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Singlet pairs, covalent bonds, superexchange, and superconductivity
Volume 136, number 3
PHYSICS LETTERS A
27 March 1989
SINGLET PAIRS, COVALENT BONDS, SUPEREXCHANGE,
AND SUPERCONDUCTIVITY
J.E. HIRSCH Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA Received 27 January 1989; accepted for publication 9 February I989 Communicated by A.A. Maradudin
The possibility of superconductivity arising from interactions involving nearest neighbor singlet pairs is discussed. It is shown that the strong coupling effective Hamiltonian for the Hubbard model only admits extended s-wave solutions, and it is argued that those are suppressed by the Hubbard CJ.Neglecting hopping terms in the effective Hamiltonian involving three sites while keeping nearest neighbor superexchange and/or density interactions introduces spurious d- and p-wave solutions. A related but more fundamental interaction describing covalent bonds between polarizable ions can give rise to extended s-wave superconductivity, and the relationship with the resonating valence bond picture is discussed.
The possibility of superconductivity arising from condensation of nearest neighbor singlet pairs induced by Coulomb interactions was suggested in 1985 [ 1 ] in connection with heavy fermion systems. In 1987 Anderson proposed [ 2 1, as an explanation for high T, superconductivity in oxides, that single pairs (valence bonds ) pre-existing in a half-filled “resonating valence bond”’ (RVB) insulator [ 31 would condense into a superconducting state when the system was doped away from half filling. Mean field studies [ 45 ] of the effective strong coupling Hamiltonian for the single band Hubbard model [ 6,1] appeared to support this point of view, and the possibility of s- or d-wave superconductivity induced by superexchange-like interactions has been extensively discussed recently [ 7- Ill. This possibility, however, has not been conclusively established in small cluster studies, and a disturbing outstanding point is that different Monte Carlo simulation studies [ 12,131 have reached different conclusions on the possibility of dawave superconductivity in the repulsive Hubbard model. In this paper we shed some light on these issues. We show that the strong coupling effective Hamiltonian for the Hubbard model admits, if anything, only s-wave symmetry superconductivity, and argue that the large Hubbard U disallows this possibility. We show that spurious d- and even p-wave solutions
can arise if one neglects pieces of the effective Hamiltonian. We then show that the interaction terms in the effective Hamiltonian originate from a more fundamental interaction, modulated hopping by the presence of other particles. This interaction is identically zero in the single band Hubbard model but can arise if coupling to other atomic electrons is included [ 14,151 and give rise to extended s-wave superconductivity. The Hubbard Hamiltonian is given by HE-t
C (czcjo+h.c.)+UC
nitnii-px i
ni, 10 (1)
and we consider for definiteness a two-dimensional square lattice with nearest neighbor hopping (our discussion applies equally to three dimensions). Elimination of doubly occupied sites in perturbation theory in t/U yields an effective interaction of the form [6]
v=
If. 1 (ai’bj-ninj) u
-
$(5, (Ci+,Q,-ucjo+h.c.) 0
-
$
(2)
I
c7
In the last two terms, i, 1, j denote sites that can be
0375-9601/89/S 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
163
Volume
136, number
3
PHYSICS
reached successively by two nearest neighbor hops. We consider the part of the interaction that enters in a reduced Hamiltonian for Cooper pairs: Vred= &
(4) Using this interaction in the BCS equation for T,,
’ -;2’*.) , k'
it is clear that only extended s-wave solutions can exist: k,v+cos k,) ,
(6)
with T, determined by (cos k:+cosk;)* h’
1--2f(tk’)
2t
. k’
(7)
However, because in the effective Hamiltonian doubly occupied sites are not allowed, we should add to the interaction eq. (4) a momentum independent repulsion u’ that strictly speaking tends to infinity. The gap in that case has the functional form A,=A(cos k,+cos k,“+c)
(o,v3-j-n,nj).
u
The pair interaction in this case is v,,. = - s
(cos k, cos k: + cos ky cos k;),
&=d(COs
k,-cos
k,) ,
(10)
(11)
as is easily verified from eq. ( 5 ). This solution is insensitive to the presence of the double occupancy constraint u’-+co and gives highest T, close to the half-filled band, as the effective interaction in the dwave channel is strongest there [ 7-91. Gros et al. [ 12 ] found d-wave superconductivity in Monte Carlo studies with the interaction eq. (9), while the present author and coworkers found no superconductivity in numerical studies of the full Hubbard model [ 13 1. The present analysis clarifies the origin of the discrepancy and shows that the d-wave state found by Gros et al. is spurious and originates in the neglect of the three-site terms in the interaction eq. (2). Further neglecting the density-density interaction gives rise to the so-called “J-t model” with interaction (12)
(8)
but because of its symmetry it cannot avoid the onsite double occupancy restriction u’ -+o(, and yields no solution to the gap equation (5) in that limit. This implies that there is no superconductivity in the strong coupling Hubbard Hamiltonian at the mean field level, contrary to various suggestions in the literature [4,5,7-l 11. We believe that this conclusion will hold beyond the mean-field level. One can be misled into finding superconductivity in the Hubbard model in the following two ways. Neglecting the “double occupancy constraint” u’ -+03 in the strong coupling Hamiltonian will lead to extended s-wave superconductivity, as discussed above. Neglecting various terms in the effective interaction eq. (2) will lead to spurious d- and p-wave solutions. 164
1989
which admits, in addition to extended s-wave, a dwave solution of the form
vkVkk.=-~(cosk,+COSk,)(COSk:+COSk;).
I=~;~
27 March
Consider first neglecting the interactions involving three sites, so that the interaction is
(3)
After Fourier transforming and carefully keeping all terms in the effective interaction we obtain the remarkably simple answer
&=d(cos
A
v= t’ 1
v~k’C~CtklC_k.iC~.t .
Ak = - ; -& Vkk,Ak. h'
LETTERS
v,,. = - %[3(
cos k, cos k: + cos kv cos k; )
- (sin k, sin k: + sin ky sin k;) 1,
(13)
which again gives rise to an extended s-wave solution that is suppressed as u’+cr, and a spurious dwave solution. Finally, if we keep only the densitydensity interaction from eq. (2 ) V=-$
2 ninj
we
have
(14)
Volume
136, number
PHYSICS
3
2
V,,. = - $
(cos k, cos k: + cos ky cos k;
+ sin k, sin k: + sin k, sin k;) ,
(15)
which admits extended s-wave, d-wave and p-wave solutions, as discussed first by Okhawa and Fukuyama [ 16 ] and recently by Micnas et al. [ 17 1. Once again, U’ + 00 will suppress extended s-wave and not affect d- and p-wave. Our discussion shows that interactions that arise from superexchange processes will not give rise to superconductivity in a single band model. Whenever a superexchange interaction of the form eq. (9) is generated from second order processes we will necessarily also generate the other terms in the interaction eq. ( 2 ) . Then, the only possibly symmetry solution is an extended s-wave state that will be suppressed by the very same interaction that gave rise to the superexchange in the first place (on-site repulsion). We do not believe that inclusion of other bands, as in a two-dimensional Cu-0 model [ 18 1, would change this conclusion. Although here the twosite terms can have a different amplitude than the three-site terms, the presence of Hubbard U’s on Cu and 0 will always raise the energy of the states eliminated in perturbation theory increasingly with the Us, leading again to absence of pairing. However, inclusion of other degrees of freedom like a modulation oft by lattice displacement could change these conclusions. The fact that the effective interaction eq. (2) only admits s-wave symmetry solutions is not accidental. The reason is that it can be obtained in second order perturbation theory from a more fundamental interaction, a modulation of the intersite hopping by the presence of particles on these sites: V=-At
C (ci’,Cj~+h.C.)(n,,_,+nj,_,) (U> CT
a
(16)
Here, At indicates the difference in hopping amplitude for a particle when another particle is present in one of the two sites involved, relative to the hopping amplitude in the absence of other particles. Eliminating doubly occupied sites from the Hilbert space leads to an interaction of the form eq. (2) with (t+ A.t) replacing t. The reason for s-wave only is that eq. ( 16) and the kinetic energy have the same sign
LETTERS
A
27 March
1989
for hopping in the x and y direction; if we took opposite sign for hopping in the x and y directions it would give rise to only d-wave solutions. Taking different pieces of the interaction eq. (2) and neglecting others is equivalent to taking combinations of the interaction eq. ( 16 ) with equal and different sign of hopping in the x and y directions, an unphysical situation. In the single band Hubbard model the more fundamental interaction eq. ( 16) is identically zero: there is a priori the same amplitude for a particle to hop to a nearest neighbor site whether another particle of opposite spin is there or not. However, we have recently shown [ 141 that such an interaction can arise for hole conduction through oxygen anions when coupling to a degree of freedom describing the deformation of the outer electron cloud of the ion by the presence of the conducting hole is included [ 15 1. Zawadovski has recently discussed a similar interaction [ 19 1. More generally, such an interaction always arises in the derivation of a Hubbard-like model (Hubbard’s [ 201 (ii 11/r] ij) term) but is usually neglected. The sign is as in eq. ( 16) if the operators describe holes rather than electrons. The interaction eq. ( 16) has a fundamental feature: its sign depends on the phases of the wave function. For an isolated pair of sites, it implies that the energy of the bonding state of a hole is lower if a hole of opposite spin is present on that bond, while the opposite occurs in the antibonding state. If we talk about electrons rather than holes, the terms bonding and antibonding are interchanged. Eq. ( 16) with a Hubbard U on the sites simply describes the holes in a covalent bond between two polarizable ions. In a lattice with a non-stochiometric number of particles, these hole singlet pairs will delocalize. The interaction eq. ( 16) in momentum space (with a Hubbard U included) is Vkk,=-4At(coskx+cosky+cosk:+cosk;)+U, (17)
and it can give rise only to extended s-wave superconductivity, as discussed in ref. [ 141. The attractive part of the interaction at the Fermi surface is largest for the empty band, R= k’ = 0, and goes to zero at the half-filled band (k+ k’ = xc). This implies that superconductivity, if it exists, will be restricted to low hole concentration. 165
Volume 136,.number 3
PHYSICS LETTERS A
In summary, we have exposed the difficulties involved with superconductivity through superexchange interactions in a single band model and argued that condensation of nearest neighbor single pairs can only occur for holes rather than electrons, and particularly in cases where a strong interaction with a local degree of freedom exists as in the case of holes in unstable anions with tilled shells [ 15 1. This delocalization and condensation of singlet pairs is very different from Anderson’s RVB picture: it occurs near the empty band for holes (filled band for electrons) as opposed to near the half-filled band, and not in a single band Hubbard model. In addition, the normal state is likely to be a Fermi liquid of heavy quasiparticles [ 15 1. We expect this physical picture to apply to the high T,oxides and to Cd!3 under certain conditions [ 2 11. This work was supported by NSF Grant DMR-8517756 and contributions from AT&T Bell Laboratories. I have greatly benefitted from interactions with F. Mars&ho.
References [ 1] J.E. Hirsch, Phys. Rev. Lett. 54 ( 1985 ) 13 17.
166
27 March 1989
[2] P.W. Anderson, Science 235 (1987) 1196. [3] P.W. Anderson, Mater. Res. Bull. 8 (1973) 153. [4] G. Baskaran, 2. Zou and P.W. Anderson, Solid State Commun. 63 (1987) 973. [ 51 A. Ruckenstein, P. Hirschfeld and J. Appel, Phys. Rev. B 36 (1987) 857. [6] K.A. Chao, J. Spalek and A.M. Oles, J. Phys. C 10 (1977) L271. [ 71 M. Inui et al., Phys. Rev. B 37 ( 1988) 2320. [8] G. Kotliar, Phys. Rev. B 37 (1988) 3664. [9] F. Ohkawa, J. Phys. Sot. Japan 56 (1987) 2267. [ lo] R.H. Parmenter, Phys. Rev. Lett. 59 (1987) 923. [ 111 M. Cyrot, Solid State Commun. 62 (1987) 821; 63 (1987) 1015. [12]G.Gros,R.JoyntandT.Rice,Z.Phys.B68 (1987)425. [ 131 J.E. Hirsch and H.Q. Lin, Phys. Rev. B 37 (1988) 5070; H.Q. Lin, J.E. Hirsch and D.J. Scalapino, Phys. Rev. B 37 (1988) 7359. [ 141 J.E. Hirsch and F. Masiglio, UCSD preprint, December 1988. [ 151 J.E. Hirsch, Phys. Lett. A 134 (1989) 451; J.E. Hirsch and S. Tang, UCSD preprint, December 1988. [ 161 F. Okhawa and H. Fukuyama, J. Phys. Sot. Japan 53 (1984) 4344. [ 171 R. Micnaset al., Phys. Rev. B 37 (1988) 9410. [ 181 V.J. Emery, Phys. Rev. Lett. 58 (1987) 2794. [ 19]A. Zawadowski, Phys. Rev. Lett. 59 (1987) 469; and preprint. [ 201 J. Hubbard, Proc. R. Sot. A 276 ( 1963) 238. [2l]P.J.Coteetal.,Appl.Phys.Lett.38 (1981)927.
PHYSICS LETTERS A
27 March 1989
SINGLET PAIRS, COVALENT BONDS, SUPEREXCHANGE,
AND SUPERCONDUCTIVITY
J.E. HIRSCH Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA Received 27 January 1989; accepted for publication 9 February I989 Communicated by A.A. Maradudin
The possibility of superconductivity arising from interactions involving nearest neighbor singlet pairs is discussed. It is shown that the strong coupling effective Hamiltonian for the Hubbard model only admits extended s-wave solutions, and it is argued that those are suppressed by the Hubbard CJ.Neglecting hopping terms in the effective Hamiltonian involving three sites while keeping nearest neighbor superexchange and/or density interactions introduces spurious d- and p-wave solutions. A related but more fundamental interaction describing covalent bonds between polarizable ions can give rise to extended s-wave superconductivity, and the relationship with the resonating valence bond picture is discussed.
The possibility of superconductivity arising from condensation of nearest neighbor singlet pairs induced by Coulomb interactions was suggested in 1985 [ 1 ] in connection with heavy fermion systems. In 1987 Anderson proposed [ 2 1, as an explanation for high T, superconductivity in oxides, that single pairs (valence bonds ) pre-existing in a half-filled “resonating valence bond”’ (RVB) insulator [ 31 would condense into a superconducting state when the system was doped away from half filling. Mean field studies [ 45 ] of the effective strong coupling Hamiltonian for the single band Hubbard model [ 6,1] appeared to support this point of view, and the possibility of s- or d-wave superconductivity induced by superexchange-like interactions has been extensively discussed recently [ 7- Ill. This possibility, however, has not been conclusively established in small cluster studies, and a disturbing outstanding point is that different Monte Carlo simulation studies [ 12,131 have reached different conclusions on the possibility of dawave superconductivity in the repulsive Hubbard model. In this paper we shed some light on these issues. We show that the strong coupling effective Hamiltonian for the Hubbard model admits, if anything, only s-wave symmetry superconductivity, and argue that the large Hubbard U disallows this possibility. We show that spurious d- and even p-wave solutions
can arise if one neglects pieces of the effective Hamiltonian. We then show that the interaction terms in the effective Hamiltonian originate from a more fundamental interaction, modulated hopping by the presence of other particles. This interaction is identically zero in the single band Hubbard model but can arise if coupling to other atomic electrons is included [ 14,151 and give rise to extended s-wave superconductivity. The Hubbard Hamiltonian is given by HE-t
C (czcjo+h.c.)+UC
nitnii-px i
ni, 10 (1)
and we consider for definiteness a two-dimensional square lattice with nearest neighbor hopping (our discussion applies equally to three dimensions). Elimination of doubly occupied sites in perturbation theory in t/U yields an effective interaction of the form [6]
v=
If. 1 (ai’bj-ninj) u
-
$(5, (Ci+,Q,-ucjo+h.c.) 0
-
$
(2)
I
c7
In the last two terms, i, 1, j denote sites that can be
0375-9601/89/S 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
163
Volume
136, number
3
PHYSICS
reached successively by two nearest neighbor hops. We consider the part of the interaction that enters in a reduced Hamiltonian for Cooper pairs: Vred= &
(4) Using this interaction in the BCS equation for T,,
’ -;2’*.) , k'
it is clear that only extended s-wave solutions can exist: k,v+cos k,) ,
(6)
with T, determined by (cos k:+cosk;)* h’
1--2f(tk’)
2t
. k’
(7)
However, because in the effective Hamiltonian doubly occupied sites are not allowed, we should add to the interaction eq. (4) a momentum independent repulsion u’ that strictly speaking tends to infinity. The gap in that case has the functional form A,=A(cos k,+cos k,“+c)
(o,v3-j-n,nj).
u
The pair interaction in this case is v,,. = - s
(cos k, cos k: + cos ky cos k;),
&=d(COs
k,-cos
k,) ,
(10)
(11)
as is easily verified from eq. ( 5 ). This solution is insensitive to the presence of the double occupancy constraint u’-+co and gives highest T, close to the half-filled band, as the effective interaction in the dwave channel is strongest there [ 7-91. Gros et al. [ 12 ] found d-wave superconductivity in Monte Carlo studies with the interaction eq. (9), while the present author and coworkers found no superconductivity in numerical studies of the full Hubbard model [ 13 1. The present analysis clarifies the origin of the discrepancy and shows that the d-wave state found by Gros et al. is spurious and originates in the neglect of the three-site terms in the interaction eq. (2). Further neglecting the density-density interaction gives rise to the so-called “J-t model” with interaction (12)
(8)
but because of its symmetry it cannot avoid the onsite double occupancy restriction u’ -+o(, and yields no solution to the gap equation (5) in that limit. This implies that there is no superconductivity in the strong coupling Hubbard Hamiltonian at the mean field level, contrary to various suggestions in the literature [4,5,7-l 11. We believe that this conclusion will hold beyond the mean-field level. One can be misled into finding superconductivity in the Hubbard model in the following two ways. Neglecting the “double occupancy constraint” u’ -+03 in the strong coupling Hamiltonian will lead to extended s-wave superconductivity, as discussed above. Neglecting various terms in the effective interaction eq. (2) will lead to spurious d- and p-wave solutions. 164
1989
which admits, in addition to extended s-wave, a dwave solution of the form
vkVkk.=-~(cosk,+COSk,)(COSk:+COSk;).
I=~;~
27 March
Consider first neglecting the interactions involving three sites, so that the interaction is
(3)
After Fourier transforming and carefully keeping all terms in the effective interaction we obtain the remarkably simple answer
&=d(cos
A
v= t’ 1
v~k’C~CtklC_k.iC~.t .
Ak = - ; -& Vkk,Ak. h'
LETTERS
v,,. = - %[3(
cos k, cos k: + cos kv cos k; )
- (sin k, sin k: + sin ky sin k;) 1,
(13)
which again gives rise to an extended s-wave solution that is suppressed as u’+cr, and a spurious dwave solution. Finally, if we keep only the densitydensity interaction from eq. (2 ) V=-$
2 ninj
we
have
(14)
Volume
136, number
PHYSICS
3
2
V,,. = - $
(cos k, cos k: + cos ky cos k;
+ sin k, sin k: + sin k, sin k;) ,
(15)
which admits extended s-wave, d-wave and p-wave solutions, as discussed first by Okhawa and Fukuyama [ 16 ] and recently by Micnas et al. [ 17 1. Once again, U’ + 00 will suppress extended s-wave and not affect d- and p-wave. Our discussion shows that interactions that arise from superexchange processes will not give rise to superconductivity in a single band model. Whenever a superexchange interaction of the form eq. (9) is generated from second order processes we will necessarily also generate the other terms in the interaction eq. ( 2 ) . Then, the only possibly symmetry solution is an extended s-wave state that will be suppressed by the very same interaction that gave rise to the superexchange in the first place (on-site repulsion). We do not believe that inclusion of other bands, as in a two-dimensional Cu-0 model [ 18 1, would change this conclusion. Although here the twosite terms can have a different amplitude than the three-site terms, the presence of Hubbard U’s on Cu and 0 will always raise the energy of the states eliminated in perturbation theory increasingly with the Us, leading again to absence of pairing. However, inclusion of other degrees of freedom like a modulation oft by lattice displacement could change these conclusions. The fact that the effective interaction eq. (2) only admits s-wave symmetry solutions is not accidental. The reason is that it can be obtained in second order perturbation theory from a more fundamental interaction, a modulation of the intersite hopping by the presence of particles on these sites: V=-At
C (ci’,Cj~+h.C.)(n,,_,+nj,_,) (U> CT
a
(16)
Here, At indicates the difference in hopping amplitude for a particle when another particle is present in one of the two sites involved, relative to the hopping amplitude in the absence of other particles. Eliminating doubly occupied sites from the Hilbert space leads to an interaction of the form eq. (2) with (t+ A.t) replacing t. The reason for s-wave only is that eq. ( 16) and the kinetic energy have the same sign
LETTERS
A
27 March
1989
for hopping in the x and y direction; if we took opposite sign for hopping in the x and y directions it would give rise to only d-wave solutions. Taking different pieces of the interaction eq. (2) and neglecting others is equivalent to taking combinations of the interaction eq. ( 16 ) with equal and different sign of hopping in the x and y directions, an unphysical situation. In the single band Hubbard model the more fundamental interaction eq. ( 16) is identically zero: there is a priori the same amplitude for a particle to hop to a nearest neighbor site whether another particle of opposite spin is there or not. However, we have recently shown [ 141 that such an interaction can arise for hole conduction through oxygen anions when coupling to a degree of freedom describing the deformation of the outer electron cloud of the ion by the presence of the conducting hole is included [ 15 1. Zawadovski has recently discussed a similar interaction [ 19 1. More generally, such an interaction always arises in the derivation of a Hubbard-like model (Hubbard’s [ 201 (ii 11/r] ij) term) but is usually neglected. The sign is as in eq. ( 16) if the operators describe holes rather than electrons. The interaction eq. ( 16) has a fundamental feature: its sign depends on the phases of the wave function. For an isolated pair of sites, it implies that the energy of the bonding state of a hole is lower if a hole of opposite spin is present on that bond, while the opposite occurs in the antibonding state. If we talk about electrons rather than holes, the terms bonding and antibonding are interchanged. Eq. ( 16) with a Hubbard U on the sites simply describes the holes in a covalent bond between two polarizable ions. In a lattice with a non-stochiometric number of particles, these hole singlet pairs will delocalize. The interaction eq. ( 16) in momentum space (with a Hubbard U included) is Vkk,=-4At(coskx+cosky+cosk:+cosk;)+U, (17)
and it can give rise only to extended s-wave superconductivity, as discussed in ref. [ 141. The attractive part of the interaction at the Fermi surface is largest for the empty band, R= k’ = 0, and goes to zero at the half-filled band (k+ k’ = xc). This implies that superconductivity, if it exists, will be restricted to low hole concentration. 165
Volume 136,.number 3
PHYSICS LETTERS A
In summary, we have exposed the difficulties involved with superconductivity through superexchange interactions in a single band model and argued that condensation of nearest neighbor single pairs can only occur for holes rather than electrons, and particularly in cases where a strong interaction with a local degree of freedom exists as in the case of holes in unstable anions with tilled shells [ 15 1. This delocalization and condensation of singlet pairs is very different from Anderson’s RVB picture: it occurs near the empty band for holes (filled band for electrons) as opposed to near the half-filled band, and not in a single band Hubbard model. In addition, the normal state is likely to be a Fermi liquid of heavy quasiparticles [ 15 1. We expect this physical picture to apply to the high T,oxides and to Cd!3 under certain conditions [ 2 11. This work was supported by NSF Grant DMR-8517756 and contributions from AT&T Bell Laboratories. I have greatly benefitted from interactions with F. Mars&ho.
References [ 1] J.E. Hirsch, Phys. Rev. Lett. 54 ( 1985 ) 13 17.
166
27 March 1989
[2] P.W. Anderson, Science 235 (1987) 1196. [3] P.W. Anderson, Mater. Res. Bull. 8 (1973) 153. [4] G. Baskaran, 2. Zou and P.W. Anderson, Solid State Commun. 63 (1987) 973. [ 51 A. Ruckenstein, P. Hirschfeld and J. Appel, Phys. Rev. B 36 (1987) 857. [6] K.A. Chao, J. Spalek and A.M. Oles, J. Phys. C 10 (1977) L271. [ 71 M. Inui et al., Phys. Rev. B 37 ( 1988) 2320. [8] G. Kotliar, Phys. Rev. B 37 (1988) 3664. [9] F. Ohkawa, J. Phys. Sot. Japan 56 (1987) 2267. [ lo] R.H. Parmenter, Phys. Rev. Lett. 59 (1987) 923. [ 111 M. Cyrot, Solid State Commun. 62 (1987) 821; 63 (1987) 1015. [12]G.Gros,R.JoyntandT.Rice,Z.Phys.B68 (1987)425. [ 131 J.E. Hirsch and H.Q. Lin, Phys. Rev. B 37 (1988) 5070; H.Q. Lin, J.E. Hirsch and D.J. Scalapino, Phys. Rev. B 37 (1988) 7359. [ 141 J.E. Hirsch and F. Masiglio, UCSD preprint, December 1988. [ 151 J.E. Hirsch, Phys. Lett. A 134 (1989) 451; J.E. Hirsch and S. Tang, UCSD preprint, December 1988. [ 161 F. Okhawa and H. Fukuyama, J. Phys. Sot. Japan 53 (1984) 4344. [ 171 R. Micnaset al., Phys. Rev. B 37 (1988) 9410. [ 181 V.J. Emery, Phys. Rev. Lett. 58 (1987) 2794. [ 19]A. Zawadowski, Phys. Rev. Lett. 59 (1987) 469; and preprint. [ 201 J. Hubbard, Proc. R. Sot. A 276 ( 1963) 238. [2l]P.J.Coteetal.,Appl.Phys.Lett.38 (1981)927.