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Magnetism in the large-U single-band Hubbard model
ELSEVIER
SyntheticMetals 80 (1996)
159-165
Magnetism in the large- U single-band Hubbard model Y.R. Wang, M.J. Rice Xerox
Webster Research
Center, 0114-410,
Webster, NY 14580,
USA
Received1 April 1996;accepted15April 1996
Abstract
Usingan exactexpressionof the r-J modelin termsof fermionicholonandlocalspinoperators,we investigatemagnetism in thelarge-U limit of the Hubbardmodelasa functionof doping(xf awayfrom half-filling. The characteristicdopantconcentrationx, belowwhichthe localspinsremainantiferromagnetically correlated,andabovewhichtheyareferromagnetically coupled,isestimated. In theantiferromagnetic regimexx,, andfor a three-dimensional lattice, themodelexhibitscoexistence of chargeitinerancyandlocalizedmagnetic momentbehavior.At low temperature Tthemagnetization followstheBlochT3’2law, whileabovetheCurietemperature (T,) theparamagnetic susceptibilityfolIowsthe Curie-Weisslaw. The Bohr magnetonnumberis 1--x and,hence,noninteger.For T-=zT, the specificheatvaries asC,/T=A +BT”2, wherethe first termis contributedby itinerantholonsandthesecondby spinreversals.Theseresultsareconsistentwith theobservedferromagnetism in the3dtransitionmetals,suggesting thatthelarge-UHubbardmodelsharesmanyimportantphysicalproperties with the intermediate-umodel.However,we point out that thereare significantdifferencesbetweenthe large-Uferromagnetism andthe Stoner-Slaterbandferromagnetism, andthatthesedifferencescanbemeasured by spinpolarizationexperimentsat low temperature. Keywards:Magnetism; Hubbard model;Models;Theoretical study
1. Introduction
It is a great pleasurefor us to presentthis paper to honor ProfessorAlan J. Heegeron the occasionof his 60th birthday. Since we expect that most of the contributions to honor him in this volume will be on the subject of conjugatedpolymers and the organic solid state, we felt that it would be fitting to write on a subjectrelated to hisgroup’s work on investigating the elementary excitations of the highly correlated electron systemin the cupric high-temperaturesuperconductors[ 11. The nature of magnetism in a highly correlated electron systemhasbeena constantchallengeto the solidstatetheorist. A simple, yet non-trivial model for describingelectron correlation is the Hubbard model [ 21. From the standpoint of magnetism,the central issuesto be addressedin the study of this model are (1) the coexistence of charge itinerancy and localized magnetic moments,and (2) the interplay between ferromagnetism and antiferromagnetism. Anderson [ 31 pointed out that in the large-U limit and for the half-filled case(one electron per lattice site), the model resultsin antiferromagnetic coupling betweenthe spins.On the other hand, Nagaoka [4] showed that in the infinite-U limit and for a bipartite lattice, a small amount of doping away from hatf0379-6779/96/$15,00 Q 1996Elsevier Science S.A.All rightsreserved PffSO379-6779(96)03697-l
filling gives a ferromagnetic ground statewith maximum total spin, while the addedcharge (electron or hole) is delocalized with a bandwidth equal to that of the non-interacting case. From theseresults it can be inferred [51 that for the caseof a finite value of U, but a value of U still large comparedwith the electron hopping energy f, antiferromagnetic coupling will persistup to somecharacteristicdoping concentrationX, above which ferromagnetic coupling takes over. Recently, we have derived [ 61 an exact expressionof the Hamiltonian +TfWJof the r-J model, i.e., the Hubbard model in the largeU limit, in terms of fermionic holon ( ei) and local spin l/2 (Si) operators,Importantly, this expressiondoesnot require the imposition of a constraint between the spin and holon operators, the bane of previous formulations, and, in consequence, rendersthe t-J model amenableto analysis. While the result of Anderson is already built into the t-Jmodel, the more subtle result of Nagaoka is immediately obtainedfrom the new expression. Encouragedby the latter progress,we considerin thispaper the consequencesfor magnetismthat result from applying to the exact expressionfor XrmJ a mean-field decoupling procedure that we have employed in a previous discussion[7]. An estimation of the characteristic dopant concentration X,
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above which ferromagnetic coupling is favored is first carried out. We then consider the physical features of the two regimes xx,. In agreement with earlier work [ 81, the salient physical feature of the regime n x,, the holons are found to be strongly itinerant while the spins exhibit localized properties. In particular, in 3D, the magnetization M(T) at temperatures T< T, where T, is the Curie temperature, follows the Bloch law 6M( T) a T3’2, while above T, the paramagnetic susceptibility follows the Curie-Weiss law. The Bohrmagneton number as deduced from M(O) is (1 -x) and, hence, non integer. Interestingly, the low-temperature specific heat C, consists both of a linear-T contribution, arising from the itinerant holons, and a T3’2 contribution due to spin-reversal. Consequently, at low temperature, CJT is a linear function of T1’2. Thus for n > xc, the one-band Hubbard model in the large-U l&it leads to coexistence of charge itinerancy and localized magnetic moment behavior. We comment on the relevance of these results to several types of highly correlated 3d-electron systems.
2. Theory
For the case of hole doping (away from half-filling), the exact expression of X,-J, in terms of the holon and local spin operators is [ 61
x si.sJ-; (
1
(l+eJ
(1)
where J= 4?/ LJ.In Eq. ( 1) , i runs over all lattice sites (i = 1, 2 , . . ., N, N+ co), while j runs over the z nearest neighbors of
i. The fermionic holon operator ei creates a hole at site i, and the local spin operator Si satisfies the commutation relations of spin l/2, i.e., [ Si+, 5”: ] = 2$si,, [ 3, S,* ] = + SL*6,, and S+S; + S;$’ = 1. Notice that the true spin operator, Si, is related to the local spin operator by
Metals
80 (1996)
159-165
A similar expression for the case of electron doping can also be derived (see Appendix), and is
where 4 is a fermionic operator which creates a ‘doublon’, i.e., a doubly occupied site at i. The electron number at site i i_sni = 1 + dd,, and the true spin operator at site i is given by Sj= ( 1 - d;tdi)Si. Also, reflecting the particle-hole symmetry of the Hubbard model, the sign of r in Eq. (2) is opposite to that which occurs in Eq. ( 1) . In the limit J + 0, Eqs. ( 1) and (4) produce the Nagaoka theorem [ 41. For the case of a bipartite lattice, small concentrations of holes or electrons added to the half-filled band lead to a ferromagnetic ground state with the maximum total spin ( l/2 + 2Si.Sj= l), while the added charges are delocalized with a bandwidth equal to that of the non-interacting case. This follows from the bound, - 1 < l/2 + 25,. S, 5 I, and the property that the density of fermion states of anearestneighbor tight-binding band of a bipartite lattice is symmetric about the center of the band. For the bipartite lattice, the expressions of Eqs. (1) and (4) generalize this theorem to arbitrary concentrations of added charge whenever a meanfield decoupling of the holon and the local spin operators is appropriate. For finite values of J and for the bipartite lattice, 2 explicitly describes a competition between ferromagnetism, favored by the hopping term, and antiferromagnetism, favored by the Heisenberg term. The role of doping away from half-filling in frustrating antiferromagnetic spin ordering is immediately understood from Eqs. ( 1) and (4). For sufficiently large dopant concentrations, ferromagnetic coupling between the local spins will dominate. The characteristic doping concentration (x,) above which this occurs can be estimated by applying to Z a mean-field decoupling procedure that we have previously introduced [ 71. For convenience, in the following we will only discuss the case of hole doping away from the half-filled baud. The extension to the case of electron doping on the basis of Eq. (4) is straightforward. Following-&procedure in [ 71, we obtain: ~t-j = &“h + ~~
(5) m (7)
where rare the Pauli matrices, and CL is the creation operator of an electron at site i. The number of electrons at site i, ni, is given by
&=t( 1/2+2(si*sj),)
02
Y.R. Wang, M.J. Rice /Synthetic
Vjj=J( (si*sj),-
l/4)
(9)
.To= 4t( eieJ)h - J( ( 1 - efei) ( 1 - ejej) )h
(10)
(. . .)r, and (. . .)s denote ensemble averages taken withrespect to the decoupled holon and spin Hamiltonians Zr, and Z”,, respectively. This decoupling procedure is valid only if higher order corrections beyond the decoupling are small. For a 3D lattice with a ferromagnetic ground state, where the spin wave representation of spin fluctuations is appropriate, we can show that the effective interaction between the holons mediated by the spin fluctuations is proportional to Tat low temperature and, therefore, vanishes as T+ 0. This result can be easily understood since for a ferromagnetic ground state, the term Uniirnii in the Hubbard model vanishes at T=O, and the electrons (or equivalently, the holons) behave exactly as spinless fermions. For the other cases, the justification of the decoupling procedure requires more study. Eqs. (8)-( 10) represent a set of self-consistent equations for the parameters entering Zh and Ps. These are the holon hopping amplitude &, the exchange interaction jij between the local spins, and the nearest neighbor holon-holon interaction V,. Although we shall point out the relevance of polaron effects we shall restrict our ex@cit analysis in the present paper to spatially uniform situations for which the latterquantities are constants, independent of i andj. The characteristic concentration X, may be estimated by noting that for a tight-binding band with a constant density of states the holon bond order (ele;), = -x( 1 -x) at T= 0. Consequently, from Eq. ( lo), we have Jv=Jeff=4fx(
1 -x) -J( 1 -x)2(1 -2)
(11)
The condition Jcfi= 0 yields x, = Jl(4t + J) for J$ << 1. This result is similar to a result previously obtained by Inui et al. I:91. At finite temperature T, Jeff is, in principle, temperature dependent, but this temperature dependence is only important if the holon Fermi degeneracy temperature is not large by comparison to T. For x
(12)
where @kdenotes the energy of a magnon of wavevector k, yk = ( 1/z) & exp( ik ’ S), and 6 denotes a nearest-neighbor vector. For the simple cubic lattice,t,/t= -0.097. Eq. (12)
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80 (1996)
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applies also for the 1D and 2D AFM spin wave ground states which, however, are inaccurate [ 121 for S = l/2. The salient physical feature of the regime .r x,, Jeff is positive and (Si. Sj)o = l/4, leading to a holon hopping amphtude th = t at T= 0. Since in ID and 2D there is no long-range ferromagnetic order at any finite temperature, in this paper we will restrict our attention to the case of 3D only. In this case the Curie temperature for the ferromagnetic transition, calculated from the Weiss molecular field theory, is just (13)
kBTc = zJcff/8
Here and below we shall assume that the holon Fermi energy is much larger than the relevant temperature, so that Jeff can be considered as temperature-independent. While this result is the same as that of the traditional Heisenberg model, there are significant differences for other properties between the present local spin description and the Heisenberg localized spin description. These differences primarily come from the factor that in the present th_eorythe local spins, Sj, are only auxiliary to the true spins, Sj = ( 1 - e:eJS,. For example, the magnetization of the system below the Curie temperature is M(21=((1-ejei>)~g~~,(~),=(l-X)M,(T)
(14)
where M, ( r) is the ‘magnetization’ of the local spins. Using the Weiss molecular field approximation, M,(T) satisfies 1131 &&(T)IM,(O)
=n~~=tanh(ti~~fm,/SkJ)
(15)
where&&(O) =Ng& (S= l/2). Comparing Eqs. (14) and (15) we see that the zero-temperature magnetization, M( 0) = ( 1 -x)M,( O), decreases with increasing doping concentration. More importantly, in our case M(0) is generally not an integer number of the Bohr magneton as in the traditional Heisenberg model, although the saturation magnetization, M(r) lM( 0), satisfies the Brillouin function with s= l/2. In the paramagnetic state with an applied magnetic field B,
the effective spin Hamiltonian is
Y.R. Wang, M.J. Rice /Synthetic
162
i
i
The paramagnetic susceptibility is then easily obtained as x(T)=Ce,f(T-TJ;
T>Tc
(17)
where Ceff= (1 -x)~C, and C=N(gy,)‘/4k, is the Curie constant. We note that, in general, it is not the real number of spins N, = ( 1 - x)N which enters C,, but ( 1 - x) N,. The factor ( 1 -x>’ can be understood by comparing Eqs. ( 14) and (16) which show that compared with the traditional Heisenberg model the effective Bohr magneton entering the highly correlated problem is ( 1 -x> pr+ The Weiss molecular field approximation is not accurate at low temperature or near the transition. This is because the approximation does not include spin correlation and collective excitation. The collective spin excitations of Z”s at low temperature are spin waves. Their excitation spectrumis [ 131 J
hok= (T11 2-c
cos(k.6)
6
1
where S is the nearest-neighbor vector. The deviation of the magnetization from the saturation value, 6M= M( 0) --M(T), due to the thermal excitation of the spin waves is [ 131 &W/M(O)
= (2/Q)(kBT/2TJ,n)3’2[(3/2);
T
(19)
where Q = 1,2 and4for simple ( s.c.), body-centered (b.c.c.) and face centered cubic (f.c.c.) crystals, respectively. Thus, the deviation of the low-temperature magnetization follows the Bloch law. The thermal excitation of the spin waves also contributes to the specific heat of the system. This contribution is also proportional to T3’* at low temperature [ 131, i.e.: C,=NkB(15/4Q)(kBT/2d,ff)3’2{(5/2) =0.964Nk,[SM(T)/M(O)];
TKT,
(20)
Since the holons are itinerant, they contribute a specific heat that is linear in temperature at low temperature. Thus, the total specific heat of the system at low temperature is Cv=C,,+C,=AT+BT3’*; C,= (1?/3)p(+)~T
T-=zT,
(21) (22)
where p( er) is the holon density of states at the holon Fermi energy. We note that Eq. (21) predicts that C,/T is a linear function of T”* at low temperature. The above results show that the properties of the large-U Hubbard model in the ferromagnetic coupling regime exhibit both itinerancy and localized spin characteristics. Interestingly, they are typical of the observed properties of metallic ferromagnets, such as those of the transition metals, Fe, Co and Ni [ 141. For these metals, both a localized spin picture [ 151 and an itinerant electron picture [ 161 have long been proposed. Either of the pictures is successful in explaining many properties, but each has difficulties in explaining some
Metals
80 (1996) 159-165
particular property. The localized spin theory has difficulties in explaining the properties attributed to itinerancy, such as the non-integer number of Bohr magnetons, and the large electronic (linear-T) specific heat, contributed by the 3d electrons. The itinerant electron theories, such as the StonerSlater band theories, have difficulties in obtaining the localized spin properties, such as the Bloch law 6M( T) a T3’*, the T3’* component of the low-temperature specific heat, and the Curie-Weiss paramagnetic susceptibility above T,. The latter is understood as due to the lack of inclusion of electron correlation in the band theory. It is generally believed that the Hubbard model, in its multiband form, is appropriate for describing the 3d electrons of the transition metals, and that its correct solution may bridge the views of the two extreme limits. lhis is supported by more recent calculations which obtain the Curie-Weiss law with the inclusion of electron correlation beyond the usual random phase approximation (RPA) 1171. Theoretical estimation [ 181 indicates that U- W for the transition metals, where w-is the bandwidth of the 3d states. If the Hubbard model with intermediate U is appropriate for the transition metals, the coincidence of our qualitative results with the properties of the transition metals suggests that the large-l/Hubbard model in the ferromagnetic coupling regime shares important physical properties with the intermediate U model. Thus, the intermediate-U model may be solved either starting from the small-U limit, as given by the band theory, or from the large-U limit, as described above. Starting from the large-U limit may be advantageous since the essential physical properties are already obtained at the mean-field level of approximation. There are significant differences between the large-U ferromagnetism, described in this work, and the Stoner-Sister band theory of ferromagnetism. At T = 0, the large47 ferromagnet has all the electron spins of the singly occupied sites parallel to each other (Si.Sj= l/4). It is clear from Eq. (1) or (4) that the electron Fermi surface in this case can be determined directly from the holon (or doublon) Fermi surface. More precisely, the large-V ferromagnet at T = 0 has a Fermi surface spanned only by one Fermi wave vector. In the Stoner-Slater band theory of ferromagnetism, the Fermi surface at T= 0 is spanned by two Fermi wave vectors, one for spin-up (k,? ) and one for spin-down (k,, ). Closely related to this difference is the difference in the spin population between the two kind of ferromagnetisms. In the case of hole doping (i.e., less than one electron per site) and at T= 0, &he large-U ferromagnet has zero spin-down population. The large-U ferromagnet in the case of electron doping at T=O has a finite but minimized spin-down population equal to the number of doubly occupied sites (see Eqs. (A19) and (A20) ). The Stoner-Slater band ferromagnet, however, has a finite spin-down (and spin-up) population independent of whether the system is hole doped or electron doped, Spinpolarization experiments [ 19-221, which can investigate these differences, are therefore capable of distinguishing the large-U ferromagnetism from the Stoner-Slater band ferromagnetism.
Y.R. Wang,
M.J.
Rice/Synthetic
The coexistence of the charge itinerancy and the localized moments is most prominently reflected in the low-temperature specific heat. The contribution from the localized moments can be estimated from 6M( T) /M(O) as given in Eq. (20). For Ni, 6M(T)lM(O) ~7.5 X 10-6T3’2, and for Fe, 6M( T) /M( 0) = 3.4 X 10w6T3” at low temperature [ 131. Eq. (20) then gives the estimate C, ~0.06 X T3’2 mJ/ ( deg.3’2 mol) for Ni, and C,- 0.027 X T3” mJ/(deg.3’2 mol) for Fe. Early experimental measurement [ 23 ] between 1.2 and 4.2 K indeed found the T3’2 contribution, giving C,=O.O26 N 0.038T3’2 mJ/(deg.3’2 mol) for Ni and C,=0.021-0.048T3’2 mJ/(deg.3’2 mol) for Fe. The experimental linear-T terms are found to be 7,OT and 4.7T mJ/(deg. mol) for Ni and Fe, respectively. The measurements are in qualitative agreement with the prediction of Eq. (2 1). It would be interesting to measure the low-temperature specific heat more accurately. Measurements below 1 K are particularly desirable, since the phonon contribution is significantly reduced below 1 K for these materials. NiO and CuO are known large-U systems classified as charge transfer gap systems in the Zaanen, Sawatzky and Allen Scheme, but it may nonetheless be possible to represent the low-energy-scale physics of these oxides by an effective large-U Hubbard model or f-J model [ 241 as discussed by Zhang and Rice 1251, It would be interesting to see if polaronic characteristics can be observed in these materials. Finally, it is generally believed that the large-U Hubbard model is applicable to the copper-oxide superconductors [ 261. Experimentally, it has been shown that two regimes exist, the underdoped regime and the overdoped regime. The superconductivity transition temperature appears to be optimized near the borderline between the two regimes. We speculate that the underdoped and the overdoped regimes correspond, respectively, to the antiferromagnetic and fet-romagnetic coupling regimes discussed here. Thus, in the underdoped regime we expect that the holons are polaronic, and that in the overdoped regime the holons are well delocalized, exhibiting more conventional metallic behavior. This is in accordance with many experimental observations, In particular, in the underdoped regime the effect of doping on the quasi-2D spin excitation spectrum may be approximated as that of a Heisenberg antiferromagnet defected by a static distribution of polarons as we have discussed above. That lattice distortion is involved in the polaron formation has been shown by the experiments of Professor Heeger’s group [ I] and by Falck et al. [ 271. In the Wigner-Jordan ( WJ) spinless fermion representation 1281 of the local spin operators, the static distribution of the holons gives rise to a density of zeroenergy excitations in the excitation spectrum of the WJ fermions. Using this excitation spectrum, it has been shown that many of the observed normal state properties of the copperoxide superconductors can be derived from the t-J model [ 291. In particular, the linear-T &-plane resistivity and the unusual doping and temperature dependence of the d.c. susceptibility are the consequence [29] of the zero-energy WJ fermion excitations. As the doping concentration increases to
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80 (1996)
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163
values close to or beyond the characteristic concentration x,, the holons become itinerant according to our discussion above, The density of zero-energy WJ fermion excitations is expected to be reduced, and the scattering between the holon and the WJ fermion is expected to be more conventional, giving rise to a T2 dependence of the resistivity. These and the magnetic properties in the overdoped regime requiremore experimental investigation. It would be interesting to see if characteristics of ferromagnetic coupling can be found.
Acknowledgements
The authors are grateful to M. Franz for many discussions on the subject of electron doping, M. van Veenendaal for a useful communication, and G.A. Sawatzky for several comments. Appendix Derivation
of the t-J Hamiltonian
in Eq, (4)
Since the derivation of the expression in Eq. (4) for electron doping has several subtle differences from that of hole doping [ 61 in Eq. ( 1)) in this Appendix we present our derivation of Eq. (4). We first define the Hubbard operator at a lattice site i to be X,: = ni, - &, which creates a doubly occupied site, I i,d), if the site is occupied by an electron with spin - g. $e satisfies the following equations: g&,-CT)=
li,d)
(Al)
Xi, I i,d) = I i, - a)
(A21
~~Ii,a)=~~Ii,d)=Xi,Ii,fa)=O
(A31
In order to express the t-J Hamiltonian in terms of {Xi,}, it is necessary to first express the electron density, ni, and the true spin operator, g,, in terms of {X,,}. This task is much easier for the case of hole doping because of the singleoccupancy constraint (nioni, _ (T= 0). For the case of electron doping, we now have ( 1 - ltif ) ( 1 - ni i ) = 0. This gives ni=niT+n,l=l+nirni,=l+~~Xir=l+Xj,Xi,
(A4)
Similarly, we obtain 2~=njt ‘Xi&$ Sj+ =CTTCjL
--nii =njT( 1 -nip)
-~~ji( l-n,t)
-xirzt Cnj~C~~CjJlti~
f-45) = -Xii$i
(A6)
From these relations, we can obtain the spin density ni,. As in Ref. [S], we next map the electron Hilbert space with the constraint of minimum double occupation to the tensor-product Hilbert space of the doublon (a doubly occupied site) and spin states, i.e., I i,cr) + I i,O>, I i,(~)~. A doubIy
Y.R. Wang, M.J. Rice /Synthetic
164
occupied site can have two lid)+ li,l)dl T)SOr lid) + two mappings, we can define operators, Xi, ( + ) and X,,( TQ,(+>=di’&+;
independent mappings, i.e., Ii, l)d I J,)$. Depending on the two sets of auxiliary Hubbard ) . They are given by
K,(+)=&1/2+S;)
$,(-)=4(1/2--q);
$J
-)
(A7) =Gsy
(A81
One can verify, with the respective mapping of I i,d), that both {GW( + ) } amd (zU( - ) } given above satisfy conditions (Al), (A2) and (A3). According to Ref. [5], the Hubbard operators, independent of the mapping of the doubly occupied sites ‘, can be obtained in terms of X,,( + ) and Xia(-):
~~j~,=~
(Aij+B,~~+SjB,+Fi~~-S:F,)
(A91
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80 (1996)
159-165
The ‘ - ’ signsin Eq. (A18) account for the anticommutation relation betweenthe electron operatorswith different spins,i.e., Xi,$p Ii, .J) = - Ii, t ). Finally, it is easyto show that theseresults satisfy all commutation relations and preserve the time-reversal symmetry of the Hubbard operators. The t-J Hamiltonian in Eq. (4) follows from Eqs. (A14), (AlS) and (A21).
References [l] G.Yu, C.H. Lee and A.J. Heeger, Phys. Rev. Left., 67 (1991) 2581; G. Yu et al., Physica C, 190 (1992) 563; G. Yu et al., Phys. Rev. B. 45 (1992) 4964. [2] J. Hubbard, Proc. R. Sue. London, Ser. A, 276( 1963) 238;277( 1963) 237;285(1965)542.
where
P.W. Anderson, Phys. Rev., 115 (1959) 2. [4j Y. Nagaoka, Phys. Rev. B, 147 (1966) 392. [5] N.F. Mott, Metal-insulator Transitions, Taylor and Francis, London, 1974, Ch. 4, p. 124. [6] Y.R.WangandM.J.Rice,Phys.Rev.B,49(1994)436O;G.Khaliuliii, Pis’ma Zh. Eksp. Tear. Fiz., 52 (1990) 999 (JL?TP Lea.. 52 ( 1990) 389); G. Khaliullin and P. Horsch, Phys. Rev. B,47 ( 1993) 463. [7] M.J. Rice and Y.R. Wang, Phys. Rev. B, 48 (1993) 12 921, [S] L.N. Bulaevskii and D.I. Khomskii, Sov.Phys. JEEP, 25 ( 1967) 1067; L.N. Bulaevskii, E.L. Nagaev and D,l. Khomskii, Sov.Phys. JETP, 27 ( 1968) 839; W.F. Brinkman and T.M. Rice, Phys. Rev. B, 2 (1970) 4302. For the 2D problem, see: G. Martinez and P. Horsch, Phys. Rev. B, 44 ( 1991) 317. For a recent review, also see: Yu Lu et al., Chin. J. Phys., in press. [9] M. Inui, S. Doniach and M. Gabay, Physira C, 153-155 (1988) [3]
Aij=Nr<
+ >XjrjC + ) +HmC - )X,ut( - >
(AlO)
Bfj=$q(
+ )Xj,t( + ) -xit,(
C-411)
- )Xjfl,( - )
Fij = ddj (PiST + Q,S,T ) ; i #j
-
FiiSf - SfFf, = 4di( as: + bS,’ )
(A13)
The functions Pi and Qi and the coefficients a and b are to be determined by the requirement that Eq. (A9) reproduces all matrix elements. For example, qCXj, I i, - r) lj,d) = I i,d) Ij, - u>. Applying Eq. (A9), we obtain:
1277.
[lo] H. Bethe, Z. Phys., 71 ( 1931) 205; L. Hulthen, Arr. Mar. Asrron. Fyz.,
~,X,,=~djE(112-S~)(1/2-Sjl)+S:S~];
26A (1938).
i#j $,X,,=4dj[(1/2+Sf)(1/2+5”)
X&
[ 111 S. Liang, B. Doucot and P.W. Anderson, Phys. Rev. Lert., 61 ( 1988)
tA15)
365. I121 Y.R. Wang and M.J. Rice, Phys. Rev. B, 45 [1992) 5045. 1131 C. Kittel, Introduction toSolidSrate Physics, Wiley, New York, 1976, Ch. 15, p. 457; J. Callaway, Quantum Theory of the Solid State,
+S;ST];
i#j xit?Xi, =~lXil
(A14)
=~di
(A161
= (l -4d,)S;S:;
XilXJJ = (1-4di)S+S;
(A17)
XiLXJT=-(l-d,td,)S’; Xi,xY,=-(l-~d,)S;
(A18)
nit =XiJ-$s
+xJXiJ
= (1 -4di)SzS;
+Gd;
(Al%
niL =Xir$t
+gtXi,
= (1s4di)S;S:
+ddi
t-420)
ni=l+ddi;
??j=(l-~di)Si
t-421)
’ Note that all doubly occupied sites are assigned to have the same spin state, although this spin state can be arbitrarily chosen as I T >, I L >, or any linear combination of the two, such as ( I T ) i- / i >) /fi. For details, see: Ref. [5].
Academic Press,New York, 1976, Ch. 2, p. 87. R.M. Bozorth, Ferromagnetism, Part III, VanNostrand,Toronto, 1968, p. 423. [ 151 J.H. Van Vleck, Rev. Mod. Phys., 25 ( 1953)QO; J,&i&l, G. Leman and M. Olszewski, J. Appl. Phys., 32 ( 1961 j 3258. [ 161 EC. Stoner, Proc. R. Sot. London, Ser. A, 165 ( 1938) 372; J.C.Slater, Phys. Rev., 82 (1951) 538.; E.P. Wohlfarth, Rev. Mod. Phys., 25 (1953) 211. [ 171 K.K. Murata and S. Doniach, Phys. Rev, L&t., 29 (1972) 285; T. Moriya and Y. Takahashi, J. Phys. Sot. Jpn., 45 (1978) 397. For a review, see: T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism, Springer, Berlin, 1985. [IS] B.N. COX, M.A, Coulthard and P. Lloyd, J. Phys. F: Met. Phys., 4 (1974) 807; 0. Gunnarsson, J. Phys. F: Met. Phys., 6 ( 1976) 587; G. Stollhoff, A.M. Oles and V. Heine, Phys. Rev. B, 41 (1990) 7028; M.M. Steiner, R.C. Albers and L.J. Sham, Phys. Rev., 45 (1992) 13 272. I191 C.J. Maetz et al., Phys. Rev. Len., 48 (1982) 1686. [20] H. Hopster et al., Phys. Rev. Lett., 51 (1983) 829, I211 J. Kirschner and E. Langenbach, Solid Srare Commun., 66 (l!X8) 761. [22] P. Genoud, A.A. Manuel, E. Walker and M. Peter, J. Phys.: Condens. Matter, 3 (1991) 4201. [ 141
Y.R. Wang, M.J. Rice/Synthetic
1231 M, Dixon, F.E. Hoare, T.M. Holden and D.E. Moody, Proc. R. Sot. London, Ser. A, 285 (1965) 561. See also: P.J. Meschter, J.W. Wright, C.R. Brooks and T.G. Kollie, .L Phys. Gem. Solids, 42 (1981) 861 and Refs. therein. [241 J. Zaanen, G.A. Sawatzky and J.W. Allen, Phys. Rev. La, 55 (1985) 418; J. Bela, A.M. Oles and J. Zaanen, Phys. Rev. Mt., 72 (1994) 2600. 1251 F.C. Zhang and T.M. Rice, Phys. Rev. B, 37 (1988) 3759.
Metals 80 (1996)
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16.5
[26] P.W. Anderson, Science, 235 (1987) 1196; J.R. Schrieffer, X.G. Wen and S.C. Zhang, Phys. Rev. &II., 60 (1988) 944;F.C. ZhangandT.M. Rice, Phys. Rev. B, 37 (1988) 3759. [27] J.P. Falck et al., Phys. Rev. B, 48 (1993) 4043, 1281 Y.R. Wang, Phys. Rev. B, 43 (1991) 3786; J. Am#jom and G. Semenoff, Phys. Letr. E, 226 ( 1989) 107; E. Fradkin, Phys. Rev. Lat., 63 (1989) 332; E.J. Mele, Phys. Ser., 27 (1989) 82. [29] Y.R. Wang, J. Wu and M. Franz, Phys. Rev. B, 47 (1993) 12 140; Y.R. Wang and M, Franz, to be published.
SyntheticMetals 80 (1996)
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Magnetism in the large- U single-band Hubbard model Y.R. Wang, M.J. Rice Xerox
Webster Research
Center, 0114-410,
Webster, NY 14580,
USA
Received1 April 1996;accepted15April 1996
Abstract
Usingan exactexpressionof the r-J modelin termsof fermionicholonandlocalspinoperators,we investigatemagnetism in thelarge-U limit of the Hubbardmodelasa functionof doping(xf awayfrom half-filling. The characteristicdopantconcentrationx, belowwhichthe localspinsremainantiferromagnetically correlated,andabovewhichtheyareferromagnetically coupled,isestimated. In theantiferromagnetic regimex
1. Introduction
It is a great pleasurefor us to presentthis paper to honor ProfessorAlan J. Heegeron the occasionof his 60th birthday. Since we expect that most of the contributions to honor him in this volume will be on the subject of conjugatedpolymers and the organic solid state, we felt that it would be fitting to write on a subjectrelated to hisgroup’s work on investigating the elementary excitations of the highly correlated electron systemin the cupric high-temperaturesuperconductors[ 11. The nature of magnetism in a highly correlated electron systemhasbeena constantchallengeto the solidstatetheorist. A simple, yet non-trivial model for describingelectron correlation is the Hubbard model [ 21. From the standpoint of magnetism,the central issuesto be addressedin the study of this model are (1) the coexistence of charge itinerancy and localized magnetic moments,and (2) the interplay between ferromagnetism and antiferromagnetism. Anderson [ 31 pointed out that in the large-U limit and for the half-filled case(one electron per lattice site), the model resultsin antiferromagnetic coupling betweenthe spins.On the other hand, Nagaoka [4] showed that in the infinite-U limit and for a bipartite lattice, a small amount of doping away from hatf0379-6779/96/$15,00 Q 1996Elsevier Science S.A.All rightsreserved PffSO379-6779(96)03697-l
filling gives a ferromagnetic ground statewith maximum total spin, while the addedcharge (electron or hole) is delocalized with a bandwidth equal to that of the non-interacting case. From theseresults it can be inferred [51 that for the caseof a finite value of U, but a value of U still large comparedwith the electron hopping energy f, antiferromagnetic coupling will persistup to somecharacteristicdoping concentrationX, above which ferromagnetic coupling takes over. Recently, we have derived [ 61 an exact expressionof the Hamiltonian +TfWJof the r-J model, i.e., the Hubbard model in the largeU limit, in terms of fermionic holon ( ei) and local spin l/2 (Si) operators,Importantly, this expressiondoesnot require the imposition of a constraint between the spin and holon operators, the bane of previous formulations, and, in consequence, rendersthe t-J model amenableto analysis. While the result of Anderson is already built into the t-Jmodel, the more subtle result of Nagaoka is immediately obtainedfrom the new expression. Encouragedby the latter progress,we considerin thispaper the consequencesfor magnetismthat result from applying to the exact expressionfor XrmJ a mean-field decoupling procedure that we have employed in a previous discussion[7]. An estimation of the characteristic dopant concentration X,
160
Y.R. Wang, M.J. Rice/Synthetic
above which ferromagnetic coupling is favored is first carried out. We then consider the physical features of the two regimes x
2. Theory
For the case of hole doping (away from half-filling), the exact expression of X,-J, in terms of the holon and local spin operators is [ 61
x si.sJ-; (
1
(l+eJ
(1)
where J= 4?/ LJ.In Eq. ( 1) , i runs over all lattice sites (i = 1, 2 , . . ., N, N+ co), while j runs over the z nearest neighbors of
i. The fermionic holon operator ei creates a hole at site i, and the local spin operator Si satisfies the commutation relations of spin l/2, i.e., [ Si+, 5”: ] = 2$si,, [ 3, S,* ] = + SL*6,, and S+S; + S;$’ = 1. Notice that the true spin operator, Si, is related to the local spin operator by
Metals
80 (1996)
159-165
A similar expression for the case of electron doping can also be derived (see Appendix), and is
where 4 is a fermionic operator which creates a ‘doublon’, i.e., a doubly occupied site at i. The electron number at site i i_sni = 1 + dd,, and the true spin operator at site i is given by Sj= ( 1 - d;tdi)Si. Also, reflecting the particle-hole symmetry of the Hubbard model, the sign of r in Eq. (2) is opposite to that which occurs in Eq. ( 1) . In the limit J + 0, Eqs. ( 1) and (4) produce the Nagaoka theorem [ 41. For the case of a bipartite lattice, small concentrations of holes or electrons added to the half-filled band lead to a ferromagnetic ground state with the maximum total spin ( l/2 + 2Si.Sj= l), while the added charges are delocalized with a bandwidth equal to that of the non-interacting case. This follows from the bound, - 1 < l/2 + 25,. S, 5 I, and the property that the density of fermion states of anearestneighbor tight-binding band of a bipartite lattice is symmetric about the center of the band. For the bipartite lattice, the expressions of Eqs. (1) and (4) generalize this theorem to arbitrary concentrations of added charge whenever a meanfield decoupling of the holon and the local spin operators is appropriate. For finite values of J and for the bipartite lattice, 2 explicitly describes a competition between ferromagnetism, favored by the hopping term, and antiferromagnetism, favored by the Heisenberg term. The role of doping away from half-filling in frustrating antiferromagnetic spin ordering is immediately understood from Eqs. ( 1) and (4). For sufficiently large dopant concentrations, ferromagnetic coupling between the local spins will dominate. The characteristic doping concentration (x,) above which this occurs can be estimated by applying to Z a mean-field decoupling procedure that we have previously introduced [ 71. For convenience, in the following we will only discuss the case of hole doping away from the half-filled baud. The extension to the case of electron doping on the basis of Eq. (4) is straightforward. Following-&procedure in [ 71, we obtain: ~t-j = &“h + ~~
(5) m (7)
where rare the Pauli matrices, and CL is the creation operator of an electron at site i. The number of electrons at site i, ni, is given by
&=t( 1/2+2(si*sj),)
02
Y.R. Wang, M.J. Rice /Synthetic
Vjj=J( (si*sj),-
l/4)
(9)
.To= 4t( eieJ)h - J( ( 1 - efei) ( 1 - ejej) )h
(10)
(. . .)r, and (. . .)s denote ensemble averages taken withrespect to the decoupled holon and spin Hamiltonians Zr, and Z”,, respectively. This decoupling procedure is valid only if higher order corrections beyond the decoupling are small. For a 3D lattice with a ferromagnetic ground state, where the spin wave representation of spin fluctuations is appropriate, we can show that the effective interaction between the holons mediated by the spin fluctuations is proportional to Tat low temperature and, therefore, vanishes as T+ 0. This result can be easily understood since for a ferromagnetic ground state, the term Uniirnii in the Hubbard model vanishes at T=O, and the electrons (or equivalently, the holons) behave exactly as spinless fermions. For the other cases, the justification of the decoupling procedure requires more study. Eqs. (8)-( 10) represent a set of self-consistent equations for the parameters entering Zh and Ps. These are the holon hopping amplitude &, the exchange interaction jij between the local spins, and the nearest neighbor holon-holon interaction V,. Although we shall point out the relevance of polaron effects we shall restrict our ex@cit analysis in the present paper to spatially uniform situations for which the latterquantities are constants, independent of i andj. The characteristic concentration X, may be estimated by noting that for a tight-binding band with a constant density of states the holon bond order (ele;), = -x( 1 -x) at T= 0. Consequently, from Eq. ( lo), we have Jv=Jeff=4fx(
1 -x) -J( 1 -x)2(1 -2)
(11)
The condition Jcfi= 0 yields x, = Jl(4t + J) for J$ << 1. This result is similar to a result previously obtained by Inui et al. I:91. At finite temperature T, Jeff is, in principle, temperature dependent, but this temperature dependence is only important if the holon Fermi degeneracy temperature is not large by comparison to T. For x
(12)
where @kdenotes the energy of a magnon of wavevector k, yk = ( 1/z) & exp( ik ’ S), and 6 denotes a nearest-neighbor vector. For the simple cubic lattice,t,/t= -0.097. Eq. (12)
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80 (1996)
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161
applies also for the 1D and 2D AFM spin wave ground states which, however, are inaccurate [ 121 for S = l/2. The salient physical feature of the regime .r
kBTc = zJcff/8
Here and below we shall assume that the holon Fermi energy is much larger than the relevant temperature, so that Jeff can be considered as temperature-independent. While this result is the same as that of the traditional Heisenberg model, there are significant differences for other properties between the present local spin description and the Heisenberg localized spin description. These differences primarily come from the factor that in the present th_eorythe local spins, Sj, are only auxiliary to the true spins, Sj = ( 1 - e:eJS,. For example, the magnetization of the system below the Curie temperature is M(21=((1-ejei>)~g~~,(~),=(l-X)M,(T)
(14)
where M, ( r) is the ‘magnetization’ of the local spins. Using the Weiss molecular field approximation, M,(T) satisfies 1131 &&(T)IM,(O)
=n~~=tanh(ti~~fm,/SkJ)
(15)
where&&(O) =Ng& (S= l/2). Comparing Eqs. (14) and (15) we see that the zero-temperature magnetization, M( 0) = ( 1 -x)M,( O), decreases with increasing doping concentration. More importantly, in our case M(0) is generally not an integer number of the Bohr magneton as in the traditional Heisenberg model, although the saturation magnetization, M(r) lM( 0), satisfies the Brillouin function with s= l/2. In the paramagnetic state with an applied magnetic field B,
the effective spin Hamiltonian is
Y.R. Wang, M.J. Rice /Synthetic
162
i
i
The paramagnetic susceptibility is then easily obtained as x(T)=Ce,f(T-TJ;
T>Tc
(17)
where Ceff= (1 -x)~C, and C=N(gy,)‘/4k, is the Curie constant. We note that, in general, it is not the real number of spins N, = ( 1 - x)N which enters C,, but ( 1 - x) N,. The factor ( 1 -x>’ can be understood by comparing Eqs. ( 14) and (16) which show that compared with the traditional Heisenberg model the effective Bohr magneton entering the highly correlated problem is ( 1 -x> pr+ The Weiss molecular field approximation is not accurate at low temperature or near the transition. This is because the approximation does not include spin correlation and collective excitation. The collective spin excitations of Z”s at low temperature are spin waves. Their excitation spectrumis [ 131 J
hok= (T11 2-c
cos(k.6)
6
1
where S is the nearest-neighbor vector. The deviation of the magnetization from the saturation value, 6M= M( 0) --M(T), due to the thermal excitation of the spin waves is [ 131 &W/M(O)
= (2/Q)(kBT/2TJ,n)3’2[(3/2);
T
(19)
where Q = 1,2 and4for simple ( s.c.), body-centered (b.c.c.) and face centered cubic (f.c.c.) crystals, respectively. Thus, the deviation of the low-temperature magnetization follows the Bloch law. The thermal excitation of the spin waves also contributes to the specific heat of the system. This contribution is also proportional to T3’* at low temperature [ 131, i.e.: C,=NkB(15/4Q)(kBT/2d,ff)3’2{(5/2) =0.964Nk,[SM(T)/M(O)];
TKT,
(20)
Since the holons are itinerant, they contribute a specific heat that is linear in temperature at low temperature. Thus, the total specific heat of the system at low temperature is Cv=C,,+C,=AT+BT3’*; C,= (1?/3)p(+)~T
T-=zT,
(21) (22)
where p( er) is the holon density of states at the holon Fermi energy. We note that Eq. (21) predicts that C,/T is a linear function of T”* at low temperature. The above results show that the properties of the large-U Hubbard model in the ferromagnetic coupling regime exhibit both itinerancy and localized spin characteristics. Interestingly, they are typical of the observed properties of metallic ferromagnets, such as those of the transition metals, Fe, Co and Ni [ 141. For these metals, both a localized spin picture [ 151 and an itinerant electron picture [ 161 have long been proposed. Either of the pictures is successful in explaining many properties, but each has difficulties in explaining some
Metals
80 (1996) 159-165
particular property. The localized spin theory has difficulties in explaining the properties attributed to itinerancy, such as the non-integer number of Bohr magnetons, and the large electronic (linear-T) specific heat, contributed by the 3d electrons. The itinerant electron theories, such as the StonerSlater band theories, have difficulties in obtaining the localized spin properties, such as the Bloch law 6M( T) a T3’*, the T3’* component of the low-temperature specific heat, and the Curie-Weiss paramagnetic susceptibility above T,. The latter is understood as due to the lack of inclusion of electron correlation in the band theory. It is generally believed that the Hubbard model, in its multiband form, is appropriate for describing the 3d electrons of the transition metals, and that its correct solution may bridge the views of the two extreme limits. lhis is supported by more recent calculations which obtain the Curie-Weiss law with the inclusion of electron correlation beyond the usual random phase approximation (RPA) 1171. Theoretical estimation [ 181 indicates that U- W for the transition metals, where w-is the bandwidth of the 3d states. If the Hubbard model with intermediate U is appropriate for the transition metals, the coincidence of our qualitative results with the properties of the transition metals suggests that the large-l/Hubbard model in the ferromagnetic coupling regime shares important physical properties with the intermediate U model. Thus, the intermediate-U model may be solved either starting from the small-U limit, as given by the band theory, or from the large-U limit, as described above. Starting from the large-U limit may be advantageous since the essential physical properties are already obtained at the mean-field level of approximation. There are significant differences between the large-U ferromagnetism, described in this work, and the Stoner-Sister band theory of ferromagnetism. At T = 0, the large47 ferromagnet has all the electron spins of the singly occupied sites parallel to each other (Si.Sj= l/4). It is clear from Eq. (1) or (4) that the electron Fermi surface in this case can be determined directly from the holon (or doublon) Fermi surface. More precisely, the large-V ferromagnet at T = 0 has a Fermi surface spanned only by one Fermi wave vector. In the Stoner-Slater band theory of ferromagnetism, the Fermi surface at T= 0 is spanned by two Fermi wave vectors, one for spin-up (k,? ) and one for spin-down (k,, ). Closely related to this difference is the difference in the spin population between the two kind of ferromagnetisms. In the case of hole doping (i.e., less than one electron per site) and at T= 0, &he large-U ferromagnet has zero spin-down population. The large-U ferromagnet in the case of electron doping at T=O has a finite but minimized spin-down population equal to the number of doubly occupied sites (see Eqs. (A19) and (A20) ). The Stoner-Slater band ferromagnet, however, has a finite spin-down (and spin-up) population independent of whether the system is hole doped or electron doped, Spinpolarization experiments [ 19-221, which can investigate these differences, are therefore capable of distinguishing the large-U ferromagnetism from the Stoner-Slater band ferromagnetism.
Y.R. Wang,
M.J.
Rice/Synthetic
The coexistence of the charge itinerancy and the localized moments is most prominently reflected in the low-temperature specific heat. The contribution from the localized moments can be estimated from 6M( T) /M(O) as given in Eq. (20). For Ni, 6M(T)lM(O) ~7.5 X 10-6T3’2, and for Fe, 6M( T) /M( 0) = 3.4 X 10w6T3” at low temperature [ 131. Eq. (20) then gives the estimate C, ~0.06 X T3’2 mJ/ ( deg.3’2 mol) for Ni, and C,- 0.027 X T3” mJ/(deg.3’2 mol) for Fe. Early experimental measurement [ 23 ] between 1.2 and 4.2 K indeed found the T3’2 contribution, giving C,=O.O26 N 0.038T3’2 mJ/(deg.3’2 mol) for Ni and C,=0.021-0.048T3’2 mJ/(deg.3’2 mol) for Fe. The experimental linear-T terms are found to be 7,OT and 4.7T mJ/(deg. mol) for Ni and Fe, respectively. The measurements are in qualitative agreement with the prediction of Eq. (2 1). It would be interesting to measure the low-temperature specific heat more accurately. Measurements below 1 K are particularly desirable, since the phonon contribution is significantly reduced below 1 K for these materials. NiO and CuO are known large-U systems classified as charge transfer gap systems in the Zaanen, Sawatzky and Allen Scheme, but it may nonetheless be possible to represent the low-energy-scale physics of these oxides by an effective large-U Hubbard model or f-J model [ 241 as discussed by Zhang and Rice 1251, It would be interesting to see if polaronic characteristics can be observed in these materials. Finally, it is generally believed that the large-U Hubbard model is applicable to the copper-oxide superconductors [ 261. Experimentally, it has been shown that two regimes exist, the underdoped regime and the overdoped regime. The superconductivity transition temperature appears to be optimized near the borderline between the two regimes. We speculate that the underdoped and the overdoped regimes correspond, respectively, to the antiferromagnetic and fet-romagnetic coupling regimes discussed here. Thus, in the underdoped regime we expect that the holons are polaronic, and that in the overdoped regime the holons are well delocalized, exhibiting more conventional metallic behavior. This is in accordance with many experimental observations, In particular, in the underdoped regime the effect of doping on the quasi-2D spin excitation spectrum may be approximated as that of a Heisenberg antiferromagnet defected by a static distribution of polarons as we have discussed above. That lattice distortion is involved in the polaron formation has been shown by the experiments of Professor Heeger’s group [ I] and by Falck et al. [ 271. In the Wigner-Jordan ( WJ) spinless fermion representation 1281 of the local spin operators, the static distribution of the holons gives rise to a density of zeroenergy excitations in the excitation spectrum of the WJ fermions. Using this excitation spectrum, it has been shown that many of the observed normal state properties of the copperoxide superconductors can be derived from the t-J model [ 291. In particular, the linear-T &-plane resistivity and the unusual doping and temperature dependence of the d.c. susceptibility are the consequence [29] of the zero-energy WJ fermion excitations. As the doping concentration increases to
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80 (1996)
159-165
163
values close to or beyond the characteristic concentration x,, the holons become itinerant according to our discussion above, The density of zero-energy WJ fermion excitations is expected to be reduced, and the scattering between the holon and the WJ fermion is expected to be more conventional, giving rise to a T2 dependence of the resistivity. These and the magnetic properties in the overdoped regime requiremore experimental investigation. It would be interesting to see if characteristics of ferromagnetic coupling can be found.
Acknowledgements
The authors are grateful to M. Franz for many discussions on the subject of electron doping, M. van Veenendaal for a useful communication, and G.A. Sawatzky for several comments. Appendix Derivation
of the t-J Hamiltonian
in Eq, (4)
Since the derivation of the expression in Eq. (4) for electron doping has several subtle differences from that of hole doping [ 61 in Eq. ( 1)) in this Appendix we present our derivation of Eq. (4). We first define the Hubbard operator at a lattice site i to be X,: = ni, - &, which creates a doubly occupied site, I i,d), if the site is occupied by an electron with spin - g. $e satisfies the following equations: g&,-CT)=
li,d)
(Al)
Xi, I i,d) = I i, - a)
(A21
~~Ii,a)=~~Ii,d)=Xi,Ii,fa)=O
(A31
In order to express the t-J Hamiltonian in terms of {Xi,}, it is necessary to first express the electron density, ni, and the true spin operator, g,, in terms of {X,,}. This task is much easier for the case of hole doping because of the singleoccupancy constraint (nioni, _ (T= 0). For the case of electron doping, we now have ( 1 - ltif ) ( 1 - ni i ) = 0. This gives ni=niT+n,l=l+nirni,=l+~~Xir=l+Xj,Xi,
(A4)
Similarly, we obtain 2~=njt ‘Xi&$ Sj+ =CTTCjL
--nii =njT( 1 -nip)
-~~ji( l-n,t)
-xirzt Cnj~C~~CjJlti~
f-45) = -Xii$i
(A6)
From these relations, we can obtain the spin density ni,. As in Ref. [S], we next map the electron Hilbert space with the constraint of minimum double occupation to the tensor-product Hilbert space of the doublon (a doubly occupied site) and spin states, i.e., I i,cr) + I i,O>, I i,(~)~. A doubIy
Y.R. Wang, M.J. Rice /Synthetic
164
occupied site can have two lid)+ li,l)dl T)SOr lid) + two mappings, we can define operators, Xi, ( + ) and X,,( TQ,(+>=di’&+;
independent mappings, i.e., Ii, l)d I J,)$. Depending on the two sets of auxiliary Hubbard ) . They are given by
K,(+)=&1/2+S;)
$,(-)=4(1/2--q);
$J
-)
(A7) =Gsy
(A81
One can verify, with the respective mapping of I i,d), that both {GW( + ) } amd (zU( - ) } given above satisfy conditions (Al), (A2) and (A3). According to Ref. [5], the Hubbard operators, independent of the mapping of the doubly occupied sites ‘, can be obtained in terms of X,,( + ) and Xia(-):
~~j~,=~
(Aij+B,~~+SjB,+Fi~~-S:F,)
(A91
Metals
80 (1996)
159-165
The ‘ - ’ signsin Eq. (A18) account for the anticommutation relation betweenthe electron operatorswith different spins,i.e., Xi,$p Ii, .J) = - Ii, t ). Finally, it is easyto show that theseresults satisfy all commutation relations and preserve the time-reversal symmetry of the Hubbard operators. The t-J Hamiltonian in Eq. (4) follows from Eqs. (A14), (AlS) and (A21).
References [l] G.Yu, C.H. Lee and A.J. Heeger, Phys. Rev. Left., 67 (1991) 2581; G. Yu et al., Physica C, 190 (1992) 563; G. Yu et al., Phys. Rev. B. 45 (1992) 4964. [2] J. Hubbard, Proc. R. Sue. London, Ser. A, 276( 1963) 238;277( 1963) 237;285(1965)542.
where
P.W. Anderson, Phys. Rev., 115 (1959) 2. [4j Y. Nagaoka, Phys. Rev. B, 147 (1966) 392. [5] N.F. Mott, Metal-insulator Transitions, Taylor and Francis, London, 1974, Ch. 4, p. 124. [6] Y.R.WangandM.J.Rice,Phys.Rev.B,49(1994)436O;G.Khaliuliii, Pis’ma Zh. Eksp. Tear. Fiz., 52 (1990) 999 (JL?TP Lea.. 52 ( 1990) 389); G. Khaliullin and P. Horsch, Phys. Rev. B,47 ( 1993) 463. [7] M.J. Rice and Y.R. Wang, Phys. Rev. B, 48 (1993) 12 921, [S] L.N. Bulaevskii and D.I. Khomskii, Sov.Phys. JEEP, 25 ( 1967) 1067; L.N. Bulaevskii, E.L. Nagaev and D,l. Khomskii, Sov.Phys. JETP, 27 ( 1968) 839; W.F. Brinkman and T.M. Rice, Phys. Rev. B, 2 (1970) 4302. For the 2D problem, see: G. Martinez and P. Horsch, Phys. Rev. B, 44 ( 1991) 317. For a recent review, also see: Yu Lu et al., Chin. J. Phys., in press. [9] M. Inui, S. Doniach and M. Gabay, Physira C, 153-155 (1988) [3]
Aij=Nr<
+ >XjrjC + ) +HmC - )X,ut( - >
(AlO)
Bfj=$q(
+ )Xj,t( + ) -xit,(
C-411)
- )Xjfl,( - )
Fij = ddj (PiST + Q,S,T ) ; i #j
-
FiiSf - SfFf, = 4di( as: + bS,’ )
(A13)
The functions Pi and Qi and the coefficients a and b are to be determined by the requirement that Eq. (A9) reproduces all matrix elements. For example, qCXj, I i, - r) lj,d) = I i,d) Ij, - u>. Applying Eq. (A9), we obtain:
1277.
[lo] H. Bethe, Z. Phys., 71 ( 1931) 205; L. Hulthen, Arr. Mar. Asrron. Fyz.,
~,X,,=~djE(112-S~)(1/2-Sjl)+S:S~];
26A (1938).
i#j $,X,,=4dj[(1/2+Sf)(1/2+5”)
X&
[ 111 S. Liang, B. Doucot and P.W. Anderson, Phys. Rev. Lert., 61 ( 1988)
tA15)
365. I121 Y.R. Wang and M.J. Rice, Phys. Rev. B, 45 [1992) 5045. 1131 C. Kittel, Introduction toSolidSrate Physics, Wiley, New York, 1976, Ch. 15, p. 457; J. Callaway, Quantum Theory of the Solid State,
+S;ST];
i#j xit?Xi, =~lXil
(A14)
=~di
(A161
= (l -4d,)S;S:;
XilXJJ = (1-4di)S+S;
(A17)
XiLXJT=-(l-d,td,)S’; Xi,xY,=-(l-~d,)S;
(A18)
nit =XiJ-$s
+xJXiJ
= (1 -4di)SzS;
+Gd;
(Al%
niL =Xir$t
+gtXi,
= (1s4di)S;S:
+ddi
t-420)
ni=l+ddi;
??j=(l-~di)Si
t-421)
’ Note that all doubly occupied sites are assigned to have the same spin state, although this spin state can be arbitrarily chosen as I T >, I L >, or any linear combination of the two, such as ( I T ) i- / i >) /fi. For details, see: Ref. [5].
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