- Email: [email protected]
Electron correlations in high-Tc superconductors
Physica B 206 & 207 (1995) 641-644
ELSEVIER
Electron correlations in high-T c superconductors S. Maekawa*, Y. Ohta Department of Applied Physics, Nagoya University, Nagoya 464-01, Japan
Abstract The electronic structure and excitation spectra in high-Tc superconducting cuprates are examined by using the exact diagonalization method on finite-size clusters. The low-energy electronic states are mapped onto the two-dimensional tJ model. Several excitation spectra in the tJ model are derived as functions of electron concentration and parameter (J/t) value where J is the antiferromagnetic exchange interaction between Cu spins and t is the hopping parameter of charge carriers introduced into the cuprates. The spectra are found to be in accord with those calculated in BCS theory with d(x 2- y2)-wave pairing.
1. Introduction
2. Crystal structure and electronic structure
The discovery of high-Tc superconducting cuprates [l] has renewed the physics of strongly correlated electron systems. The cuprates are unique among the systems [2,3]. The parent compounds of the superconductors are antiferromagnetic insulators of chargetransfer type. Upon doping of carriers, they have both metallic and ionic characters which are brought about by the layered structure. The physical properties are controlled by carrier concentration. In addition, quantum effects of spins are enhanced by the two-dimensional structure and the spin value (S = 1/2) of Cu ions. Here, we examine the electronic structure and derive the low-energy excitations. We utilize the exact diagonalization method on finite-size clusters. The low-energy electronic states are mapped onto the twodimensional tJ model. Several excitation spectra are obtained in the tJ model. We find that the spectra are in accord with those of BCS theory with d(x 2 - y 2 ) wave pairing. The characteristic features of the cuprates among strongly correlated electron systems are discussed.
The cuprates have a layered structure. Let us consider La 2 xSrxCuO4, as an example. Doping of hole-carriers in the CuO 2 planes is made by replacing La 3÷ ions by Sr z÷ ions which are located between the CuO 2 planes. The doping is possible because the cuprates are ionic crystals; i.e., the CuO z planes with negative charge carry doped holes with positive charge. We note that the structure of the CuO 2 planes is stable against the doping. It has been observed that the Madelung potentials acting on the doped holes govern the superconducting transition temperature [4]. The cuprates are charge-transfer type compounds, and the energy difference between Cu and O ions, A, is small. Therefore, the antiferromagnetic exchange interaction between neighboring Cu spins, J, is strong 4 3 as is seen in the perturbational expression, J = tpd/A , where tpd denotes the hopping parameter of holes between Cu and O ions and Ud the Coulomb repulsive interaction on a Cu site. In addition, quantum effects of spins are enhanced in the cuprates by the spin value (S = 1/2) of Cu ions in two-dimensions. As discussed in the next section, these are of crucial importance for high-Tc superconductivity. It has been shown in a numerical study [5] that the charge transfer gap is given by the energy difference
* Corresponding author.
0921-4526/95/$09.50 ~) 1995 Elsevier Science B.V. All rights reserved SSD1 0921-4526(94)00544-3
S. Maekawa, Y. Ohta / Physica B 206 & 207 (1995) 641-644
642
between the upper Hubbard band caused by the correlation U d and the singlet state made by the holes on O and Cu sites, the so-called Zhang-Rice singlet [6]. When holes are introduced into the CuOe planes, the Fermi level shifts to the singlet states [7] so that the states carry low-energy excitations and transport properties. Considering these many-body effects, it is convenient to map the low-energy states onto the two-dimensional tJ model given by the Hamiltonian,
n=-t
~
(Si.Sj-¼ninj)
(c~cj + h . c . ) + J ~
(1)
(ij)
(ij)¢r
where t is the hopping parameter of electrons and c~ is the projected annihilation operator with spin ~r at site i allowing no doubly occupied sites, $~ is the spin operator at site i and n~=n~$ +n~r is the number operator of electrons at site i. The numerical calculation on finite-size clusters gives that t ~ 0.24 eV and J ~ 0 . 1 eV in La2CuO 4 [8,9]. In the next section, we study several excitation spectra and the superconductivity in the tJ model.
potential /z and the renormalized hopping parameter teef. Then, we have F(k,to)= F , ~ ( t o - Ek) and F k --zkA~/2E, where z k is the wave function renormalization constant due to the electron correlation. In Fig. 1, F(k,to) calculated by the exact diagonalization method on 16-site clusters with N = 6 and 8 is plotted by solid lines. We find that the low-energy peak near the Fermi momentum k F is pronounced whereas the peaks at higher energies are small. The sign of F(k, to) changes under rotation by ~r/2 (not shown in the figure) and the value vanishes along the
×'°-1
......
'
k=(nt2.0)
k = ( n" ft.
"
,0)
~... k=( ~, ,r/2)
3. Low-energy excitations and superconductivity
X10-
The two-dimensional tJ model given by Eq. (1) has been studied extensively by several authors. However, many problems still remain to be solved. In the following, we examine the low-energy excitation spectra and the superconducting properties by using the exact diagonalization method on 4 x 4- and ~/18 × X/18-site clusters. This unbiased numerical method offers considerable information to check theoretical approximations and various physical pictures. In order to study the Bogoliubov quasiparticle excitations, the one-particle anomalous Green's function is introduced [10]: + Eo
c 2 , Iq, o~>
(2)
where c*ktr is the Fourier transform of oil , I ,g) is the ground state of N-electrons with energy E0N. E 0 is the ground state energy chosen as E o = (EoN+2 + E~)/2. The spectral function F(k,to)= - ( 1 H r ) Im G(k, to + irl) with rl---~ + 0 and its frequency integral F~ = (g'oN+21C*T c*+~k~ I ~ ) are defined. These quantities are related to the superconducting gap function A~ when they are mapped onto the BCS theory as follows: the energy of the Bogoliubov quasiparticle is written as E, = (A 2 + ~:2)~/2 where ~ is the single-particle energy ~r = _2t~n(cos k~ + c o s k y ) - / ~ with the chemical
(b) k = ( ,r •2,0)
0
~5 0
__L
k = ( n- ,0)
5 k=(
n , tr/2)
0 o i
xlo-
2 -
,
•
4
'
~
,
•
,
5
•
(c) ,
k=( n-/2,0)
0
.~L..'L,..~
a(
1
G(k, to) = (~og+~lc2, to + in - n
.~
5
k=(,r,0)
o 5
kffi(~, 7r12)
~It
Fig. 1. Bogoliubov quasiparticle spectrum F(k, ~) for the 16-site cluster with doping between 8 and 10 holes at (a) J/t = 0.8, (b) 1.5, and (c) 2.5. The sign follows the d(x 2 - y2)_ wave symmetry and the spectrum vanishes along the kx = ky lines, k F is located at 0r/2, 0). The dotted lines show the results of the BCS theory with parameter values t~,/t = 0.55, z k = 0.8 and (a) Ad/t = 0.11, (b) 0.26, and (c) 0.50. The value 71/t = 0.15 is used.
S. Maekawa, Y. Ohta
/
Physica B 206 & 207 (1995) 641-644
kx=-z--ky
lines, indicating d(x2-y2)-wave pairing given by Ak = Ad(COS k x - c o s ky) [11]. The results in the exact diagonalization method were analysed by the BCS theory. The dotted lines show the results of the BCS theory evaluated by taking A d and teff to be adjustable parameters and choosing p. so that ~k = 0 at k F. We have examined F(k, to) of 16- and 18-site clusters in various parameter values and doping regions, and obtained that A d scales with J and has the value Ad=O.15J--O.27J in a wide region between quarter and half fillings until reaching the region of phase separation. The effective hopping parameter and renormalization factor are also strongly dependent on the J/t values and fillings. The one-particle excitation spectrum A(k,to) which gives angular-resolved electron-removal (PES) and electron-addition (BIS) spectroscopies is shown in Fig. 2. We observe that with increasing J/t a gap-like feature appears at k = (2~r/3, 0) which may be interpreted as a superconducting gap. The spin excitation spectrum S(q, to) is shown and compared with the results of BCS theory in Fig. 3. We find that S(q, to) from BCS theory with parameter values determined in
(~)
i
,
,
,
i
,
I
(a)
i
,
i
q=(*,')
~
i
,
~:~"~" ~. ..... .... :JL 1
q=(2x/3,2x/3) q=(*/3,
q=(O,O)
q=(O,O)
(b)
A. q = ( ~ ' ' )
q=(2x/3,2./3)
q=(2*/3,2./3)
q=(x/3,,/3)
q=(x/3,*/3) q=(O,O) I ~
Ili
. . . .
q=(2x/3,O)
~
~ (f)
(c)
q=(a,~-/3)
~
A
2x/3)
q=(o,o) q=(2x/3,
q=(~, ~) q=(~'~/a) q=(21r/3,2x/3) q=(x/S, x/3)
q=(~r/3,~r/3)
(';)1
~,,
(e)
q=(~,x)
--
--I
(o,o)
~ ....
q=(2x/3,0)
'
8 ~ "
r/3)
q=(x,*/3)
q=(2x/3,0)
(~,o)/
~
,,_, ....
_.•
O)
q=(O,O)
'
__
q=(x/3,~/3)
q=(~,xl3)
_._A
, |
2x 2x. I
A. q = ( ~ ' x )
q=(2n/3,2x/3)
q=(2x/3,
4
I 1..'~
"
,
"(d)]
q=(~,~/3)
q=(lr, x/3)
q=(2x/3, I
643
q=(O, O) O)
•~ 0
q=(2,/3,o) 2
4 w/t
6
Fig. 3. Spin excitation spectrum S(q, to) for the 18-site cluster with 6 holes at (a) J/t = 0.4, (b) 1.5, and (c) 2.5 is shown in the left panels. The spectrum given by the BCS theory is shown in the right panels for comparison where we use the parameter values te,/t = 0.45 and (d) Ad/t = 0.05, (e) 0.285, and (f) 0.40.
/
(b)
iI ,~JL~ g
c 2" o) 7'°__ (.,3) 2~r 2~
(g,g)
the analyses of F(k, to) is consistent with that by the exact diagonalization method; the opening of the superconducting gap is clearly observed at momentum transfer q = (2~r/3,2ar/3) and the size of the gap increases with increasing J/t value. The spectral weight distributions are also in overall agreement with those of BCS theory.
(o,o) •
.
t
-5
. . . .
I
0 ~/t
. . . .
i
5
,
,
Fig. 2. One-particle spectral function A(k, to) for the 18-site cluster with 6 holes at (a) J/t = 0.4 and (b) 1.0. to is measured from the 6-hole ground-state energy, k is shown in the panel, and the value ~l/t = 0.05 is used.
4.
Conclusion
We have studied the electronic structure and superconductivity in high-Tc superconducting cuprates which are typical strongly correlated electron systems. The cuprates have a two-dimensional character and
644
S. Maekawa, Y. Ohta / Physica B 206 & 207 (1995) 641-644
strong quantum effects of spins. The low-energy states were mapped onto the tJ model, in which the superconductivity was examined. It was shown that the superconductivity in the tJ model is in accord with that in BCS theory. The pairing occurs in the d(x 2 - y2)_ wave channel and the energy gap has a size Aa = 0 . 1 5 J - 0 . 2 7 J in a wide region between quarter and half fillings. The large J value is caused by the fact that the levels of holes on Cu and O ions are close in energy, and suggests the high-Tc superconductivity. These characteristic features of the electronic structure as well as the variety of the materials provide a unique opportunity for further study of strong electron correlations.
Acknowledgements The results presented in Section 3 are based on the work in collaboration with R. Eder and T. Shimozato. This work was supported by Priority-Areas Grant from the Ministry of Education, Science, and Culture of Japan. S.M. acknowledges financial support by the Kurata Foundation. Computations were partly carried out in the Computer Center of Institute for Molecular Science, Okazaki National Research Institutes.
References [1] J.O. Bednorz and K.A. Miiller, Z. Phys. B 64 (1986) 189. [2] See, for example, H. Fukuyama, S. Maekawa and A.E Malozemoff (eds.), Strong Correlation and Superconductivity, Springer Series in Solid State Sciences 89 (Springer, Berlin, 1989). [3] See, for example, S. Maekawa and M. Sato (eds.), Physics of High Temperature Superconductors, Springer Series in Solid State Sciences 106 (Springer, Berlin, 1992). [4] Y. Ohta, T. Tohyama and S. Maekawa, Phys. Rev. B 43 (1990) 2968. [5] Y. Ohta, T. Tohyama and S. Maekawa, Phys. Rev. Lett. 66 (1991) 1228. [6] F.C. Zhang and T. M. Rice, Phys. Rev. B 37 (1988) 3759. [7] T. Tohyama and S. Maekawa, Physica C 191 (1992) 183. [8] L.H. Tjeng, H. Eskes and G.A. Sawatzky, in Ref. [2], p. 66. [9] S. Maekawa, J. Inoue and T. Tohyama, in Ref. [2], p. 66. [10] Y. Ohta, T. Shimozato, R. Eder and S. Maekawa, Phys. Rev. Lett., 73 (1994) 324. [11] E. Dagotto and J. Riera, Phys. Rev. Lett. 2(1 (1993) 682.
ELSEVIER
Electron correlations in high-T c superconductors S. Maekawa*, Y. Ohta Department of Applied Physics, Nagoya University, Nagoya 464-01, Japan
Abstract The electronic structure and excitation spectra in high-Tc superconducting cuprates are examined by using the exact diagonalization method on finite-size clusters. The low-energy electronic states are mapped onto the two-dimensional tJ model. Several excitation spectra in the tJ model are derived as functions of electron concentration and parameter (J/t) value where J is the antiferromagnetic exchange interaction between Cu spins and t is the hopping parameter of charge carriers introduced into the cuprates. The spectra are found to be in accord with those calculated in BCS theory with d(x 2- y2)-wave pairing.
1. Introduction
2. Crystal structure and electronic structure
The discovery of high-Tc superconducting cuprates [l] has renewed the physics of strongly correlated electron systems. The cuprates are unique among the systems [2,3]. The parent compounds of the superconductors are antiferromagnetic insulators of chargetransfer type. Upon doping of carriers, they have both metallic and ionic characters which are brought about by the layered structure. The physical properties are controlled by carrier concentration. In addition, quantum effects of spins are enhanced by the two-dimensional structure and the spin value (S = 1/2) of Cu ions. Here, we examine the electronic structure and derive the low-energy excitations. We utilize the exact diagonalization method on finite-size clusters. The low-energy electronic states are mapped onto the twodimensional tJ model. Several excitation spectra are obtained in the tJ model. We find that the spectra are in accord with those of BCS theory with d(x 2 - y 2 ) wave pairing. The characteristic features of the cuprates among strongly correlated electron systems are discussed.
The cuprates have a layered structure. Let us consider La 2 xSrxCuO4, as an example. Doping of hole-carriers in the CuO 2 planes is made by replacing La 3÷ ions by Sr z÷ ions which are located between the CuO 2 planes. The doping is possible because the cuprates are ionic crystals; i.e., the CuO z planes with negative charge carry doped holes with positive charge. We note that the structure of the CuO 2 planes is stable against the doping. It has been observed that the Madelung potentials acting on the doped holes govern the superconducting transition temperature [4]. The cuprates are charge-transfer type compounds, and the energy difference between Cu and O ions, A, is small. Therefore, the antiferromagnetic exchange interaction between neighboring Cu spins, J, is strong 4 3 as is seen in the perturbational expression, J = tpd/A , where tpd denotes the hopping parameter of holes between Cu and O ions and Ud the Coulomb repulsive interaction on a Cu site. In addition, quantum effects of spins are enhanced in the cuprates by the spin value (S = 1/2) of Cu ions in two-dimensions. As discussed in the next section, these are of crucial importance for high-Tc superconductivity. It has been shown in a numerical study [5] that the charge transfer gap is given by the energy difference
* Corresponding author.
0921-4526/95/$09.50 ~) 1995 Elsevier Science B.V. All rights reserved SSD1 0921-4526(94)00544-3
S. Maekawa, Y. Ohta / Physica B 206 & 207 (1995) 641-644
642
between the upper Hubbard band caused by the correlation U d and the singlet state made by the holes on O and Cu sites, the so-called Zhang-Rice singlet [6]. When holes are introduced into the CuOe planes, the Fermi level shifts to the singlet states [7] so that the states carry low-energy excitations and transport properties. Considering these many-body effects, it is convenient to map the low-energy states onto the two-dimensional tJ model given by the Hamiltonian,
n=-t
~
(Si.Sj-¼ninj)
(c~cj + h . c . ) + J ~
(1)
(ij)
(ij)¢r
where t is the hopping parameter of electrons and c~ is the projected annihilation operator with spin ~r at site i allowing no doubly occupied sites, $~ is the spin operator at site i and n~=n~$ +n~r is the number operator of electrons at site i. The numerical calculation on finite-size clusters gives that t ~ 0.24 eV and J ~ 0 . 1 eV in La2CuO 4 [8,9]. In the next section, we study several excitation spectra and the superconductivity in the tJ model.
potential /z and the renormalized hopping parameter teef. Then, we have F(k,to)= F , ~ ( t o - Ek) and F k --zkA~/2E, where z k is the wave function renormalization constant due to the electron correlation. In Fig. 1, F(k,to) calculated by the exact diagonalization method on 16-site clusters with N = 6 and 8 is plotted by solid lines. We find that the low-energy peak near the Fermi momentum k F is pronounced whereas the peaks at higher energies are small. The sign of F(k, to) changes under rotation by ~r/2 (not shown in the figure) and the value vanishes along the
×'°-1
......
'
k=(nt2.0)
k = ( n" ft.
"
,0)
~... k=( ~, ,r/2)
3. Low-energy excitations and superconductivity
X10-
The two-dimensional tJ model given by Eq. (1) has been studied extensively by several authors. However, many problems still remain to be solved. In the following, we examine the low-energy excitation spectra and the superconducting properties by using the exact diagonalization method on 4 x 4- and ~/18 × X/18-site clusters. This unbiased numerical method offers considerable information to check theoretical approximations and various physical pictures. In order to study the Bogoliubov quasiparticle excitations, the one-particle anomalous Green's function is introduced [10]: + Eo
c 2 , Iq, o~>
(2)
where c*ktr is the Fourier transform of oil , I ,g) is the ground state of N-electrons with energy E0N. E 0 is the ground state energy chosen as E o = (EoN+2 + E~)/2. The spectral function F(k,to)= - ( 1 H r ) Im G(k, to + irl) with rl---~ + 0 and its frequency integral F~ = (g'oN+21C*T c*+~k~ I ~ ) are defined. These quantities are related to the superconducting gap function A~ when they are mapped onto the BCS theory as follows: the energy of the Bogoliubov quasiparticle is written as E, = (A 2 + ~:2)~/2 where ~ is the single-particle energy ~r = _2t~n(cos k~ + c o s k y ) - / ~ with the chemical
(b) k = ( ,r •2,0)
0
~5 0
__L
k = ( n- ,0)
5 k=(
n , tr/2)
0 o i
xlo-
2 -
,
•
4
'
~
,
•
,
5
•
(c) ,
k=( n-/2,0)
0
.~L..'L,..~
a(
1
G(k, to) = (~og+~lc2, to + in - n
.~
5
k=(,r,0)
o 5
kffi(~, 7r12)
~It
Fig. 1. Bogoliubov quasiparticle spectrum F(k, ~) for the 16-site cluster with doping between 8 and 10 holes at (a) J/t = 0.8, (b) 1.5, and (c) 2.5. The sign follows the d(x 2 - y2)_ wave symmetry and the spectrum vanishes along the kx = ky lines, k F is located at 0r/2, 0). The dotted lines show the results of the BCS theory with parameter values t~,/t = 0.55, z k = 0.8 and (a) Ad/t = 0.11, (b) 0.26, and (c) 0.50. The value 71/t = 0.15 is used.
S. Maekawa, Y. Ohta
/
Physica B 206 & 207 (1995) 641-644
kx=-z--ky
lines, indicating d(x2-y2)-wave pairing given by Ak = Ad(COS k x - c o s ky) [11]. The results in the exact diagonalization method were analysed by the BCS theory. The dotted lines show the results of the BCS theory evaluated by taking A d and teff to be adjustable parameters and choosing p. so that ~k = 0 at k F. We have examined F(k, to) of 16- and 18-site clusters in various parameter values and doping regions, and obtained that A d scales with J and has the value Ad=O.15J--O.27J in a wide region between quarter and half fillings until reaching the region of phase separation. The effective hopping parameter and renormalization factor are also strongly dependent on the J/t values and fillings. The one-particle excitation spectrum A(k,to) which gives angular-resolved electron-removal (PES) and electron-addition (BIS) spectroscopies is shown in Fig. 2. We observe that with increasing J/t a gap-like feature appears at k = (2~r/3, 0) which may be interpreted as a superconducting gap. The spin excitation spectrum S(q, to) is shown and compared with the results of BCS theory in Fig. 3. We find that S(q, to) from BCS theory with parameter values determined in
(~)
i
,
,
,
i
,
I
(a)
i
,
i
q=(*,')
~
i
,
~:~"~" ~. ..... .... :JL 1
q=(2x/3,2x/3) q=(*/3,
q=(O,O)
q=(O,O)
(b)
A. q = ( ~ ' ' )
q=(2x/3,2./3)
q=(2*/3,2./3)
q=(x/3,,/3)
q=(x/3,*/3) q=(O,O) I ~
Ili
. . . .
q=(2x/3,O)
~
~ (f)
(c)
q=(a,~-/3)
~
A
2x/3)
q=(o,o) q=(2x/3,
q=(~, ~) q=(~'~/a) q=(21r/3,2x/3) q=(x/S, x/3)
q=(~r/3,~r/3)
(';)1
~,,
(e)
q=(~,x)
--
--I
(o,o)
~ ....
q=(2x/3,0)
'
8 ~ "
r/3)
q=(x,*/3)
q=(2x/3,0)
(~,o)/
~
,,_, ....
_.•
O)
q=(O,O)
'
__
q=(x/3,~/3)
q=(~,xl3)
_._A
, |
2x 2x. I
A. q = ( ~ ' x )
q=(2n/3,2x/3)
q=(2x/3,
4
I 1..'~
"
,
"(d)]
q=(~,~/3)
q=(lr, x/3)
q=(2x/3, I
643
q=(O, O) O)
•~ 0
q=(2,/3,o) 2
4 w/t
6
Fig. 3. Spin excitation spectrum S(q, to) for the 18-site cluster with 6 holes at (a) J/t = 0.4, (b) 1.5, and (c) 2.5 is shown in the left panels. The spectrum given by the BCS theory is shown in the right panels for comparison where we use the parameter values te,/t = 0.45 and (d) Ad/t = 0.05, (e) 0.285, and (f) 0.40.
/
(b)
iI ,~JL~ g
c 2" o) 7'°__ (.,3) 2~r 2~
(g,g)
the analyses of F(k, to) is consistent with that by the exact diagonalization method; the opening of the superconducting gap is clearly observed at momentum transfer q = (2~r/3,2ar/3) and the size of the gap increases with increasing J/t value. The spectral weight distributions are also in overall agreement with those of BCS theory.
(o,o) •
.
t
-5
. . . .
I
0 ~/t
. . . .
i
5
,
,
Fig. 2. One-particle spectral function A(k, to) for the 18-site cluster with 6 holes at (a) J/t = 0.4 and (b) 1.0. to is measured from the 6-hole ground-state energy, k is shown in the panel, and the value ~l/t = 0.05 is used.
4.
Conclusion
We have studied the electronic structure and superconductivity in high-Tc superconducting cuprates which are typical strongly correlated electron systems. The cuprates have a two-dimensional character and
644
S. Maekawa, Y. Ohta / Physica B 206 & 207 (1995) 641-644
strong quantum effects of spins. The low-energy states were mapped onto the tJ model, in which the superconductivity was examined. It was shown that the superconductivity in the tJ model is in accord with that in BCS theory. The pairing occurs in the d(x 2 - y2)_ wave channel and the energy gap has a size Aa = 0 . 1 5 J - 0 . 2 7 J in a wide region between quarter and half fillings. The large J value is caused by the fact that the levels of holes on Cu and O ions are close in energy, and suggests the high-Tc superconductivity. These characteristic features of the electronic structure as well as the variety of the materials provide a unique opportunity for further study of strong electron correlations.
Acknowledgements The results presented in Section 3 are based on the work in collaboration with R. Eder and T. Shimozato. This work was supported by Priority-Areas Grant from the Ministry of Education, Science, and Culture of Japan. S.M. acknowledges financial support by the Kurata Foundation. Computations were partly carried out in the Computer Center of Institute for Molecular Science, Okazaki National Research Institutes.
References [1] J.O. Bednorz and K.A. Miiller, Z. Phys. B 64 (1986) 189. [2] See, for example, H. Fukuyama, S. Maekawa and A.E Malozemoff (eds.), Strong Correlation and Superconductivity, Springer Series in Solid State Sciences 89 (Springer, Berlin, 1989). [3] See, for example, S. Maekawa and M. Sato (eds.), Physics of High Temperature Superconductors, Springer Series in Solid State Sciences 106 (Springer, Berlin, 1992). [4] Y. Ohta, T. Tohyama and S. Maekawa, Phys. Rev. B 43 (1990) 2968. [5] Y. Ohta, T. Tohyama and S. Maekawa, Phys. Rev. Lett. 66 (1991) 1228. [6] F.C. Zhang and T. M. Rice, Phys. Rev. B 37 (1988) 3759. [7] T. Tohyama and S. Maekawa, Physica C 191 (1992) 183. [8] L.H. Tjeng, H. Eskes and G.A. Sawatzky, in Ref. [2], p. 66. [9] S. Maekawa, J. Inoue and T. Tohyama, in Ref. [2], p. 66. [10] Y. Ohta, T. Shimozato, R. Eder and S. Maekawa, Phys. Rev. Lett., 73 (1994) 324. [11] E. Dagotto and J. Riera, Phys. Rev. Lett. 2(1 (1993) 682.