Theory of Nested d-p Model with Strong Repulsion Graduate School

Progress of Theoretical Physics, Vol. 91, No.3, March 1994

Theory of Nested d-p Model with Strong Repulsion

Taro MATSUNAMI and Minoru KIMURA

Graduate School of Natural Science and Technology Kanazawa University, Kanazawa 920-11

(Received September 22, 1993)

453

The nested d-P model with a strong repulsion on d-sites is studied. Starting with Green functions of a normal paramagnetic state in an isolated atomic limit (transfer->0), expansions with respect to residual interactions are performed and scaling equations are solved. The attraction between d-electrons mediated by doped holes plays a dominant role. The possible orders in onedimension are of charge-density wave and singlet superconductivity for small doping, and of charge and spin-density wave for large doping, respectively. In quasi-one-dimension only charge-density wave establishes three-dimensional order and the fluctuation of singlet superconductivity develops in short range.

§ 1. Introduction

Recently much attention is paid to the cupper-oxide high-temperature superconductivity. The electronic structure in the CuO-plane has been extensively studied on the basis of the two-dimensional d-P model.ll However it is safe to say that so far the common understanding has not been established yet. The difficulty comes from the strong Cu-site and Cu-0 intersite repulsion, which prevents the application of ordinary perturbation technique. One approach is the nonperturbative method such as the Fermi liquid2H> or the 1/N-expansion.5>

It is widely accepted that the nesting structure of the electronic spectrum is a conspicuous feature in the cupper-oxide superconductivity. We concentrate on this property so that we restrict ourselves to the one-dimensional and quasi-onedimensional (Q1D) nested d-P modeL The central idea of the present work is given as follows. We take the d-p intersite transfer divided by the d-P energy separation as a:sm:aU parameter. The repulsion on the d-site is treated in strong coupling limit and is included in the starting Green function in atomic limit.6>'7> The residual interactions, such as neighboring repulsion between d and p sites, can be scaled by the small parameter of the transfer and reduced to weak ones. This reduction enables us to apply the usual theory of multiplicative renormalization group (MRG) based on the perturbation expansion.8>

The Hamiltonian of the d-P modelll,s> is defined by

(1·1)

where

(1·2)

Dow

nloaded from https://academ

ic.oup.com/ptp/article/91/3/453/1902087 by guest on 24 February 2022

454 T. Matsunami and M. Kimura

(1·3)

Hz= U/J:,dtrd;rdil d;J i

(1·4)

and

(1·5)

in which the operator Pur (dur) denotes the annihilation of a P-(d-) hole at i-th site with

spin 6, and <ij> means a pair of nearest neighbor sites. Hereafter we sometimes refer to electrons instead of holes for convenience.

In § 2 a method of expansion with respect to the small parameter is formulated.

We assume that (i) the on-site repulsion, intersite repulsion and d-P energy level separation are substantially greater than the intersite transfer, i.e., Ud, Upd, Ep- Ed'P tpd,

(ii) P-sites are empty, d-sites are half-filled, and holes are doped. The essential point is that the intersite transfer scaled by the d-P level separation

(1·6)

is taken as an expansion parameter. The couplings are effectively scaled with respect to the dimensionless transfer given by Eq. (1· 6) so that the couplings are reduced as Dd~ ()( t 4

) and DPd~ CJ(tz).

In §§ 3 and 4 the one-dimensional case of the model is studied by the help of the

expansion and MRG. The greatest contribution comes from a superexchange process in which a backward scattering between d -electrons is mediated by a pair of virtual P-electrons. The fixed points of the effective couplings are sensitive to the

amount of doped hole. In case of the small doping the effective interactions Dd and DPd are scaled to infinity in attractive region. In case of the large doping Dd is scaled

to infinity in repulsive region while Upd remains in weak-coupling region. Possible ordering states are studied by examining the linear response function. Two cases of the ordering states are found, corresponding to the two different behaviors of the fixed

points of the effective interactions. In case of small doping the possible ordering state is of the charge-density wave and singlet superconductivity, where the latter is weaker. In case of the large doping the possible ordering state is of the charge and

spin density wave. In§ 5 the model is extended to the quasi-one dimensional system.9

> We assume

that the d-p intersite repulsion UPd acts between chains. The interchain coupling causes a drastic change to the structure of fixed point. The fixed point of Upd in the weak-coupling region becomes unstable and is scaled to the infinity in attractive

region passing through a sharp crossover in weak-coupling region. The possible ordering state is of the charge-density wave and singlet superconductivity where the

latter is weaker irrespective of the amount of the doping. In § 6 the weak-coupling limit is studied, to compare the strong-coupling case.

According to the hypothesis of the Luttinger liquid10> in one-dimensional electronic

Dow



Theory of Nested d-P Model with Strong Repulsion 455

system, the behavior is universal irrespective of the magnitude of interaction. In the weak-coupling limit the model is reduced to the ordinary hybridized two-band with small on-site and intersite repulsions. It is treated with the help of the conventional g-ology.8l,lll,12l The phase diagram is straightforwardly obtained.

In § 7 concluding remarks are given.

§ 2. Formulation of expansion with respect to t

The Green function in atomic limit of the d -site with the on-site repulsion Ud are obtained following Hubbard's technique6 l.7 l as

Gcrd(Ol(i€)=( 1- ~ nd)[i€-Ed]-1+ ~ nd[i€-Ed- Ud]-1, (2·1)

where nd is the density of the d -hole. We consider that we take the normal state where translational and magnetic symmetries are not broken as a starting point. The energy levels of the d -site are split because of the strong on-site repulsion. Within the range of energy and temperature under consideration the upper level is irrelevant so that we discard the second term of the r.h.s. in Eq. (2·1). The Green function of P-site is given by

(2·2)

The· intersite transfer tpd hybridizes the isolated Green functions in the atomic limit. The hybridized Green function is written as

- (. ) ( G/(i€, k) Gld(ic, k)) Gcr Z€, k = G-dP(z·€, k) d( ) '

u Gcr i€, k

where

and

The dispersion of the hybridized band is determined by

{f)cr(i€, k)-1=0.

It gives

cp(k)= ~ [ cp+ td+/ iP+ btpd(k)2 ] ,

(2·3)

(2·4)

(2·5) .

(2·6)

(2·7)

(2·8)

Dow




(2·9) .

where

LJ=Ep-€d (2·10)

and

(2·11)

The amplitude of the hybridized Green function is expanded with respect to t up to LJ ( t 2

) as

where

and

P<1(i€, k)=[i€- €p(k)]-I'

D<1(i€, k)=[i€- €d(k)]-I'

is used for convenience.

-bt(k) ) . b-b2 t(k)2 Da{Z€, k)'

(2·12)

(2·13)

(2·14)

(2·15)

We remind that only the electrons in a vicinity of the Fermi surface play a dominant role so that t (k) is represented by the value at the Fermi surface, tF= t(kF). We consider the hole-doping case, where the lower d-band is fully

occupied, thus the second term of the r.h.s. of Eq. (2 ·12) is irrelevant, hence,

The order of the elements of the Green function Eq. (2 ·16) is given by

Gl(i€, k)~ LJ( t 0),

G,f'd(i€, k)' GiP(i€, k)"-' LJ( tr)'

G<1d(iE, k)~ LJ( [2).

(2·16)

(2 ·17)

(2·18)

(2·19)

The parameter t F is t times a factor of order of unity therefore it is able to use t instead of t F as an expansion parameter.

Perturbation corrections appear as products of couplings and Green functions as follows,

(2·20)

We note that the products Eq. (2·20) are invariant up to LJ ( t 2) under the simultaneous

Dow




scaling of the couplings and the Green function Eq. (2 ·16) as

(2. 21)

"' . _ ( G/(ic,k) b-1 FF-1Gid(ic,k)) Ga{zc, k)= b-1 '[F- 1GiP(i€, k) b-2 FF-2 Gi(ic, k) · (2·22)

Now the strong couplings are reduced to weak ones as [Jd~ CJ( t 4) and Upd~ ()( P).

All of the elements of the rescaled Green functions Eq. (2 · 22) are CJ ( t 0) as

"' . (1- b f F 2 1) . G(zc, k)= 1 1

P(J(zc, k). (2·23)

For later convenience we define dimensionless couplings,

u1IJ = VFUPd(2kF) , u12J = VFUPd( t) (2·24)

and

(2·25)

where the superscripts (1) and (2) mean backward and forward scattering, respectively,8> VF is the density of states at the Fermi surface. The couplings are illustrated in Fig. 1.

The Fermi momentum depends on the amount of hole-doping as

(2·26)

in which the density of doped hole is defined by

o=nd-'1. (2·27)

Here we estimate the magnitude of the bare couplings. The intersite repulsion defined by Eq. (2·24) depends on the density of electrons through momentum dependence. For simplicity we neglect the momentum dependence of the density of states VF, reduction parameter b, and the dimensionless transfer t F, then the intersite couplings are given by

u1IJ~ Ccos2kF,

u12J~ C,

where C is a positive constant.

(2·28)

(2·29)

The bare d -site repulsion has been absorbed in the Green function in the atomic limit therefore bare couplings are taken zero as

' /

~d

Fig. 1. The bare scattering process. The solid (dashed) line denotes the .electron which has the momentum near kF(.- kF ). The black (white) vertex means the d-site (p-site).

Dow




(2·30)

§ 3.. Renormalization up to () ( t 0)

In order to elucidate the crucial role played by the superexchange process, we first examine the expansion with respect to t within the accuracy up to the lowest order LJ ( t 0

). For vertex correction up to the present order only the superexchange process logarithmically contributes, in which a backward scattering between d -electrons is mediated by a pair of virtual P-electrons as shown in Fig. 2.

It gives

-(u<ll)2ln 2yW Pd TCT ' (3·1)

where W is a cutoff, T is the temperature, and y is Euler's constant. By the help of MRG we obtain the scaling equation,

diid(!) -( (1))2 X dX - Upd , (3·2)

where xis a dimensionless temperature x= T/To, and the tilde means variables. The reference temperature may be identified as To~ tpdo.

12l The rest of couplings, ud<

2l,

u1~, and u12J, are unaffected under scaling. With the initial condition Eq. (2·30) at x=l the solution of Eq. (3·2) is obtained

as

(3·3)

The d-site backward coupling ud<rJ tends to minus infinity as temperature is lowered to zero unless u1~=0. Thus the d-site interaction mediated by the P-electron superexchange is effectively attractive and slowly diverges at zero temperature.

Now we study the instability induced by this attractive interaction by the linear response function defined as

(3·4)

where index i identifies the kind of order parameter. The response function would diverge as the system tends to order.

In one-dimensional nesting problem we must consider four kinds of instability:8l

charge-density wave (CDW), spin-density wave (SDW), singlet superconductivity (SS)

Fig. 2. The logarithmic scattering process up to C! ( {0).

and triplet superconductivity (TS). The respective order parameters are defined as follows.

QCDW(k)= ':E.d:cdq+k+2kF<J) (3 • 5) q<J

(3·6)

Dow




(3·7)

(3·8)

Applying MRG we obtain the following scaling equations for the response func-tions,

aln_fCDW X ax 2ud<1l,

aln_fSDW X ax 0'

aln%55 u (1)

X ax d '

alnxTs X ax -u/1l,

where x is defined by

-- ax x- alnT ·

(3·9)

(3·10)

(3·11)

(3 ·12)

(3·13)

No correction to the SDW response arises within the present approximation. The scaling equations (3·9)~(3·12) are solved with ud<o given by Eq. (3·3). The

response functions behave as temperature is lowered as

(3·14)

(3·15)

(3·16)

The CDW and SS responses diverge at zero temperature where the CDW response is stronger than the SS. The TS response dwindles. The possible ordering states are of CDW and SS where the former is more favorable. We note that, in the present approximation, the behavior of all the ordering states is independent of the hole doping-rate.

§ 4. Renormalization up to (') ( tz)

We proceed to the next approximation () ( t 2). In this order all of the couplings,

udOl, ud<2l, u11J and u12J, are affected by the renormalization. In contrast to the

renormalization up to () ( t 0) the results depend essentially on the hole doping-rate.

Each process which logarithmically contributes to the renormalization is shown in Fig. 3.

The scaling equations are derived as

Dow




-7-~

~--)---

--)- --r ) ~~--

rr -~:-~-!-~-)-' ) 1 ) -7-w->--

.:.tr?\· > )

-~9~-- -:-iEi~--) 4 > ',e

-~~r) --}-- -~-~>--)- > ) -'->

- - - ~----- -...:..- ~--- -

Fig. 3. The logarithmic scattering processes up to E! ( t 2).

(4·1)

aud<2> __ _l_( -<1l)2 -<2>+_1_ -<1>( -<2l)2 _ _l_( -<2>)3

X ax - 2 U Pd U Pd 2 U Pd U Pd 2 U Pd , (4·2)

a -(1) Upd -( -(1))2+ -(1) -(2)

X ax - Upd UpdUPd, (4·3)

au12J -l( -n>)2 X ax -2 Upd . (4 ·4)

We numerically analyze the scaling equations under the initial conditions given in Eqs. (2 · 28) ~ (2 · 30 ). Figure 4 shows the scaling trajectory of Upd on the u1~- u12J plane.

Dow




0.1

-0.1

Fig. 4. The trajectory of Upd on the uJ,~-uJ,2J plane. a: 8<8c. b: 8>8c.

It is seen that there are two distinct types of flows of trajectory of UPd. The branch into the two flows depends on the doping rate, for instance the critical doping O'c~0.48 for C=O.l. In case of the small doping (o< oc) the coupling is scaled to infinity in attractive region passing through· a sharp crossover about the weak-coupling region. In case of the large doping (o>oc) the coupling is scaled to the weak fixed point lying on the u12J-axis.

Corresponding to the two types of flows of trajectory of Upd, there are two distinct types of the flows of trajectory of ud. In case of the small doping Ud

attractively diverges at finite temperatures.· In case of the large doping ud is scaled to the repulsion and diverges to the positive infinity at zero temperature. The trajectory of the Ud on the ud.(l)-ud(2l

plane is shown in Fig. 5. The transition to ordered states

from the normal state is recognized by

-0.5

0.5 (1) ud

0

a

-0.5

d 0.5

c

b

Fig. 5. The trajectory of ud on the ud<'>-ud<2> plane. Initial conditions. a: -0.1 < uJ,~<: -0.073, uJ,ZJ =0.1. b: uJ,~= -0.065, u),2J=O.l. c:. uJ,~ = -0.06, uj,2J=0.1. d: uJ,~= -0.055, uj,2J=0.1. Hole-doping rate. a: CJ<CJc. b~d: CJ>CJc.

0.---.---.----r---.----.---.

><: -50 E

CDW

(SS)

-- NORMAL ::,....~,,

' ' ' \ \ \ \ I I I I I I I I I I

- 1 0°o~--~-o~.=2--~~o~.4~~--~o.6

8

Fig. 6. The hole-doping dependence of the critical temperature. We put C=O.l in Eqs. (3·3) and (3·4). The solid line is of the one-dimensional case. The dashed line is of the quasi-one· dimensional case, where the ratio of the interchain and intrachain couplings is 0.01 and N = 5. The ordered state is of CD\Y and SS where the latter is weaker, in one-dimesnsional case. In QlD case the ordered state is only of CDW.

the divergence of the effective couplings, where the responses simultanyously diverge. Figure 6 shows the phase diagram in the o- T plane. ·

The responses behave in characteristically different way according to the two distinct sets of fixed points of the effective couplings.

Dow




Table I. The fixed points of effective couplings. In case of small doping the divergence of ud<1

> is dominant.

small doping (8< 8c) large doping (8>8c)

uJ,~ -CXJ 0

uJ,?J -CXJ positive definite

Ud(l) -CXJ negative definite

Ud(2 ) -CXJ +oo

In case of the small doping (o< oc) udnJ plays a dominant role at low temperatures as is seen in Fig. 5. It diverges at finite temperatures (Fig. 6). In this case Eqs. (3·9)~(3·12) show that the possible ordering state is of CDW and SS where the former is more favorable.

In case of the large doping (o>oc) ud<2J plays a dominant role at sufficiently low

temperatures as shown in Fig. 5. Its behavior is governed by the fixed point of u1?;1 as

The scaling equations for the responses are obtained as follows,

alnf"CDW X ax - ud<2l,

alnxSDW X ax - (2) -ud ,

alnxss X ax - (2)

Ud ,

alnxTS X ax - (2)

Ud .

The response functions behave as temperature is lowered as

;rcow=exp{} (u1?;1)3(lnxY},

;rsow=exp{} (u1?;1)3(lnxY},

X55 =exp{-} (u1?;1)3(lnxY},

xTS=exp{-! (ii1?;1)3(lnx)2}.

(4·5)

(4 ·6)

(4 ·7)

(4·8)

(4·9)

(4·10)

(4·11)

(4·12)

(4·13)

Since the value at the fixed point of u1?;1 is positive, (see Table I) the CDW and SDW responses diverge at zero temperature thus the possible ordering states are of CDW and SDW.

The results in this section are summarized as follows. The dominant interaction

Dow




changes its sign according to the amount of hole-doping. In case of small doping (o< oc) the attractive ud<1

> is dominant, where the possible ordering states are of CDW and SS. In case of large doping (o>oc) the repulsive ud<2

> is dominant, where the possible ordering states are of CDW and SDW. Aside from CDW we see that SS turns to be SDW with increasing doping-rate.

§ 5. Extension to quasi-one dimension

In order to approach a more realistic case we extend the present model to Q1D system9

> including the interchain-coupling. We assume that the d-P intersite repulsion acts between one-dimensional chains but is substantially smaller than the intrachain coupling, and there is no transfer between chains. The analysis in the preceding sections are extended straightforwardly to Q1D.

The scaling equations up to () ( f2) involving the interchain coupling are obtained as

oJ-(1) N-1

X~=2u- (l)U-(1) - u- (2)U-(l) +(1-2b t- 2) .._.., cu-(1) ·)2 o:1 d Pd,O d Pd,O F £..J Pd,J , ~ j~

(5·1)

OU}2) _ 1 N-

1 -(1) )2 -(2) 1 -(1) ( -(2) 2 1 -(2) N-

1 - 2) 2 x-o:l-----2 ~ ( U Pd,J U Pd,J +-2 U Pd,O U Pd,o) --2 U Pd,O ~ ( U 1d,J) ,

uX J=O J=O (5·2)

ou1~.i _ N-1

- 1) -<1> - 1) -<2> x-o:1--- ~ Ubd,i-;Upd,;+ Ubd,iUpd,i, uX J=O

(5·3)

ou12J.i -lc -n> )2 X OX - 2 U Pd, i , (5·4)

where the subscript i in uJ;~.; and uJ;2J,; represents the distance between chains. The scaling equations (5·1)~(5·4) are numerically analyzed for N=21 under the

periodic condition. The tentatively chosen number of chains is considered to be sufficiently large.9

> The bare interchain interaction is assumed to reach only nearest neighbor chains. Through the scaling, however, the interchain interaction between far neighbors gradually develops down to the critical temperature and the true long-range transverse coupling grows abruptly just above the three-dimensional order as shown by the renormalization trajectories of the couplings in Q1D system in Fig. 7.

The scaling trajectory of the intersite coupling Upd which is renormalized under the interchain interaction is shown in Fig. 8. Comparing with the one-dimensional case as is shown in Fig. 4, fixed points in the weak-coupling region in one-dimensional case become unstable. The coupling is scaled to attractive infinity passing through a sharp crossover in the weak coupling region.

The scaling equation for the response functions is easily extended to Q1D case as,9

>

OX-.CDW X ' ox (5·5)

Dow




0 r--------------

-0.2

-0.4

1 I I I I I I I I I I

: 10-10

X

Fig. 7. The temperature-dependence of the intrachain coupling uJ,~.o (solid line) and transverse coupling uJ,~.w (dashed line). The ratio of the interchain and intrachairi couplings is 0.01.

olnx--SDW X O; 0,

X oln_f?5 _ -(1)

OX Ud,O,

where the initial condition at x=1 is

. {1 if i=O, .f/ = 0 otherwise .

0.1

-0.1 0

-0.1

Fig. 8. The trajectory of intrachain coupling Upd,o

in the u!lJ.o-uJ,>;i,o plane. The ratio of the interchain and intrachain couplings is O.L

(5·6)

(5·7)

(5·8)

(5·9)

The possible three-dimensional ordered state of Q1D is always of CDW. The SS

instability develops only in a single chain but does not establish three-dimensionally. The region of ordered state in the o- T plane is enlarged compared with the onedimensional case as shown by the dashed line in Fig. 9.

§ 6. Weak coupling case

Now we turn to the weak-coupling limit. In contrast to the previous sections we assume that the couplings are substantially small compared with the intersite transfer, i.e., Ud, UPd<. tpd. The kinetic parts of the Hamiltonian, Eqs. (1· 2) and (1· 3), are diagonalized as

(6·1)

where

Dow



Theory of Nested d-p Model with Strong Repulsion 465

in which

-il+J.tF+tpd(k)Z il+Jil2 +tpd(k)2.

(6·2)

(6·3)

(6·4)

(6·5)

The Fermi surface may locate in the hybridized P-band or in d -band so that we must study two cases separatE:;ly. When the Fermi surface lies in the P-band, the couplings are given by

(6·6)

and

(6·7)

where suffixes 1 and 2 mean the backward and forward scatterings respectively. The possible ordering state in the grgz plane is well-studied.8

> It is straightforwardly transformed to the Ud- UPd plane as shown in Fig. 9. The boundary is determined by lines lr and lz given by

lr: Udsin2 8kF+4 UPdcoskFcos2 8kF=O, (6·8)

=0. (6·9)

When the Fermi surface lies in the d -band, the couplings are given by

gl = Udcos4 8kF ud

+4 UPdcoskF:cos2 8kFsin2 8kF (6·10)

and

gz= Udcos4 8kF+4Updcos2 8k~in2 8kF. (6·11)

The phase boundaries are given by

lr: Udcos 2 8k.F+4 UPdcoskFsin2 8kF=O, (6·12)

(6·13)

Fig:'9. The possible ordering states in the ud. UPd plane. I: SDW and CDW. II: CDW and (SS). III: SS and (CDW). IV: TS and SS. Orders in parenthesis are weaker ones. The effective couplings in domains I and IV are scaled to strong ones while the effective couplings in domains II and III are scaled to weak ones.

Dow




The non-doped situation of the strong-coupling case corresponds to the quarter filled d-P hybridized band. As is seen from Fig. 9 in case of the d-hole doping the ordering state about the plus Upd-axis is CDW (SS). Therefore the intersite coupling UPd enhances the charge instability, which qualitatively agrees with the results of the strong-coupling case.

§ 7. Concluding remarks

In the present paper we studied the one-dimensional and QlD d-p model with strong repulsion. We performed expansions with respect to a dimensionless transfer i. The d-band is mixed with P-band with the factor of i at the Fermi surface.

When electrons interact at d-sites, tz is multiplied to the matrix element therefore the couplings between electrons are effectively reduced by this factor.

In the present model the backward attraction between d-holes mediated by a pair of virtual P-electron superexchange plays a dominant role. Physically, the charge instability at P-sites overcomes the large Coulomb repulsion at d -sites. Considering the nesting nature of the Fermi surface in the two-dimensional CuO-plane, we think the superexchange process also plays a crucial role in the cupper-oxide superconductors.

At high temperatures the effective intersite repulsion Upd is larger than the residual d -site repulsion ud. In one dimension we see that in case of the small doping the fixed points of Upd are found in the infinity, and in case of the large doping in the weak-coupling region. According to Zimanyi and Bedell's study/3

> of two dimensional Luttinger liquid, the dependence of fixed points to doping suggests the transition from the superconductivity to the marginal Fermi liquid with increasing doping.

At sufficiently low temperatures Ud plays a dominant role. There are two fixed points corresponding to the fixed points of Upd. In case of the large doping Ud is scaled to the infinity of repulsive region and diverges at zero temperature. It means that the charge fluctuation is suppressed in case of the large doping. In this case possible ordering states are CDW and SDW.

In QlD case only the order of CDW is established in the transversedirection. It comes from the negligence of the process which induces the transversal Cooper-type scattering.9

> In the intrachain the short range order of SS is enhanced.

Acknowledgements

One of the authors (T.M.) would like to thank Dr. Y oshiaki Ono for suggesting this problem and encouragement.

References

1) V. J. Emery, Phys. Rev. Lett. 58 (1987), 2794. 2) C. M. Varma, P. B. Littlewood and Schmitt-Rink, Phys. Rev. Lett. 63 (1989), 1996. 3) A. Virosztek and J. Ruvalds, Phys. Rev. B42 (1990), 4064. 4) H. Kohno and K. Yamada, Prog. Theor. Phys. 85 (1991), 13. 5) T. Matsuura, K. Miura, Y. Ono, D. Hirashima and Y. Kuroda, Prog. Theor. Phys. 85 (1991), 13. 6) J. Hubbard, Proc. R. Soc. London A276 (1963), 238.

Dow




7) J. Hubbard, ibid. A285 (1964), 542. 8) ]. S6lyom, Adv. Phys. 28 (1979), 201. 9) T Matsunami and M. Kimura, Prog. Theor. Phys. 86 (1991), 1177.

10) F. D. M. Haldane, J. of Phys. C14 (1981), 2585. 11) G. Tuzuki and M. Kimura, Synth. Met. 48 (1992), 7. 12) N. Mitani and S. Kurihara, Phys. Rev. B48 (1993), 3356. 13) G. T. Zimanyi and K. S. Bedell, Phys. Rev. Lett. 66 (1991), 228.

Dow



Documents

Theory of Nested d-p Model with Strong Repulsion Graduate School