)

2

E0 (φ)
with D = Nπu ∂ ∂φ
, where E0 (φ) is the total energy of the ground state as a function of magnetic
2
flux Nu φ[31]. Here, the flux is imposed by introducing the following gauge transformation: cmσ † →eimφ/Na c†mσ for an arbitrary site m. When the charge gap vanishes in the thermodynamic limit, theuniform charge susceptibility χc is obtained from

χc =

4/Nu
,
E0 (Ne + 2, Nu ) + E0 (Ne − 2, Nu ) − 2E0 (Ne , Nu )(3)

where E0 (Ne , Nu ) is the ground state energy of a system with Nu unit sites(cells) and Ne electrons.
The filling n is defined by n = Ne /Nu , where Nu is the total number of unit cells or sites. The valuesof D and χc are calculated from the ground state energy of the finite size system through eq.(2) and(3).
It is noted that the situation is complicated for two-band models. When both bands have been
occupied by electrons simultaneously, Fermi points appear in both bands. In this case, the electronic
state of the low energy is completely changed and we can not use the above relations[35, 36, 37, 38].
On the other hand, if the electron density is sufficiently small and/or the band splitting is sufficiently
large, electrons occupy only lower band. In this case, Fermi level exits in only lower band and the low
energy effective Hamiltonian is considered to be equivalent to that of the single band model. Therefore,
we can adopt the above Luttinger liquid relations of the single band model for even two-band model.
We numerically diagonalize the model Hamiltonian and obtain the value of Kρ from the groundstate energy of finite size systems using the standard Lanczos algorithm. In addition to the numerical
diagonalization method, we use mean-field(MF) approximation to calculate the ground state energy
and obtain the approximate value of Kρ of the infinite system. This approximation has been confirmed
to be reliable at least in the weak coupling regime[39].

3Multi-orbital Hubbard Model

Previous works[17, 18, 19, 20, 21, 22, 23, 24] have studied the multi-orbital Hubbard model inthe case of the crystal-field splitting ∆ = 0. In one dimension, some rigorous results are shown in thestrong coupling limit U → ∞: the ground state is a fully polarized ferromagnetism for 0 < n < 2
except for n = 1 when U 0 and J = J 0 are positive and finite[21]. The numerical result suggests thatthe ferromagnetism is stable also for n = 1 in the strong coupling region[23]. In infinite dimensions,the dynamical mean-field theory shows the existence of the ferromagnetism in the same parameterregion observed in one dimension[22]. They revealed that Hund’s rule coupling J plays a crucial rolein ferromagnetism. However, possible mechanisms of superconductivity and the effect of crystal-field
splitting ∆ was not considered in these studies.
In this section, we examine ferromagnetism and related superconductivity in the multi-orbital
Hubbard model, while particularly paying attention to the effect of the interplay between J and∆[40, 41, 42].

－3－3

Kazuhiro Sano and Yoshiaki Ōno

3.1
Model Hamiltonian
We consider the following Hamiltonian for the one-dimensional multi-orbital Hubbard model:
X †X
H = −t(ci,m,σ ci+1,m,σ + h.c.) + U
ni,m,↑ ni,m,↓
i,m,σ

X

i,m

ni,u,σ ni,l,−σ + (U 0 − J)X

∆X(ni,u,σ − ni,l,σ )
2 i,σ

+

U0

−

X †
X †
J
(ci,u,↑ ci,u,↓ c†i,l,↓ ci,l,↑ + h.c.) − J 0(ci,u,↑ c†i,u,↓ ci,l,↑ ci,l,↓ + h.c.),i,σni,u,σ ni,l,σ +

i,σ

i,m

(4)

i,m

where c†i,m,σ stands for the creation operator of an electron with spin σ in the orbital m (= u, l) at
site i and ni,m,σ = c†i,m,σ ci,m,σ . Here, t represents the hopping integral between the same orbitals andwe set t = 1 in this study.
The interaction parameters U , U 0 , J and J 0 stand for the intra- and inter-orbital direct Coulombinteractions, the exchange (Hund’s rule) coupling and the pair-transfer, respectively. ∆ denotes theenergy difference between the two atomic orbitals, that is, crystal-field splitting. For simplicity, we
impose the relations, J = J 0 and U = U 0 + 2J; the latter holds exactly in 3d-orbitals for ∆ = 0 andis a good approximation for ∆ 6= 0. The model in eq. (4) is schematically represented by Fig. 1(a).
We numerically diagonalize the above model Hamiltonian up to 9 sites. In the noninteracting case(U = U 0 = J = 0), the Hamiltonian eq. 4 yields a dispersion relation ²± (k) = −2t cos(k) ± ∆2 , where
k is the wave vector and ²+ (k) (²− (k)) represents the upper (lower) band energy. This band structureis schematically represented by Fig. 1(b).
(a)
J U’

(b)

ε+(k)

U

t

∆

J’

∆

−

ε (k)

Ek

F

t

Figure 1: Schematic diagrams of (a) the model Hamiltonian and (b) the band structure in the noninteracting
case.

Figure 2: Schematic diagram of
Moebius boundary condition at∆ = 0.

3.2
Result for degenerate caseWe first exhibit the result at ∆ = 0 and compare it with the previous works. At ∆ = 0, twobands are doubly degenerate and the finite size effect of the ordinary periodic boundary condition isfairly large. To reduce it, we use the Moebius boundary condition, which has been already used in
t-J ladder models[43]. The Moebius boundary condition as shown in Fig.2, is a kind of the periodic
boundary condition and efficiently reduces the finite size effect of the systems than the ordinary
periodic boundary condition.
Figure 4 shows the phase diagram of the ground state for the Nu = 8 system at n = 0.5 (4electrons/8sites) with the result of the DMRG method by Sakamoto et al.[23] on the U 0 vs. J parameter
plane. A ferromagnetic ground state with full spin polarization(S = max) appears around J ' U 0> 3. The parameter regions J/U 0 À 1 and J/U 0 ¿ 1 are paramagnetic(S = 0). No partiallyfor U 0 ∼polarized state was found at this filling. Figure also shows that the difference between our result andthe DMRG result is small. The present result is similar to those by previous studies employing theperiodic or the anti-periodic boundary conditions, but the phase boundaries of their ferromagnetic
phases seem to depend on the system size greatly. We also indicate the phase boundary of the ferromagnetic phase by using the mean-field(MF) approximation in Fig.4. In this approximation, theground-state energy E0 is calculated as
E0

=
=

<H>
X

²± (k) + U Nu

k<kF ,m,σ+(U 0 − J)Nu

X

< nm,↑ >< nm,↓ > +U 0 Nu

σ< nu,σ >< nl,−σ >

σ

m

X

X

< nu,σ >< nl,σ > +X∆(< nu,σ > − < nl,σ >),
Nu
2
σ

－4－
4

(5)

Ferromagnetism and Superconductivity in Two-band Hubbard Models

15

1

n=0.5

n=0.5
∆=0

∆=0

MF

J=U’

5

S=max

J=2U’
S=0

Jc

Jc
2

Sakamoto et al.

JKρ

J=U’/2

0.5

0.25
0

S=Max

10

0.75

U’

0
0

4

Figure 3: Kρ as a function of U 0 in thecases of J = U 0 /2, J = U 0 and J =2U 0 at n = 0.5(4electrons/8sites). Jc
indicates the critical point of the singlet
ground state changing into the fully polarized ferromagnetic(S=max) ground state.

SC

S=0 (Kρ<0.5)

Kρ=0.5MF(ferro)

5

U’

10

15

Figure 4: Phase diagram of the ferromagnetic state on the U 0 −J parameter plane for
n = 0.5(4electrons/8sites) at ∆ = 0. The
dashed line represents the phase boundaryof the ferromagnetic state obtained by MF
approximation. The solid triangle shows thepoint of Kρ = 0.5 on the U 0 -axis.

where <>P
represents the expectation value of operators
P by the noninteracting ground state. Here, theterms −J i,m (c†i,u,↑ ci,u,↓ c†i,l,↓ ci,l,↑ + h.c.) and −J 0 i,m (c†i,u,↑ c†i,u,↓ ci,l,↑ ci,l,↓ + h.c.) are omitted since
the expectation values of these terms are equal to zero. The result indicates that the ferromagnetic
phase appears all over the region except the near the origin. Comparing with the numerical results,
it seems to overestimate the ferromagnetic region.
Figure 5 shows the value of Kρ as a function of U 0 for J = 0 at quarter filling n = 1(4electrons/4sites).
We also show the result of the Green’s function Monte Carlo method obtained by Assaraf et al.[44]and the MF approximation for Kρ [39], which are represented by the broken and the dashed lines in
fig.5, respectively. Here, Kρ of the MF approximation is estimated by the Luttinger liquid relation
as well as Numerical diagonalization method and the ground state is restricted to the singlet state,that is, the condition < nm,↑ >=< nm,↓ > is imposed. It shows good agreement with the numericalresult in the weak coupling regime, ensuring the small finite-size effect of the numerical calculation.
As U 0 increases, Kρ decreases from unity to ∼ 0.42. For J = 0, the model Hamiltonian is equivalent
to the SU(4) Hubbard model and the metal-insulator(MI) transition occurs at Kρ = 0.5[44]. WhenU 0 is larger than a critical value Uc0 ∼ 3, the MI transition is expected. Sakamoto et al. show then-dependence of the chemical potential at quarter filling and the existence of large charge gap inthe strong coupling region. Although the finite size effect does not allow the correct estimation of
the transition point, they claim that the metal-insulator transition occurs at finite positive value ofU 0 = J ∼ 3.
Figure 6 gives the value of Kρ as a function of U 0 in several conditions of J = U 0 /2, J = U 0 and
J = 2U 0 at n = 1. As U 0 increases, Kρ decreases, while it increases for a large U 0 in the condition
J = 2U 0 . In the region Kρ > 0.5, the SC correlation is expected to be the most dominant comparedwith the CDW and SDW correlations. To confirm the SC state, we calculate the lowest energy ofthe singlet state E0 (φ) as a function of an external flux φ. As shown in Fig.7, the anomalous flux
quantization occurs in the region Kρ > 0.5, where the signature of it increases with J. It indicatesthat the SC state is surely realized in this parameter region.
In Figure 8, we give the phase diagram of the ground state for Nu = 4 at the electron density
n = 1(4electrons/4sites) with the result of DMRG method[23] on the U 0 vs. J parameter plane. A
complete ferromagnetism with S=max appears around J ' U 0 in the strong coupling regime. This
result is in good agreement with the DMRG result obtained by Sakamoto et al.[23]. The dashed line
presents the boundary between the ferromagnetic state and the singlet state by using the mean-fieldapproximation. It seems to be fairly overestimate for the ferromagnetic region. It also indicates that－5－5Kazuhiro Sano and Yoshiaki Ōno

0.4

1

1

∆=0, J=0
n=1

2

4

U’

Figure 5: Kρ as a function of
U 0 in the cases of J = 0 for
n = 1.0 (4electrons/4sites) at∆ = 0. The dashed line represents a MF estimation for Kρ .
The broken line indicates the
GFMC result obtained by Assaraf et al.

0.8
1.0

J=U’

0.5

S=0
J=0.5U’

0
0

0.2

J=2U’

Assaraf et al.

∆=0

∆E

Kρ

Kρ
0.5

0.6

n=1

0.75

MF

J=U’=0.0

n=1
∆=0

0.25
0

2

Jc

4
U’

S=max

1.2

Figure 6: Kρ as a function of
U 0 in the cases of J = U 0 /2,
J = U 0 and J = 2U 0 for n =1.0 (4electrons/4sites) at ∆ =
0.

1.4

0

6

0

2π

Figure 7: The energy difference E0 (φ) − E0 (0) as a function of an external flux φ at
n = 1.0(6electrons/6sites) at∆ = 0.

the ferromagnetic region is smaller than the case of n = 0.5. In this ferromagnetic region, Sakamoto
et al. also claimed that the system shows the triplet SC for J > U 0 , while it becomes insulator
for J < U 0 . The SC phase boundary in the ferromagnetic region is smoothly connected to that in
the paramagnetic region as shown in Fig. 8. These results tell us that the Hund’s rule coupling
J plays important roles not only for the ferromagnetism but also for the superconductivity. In theparamagnetic state (S=0), we plot the phase boundary separating the SC region with Kρ > 0.5
and the insulating region with Kρ < 0.5. As mentioned above, this SC phase is confirmed by theanomalous flux quantization, while this quantization disappears in the insulating region with large U 0(not shown). Within the bosonaization method, the SC state has been identified as the triplet SC
with spin gap[25]. However, very recent work claims that the relevant symmetry of the paring is not
15

n=1

1

S=0 Kρ>0.5(spin gapped)

10

(SC)

n=8/6
∆=0

(ferro SC)

S=max

Kρ

J

S=max
(INSULATOR)

MF

5

J=U’

0.5

(SC)

S=0 Kρ<0.5

MF

J=0.5U’(spin gapped)(INSULATOR)(Ferro)005Kρ=0.5

J=2U’

S=0

U’

10

15

0

2

U’

4

S=max
Jc
6

Figure 9: Kρ as a function of U 0 for n = 4/3(8electrons/6sites) at ∆ = 0.

Figure 8: Phase diagram of the ferromagnetic state on the U 0 − J parameter plane
for n = 1(4electrons/4sites) at ∆ = 0. The
dashed line represents the phase boundaryof the ferromagnetic state obtained by MF
approximation. The solid triangle shows thepoint of Kρ = 0.5 on the U 0 -axis.

－6－6Ferromagnetism and Superconductivity in Two-band Hubbard Models0

p−0 wave like triplet but 0 dxy −0 like singlet by using the bosonaization method and DMRG method
within the model at U 0 = 0[26].
To examine the case of over the quarter-filling(n=1), we calculate the value of Kρ as a function of
U 0 in several conditions of J = U 0 /2, J = U 0 and J = 2U 0 at n = 4/3 (8electrons/6sites). As shownin Figure 9, it seems to be similar to the case of (n ≤ 1. This result suggests that the overall aspectof the electronic state dose not much depend on n.
3.3
Case for finite ∆Next, we consider the case of ∆ > 0. When the lowest energy of the upper band, ²+ (0), is larger thanthe Fermi energy, EkF , electrons occupy only the lower band with kF = πn2 and the model is regarded
as a ”single component” electron system. Hereafter, we mainly treat the case with ²+ (0) > EkF . In
this case, we can not adopt the Moebius boundary condition, since the symmetry of inter-band isbroken by finite ∆. In the case n < 1, we find the following boundary conditions are suitable, thatis, the periodic(antiperiodic) boundary condition for the lower(upper) band at Ne = 4m + 2 and theantiperiodic(periodic) boundary condition for the upper(lower) band at Ne = 4m, where Ne is thetotal electron number and m is an integer. This boundary condition can be regard as an extension of
the Moebius boundary condition for the finite ∆.
For n > 1, on the other hand, our experience indicates that the usual boundary conditions, that is,
the periodic boundary condition at Ne = 4m + 2 and the antiperiodic boundary condition at Ne = 4m
for the both bands, are little better. This choice of the boundary conditions is rather empirical, but
it seems to lead better results.
1.5

0.6

ferro(S=2)

n=2/3
∆=1.2

∆=1.0

Kρ

1.2

1.4

1

0

S=2

S=1

S=1
S=1
1.6

n=2/3
1

E(Φ)−E(0)

1.25

0.75

J(=U’)=0.4

2
J(=U’)

0.4

1.8

0.2

MF

0

0.8
1.0

1.3
0

3

Figure 10: Kρ as a function of J(= U 0 )for n = 2/3(6electrons/9sites) at ∆ =1.0, 1.2, 1.4, 1.6, and 1.8. The down arrows
indicate the critical points of J(= U 0 ). The
dashed line represents a weak coupling estimation for Kρ .

Φ/2π

1

Figure 11: The energy difference E0 (φ) −E0 (0) as a function of an external flux φ
for J(= U 0 ) =0.4,0.8,1.0 and 1.3 at n =2/3(6electrons/9sites) and ∆ = 1.2.

Figure 10 shows the value of Kρ as a function of J(= U 0 ) for several values of ∆ at the electron
density n = 2/3(6electrons/9sites). As J increases, Kρ decreases for a small J, while it increases fora large J, and then becomes larger than unity. Since we consider the single component system, theSC correlation is expected to be the most dominant compared with the CDW and SDW correlationsin the region Kρ > 1. To confirm the superconducting state, we calculate the lowest energy of thesinglet state E0 (φ) as a function of an external flux φ. As shown in Fig. 11, the anomalous flux
quantization occurs clearly at J ∼ 1.3, where Kρ is about 1.1. When J = 0.4, Kρ is less than unity
and the anomalous flux quantization is not found.
When J is larger than a certain critical value, the ground state changes into the partially ferromagnetic state with S=1 or S=2 from the singlet state. In Fig. 10, the broken line represents theMF approximation for Kρ . This approximation depends upon only the noninteracting ground state
where the lower band is exclusively occupied, as shown in Fig. 1. Therefore, the effect of the upper－7－7

Kazuhiro Sano and Yoshiaki Ōnoband is omitted and the result is independent of ∆. This approximation breaks down when the effectof the upper band becomes crucial. In this regime, Kρ rapidly increases with increasing J, and finallybecomes larger than unity indicating the superconducting state. The critical values of J with Kρ = 1
are Jc ≈ 0.3, 1.0, 2.4 and 4.0 for ∆ = 1.0, 1.2, 1.6 and 1.8, respectively.
In Fig.12, we show the phase diagram of the superconducting state with Kρ > 1 together withthe partial ferromagnetic state with S = 1 and S = 2 on the U 0 vs. J parameter plane forn = 2/3(6electrons/9sites) at ∆ = 1.2, where the S = 2 state is depicted by the bunch of thinsolid circles(shadowed region). The superconducting phase appears near the partially polarized ferromagnetic region. It extends from the attractive region with J < 0 and U 0 < 0 to the realistic parameterregion for 3d transition-metals with U 0 > J > 0. We have confirmed that the superconducting region
increases as ∆ decreases, as shown in Fig.10.
Figure 13 shows the global phase diagram on the U 0 − J plane for n = 4/3(8electrons/6sites)> J > ∆, the fully polarized ferromagnetism with S = S
at ∆ = 4.0. When U 0 ∼∼max appears. It< ∆. Inset shows theaccompanies the partially polarized ferromagnetism with 0 < S < Smax for J ∼magnification of the phase diagram near the origin. As well as n < 1, the superconducting statewith Kρ > 1 appears near the partially polarized ferromagnetic region and it extends to the realistic
parameter region with U 0 > J > 0. The configuration of the partially polarized ferromagnetic state
and the superconducting state is similar to the case n < 1. When J exceeds ²+ (0) − EkF , the interband excitation develops and the occupation number of the upper band increases, which results ina large orbital fluctuation accompanied by the fluctuation between the low-spin and the high-spinstates. The mechanism of superconductivity may be related to this orbital fluctuation. We note thatsuperconductivity is also observed for the J 0 = 0 (J 6= 0) case in contrast to the previous study[45, 46],
where the pair-transfer J 0 is crucial to superconductivity. The existence of the partially polarized
ferromagnetism for ∆ > 0 has been reported in a different type of two-band Hubbard model[40].
In Fig. 14, we show the doping dependence of the critical values of J for the superconductivity
and the ferromagnetism at ∆ = 4 with J = U 0 , where we use n=4/3, 10/7, 3/2, 8/5 and 12/7 systems.
We determine the critical values on condition that Kρ > 1 for the superconductivity and S > 0 for
the ferromagnetism. Although the finite size effect is considerably large, the phase boundary for thesuperconductivity can be approximately given by the phenomenological equation, Jc = ²+ (0) − EkF .
Here, ²+ (0) − EkF corresponds to the lowest energy of the single-particle excitation from the lower
band to the upper band. This equation was found to be a good approximation for various n > 1[41].
2

ferro

S=1

J

MF

partial ferro

S=2

4

ρ

S=1

J

S=0

J

S=0
Kρ>1
0
0

SC
2

U’

n=4/3
∆=4

Kρ=0.8

Kρ=1

S=2

2

S=1

15 1 SC(K >1)

2

S=3

5
S=2

S=4
(max)

S=2
S=1

0
SC 0
(Kρ>1)

4

Figure 12: Phase diagram of the superconducting state with Kρ > 1 and the ferromagnetic states with S = 1 and S = 2(shadowed
region depicted by the bunch of thin solid
circles) on the U 0 − J parameter plane forn = 2/3(6electrons/9sites) at ∆ = 1.2.

1 U’

S=0

Kρ<1
∆=1.2
n=2/3

10 0

MF

5

S=0 Kρ<1
10
U’

15

Figure 13: Global phase diagram on the U 0 − J
plane for n = 8/6 at ∆ = 4.0. The dashedline indicates the phase boundary of the complete(partial) ferromagnetic state obtained by
the MF approximation. Inset shows the magnification of the phase diagram near the origin.

－8－
8

Ferromagnetism and Superconductivity in Two-band Hubbard Models

Su−u,Tu−u

Sul

Suu

2

U’(=J)

ferro

7units6units5units4units

∆
Tul

1
Kρ>1
Kρ<1

0
1

∆
1.5
n

2

Sl−l,Tl−l

Figure 14: Phase diagram on the n − Jplane ∆ = 4.0. The broken line represents the phenomenological equation Jc =²+ (0) − EkF (see text) as a function ofn. The solid(dashed) line indicates thephase boundary of the complete(partial) ferromagnetic state obtained by the MF approximation.

Sll

Figure 15: Schematic diagrams of various
types of superconducting paring; Sll , Suu ,Sl−l , Su−u and Sul ( singlet paring) andTl−l(r) , Tu−u (r) and Tul (r) (triplet paring)

We also show the phase boundary of the ferromagnetic state obtained by the MF approximation. It
seems to consist with the numerical result.
3.4 Paring correlation functions
Next, we consider superconducting paring of the system, as shown in Fig. 15. In Fig.16, we show various types of superconducting paring correlation functions C(r) in detail for n = 2/3(6electrons/9sites)at ∆ = 1.0 and J(= U 0 ) = 1.04. The paring correlation functions are defined bySll (r) =Suu (r) =Sl−l (r) =Su−u (r) =Sul (r) =Tl−l (r) =Tu−u (r) =Tul (r) =1 X< c†i,l,↑ c†i,l,↓ ci+r,l,↓ ci+r,l,↑ >,
(6)
Nu i
1 X
< c†i,l,↑ c†i,u,↓ ci+r,u,↓ ci+r,u,↑ >,
(7)
Nu i
1 X
< (c†i,l,↑ c†i+1,l,↓ − c†i,l,↓ c†i+1,l,↑ )(ci+r+1↓ ci+r,l,↑ − ci+r+1,l,↑ ci+r,l,↓ ) >,
(8)
2Nu i
1 X
< (c†i,u,↑ c†i+1,u,↓ − c†i,u,↓ c†i+1,u,↑ )(ci+r+1,u,↓ ci+r,u,↑ − ci+r+1,u,↑ ci+r,u,↓ ) >,(9)
2Nu i
1 X
< (c†i,l,↑ c†i+1,u,↓ − c†i,l,↓ c†i+1,u,↑ )(ci+r+1,u,↓ ci+r,l,↑ − ci+r+1,u,↑ ci+r,l,↓ ) >, (10)
2Nu i
1 X
< (c†i,l,↑ c†i+1,l,↓ + c†i,l,↓ c†i+1,l,↑ )(ci+r+1↓ ci+r,l,↑ + ci+r+1,l,↑ ci+r,l,↓ ) >,
(11)
2Nu i
1 X
< (c†i,u,↑ c†i+1,u,↓ + c†i,u,↓ c†i+1,u,↑ )(ci+r+1,u,↓ ci+r,u,↑ + ci+r+1,u,↑ ci+r,u,↓ ) >,(12)
2Nu i
1 X
< (c†i,l,↑ c†i+1,u,↓ + c†i,l,↓ c†i+1,u,↑ )(ci+r+1,u,↓ ci+r,l,↑ + ci+r+1,u,↑ ci+r,l,↓ ) >, (13)
2Nu i

－9－
9

Kazuhiro Sano and Yoshiaki Ōno

where C(r) = Sll (r), Suu (r), Sl−l (r), Su−u (r) and Sul (r) denote the singlet paring correlation functions
on the same site in the lower orbital, on the same site in the upper orbital, between the nearest neighborsites in the lower orbital, between the nearest neighbor sites in the upper orbital, between lower
and upper orbitals on the same site, respectively. Further, Tl−l(r) , Tu−u (r) and Tul (r) are the triplet
paring correlation functions between the nearest neighbor sites in the lower orbital, between the nearest
neighbor sites in the upper orbital and between lower and upper orbitals on the same site, respectively.
The absolute value of Tu−u (r) is small, but the correlation of it is the slowest to decay. This result
Sl−ln=6/9 ∆=1.0
J(=U’)=1.04

Sll

1|R(r)|

Tl−l

−2

10
|C(r)|

Suu

Sul

102

1

2

3

4

Tu−u

0.42

r

∆=1.0

Tul

1

10

Tl−l
1

r

Figure 16: The singlet paring correlation
functions C(r) = Sll (r), Sl−l (r), Suu (r),Su−u (r), Sul (r) and the triplet correlation
functions Tl−l (r), Tu−u (r), Tul (r), respectively(see text). Here we show the absolute value of the correlation functionsat ∆ = 1 and J(= U 0 ) = 1.04 for
n=2/3(6electrons/9sites).

Sll

100

5

Suu

Sl−l

0

103

Tu−u

−6

10

∆=1.0

Tul
|R(r)|

−4

Sul

Su−u

10

Su−u
10

10

2

r

3

4

5

6

Figure 17:
The ratio of the singlet
paringcorrelation
functionsR(r) = C(r)J=1.04 /C(r)J=0.2 for Sl−l (r),Suu (r), Su−u (r), Sul (r) and that of thetriplet correlation functions for Tl−l (r),Tu−u (r), Tul (r) with the power-low r0.42 ,respectively(see text). The broken line
represents the power-low r0.42 predicted by
the Luttinger liquid relation.

seems to suggest that the relevant paring of the superconductivity is the tripletparing between lower
and upper orbitals on the same site and the ferromagnetic fluctuation near the ferromagnetic phase
may cause the paring. To clarify the behavior of the correlation functions, we calculate the ratio R(r)of the paring correlation functions at J(= U 0 ) = 1.04 and that of J(= U 0 ) = 0.2 as
R(r) =

C(r)J=1.04.
C(r)J=0.2(14)Although the correlation function C(r) decays as distance r increases, the function R(r) for relevant
paring is expected to increase with r, because the value of Kρ at J = 1.04 is larger than that at
J = 0.2, where Kρ is about at 1.4 and 0.98, respectively. Then, the behavior of R(r) is expectedto ∼ r0.42 . In fig.17, we show R(r) for Sll (r), Sl−l (r), Suu (r), Su−u (r) and Sul , (upper panel) andthe triplet paring correlation functions Tl−l (r), Tu−u (r) and Tul (r) with the power-low r0.42 predictedby the Luttinger liquid relation (lower panel), respectively. It indicates that that function R(r) for
the triplet paring, Tu−u , is much enhanced for long range paring correlation. On the other hand, theremains are not enhanced or decay as r increases. These results suggest that the paring correlation
function Tu−u (r) is most relevant paring to the superconductivity. Although the system size is too
small to compare the slope of the function R(r) with the power-low enhancement ∼ r0.42 directly, thebehavior of R(r) for Tu−u (r) seems to be roughly consistent with the result of the Luttinger liquid
relation.

4

d-p Chain ModelIn the previous works[47, 48, 49, 50, 51, 52], the present authors and many other authors studied
the one-dimensional d-p model with large on-site Coulomb repulsion Ud at Cu sites and intersite re-

－10－
10

Ferromagnetism and Superconductivity in Two-band Hubbard Models

(a)

tpp
p

∆

tpd

d

(b)
+

ε (k)
∆
−

ε (k)

Ud

Figure 18: Schematic diagrams of (a) the model Hamiltonian and (b) the band structure in thenoninteracting case.
pulsion Upd , by using the numerical method. They claimed that the superconducting (SC) correlation
is found to be dominant compared with the charge density wave (CDW) and spin density wave (SDW)
correlations and the parameter Upd is central in inducing a SC state. However, we found that if the
hopping term tpp is added to the model, the situation will be completely changed[53]. It enhances thefluctuation of charge and spin, and increases the exponent Kρ as parameter Upd does. Therefore, therepulsive interaction Upd is not always necessary for the SC state.
In this section, we reexamine the one-dimensional d-p model in the presence of the hopping term
tpp to clarify the electronic structure of the SC state, especially paying attention to the symmetry of
the paring correlation.
4.1
Model Hamiltonian
We consider the following model Hamiltonian for the Cu-O chain in the hole picture;X †X †H = tpd(piσ djσ + h.c.) + tpp(piσ pjσ + h.c.)<ij>,σ+<ij>,σ

X †
X †
X
²d
djσ djσ + ²p
piσ piσ + Udn̂dj↑ n̂dj↓ ,
j,σ

i,σ

(15)

j

where d†jσ and p†iσ stand for creation operators of a hole with spin σ in the Cu(d) orbital at site j andof a hole with spin σ in the O(p) orbital at site i, respectively, and n̂djσ = d†jσ djσ . Here, tpd stands
for the transfer energy between the nearest-neighbor d and p sites and will be set at unity (tpd =1)
hereafter in this study. The atomic energy levels of d orbital and p orbital are given by ²d and ²p ,respectively. The charge-transfer energy ∆ is defined as ∆ = ²p − ²d .
For the noninteracting case (Ud = 0), the Hamiltonian in eq. (15) is easily diagonalized to yield adispersion relationqo1n²d + ²p + 2tpp cos k ± (∆ + 2tpp cos k)2 + 16(tpd cos (k/2))2 ,(16)
E ± (k) =2
where k is a wave vector and E + (k)(E − (k)) represent an upper (lower) band energy. For tpp > 0, thewidth of the lower band E − (k) decreases with decreasing ∆ and becomes perfectly flat at ∆f lat =2tpp − t2pd /tpp . When ∆ < ∆f lat , the band bends with a peak at k = 0. To investigate the electronic
structure of the interacting case, we numerically diagonalize the Hamiltonian up to 14 sites (7 unit
cells). To carry out a systematic calculation, We use the periodic boundary condition for Nh = 4m + 2
and the antiperiodic boundary condition for Nh = 4m, where Nh is the total hole number and m is
an integer.
4.2 Critical exponent Kρ
In Fig. 19, we show the numerical results of Kρ obtained through eq. 2 as a function of Ud forn = 4/3(8holes/6units) at ∆ = 2 and tpp = 0.5. We also plot Kρ obtained through the mean field(MF) approximation, respectively. In the MF approximation, the renormalized bands Ẽ ± (k) are given
by eq. 16 in which ∆ is replaced by˜ = ∆ − 1 Ud < nd >,
∆
2

(17)

where < nd > is the hole density at a d site and is determined by solving the self-consistent equation in
the MF approximation. Using the renormalized band Ẽ − (k), we can calculate the charge susceptibility

－11－
11

Kazuhiro Sano and Yoshiaki Ōno

Kρ=

0.02
Ud=6.0Ud=6.7

0

1

6

SC

4

Kρ=1

0.01

0

Ud

Kρ

Eg(Φ)−Eg(0)2

1 Φ/2πn=8holes/6unitsMF2∆=2, tpp=0.5
0
0

8

8
0.03

2

4

Ud

0

6

∆=2, tpp=0.5

1.4

1.6
n

Figure 19: The critical exponent Kρ as afunction of Ud at ∆ = 2 and tpp = 0.5 for
n = 4/3(8holes/6units). The dashed lines
represent the results of the MF approximation (See in the text) Inset shows the energy
difference E0 (φ) − E0 (0) as a function of an
external flux φFigure 20: The SC region sandwiched between the two lines of Kρ = 1 and Kρ =∞ on the n-Ud parameter plane at ∆ =2 and tpp = 0.5. Here, we use n =8/6, 10/7, 6/4, 8/5, 10/6 and 12/7 systems.

χc and the Drude weight D. Substituting those values into eq. (2), we obtain Kρ within the MF
approximation. In the weak coupling regime, the results of the numerical diagonalization are in good
agreement with the MF approximation. It also seems to be roughly consistent with the numerical> 6.
results even in the strong coupling regime except for Ud ∼< 3, K decreases as U increases. For sufficiently large U , K increases with increasing
When Ud ∼ρddρUd and diverges at Ud ∼ 6.8 when the charge susceptibility diverges. The region where Kρ is larger than
< U < 6.8. When K > 1, the SC correlation is expected to be most dominant
unity appears at 6.5 ∼d ∼ρ
compared with the CDW and SDW correlations. Inset shows the energy difference E0 (φ) − E0 (0) as
a function of an external flux φ. It shows that the anomalous flux quantization occurs at Ud = 6.7,
when Kρ is about 2.0. While, at Ud = 6.0, Kρ ' 0.86 and the anomalous flux quantization is notfound. These result also confirms the SC phase at Ud = 6.7. The divergence of Kρ suggests thatthe effective bandwidth is close to zero. In fact, if we use the numerical value of < nd >' 0.83 with˜ ∼ −0.8 and the effective bandwidth is nearly equal to 0.1. It is
Ud = 6.7 in eq. (7), we obtain ∆very small compared with the noninteracting band. The result of the anomalous flux quantizationshows that the variation of |E0 (φ) − E0 (0)| at Ud = 6.7 is much smaller than that at Ud = 6.0. It also
indicates that the effective bandwidth is small.
Figure 20 shows the SC region where 1 < Kρ < ∞ on the n − Ud parameter plane at ∆ = 2 andtpp = 0.5. We use n = 8/6, 10/7, 6/4, 8/5, 10/6 and 12/7 systems, where numerator and denominate
correspond to numbers of electrons and unit cells, respectively. The region sandwiched between the
two lines of Kρ = 1 and Kρ = ∞ corresponds to the SC region. It is narrow near n = 1.4, however,it increases with increasing n.
4.3 Paring correlation functions
We investigate various types of superconducting paring in detail. as shown in Fig. 21. Figure 22
shows the paring correlation functions C(r) defined by
Sp−on (r) =Sd−on (r) =1 X< p†i↑ p†i↓ pi+r↑ pi+r↓ >,Nu i
1 X< d†i↑ d†i↓ di+r↑ di+r↓ >,
Nu i

－12－
12

(18)
(19)

Ferromagnetism and Superconductivity in Two-band Hubbard Models

Sd−d

Sp−on

p

∆

d

10

−4

10

|C(r)|

Sp−p, Tp−p∆dUd=6.7∆=2, tpp=0.5

Sd−on

Sd−p

Sp−p

Tp−p

10−2
10−4

p

Sp−on

−2

|C(r)|

Sd−p, Td−p

Td−p

Ud=6.7∆=2, tpp=0.5
1

Td−d
2

3

4

5

r

Sd−d, Td−d

Sd−on

Figure 21: Schematic diagrams of various
paring for d-p model.

Sp−p (r) =Sd−d (r) =Sd−p (r) =Tp−p (r) =Td−d (r) =Td−p (r) =Figure 22: The singlet paring correlation functions C(r) = Sp−on (r), Sp−p (r),Sd−on (r), Sd−d (r) and Sd−p (r) (upper
panel) and the triplet paring correlation
functions Tp−p (r), Td−d (r) and Td−p (r)(lower panel), respectively (see text). Here
we show the absolute value of the correlation
functions at ∆ = 2, tpp = 0.5 and Ud = 6.7
for n=4/3(8holes/6units).

1 X< (p†i↑ p†i+1↓ − p†i↓ p†i+1↑ )(pi+r+1↓ pi+r+1↑ − pi+r+1↑ pi+r↓ ) >,
2Nu i
1 X
< (d†i↑ d†i+1↓ − d†i↓ d†i+1↑ )(di+r+1↓ di+r+1↑ − di+r+1↑ di+r↓ ) >,
2Nu i
1 X
< (d†i↑ p†i↓ − d†i↓ p†i↑ )(pi+r↓ di+r↑ − pi+r↑ di+r↓ ) >,
2Nu i
1 X
< (p†i↑ p†i+1↓ + p†i↓ p†i+1↑ )(pi+r+1↓ pi+r+1↑ + pi+r+1↑ pi+r↓ ) >,
2Nu i
1 X
< (d†i↑ d†i+1↓ + d†i↓ d†i+1↑ )(di+r+1↓ di+r+1↑ + di+r+1↑ di+r↓ ) >,
2Nu i
1 X
< (d†i↑ p†i↓ + d†i↓ p†i↑ )(pi+r↓ di+r↑ + pi+r↑ di+r↓ ) >,
2Nu i

(20)
(21)
(22)
(23)
(24)
(25)

where Sp−on (r), Sd−on (r), Sp−p (r), Sd−d (r) and Sd−p (r) denote the singlet paring correlation functions on a same p site, on a same d site, between the nearest neighbor p sites, between the nearest
neighbor d sites and between the nearest neighbor p and d sites, respectively. Further, Tp−p (r),Td−d (r) and Td−p (r) denote the triplet paring correlation functions between the nearest neighbor psites, between the nearest neighbor d sites and between the nearest neighbor p and d sites, respectively.
Here, we show the absolute value of the correlation functions at ∆ = 2, tpp = 0.5 and Ud = 6.7 for
n=4/3(8holes/6units). The orders of the absolute value of these correlation functions are almost equal
to one another.
However, the correlation functions C(r) for Sp−on (r) and Sp−p (r) seem to decay slower thanthat of the remains. It suggests that these paring play an important role in the superconductivity.
We also examine the ratio of the paring correlation functions R(r) at Ud = 6.7 and Ud = 6.0 as
R(r) = C(r)Ud =6.7 / = C(r)Ud =6.0 , where Kρ ∼ 2.0 at Ud = 6.7 and Kρ ∼ 0.86 at Ud = 6.0. The
function R(r) for relevant paring is expected to behave as r1.1 . In fig.23, we show the ratio of theparing correlation functions R(r) for Sp−on (r), Sp−p (r), Sd−on (r), Sd−d (r) and Sd−p (r) with thepower-low r1.1 (upper panel) and that of the triplet paring correlation functions for Tp−p (r), Td−d (r)

－13－
13

Kazuhiro Sano and Yoshiaki Ōno

1.1

r

1

0.6

Sp−on

10

Sp−p

Sd−on
Sd−p
Sd−d

−1

Sd−d
Sd−on

−2

Tp−p

100
|R(r)||R(r)|

Sd−p

10

2
Tp−p

10−1

Td−p

1

Sp−p
Sp−on

10

∆=2, tpp=0.5

0.6

0

2

Td−d
3
4

5

2

1

1

Figure 23:
The ratio of the singlet
paringcorrelation
functions
R(r)=C(r)Ud =6.7 /C(r)Ud =6.0 forSp−on (r), Sp−p (r), Sd−on (r), Sd−d (r)and Sd−p (r) with the power-low r1.1
depicted the broken line (upper panel)and that of the triplet paring correlation
functions for Tp−p (r), Td−d (r) and Td−p (r)(lower panel), respectively. Here we show
the absolute value of R(r) at ∆ = 2,
tpp = 0.5 for n=4/3(8holes/6units).

Tp−p|R(r)|

2|R(r)||R(r)|∆=2, tpp=0.5

1

r0.15
2 r

Td−d

Td−p

Sp−p
3

4 5

2

3

4

5

Figure 24:
The ratio of the singlet
paringcorrelation
functions
R(r)=C(r)Ud =6.59 /C(r)Ud =4.14 forSp−on (r), Sp−p (r), Sd−on (r), Sd−d (r)and Sd−p (r)(upper panel), and that of
the triplet paring correlation functions
for Tp−p (r), Td−d (r) and Td−p (r) (lower
panel), respectively. Here we show theabsolute value of R(r) at ∆ = 2, tpp = 0.5
for n=12/7(12holes/7units). Inset shows
R(r) for Sp−p (r) and Tp−p (r) with thepower-low r0.15 .

and Td−p (r) (lower panel), respectively. It indicates that that R(r) for Sp−on (r) and Sp−p (r) aremuch enhanced at long range paring correlation and seem to be roughly consistent with the powerlow obtained by the Luttinger liquid relation. On the other hand, the remains decay as distance r
increases or are almost constant. These results suggest that the singlet paring correlation functions
Sp−on (r) and Sp−p (r) are most relevant to the superconductivity and the triplet paring is irrelevant.
On the other hand, the situation is changed for large hole density. In this case, the triplet paring correlation is enhanced as well as the singlet paring correlation. In fig.24, we show the ratioof the singlet paring correlation functions R(r) = C(r)Ud =6.59 /C(r)Ud =4.14 for Sp−on (r), Sp−p (r),Sd−on (r), Sd−d (r) and Sd−p (r) (upper panel), and that of the triplet paring correlation functions for
Tp−p (r), Td−d (r) and Td−p (r)(lower panel), respectively, where we show the absolute value of R(r)for n=12/7(12holes/7units). Here, R(r) of the relevant paring is expected to behave as r0.15 sinceKρ ∼ 1.15 at Ud = 6.59 and Kρ ∼ 1.0 at Ud = 4.14. Inset shows R(r) for Sp−p (r) and Tp−p (r) withthe power-low r0.15 . It shows that R(r) for Tp−p (r) increases with r as well as that for Sp−p (r), which
seems to be roughly consistent with the power-low r0.15 predicted by the Luttinger liquid relation.
suggests that the triplet paring becomes relevant to the superconductivity as well as the singlet paring
near n = 2.

5

d-p Model in Infinite Dimensions

Recently, some significant progress has been achieved in understanding the strongly correlated
electron systems by using the dynamical mean-field theory (DMFT)[54, 55]. In this approach, thelattice problem is mapped onto an effective impurity problem where a correlated impurity site isembedded in an effective uncorrelated medium that has to be determined self-consistently. To solve
the effective impurity problem, several methods have been applied including the iterated perturbation
theory[55], the non-crossing approximation[56], the quantum Monte Carlo (QMC) method[57], theexact diagonalization (ED) method[58] and the numerical renormalization group (NRG) method[59,60]. The DMFT becomes exact in the limit of infinite spatial dimensions (d = ∞)[54] and believed to
be a good approximation in high dimensions.

－14－
14

Ferromagnetism and Superconductivity in Two-band Hubbard Models

In the d = ∞ single-band Hubbard model at half-filling, the Mott metal-insulator transition isfound to occur as a first-order phase transition at finite temperature below a critical temperature
Tcr [55]. Below Tcr , a coexistence of the metallic and insulating solutions is found for the same valueof the on-site Coulomb interaction U in the range Uc1 < U < Uc2 [55, 60]. The magnetic phase
diagram was obtained as a function of doping, temperature and U . A commensurate antiferromagnetic
order changes to an incommensurate state at a value of the doping which depends on U [61]. The> 2 in non-bipartite lattices
ferromagnetism was observed for an intermediate interaction strength U ∼< 20 in bipartite lattices such as
such as the fcc-type lattice[62] but was completely suppressed for U ∼the hypercubic lattice[63]. The superconducting phase is absent in the d = ∞ single-band Hubbard
model[64].
The DMFT has also been applied to the multi-band Hubbard model to elucidate the effect of the
Hund’s rule coupling on the Ferromagnetism[20, 22] and the Mott transition[65, 66]. Several authors
have extensively studied the d-p model using the DMFT[58, 67, 68, 69, 70, 71, 72, 73, 74, 75]. The
Mott transition was found to occur at n = 1 (or n = 3)[67, 68, 69, 70, 71, 73, 74], where n is thetotal electron number per unit cell and given by the sum of p- and d-electron numbers: n = np + nd .
The phase diagram of the Mott transition was obtained over the whole parameter regime including
the Mott-Hubbard type (U < ∆) and the charge-transfer type (U > ∆)[69, 70]. When the carrier isdoped with n > 1, the superconductivity was observed in the charge-transfer regime[58, 67, 69], while
it was not observed in the Mott-Hubbard regime. The antiferromagnetism was also observed at andnear n = 1[75]. More recently, the present authors discussed the ferromagnetism which was found to> 2[72]. Therefore it isoccur at and near n = 2 in the intermediate interaction strength U ≈ 2∆ ∼interesting to discuss the relationship between the magnetism and the superconductivity in the d-p
model in infinite dimensions and to obtain the phase diagram as functions of the parameters such as
the interaction, the electron filling and the temperature.
5.1

Model and formulation
We consider the d-p model on a Bethe lattice with infinite connectivity z → ∞. The Hamiltonianis written as
X
X †
X †
X
H=
(ti,j d†iσ pjσ + h.c.) + ²p
pjσ pjσ + ²d
diσ diσ + Ud
ndi↑ ndi↓ ,
(26)
i,j,σ

j,σ

i,σ

i

where d†iσ and p†jσ stand for creation operators of a electron (or a hole) with spin σ in the d-orbitalt
at site i and in the p-orbital at site j, respectively. ndi,σ = d†iσ diσ . ti,j = √pdz represents the transfer
energy between the nearest neighbor site and the parameter tpd will be set to unity in the present
study. The atomic energy levels of d-orbital and p-orbital are given by ²d and ²p , respectively. The
charge-transfer energy ∆ is defined as ∆ = ²p − ²d > 0.
In the DMFT, the effective action of the impurity problem is given by
Z βZ βZ βdτ nd↑ (τ )nd↓ (τ ) −S = Ud00dτ dτ 00X−1d†σ (τ )D0σ
(τ − τ 0 )dσ (τ 0 ),

(27)

σ

where the Weiss function D0σ includes effects of the interaction at all the sites except the impurity
site. This action is derived by tracing out the fermionic degrees of freedom in the original lattice model
except the impurity site. The local Green’s function Dσ (τ −τ 0 ) = −hT dσ (τ )d†σ (τ 0 )iS is calculated withthis action. Using D0σ and Dσ , we introduce the local self-energy Σσ (iωn ) = D0σ (iωn )−1 −Dσ (iωn )−1 ,where ωn is the Matsubara frequency, ωn = (2n + 1)π/β. The self-consistency condition for the local
Green’s functions gives the relationsZiωn + µ − εpDσ (iωn ) = dεN (ε) ×,(28)(iωn + µ − εd − Σσ (iωn ))(iωn + µ − εp ) − ε2Ziωn + µ − εd − Σσ (iωn )Pσ (iωn ) = dεN (ε) ×,(29)(iωn + µ − εd − Σσ (iωn ))(iωn + µ − εp ) − ε2where µ is the chemical potential and Pσ (τ − τ 0 ) = −hT pσ (τ )p†σ (τ 0 )iS is the local Green’s function
at p-site. For the Bethe lattice with z = ∞, the density of states is given by a semi-circular function,

－15－
15

Kazuhiro Sano and Yoshiaki Ōnop
N (ε) =
4 − (ε/tpd )2 /2πtpd . Using the semi-circular density of states in eqs.(28) and (29) andeliminating Σσ (iωn ), we obtain the simple form of the self-consistency equations
D0σ (iω)−1=iωn + µ − εd − t2pd Pσ (iωn ),(30)

Pσ (iω)−1=iωn + µ − εp − t2pd Dσ (iωn ).

(31)

To calculate the local Green’s function Dσ (iωn ) for a given D0σ (iωn ), we use the numerical diagonalization method (DMFT-ED method). The self-consistency equations (30) and (31) lead a new
D0σ (iω) and we repeat the calculation of the local Green’s function. This process is iterated until thesolutions converge.
In the DMFT-ED method, we approximately solve the impurity Anderson model of a finite-sizecluster;
HAnd = ε0σ

X
σ

ndσ +

Ns
X

εlσ c†lσ clσ +

l=2,σ

Ns
X

Vlσ (d†σ clσ + c†lσ dσ ) + Ud nd↑ nd↓ ,

(32)

l=2,σ

where ε0σ is the impurity level and εlσ (l = 2, 3, ..., Ns ) are levels of the ’conduction electron’ hybridized
with the impurity by Vlσ . We regard the non-interacting Green’s function GAnd
0σ (iωn ) as the Weissfunction D0σ (iω) in the action eq.(27). Then, the interacting Green’s function GAnd(iωn ) correspondsσto the local Green’s function Dσ (iω) in the original lattice problem. Here, GAnd
0σ (iωn ) is defined by
GAnd
0σ (iωn ) =iωn − ε0σ −

1
PNs

2
Vlσ
l=2 iωn −εlσ

.

(33)

For a given D0σ (iω), we determine 2Ns −1 parameters ε0σ , εlσ , Vlσ (l = 2, 3, ..., Ns ) to make GAnd
0σ (iωn )as close to D0σ (iω) as possible. Using these parameters, we diagonalize the finite cluster of the impurity
Anderson model, and calculate GAnd
0σ (iωn ) (D0σ (iωn )). At finite temperature, the Green’s function isstraightforwardly calculated from the full set of states |ii with eigenvalues Ei according toDσ (iωn ) =1 X |hi|d†σ |ji|2(e−βEi + e−βEj ).
Z i,j iωn − Ei + Ej

(34)

Using the Lanczos method with the continued-fraction expansions, we also calculate the zero temperature Green’s functions. In this case, we replace the Matsubara frequencies by a fine grid ofimaginary frequencies, β̃ (iωn = (2n + 1)π/β̃), where the fictitious inverse temperature β̃ determines
a low-frequency cut-off.
5.2

Metal-Insulator transition at n = 2
First, we consider the paramagnetic state at zero temperature for n = nd + np = 2. In the
non-interacting case Ud = 0, the system is a band-insulator with the energy gap ∆ for ∆ 6= 0 while
it is a semimetal for ∆ = 0. Within the restricted Hartree-Fock (HF) approximation, the energy gap
is given by ∆ − Ud2nd . Then the system is metallic for Ud = 2∆, otherwise it is insulating. We notethat nd = np = 1 for Ud = 2∆ due to the particle-hole symmetry. In the inset in Fig. 25, we show
the chemical potential µ as functions of n for Ud = 7, 8, 9, 10 at ∆ = 6 and T = 0 calculated from the
DMFT-ED method with the system size Ns = 8[72]. In this calculation, the solution is restricted tothe paramagnetic state with ε0σ = ε0 , εlσ = εl and Vlσ = Vl for l = 2, 3, ..., Ns . When Ud increases
from Ud = 0, the discontinuity in the chemical potential at n = 2 decreases and finally becomes zeroat a critical value, where a transition from the band-insulator to the correlated semimetal occurs[67].
The critical values for the metal-insulator transition are plotted in Fig. 25. In contrast to the HF
approximation, the metallic state is found in the wide parameter region due to a correlation effectconsidered in the DMFT.
5.3

Ferromagnetism
At low temperature, the correlated semimetal mentioned above becomes unstable compared to aferromagnetic state. In Fig. 26, we plot the magnetization for the d-electron Md , that for the p-electron
Mp and the total magnetization M = Md + Mp as functions of the temperature T at n = 2 for Ud = 8

－16－
16

Ferromagnetism and Superconductivity in Two-band Hubbard Models

Ud=2∆

Metal

0

Ud=10

µ

5

9

−0.1

8

−0.2

7

T=0, ∆=6

1.95

0
0

n=2
T=0

Insulator

Ud

Insulator

10

2

5

2.05

n

10

∆

Figure 25: The phase boundary separating the metallic and insulating regimes as a function of Ud and∆ at n = 2 and T = 0. The inset shows the chemical potential as functions of n for Ud = 7, 8, 9, 10 at∆ = 6 and T = 0 calculated from the DMFT-ED method with Ns = 8.
and ∆ = 4 calculated from the DMFT-ED method with Ns = 6[72]. As shown in Fig. 26, Md and Mp
have opposite sign to each other. In the low temperature limit, both of Md and Mp become constant,while the sum of them M becomes zero. The feature of the ferromagnetism from the DMFT issimilar to that from the HF approximation as shown in Fig. 26. However, the transition temperature
Tc from the HF approximation is much higher than that from the DMFT (see also Fig. 29). In
Fig. 27, we plot the magnetization as functions of Ud with keeping ∆ = U2d at n = 2 and T = 0.01
calculated from the DMFT-ED method together with those from the HF approximation. When Ud
increases, the each component of the magnetization monotonically increases. In both approximations,> 2. This is a striking
the ferromagnetism is observed for the intermediate interaction strength Ud ∼contrast to the single-band Hubbard model where the ferromagnetism is observed only for the strong

Md

0.2

0.2
n=2

M(DMFT)

M

0
Mp

0.02

T (DMFT)

Md

Figure 26: The magnetization for the delectron Md , that for the p-electron Mp andthe total magnetization M = Md + Mp asfunctions of the temperature T at n = 2,
obtained from the DMFT-ED method for
Ud = 8 and ∆ = 4 with Ns = 6 and from the
HF approximation (dashed lines) for Ud = 2
and ∆ = 1.

0

0

−0.2
0

0.04

0.5

M=Md+Mp

Mp−0.1

DMFT(Ud=8)
HF(Ud=2)

−0.2

1
n=2
T=0.010.1

M=Md+Mp

0

0.2

−0.5

DMFT
HF
2

4
Ud=2∆

M(HF)

0.1 T (HF)

0

6

−1
8

Figure 27: The magnetization for the delectron Md , that for the p-electron Mp andthe total magnetization M = Md + Mp asfunctions of Ud with ∆ = Ud /2 at n = 2
and T = 0.01, obtained from the DMFTED method and from the HF approximation(dashed lines).

－17－
17

Kazuhiro Sano and Yoshiaki Ōno> 20 in bipartite lattices within the DMFT[63]. We note that the value of thecoupling region U ∼magnetization from the DMFT is about five times smaller than that from the HF for the same valuesof Ud and ∆.
Fig. 28 shows the total magnetization M as a function of the electron filling n. When n decreases,
M continuously becomes zero at a critical value of n for high temperatures (see for T = 0.025), while itdiscontinuously becomes zero for low temperatures (see for T = 0.01). The similar properties are also
observed within the HF approximation as shown in Fig. 28. At low temperatures, however, the HF
approximation also predicts a metastable state where M decreases with increasing n and continuously
becomes zero at a critical n (see for T = 0.025). By calculating the thermodynamic potential, we findthat the phase separation of the ferromagnetic state and the paramagnetic state occurs at the lowtemperatures (see Fig. 29).
Fig. 29 shows the transition temperature for the ferromagnetism Tc as a function of n for several
values of Ud (= 2∆)[72]. Tc monotonically decreases with decreasing n. The closed circles show
the second-order phase transition, while the open circles show the discontinuous transition as seenin Fig. 28. Within the HF approximation, the second-order phase transition occurs at the hightemperature (solid line), while the phase separation occurs at the low temperature (area between thedotted lines). In the DMFT, it is difficult to calculate the thermodynamic potential directly fromthe local Green’s function. But we may expect that, within the DMFT, the similar phase separation
takes place at low temperatures where the magnetization shows a discontinuous transition as shownin Fig. 28.

5.4 Superconductivity
Finally, we discuss the superconductivity in the d-p model (26). In infinite dimensions, the on-siteparing susceptibility χ of this model is given by[55, 67]
Z
1 β X
dτ
< T di↑ (τ )di↓ (τ )d†j↓ (0)d†j↑ (0) >
N 0
ij
X
−1/2
= T
[α
{I − Λ}−1 · Λ · α−1/2 ]ν,ν 0 ,

χ =

(35)

ν,ν 0

0.4

DMFT(Ud=8)
HF(Ud=2)

T=0.025

0.04

HF

Tc /5 (Ud=2)

0.3
0.03

M

0.2

0.1

Ud=8

Tc

T=0.01

0.02

T=0.1

T=0.025

Ud=4

0.01

Ud=2

PS

0

1.6

n

1.8

0

2

Figure 28: The total magnetization M as afunction of the electron number n, obtained
from the DMFT-ED method for Ud = 8 and∆ = 4 at T = 0.01 (closed circles), 0.025(open circles), and from the HF approximation for Ud = 2 and ∆ = 1 at T = 0.025(dotted line), 0.1 (dashed line).

1.6

n

1.8

2

Figure 29: The transition temperature Tc
for the ferromagnetism as a function of theelectron filling n, obtained from the DMFTED method (closed and open circles) for
Ud = 2∆ = 8, 4, 2, and from the HF approximation (solid and dotted lines) for Ud =2∆ = 2.

－18－
18

Ferromagnetism and Superconductivity in Two-band Hubbard Models

with= t4pd |P (iν)|[χ̃loc ]ν,ν 0 |P (iν 0 )|,Λν,ν 0αν,ν 0=(36)

t4pd |P (iν)|2 δν,ν 0 ,(37)

where, χ̃loc is the local paring susceptibility at a d-site given by[χ̃loc ]ν,ν 0 = T

2

Z β

Z β
dτ1

0

Z β
dτ2

0

Z β
dτ3

0

0

0

dτ4 eiν(τ1 −τ2 ) eiν (τ3 −τ4 ) < T d↑ (τ1 )d↓ (τ2 )d†↓ (τ3 )d†↑ (τ4 ) > . (38)

To calculate χ̃loc within the DMFT-ED method, we use a spectral representation of r.h.s. in eq.(38)by inserting a complete set of eigenstates |ii.
When the largest eigen value of Λ, λmax , approaches unity, the paring susceptibility diverges. It
signals the transition into the superconducting state from the normal state. In Fig.30, we show thevalue of λmax as a function of the temperature T for Ud = 8 and ∆ = 4 at n = 1.3 and 1.7 obtained
from the DMFT-ED with the system size Ns = 5. The value of λmax increases with decreasing T
and exceeds unity at a certain critical temperature. For n = 1.3, the critical temperature for the
spin-singlet pairing, TSS , is higher than that for the spin-triplet pairing, TT S , while, for n = 1.7, TT S
is higher than TSS [76]. This tendency is consistent with the zero-temperature DMFT-ED result withthe system size Ns = 6 − 8[58].
We note that the triplet superconductivity is on-site pairing and the gap function is odd in the Matsubara frequency as first proposed by Berezinskii in the superfluid 3 He[77]. In the present calculation,
the eigen function corresponding to the largest eigen value of Λ for the triplet-pairing, is proportional
to the gap function at the critical temperature, and is confirmed to be odd in the Matsubara frequency.
In Fig.31, the superconducting transition temperature for the triplet pairing TT S and that for the
singlet pairing TSS , obtained from the DMFT-ED method mentioned above, are plotted as functionsof n for Ud = 8 and ∆ = 4. We also plotted the transition temperature for the ferromagnetismTc (see Fig. 29) together with that for the antiferromagnetism TN calculated from the DMFT-ED< 1.4 nearmethod with Ns = 6. As seen in Fig.31, the singlet superconductivity is realized for n ∼the antiferromagnetic phase, where the antiferromagnetic fluctuation is considered to be responsible> 1.4 nearfor the singlet pairing. On the other hand, the triplet superconductivity is realized for T ∼the ferromagnetic phase, where the ferromagnetic fluctuation is considered to be responsible for the< n < 1.5. The
triplet pairing. Significantly, a reentrant superconducting transition is observed for 1.4 ∼∼reentrant transition has also been observed in the d = ∞ periodic Anderson model[78]. Therefore, the
0.04

1.5

Ud=8, ∆=4

Ud=8, ∆=4

n=1.3

0.03

n=1.7

TTS

1

T

λmax

Tc

0.02

TN

TSS

0.01

tripletsinglet
0.5
0

0.02
T

0
1

0.04

Figure 30: The largest eigen value λmax as
a function of the temperature T for Ud = 8
and ∆ = 4 at n = 1.3 and 1.7 obtained fromthe DMFT-ED with Ns = 5.

1.5
n

2

Figure 31: Transition temperatures for the
ferromagnetism Tc , the spin-triplet superconductivity TT S , the spin-singlet superconductivity TSS and the antiferromagnetism
TN , obtained from the DMFT-ED for Ud =8 and ∆ = 4.

－19－
19

Kazuhiro Sano and Yoshiaki Ōno

reentrant superconducting transition is considered to be a specific feature of two-band type models
including d-p model and the periodic Anderson model.

6

Summary and Discussion

We have investigate the ferromagnetism and the related superconductivity of the Hubbard modelwith two-fold orbital degeneracy and the d-p model with paying attention to the effect of the interplay
between the Coulomb interactions and the band splitting. To obtain reliable results beyond theperturbative or the mean-field like approximations, we use the numerical diagonalization method forthe one-dimensional models and the dynamical mean-field theory for the infinite-dimensional model.
For one-dimensional models, we calculate the critical exponent Kρ based on the Luttinger liquid
theory and the paring correlation functions of the ground state.
In the one-dimensional multi-orbital Hubbard model, we have obtained various phase diagrams
including the ferromagnetic and the superconducting state on the U 0 − J parameter plane. The> J > ∆. For
fully polarized ferromagnetism has been found in the strong coupling regime with U 0 ∼∼1 < n < 2, the ferromagnetism is metallic and mainly caused by the double-exchange mechanism[23].
Crystal-field splitting destroys the fully polarized ferromagnetism resulting in a partially polarized< ∆. In the vicinity of the partially polarized ferromagnetism, we have found the triplet
one for J ∼superconducting phase, when J exceeds the lowest energy of the inter-band excitation. It extends to
the realistic parameter region for 3d transition-metals with U 0 > J.
Sakamoto et al.[23] claimed that the metallic ferromagnetism for ∆ = 0 appears in a similar parameter region in any dimension by comparing the results from one dimension with those from infinite
dimensions. It is natural to think that this ferromagnetism will appear in two and three dimensions
even in the presence of ∆. Then, we expect that a partially polarized (weak) ferromagnetism appears in real materials, in which Hund’s rule coupling and crystal-field splitting compete with eachother, such as in cobalt oxides. In fact, a weak ferromagnetism has been observed in the layered
Na0.75 CoO2 [13] as well as in the perovskite R1−x Ax CoO3 [79]. The orbital degeneracy of 3d electrons
is considered to play a crucial role in Nax CoO2 as well as in La1−x Srx CoO3 [15].
In our calculation, we can not find any sign of the superconductivity near the fully polarized
ferromagnetic state at the realistic parameter region U 0 > J. It suggests that not the complete
ferromagnetic state but the weak ferromagnetic state can be a key to the superconductivity of realmaterials. The competition between Hund’s rule coupling and crystal-field splitting causes the large
orbital fluctuation, accompanied by the fluctuation between the low-spin and the high-spin states at
each Co ion. This fluctuation is expected to mediate the superconductivity near the weak ferromagnetism. Since the orbital fluctuation has a local character, the mechanism for the superconductivity
could be common in all dimensions. We therefore think that exploration of the superconductivity inthe vicinity of the weak ferromagnetism in the perovskite R1−x Ax CoO3 [79] may be promising.
In the one-dimensional d-p model, we have calculated the critical exponent Kρ and the various
types of the paring correlation functions. Using the Luttinger liquid relations, we have found that the
SC correlation is dominant in the parameter region where the renormalized band becomes almost flat.
Since the narrowing of the effective bandwidth leads to the enhancement of the fluctuation, it can cause
the superconductivity. The behavior of the paring correlation functions suggests that the singlet paring
on a same p site and between the nearest neighbor p sites are most relevant to the superconducting
state near half-filling. This result can be interpreted that the antiferromagnetic fluctuation meditates
the singlet paring as similar as the t-J model. However, in large doping case n = 12/7, the triplet
paring between the nearest neighbor p sites becomes relevant to the superconductivity as well as the
singlet paring between the nearest neighbor p sites. This result suggests that the low energy physics
of the d-p model can not well described by the t-J model near n = 2. We think that the existence ofmulti-band might be crucial to understand the triplet paring superconductivity.
We also examine infinite-dimensional d-p model based on the dynamical mean-field theory. The
dynamical mean-field theory becomes exact in the limit of infinite spatial dimensions and believed to
be a good approximation in high dimensions. The result is expected to be complementary to the resultof the one-dimensional d-p model. We obtain the phase diagrams of the metal-insulator transition on
the ground state at half-filling and quarter-filling. We also calculate the magnetization and the pairing
susceptibility to obtain the transition temperatures for the ferromagnetism and the superconductivity.
< 1.4 near the antiferromagnetic phase,
It shows that the singlet superconductivity is realized for n ∼where the antiferromagnetic fluctuation is considered to be responsible for the singlet pairing. On

－20－
20

Ferromagnetism and Superconductivity in Two-band Hubbard Models> 1.4 near the ferromagnetic phase.
the other hand, the triplet superconductivity is realized for T ∼We think that the ferromagnetic fluctuation is responsible for the triplet pairing as well as the onedimensional case near n = 2.
Ferromagnetism and superconductivity has been studied for a long time as a central argument initinerant electron systems. In this paper, we have presented the electron correlation and the band
splitting to be crucial for the ferromagnetism and the related superconductivity in the two types of thetwo-band Hubbard models. We hope that our work would yield an insight into the deep relationshipbetween these interesting phenomena.

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