PHYSICAL REVIEW B VOLUME 56, NUMBER 20 15 NOVEMBER 1997-II

Interplay of and in the orbitally degenerate

Raymond Fre´sard* Department, Shimane University, Nishikawatsu-cho 1060, Matsue 690, Shimane, Japan and Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08855-0849

Gabriel Kotliar Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08855-0849 ͑Received 18 December 1996͒ A slave-boson representation for the degenerate Hubbard model is introduced. The location of the metal- to- transition that occurs at commensurate densities is shown to depend weakly on the band degen- eracy M. The relative weights of the Hubbard subbands depend strongly on M, as well as the magnetic properties. It is also shown that a sizable Hund’s rule coupling is required in order to have a ferromagnetic instability appearing. The metal-to-insulator transition driven by an increase in temperature is a strong function of it. ͓S0163-1829͑97͒02144-9͔

There has been dramatic progress in our understanding of rise to the Fermi disappears. Such a coherent- the Mott transition in the last few years. Careful experimen- incoherent transition is observed experimentally in V2O3 tal studies of systems in the vicinity of the Mott transition ͑Refs. 6 and 7͒ and can also be obtained out of the slave- have been carried out.1 Two new theoretical tools, slave- boson mean-field theory.8 9,10 boson mean-field theories ͑see, for instance, Fre´sard and Compared to the variational wave-function approach Wo¨lfle2 and references therein͒, and the limit of infinite di- our formalism is more flexible since, as we demonstrate in mensions have been adapted to its study. For a review see this paper, it allows us to calculate a variety of quantities that Georges et al.3 Most of the modern work has focused on the are not easily accessible to the variational approach, as a single-band Hubbard model. Now that both the metallic and function of the correlation strength and . It can also be the Mott insulating states of the ͑doped͒ titanate and vanadi- improved systematically by performing a loop expansion ates have been studied experimentally1,4 ͑corresponding to around the saddle point. Our main results are as follows: 3d1 and 3d2 configurations in the Mott insulating state͒, ͑a͒ Low-energy single-particle quantities such as the criti- there is a need for a theoretical framework allowing for un- cal value of the interaction strength of the transition, the derstanding the Mott transition for arbitrary degeneracy and residue, and the single-particle Mott-Hubbard density. This paper is aimed at providing such a technique gap depend only weakly on degeneracy. This justifies the 11 and applying it to a variety of quantities that cannot be ob- agreement between theory of Rozenberg et al. and experi- tained easily using alternative approaches. Most of the re- ment on orbitally degenerate systems. sults are obtained in a closed analytical form, allowing for a ͑b͒ The relative weights of the Hubbard bands depend 12 qualitative understanding of the physical situation. strongly on degeneracy, in agreement with other methods. In this work we investigate the effect of strong Coulomb ͑c͒ The coherence temperature decreases with increasing interaction in systems with orbital degeneracy. Such a situ- band degeneracy. ation is realized in virtually all transition-metals and transi- ͑d͒ The magnetic properties, in particular the magnetic tion metal . These systems contain d in cubic susceptibility and its associated Landau parameter in the or trigonal environments. The crystal field can only lift par- paramagnetic and the magnetic phase diagram, are tially the degeneracy of the d bands, down to two as is the strongly modified with respect to the one-band case. The Hamiltonian describing the low-energy properties of case of V2O3 ͑Ref. 5͒ or three as in LaTiO3. Our goal is to understand how degeneracy affects the behavior of the dif- these systems is commonly written as ferent physical quantities near the Mott transition. To carry out the investigation we extend the slave-boson technique, † Hϭ ͚ ti, jci,␳,␴c j,␳,␴ϩU3͚ ni,␳, ni,␳, which has been very successful in the study of the Mott i, j,␴,␳ i,␳ ↑ ↓ transition, to the orbitally degenerate case. A remarkable feature of the dynamical mean-field solu- ϩU1 ͚ ni,␳, ni,␳Ј, ϩU ͚ ni,␳,␴ni,␳Ј,␴ , tion to the large-dimension limit of the Hubbard model is the i,␳ЈÞ␳ ↑ ↓ i,␴,␳ЈϽ␳ metal-to-insulator transition that occurs in the vicinity of the ͑1͒ Mott transition under an increase of the temperature.3 In the metallic phase the spectrum of the one- Green’s where ␴ is a spin index for the up and down states while ␳ is function consists of two incoherent excitation branches and labeling the M bands. U3 describes the on-site interaction one coherent quasiparticle peak, which is precisely absent in term between two particles in the same band but with oppo- the insulating state. There is thus a coherence temperature site spin. U1 relates to a pair of particles with opposite spin Tcoh at which the coherence of the interacting system giving and different band index. U finally concerns the case of

0163-1829/97/56͑20͒/12909͑7͒/$10.0056 12 909 © 1997 The American Physical Society 12 910 RAYMOND FRE´ SARD AND GABRIEL KOTLIAR 56

Bose fields in ͑3͒ to be given by their classical values, but by introducing normalization factors L␣ and R␣ ͑Refs. 14 and 2͒ as

2M † z␣ϭ ͚ ͚ mϭ1 ␣1Ͻ•••Ͻ␣mϪ1 ϫ␺†͑m͒ L R ␺͑mϪ1͒ , ␣,␣1 ,...,␣mϪ1 ␣ ␣ ␣1,...,␣mϪ1

␣iÞ␣, ͑4͒ where

2MϪ1 Ϫ1/2 R ϭ 1Ϫ ␺†͑m͒ ␺͑m͒ , ␣ ͚ ͚ ␣1 ,...,␣m ␣1,...,␣m ͫ mϭ0 ␣1 Ͻ Ͻ␣m ͬ •••

␣iÞ␣,

2M FIG. 1. Inverse effective mass in the two-band model as a func- tion of density for several values of U. L␣ϭ 1Ϫ ͚ ͫ mϭ1 equal spin. In the two-band model rotational symmetry re- Ϫ1/2 13 ϫ ␺†͑m͒ ␺͑m͒ . quires UnϭUϩnJ. The relationship between the couplings ͚ ␣,␣1 ,...,␣mϪ1 ␣,␣1,...,␣mϪ1 ␣1Ͻ Ͻ␣mϪ1 ͬ is derived and discussed in the Appendix for a model that is ••• relevant to the titanates. The description with only three dif- ͑5͒ ferent interaction strengths is a simplified version of the full Namely, L normalizes to one the probability that no elec- problem with completely general exchange interaction. In ␣ tron in state ␣ is present on a site before one such electron that case the coupling constants would not be completely ͉ ͘ hops on that particular site, and R makes sure that it hap- independent from one another, and relationships between ␣ pened. Clearly the eigenvalues of the operators L and R them would follow from rotational symmetry as well. Here ␣ ␣ are one in the physical subspace. Now, the redundant degrees we consider a generic model where the number of indepen- of freedom are projected out with the constraints dent parameters is kept small for simplicity. Taking J finite accounts for the Hund’s rule coupling. 2M As for any model with on-site interaction, a slave-boson f † f Ϫ ␺†͑m͒ ␺͑m͒ ϭ0, ␣ ␣ ͚ ͚ ␣,␣1 ,...,␣mϪ1 ␣,␣1,...,␣mϪ1 representation can be introduced, mapping all the degrees of mϭ1 ␣1Ͻ•••Ͻ␣mϪ1 freedom onto bosons. We can rewrite any atomic state with 2M the help of a set of pseudofermions ͕f ␣͖ and slave bosons (m) (m) †͑m͒ ͑m͒ ␺ ,..., ␺ ,..., Ϫ1ϭ0. ͑6͒ ͕␺␣ ,...,␣ ͖(0рmр2M). ␺␣ ,...,␣ is the slave boson as- ͚ ͚ ␣1 ␣m ␣1 ␣m 1 m 1 m mϭ0 ␣1Ͻ•••Ͻ␣m sociated with the atomic state consisting of m electrons in We obtain the Lagrangian at Jϭ0as states ͉␣1 ,...,␣m͘ where ␣ is a composite spin and band index. By construction it is symmetric under any permutation of two indices, and zero if any two indices are equal. We can † ␶Ϫ␮ϩi␭i,␣͒fi,␣Ϫi⌳iץ͑␣,Lϭ͚fi now write the creation operator of a physical electron in i,␣ terms of the slave particles as †͑m͒ m ␶ϩi⌳iϩUץ ϩ ␺ † ˜ † † ͚ ͚ i,␣1 ,...,␣m ͩ2ͪ c ϭz f . ͑2͒ i,m ␣1Ͻ Ͻ␣m ͫ ␣ ␣ ␣ ••• † m ˜z ␣ describes the change in the boson occupation numbers Ϫi ␭ ␺͑m͒ ϩ t z† f† z f . when an electron in state ␣ is created as ͚ i,␣j i,␣ , ,␣ ͚ i,j i,␣ i,␣ j,␣ j,␣ jϭ1 ͬ 1 ••• m i,j,␣ 2M ͑7͒ ˜z † ϭ ␺†͑m͒ ␺͑mϪ1͒ , ␣ ͚ ͚ ␣,␣1 ,...,␣mϪ1 ␣1,...,␣mϪ1 mϭ1 ␣1Ͻ•••Ͻ␣mϪ1 We now proceed to the mean-field theory, and we inves- tigate the paramagnetic, paraorbital saddle point. The latter is

␣iÞ␣. ͑3͒ obtained after integrating out the fermions, and setting all bosonic fields to their classical value. The Mott transition ˜ † The operators z ␣ in Eq. ͑3͒ describe the change in the that occurs at commensurate density n is best discussed by slave-boson occupation as a many-channel process. In order projecting out occupancies that are larger than nϩ1 and to recover the correct noninteracting limit at mean-field smaller than nϪ1 ͑if any͒. The constraints allows for elimi- level, one has to observe that the classical probability for nating the variables ␺(nϪ1) and ␺(n) to obtain the grand po- these processes to happen is not simply given by taking the tential at n as 56 INTERPLAY OF MOTT TRANSITION AND . . . 12 911

2 2 2 2 ͑n,M ͒2 2 ⍀͑D͒ϭ͑1Ϫ2D ͒D ͑ͱbn,Mϩͱcn͒ ⑀0 m ͑ͱbn,Mϩͱcn͒ Uc ϪU ϭz2ϭ . ͑11͒ 8 ͑n,M ͒2 n m* Uc ϩU D2ϩ Ϫ␮␳, ͑8͒ ͫ ͩ 2ͪͬ 2 2 Due to the particular form of the coefficients b and c the with ⑀0ϵ2M͐d⑀⑀␳(⑀)fF(z ⑀ϩ␭0Ϫ␮), D 2M (nϩ1)2 dependence on the band degeneracy is weak. The critical ϵ(nϩ1)␺ , bn,Mϵ(2MϪnϩ1)/(2MϪn), and interaction strength increases with M so the quasiparticle cnϵ(nϩ1)/n. Minimizing Eq. ͑8͒ with respect to D yields a residue Z increases slightly with M. For small values of U critical interaction strength at which D vanishes. It reads ͑which we treated without projecting out higher occupan- cies͒, Z decreases with increasing M. So there is a crossover U͑n,M ͒ϭϪ⑀ ͑ͱb ϩͱc ͒2, ͑9͒ c 0 n,M n value of the interaction strength beyond which the system which reproduces the results of the Gutzwiller becomes more metallic with increasing M.15 As a function of approximation.9,10 This locates the Mott transition. Restrict- the hole doping ␦, the quasiparticle residue vanishes for ␦ (n,M) ing ourselves to a flat we can relate the going to 0 above Uc as critical interaction strength to the band width W. We obtain

nW ␦ ͉␦͉ U͑n,M ͒ϭ ͑2MϪn͒͑ͱb ϩͱc ͒2. ͑10͒ Zϭ ͑bn,MϪcn͒ϩ ͓͑bn,Mϩcn͒ͱ1ϩ4␸n,M c 4M n,M n 2 2 10 Its band-degeneracy dependence is fairly weak. The effec- ϩ4ͱbn,Mcn␸n,M͔, ͑12͒ tive mass of the diverges at the Mott transi- tion. We obtain where we introduced

͑n,M ͒2 4 Uc bn,Mcn /͑ͱbn,Mϩͱcn͒ ␸ ϵ . ͑13͒ n,M ͑n,M ͒ ͑n,M ͒ 2 ͑UϪUc ͒͑UϪUc ͓͑ͱbn,MϪͱcn͒/͑ͱbn,Mϩͱcn͒ ͔͒ The expression of the quasiparticle residue consists of two contributions that are either symmetric or antisymmetric with respect to particle or hole doping. The antisymmetric contribution vanishes for nϭM as a consequence of the particle-hole symmetry. The asymmetry of Z on particle or hole doping is seen to increase under an increase of ͉nϪM͉. For nϽM, Z(␦) vanishes more slowly for hole doping than for particle doping. Increasing the degeneracy for fixed n, or increasing the degeneracy at nϭM, makes the rate at which Z(␦) vanishes smaller. Increasing the interaction strength has the same consequence. As an example we calculate the effective mass for the two-band model without projecting out higher occupancies and show it on Fig. 1. Interestingly we also obtain a Mott gap. Indeed the number of quasiparticles is a continuous function of their chemical potential ␮Ϫ␭0/2. However, the saddle-point equations show that the Lagrange multiplier ⌳ jumps when going through the Mott gap, which implies that ␭ is going to jump as well, and so does ␮. As a result we obtain the Mott gap ⌬ϵlim␦ 0Ϫ␮(␦)Ϫlim␦ 0ϩ␮(␦)as → → ͑n,M͒ ͑n,M͒ 2 ⌬ϭͱ͑UϪUc ͕͒UϪUc ͓͑ͱbn,MϪͱcn͒/͑ͱbn,Mϩͱcn͔͒ ͖. ͑14͒

(n,M) In the limit of UӷUc , the Mott gap is given by U, trend is clearly reproduced and the quantitative agreement is while it closes at U(n,M) as ⌬ϳU(n,M)ͱU/U(n,M)Ϫ1, the very satisfactory. According to the above discussion, includ- c c c ing the threefold degeneracy of the d band in order to ac- square-root behavior being typical of slave-boson mean-field 17 theories. It can be read from Fig. 2 where it is compared to count for the experimental situation only makes a small the one-band case as obtained by Lavagna.16 Clearly going difference as compared to the nondegenerate case. We now turn to the Hund’s rule coupling dependence and from one band to two bands does not imply a big difference treat as an example the two-band model around the nϭ1 in the Mott gap. Indeed we obtain that /U(n,M) is indepen- ⌬ c Mott insulating lobe. At ␳ϭ1 the grand potential at the dent of M at nϭM, while for fixed n it depends very weakly saddle point reads on M. Our result can be compared to experimental data. For the 4 17 ⍀ϭ ⑀ ͑1Ϫ2r2͓͒rϩ͑d ϩd ϩ⌬ ͒/ͱ2͔2ϩ͑Uϩ3J͒⌬2 series LaxY1ϪxTiO3, Okimoto et al. measured how the gap 3 0 0 x 0 0 depends on the bandwidth. Assuming ͑for large ratio U/W) 2 2 ⌬ϳUϪW we obtain out of their data Uϭ3.2 eV. Inserting ϩ͑UϩJ͒dx ϩUd0Ϫ␮␳, ͑15͒ this and the experimental value of (U/W)cϳ1.3 into Eq. ͑14͒ (2) (2) (2) (2) we can compute ⌬/W for Mϭ3 as a function of W/U and with d0ϵ(␺ , ϩ␺ , )/ͱ2, dxϵ(␺ , ϩ␺ , )/ͱ2, (2) (2) ↑ ↑ ↓ ↓2 2 2 2 ↑ ↓ ↓ ↑ compare it with experiment on Fig. 3. The experimental ⌬0ϵ(␺ ,0ϩ␺0, )/ͱ2, r ϵd0ϩdxϩ⌬0 , and ␭ϵ͚␣␭␣/2, ↑↓ ↓↑ 12 912 RAYMOND FRE´ SARD AND GABRIEL KOTLIAR 56

Another intriguing feature of transition-metal oxides such as V2O3 is the metal-to-insulator transition that occurs in the vicinity of the tricritical point under an increase of temperature.6,7 It has recently been interpreted11 as the tran- sition from a Fermi liquid with finite quasiparticle residue Z to an insulator with Zϭ0. This interpretation has been ob- tained in the framework of the dynamical mean-field ap- proximation, which yields equations that have two different solutions, one that resembles a Fermi liquid with a finite Z and the other, which is totally incoherent. In the slave-boson mean-field theory and at zero temperature, the quasiparticle (M) residue Z vanishes continuously as U approaches Uc ,as given by Eq. ͑11͒. In contrast this behavior does not hold at finite temperature8 where the saddle-point equations acquire a higher degree of nonlinearity since ␧0 becomes a function of Z. Thus, for given U, they admit two solutions for UϽUc2, and only one for UϾUc2. Among the two solu- tions, one closely resembles the metallic solution with finite Z of the large d. The second one, characterized by a strongly FIG. 2. Chemical potential for nϭ1 for the one-band ͑dashed renormalized effective mass, is not really a good description line͒ and two-band ͑full line͒ models. of the insulator because it does not have the incoherent parts. Nevertheless that is really the best mean-field theory can do. and we have used the constraints to remove the variables Given the resemblance of one of the slave-boson solutions to ␺(0) and ␺(1). the metallic large-d solution it makes sense to ask at which Such an expression differs from an ordinary Ginzburg- temperature that solution ceases to exist, that is, our calcu- Landau free energy in the respect that it cannot be written as lated Tcoh . When both solutions become degenerate there a fourth-order polynomial in the variables d0, dx , and ⌬0. is a first-order metal-to-insulator transition at a critical (M) As a result, if there were to be a critical point for one field, it Uc (T): would be critical for the other ones as well. For small J we ͑M͒ ͑M͒ ͑M͒ obtain the location of the Mott transition as Uc ͑T͒ϭUc ͑0͒Ϫͱ8Uc ͑0͒Tln͑2M ͒. ͑18͒ 4 J Thus an increase in temperature may produce a metal-to- U͑2͒ ϭU͑2͒ 1Ϫ ϩO͑J/U͒2 . ͑16͒ c,͑J͒ c,͑0͒ͩ 3 U ͪ insulator transition, which is consistent with the experimen- tal situation in V2O3. Here Tcoh is given by Another regime of interest is the large-J regime. There we ͑M ͒ 2 obtain the location of the Mott transition as ͓UϪUc ͑0͔͒ T ϭ ͑19͒ coh ͑M ͒ 2 8Uc ͑0͒ln͑2M ͒ 2 8 ⑀0 ⑀0 U͑2͒ϭϪ ⑀ ͑3ϩ2ͱ2͒ 1Ϫ ϩO ͑17͒ c 3 0 ͩ 9 J ͪ ͩͩ J ͪ ͪ and thus decreases under an increase of M. In the dynamical mean-field approximation at finite temperature there is an and thus decreasing J from infinity leads to an increase of the interaction strength Uc2(T) at which the metallic solution critical interaction. ceases to exist. This quantity can also be evaluated in our slave-boson scheme and, at nϭ1, is given by

͑M ͒ ͑M ͒ 2/3 Uc2 ͑T͒ϭUc ͑0͓͒1Ϫ␣M͑T/W͒ ͔, ͑20͒

with ␣1ϳ2.53 and ␣2ϳ3.32. We now turn to the calculation of the magnetic suscepti- bility. Here we generalize the calculation of Li et al.18 to the two-band model. The linear response to an external magnetic field is obtained as a one-loop calculation of the correlation function of the slave-boson fields in the spin-antisymmetric band-symmetric channel. Three fields couple in this 15 1 (1) (2) (2) channel: ␹Ϫϵ 2 ͚␣␴␺␣ , ␹ϩϵ1/ͱ2(␺ , Ϫ␺ , ), and 1 ↑ ↑ ↓ ↓ ␬ϵ 2 ͚␣␴␭␣ , and the magnetization is expressed in terms of (1) slave bosons as ϭ4d0␹ϩϩ2␺ ␹Ϫ . The resulting sus- ceptibility arisesM as a random-phase approximation form15

␹0 ϭ FIG. 3. Dependence of the Mott gap on the bandwidth for nϭ1 ␹S a . ͑21͒ and Mϭ3. Circles: experimental data of Okimoto et al.17 1ϩF0␹0 /N͑EF͒ 56 INTERPLAY OF MOTT TRANSITION AND . . . 12 913

FIG. 4. Instability line of the paramagnetic phase for U/Jϭ10 FIG. 5. Spectral weight of the upper ͑dashed lines͒ and lower ͑dashed line͒ and U/Jϭ5 ͑solid line͒. The diamond ͑square͒ indi- ͑solid lines͒ Hubbard bands for the 1 band and 2 band models, at (M) cates the position of the Mott transition for U/Jϭ10 (U/Jϭ5). Uϭ2Uc .

We now determine the instability line of the paramagnetic each sub-band. In our language the low-energy excitations (1) phase with respect to ferromagnetism. For Jϭ0 we find no are involving the field ␺ , and the high-energy excitations (2) ferromagnetic instability even near the Mott transition, while centered around U the field ␺ . Higher-energy excitations for finite J we find that the Mott metal-to-insulator transition involving higher local occupancies are left out. We obtain the spectral weights in both bands by evaluating explicitly may be preempted by the appearance of a ferromagnetic † phase. In other words, a sufficiently strong Hund’s rule cou- the anticommutator ͕͗c␣ ,c␣͖͘ using the decomposition of pling turns a Mott insulator into a ferromagnet. Originally the physical electron operator ͓Eqs. ͑2͒ and ͑3͔͒. Accordingly † the Hubbard model was introduced in order to describe fer- the spectral weight of the Green’s function T͗c␣(␶)c␣(0)͘ † romagnetism in narrow-band systems, but it has been re- in the lower Hubbard band (WLHBϭ͕͗cL␣ ,cL␣͖͘) and the † † † cently established that the ground state is not ferromagnetic upper Hubbard band (WUHB͕͗cU␣ ,cU␣͖͘), where cL␣ (cU␣) † †(1) for any reasonable values of the parameters on the square is the contribution to c␣ , which is involving the field ␺␣ lattice.19,20 Introducing a next-nearest-neighbor hopping term †(2) (␺␣,␣ ) in Eq. ͑3͒, are given by in the Hamiltonian,21–24 which has the effect of shifting the 1 van Hove singularity of the non-interacting system away W ϭ †͑0͒ ͑0͒ϩ †͑1͒ ͑1͒ , from the center of the band, may lead to a ferromagnetic LHB ͗␺ ␺ ␺␣ ␺␣ ͘ ground state in the vicinity of the van Hove singularity, even for realistic values of the coupling strength. We find that the †͑1͒ ͑1͒ †͑2͒ ͑2͒ WUHBϭ ͗␺␤ ␺␤ ϩ␺␣␤ ␺␣␤͘ ͑22͒ ground state is much more likely to be ferromagnetic in the ␤͚Þ␣ degenerate model for finite J, as shown in Fig. 4. Our method can be applied to the calculation of dynami- and are shown in Fig. 5. Here the weights do not quite add cal quantities too. In the slave-boson method,25 the Green’s up to 1 in the two-band case because we projected out occu- function that one obtains closely resembles the exact large-d pations larger than 2. In other words, on top of the two sub- result. It namely consists of a quasiparticle resonance and bands that are considered here, there appears a second upper two Hubbard bands, which can be clearly separated. How- Hubbard band ͑centered around 3UϪ3␮, corresponding to ever, the sum rules are not quite satisfied for Nϭ2. Here we triple occupancy͒ that is becoming relevant in the particle use a decomposition that is more closely related to the Hub- doped domain. To a very good accuracy its contribution to bard operator algebra treatment, and the idea is to calculate the spectral weight is given by 1ϪWLHBϪWUHB . Clearly the spectral weights of the Hubbard operator Green’s func- the degeneracy plays an important role as the weight of the tions. We really do not know how to separate that weight upper band at nϭ1 in the strong coupling regime is given by into a lower band and a quasiparticle piece, but near half- (2MϪ1)/2M. filling the resonance has a small weight, and we can identify In summary we introduced a slave-boson representation the weight in the Hubbard bands with the spectral weight of of the degenerate Hubbard model. We obtained that the band the corresponding Hubbard operator Green’s functions. In degeneracy has a weak influence on the location of the Mott the strong coupling regime the one-particle excitation spec- transition, while the degeneracy temperature and the dynami- trum is split off into several pieces, each of them carrying cal and magnetic properties strongly depend on it. We also some fraction of the spectral weight ͑which are adding up to showed that no ferromagnetic instability occurs unless the 1 so as to fulfill the sum rule͒. The various pieces follow Hund’s rule coupling becomes sizable, yielding a generic from the discrete atomic levels, which are well separated by scenario for ferromagnetism in transition metals and multiples of U, broadened by exchange processes. Let us transition-metal oxides. In that case a ferromagnetic instabil- now determine the fraction of the spectral weight carried by ity may even shadow the Mott transition. 12 914 RAYMOND FRE´ SARD AND GABRIEL KOTLIAR 56

Note added. After this work was completed we became k rϽ aware of related work by H. Hasegawa ͓cond-mat/9612142 Jϭ r2drrЈ2drЈ R͑r͒2R͑rЈ͒2, ͚ ͵ kϩ1 ͑unpublished͒; cond-mat/9704168 ͑unpublished͔͒. k,m rϽ R.F. gratefully acknowledges financial support from the k Fonds National Suisse de la Recherche Scientifique under ͵ d⍀␸␣͑⍀͒Ym͑⍀͒␸␤͑⍀͒ Grant No. 8220-040095, as well as Rutgers University and Neuchaˆtel University for hospitality where part of this work k has been done. This work has been partially supported by the ϫ͵ d⍀Ј␸␤͑⍀Ј͒Ym͑⍀Ј͒␸␣͑⍀Ј͒, NSF under Grant No. DMR 95-29138. ͑A7͒ where R(r) and ␸ (⍀) are respectively the radial and angu- APPENDIX ␣ lar part of the wave function ␸␣(r). As usual, using, for In this appendix we derive the Hamiltonian that is rel- instance, density-functional theory, the computation of the evant to the titanates, Eq. ͑1͒. In a cubic environment, the matrix elements can be brought down to the computation of crystal field is partially lifting the degeneracy of the d band, three numbers, F0, F2, and F4. They are defined as by lowering the energy of the threefold degenerate T or- 2g 1 bital, and by raising the energy of the twofold degenerate eg 2 2 2 2 F0ϭ ͵ r drrЈ drЈ R͑r͒ R͑rЈ͒ , orbital. Here we concentrate on the T2g electrons, for which rϾ the interacting part of the Hamiltonian is given by 1 r2 2 2 Ͻ 2 2 F2ϭ r drrЈ drЈ R͑r͒ R͑rЈ͒ , ͑A8͒ H ϭ ␣␤ e2/r ␥␦ c† c† c c , 49͵ 3 int ͚ ͚ ͗ ͉ ͉ ͘ ␣,␴ ␤,␴Ј ␦,␴Ј ␥,␴ rϾ ␴,␴Ј ␣,␤,␥,␦ ͑A1͒ 4 1 rϽ F ϭ r2drrЈ2drЈ R͑r͒2R͑rЈ͒2 where the indices ␣,␤,␥,␦ label the dxy , dxz , and dyz orbit- 4 441͵ 5 als. The nonvanishing matrix elements of the Coulomb inter- rϾ action are grouped into direct and exchange interactions. and one obtains There is a diagonal direct coupling: uϭF0ϩ4F2ϩ36F4 , uϭ͗␣␣͉e2/r ͉␣␣͘, ͑A2͒ uЈϭF ϪF Ϫ4F , ͑A9͒ an off-diagonal direct coupling 0 2 4

2 5 45 uЈϭ͗␣␤͉ e /r ͉␣␤͘, ͑A3͒ Jϭ F ϩ F . 2 2 2 4 and two exchange couplings In the limit where F4ӶF2, we obtain Jϭ͗␣␤͉ e2/r ͉␤␣͘, 2JϭuϪuЈ. ͑A10͒ 2 aϭ͗␣␣͉ e /r ͉␤␤͘. ͑A4͒ Setting Explicitly, introducing the Wannier real wave functions UϵuϪ3J, ͑A11͒ ␸␣(r), we find for J: and inserting the above-found matrix elements of the Cou- 2 † † e lomb interaction in Eq. ͑A1͒ we obtain the Hamiltonian Eq. Jϭ drdrЈ␸␣͑r͒␸␤͑rЈ͒ ␸␤͑r͒␸␣͑rЈ͒. ͑A5͒ ͵ ͉rϪrЈ͉ ͑1͒ provided one neglects the two following small terms:

The reality of the wave functions implies † † HnegϭJ c␣␴c␣␴ c␤␴ c␤␴ ␣͚Þ␤ ͚ Ј Ј aϭJ. ͑A6͒ ␴Þ␴Ј Using the partial wave expansion of the Coulomb potential ϩJ c† c† c c . ͑A12͒ ͚ ͚ ␣␴ ␤␴Ј ␣␴Ј ␤␴ one obtains ␣Þ␤ ␴Þ␴Ј

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