PHYSICAL REVIEW B VOLUME 56, NUMBER 20 15 NOVEMBER 1997-II Interplay of Mott transition and ferromagnetism in the orbitally degenerate Hubbard model Raymond Fre´sard* Physics 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 1996! A slave-boson representation for the degenerate Hubbard model is introduced. The location of the metal- to-insulator 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. @S0163-1829~97!02144-9# There has been dramatic progress in our understanding of rise to the Fermi liquid 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 7! 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 therein!, 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 doping. It can also be the Mott insulating states of the ~doped! 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 state!, ~a! 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 quasiparticle 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 ~b! 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 ~c! 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- ~d! The magnetic properties, in particular the magnetic tion metal oxides. These systems contain d electrons in cubic susceptibility and its associated Landau parameter in the or trigonal environments. The crystal field can only lift par- paramagnetic phase 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. 5! 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, † H5 ( ti, jci,r,sc j,r,s1U3( ni,r, ni,r, which has been very successful in the study of the Mott i, j,s,r i,r ↑ ↓ transition, to the orbitally degenerate case. A remarkable feature of the dynamical mean-field solu- 1U1 ( ni,r, ni,r8, 1U ( ni,r,sni,r8,s , tion to the large-dimension limit of the Hubbard model is the i,r8Þr ↑ ↓ i,s,r8,r metal-to-insulator transition that occurs in the vicinity of the ~1! Mott transition under an increase of the temperature.3 In the metallic phase the spectrum of the one-electron Green’s where s is a spin index for the up and down states while r 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 0163-1829/97/56~20!/12909~7!/$10.0056 12 909 © 1997 The American Physical Society 12 910 RAYMOND FRE´ SARD AND GABRIEL KOTLIAR 56 Bose fields in ~3! to be given by their classical values, but by introducing normalization factors La and Ra ~Refs. 14 and 2! as 2M † za5 ( ( m51 a1,•••,am21 3c†~m! L R c~m21! , a,a1 ,...,am21 a a a1,...,am21 aiÞa, ~4! where 2M21 21/2 R 5 12 c†~m! c~m! , a ( ( a1 ,...,am a1,...,am F m50 a1 , ,am G ••• aiÞa, 2M FIG. 1. Inverse effective mass in the two-band model as a func- tion of density for several values of U. La5 12 ( F m51 equal spin. In the two-band model rotational symmetry re- 21/2 13 3 c†~m! c~m! . quires Un5U1nJ. The relationship between the couplings ( a,a1 ,...,am21 a,a1,...,am21 a1, ,am21 G is derived and discussed in the Appendix for a model that is ••• relevant to the titanates. The description with only three dif- ~5! 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 a tron in state a is present on a site before one such electron that case the coupling constants would not be completely u & hops on that particular site, and R makes sure that it hap- independent from one another, and relationships between a pened. Clearly the eigenvalues of the operators L and R them would follow from rotational symmetry as well. Here a a 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 2 c†~m! c~m! 50, a a ( ( a,a1 ,...,am21 a,a1,...,am21 representation can be introduced, mapping all the degrees of m51 a1,•••,am21 freedom onto bosons. We can rewrite any atomic state with 2M the help of a set of pseudofermions $f a% and slave bosons (m) (m) †~m! ~m! c ,..., c ,..., 2150. ~6! $ca ,...,a %(0<m<2M). ca ,...,a is the slave boson as- ( ( a1 am a1 am 1 m 1 m m50 a1,•••,am sociated with the atomic state consisting of m electrons in We obtain the Lagrangian at J50as states ua1 ,...,am& where a 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 † L5(fi,a~]t2m1ili,a!fi,a2iLi now write the creation operator of a physical electron in i,a terms of the slave particles as †~m! m 1 c ]t1iLi1U † ˜ † † ( ( i,a1 ,...,am S2D c 5z f . ~2! i,m a1, ,am F a a a ••• † m ˜z a describes the change in the boson occupation numbers 2i l c~m! 1 t z† f† z f . when an electron in state a is created as ( i,aj i,a , ,a ( i,j i,a i,a j,a j,a j51 G 1 ••• m i,j,a 2M ~7! ˜z † 5 c†~m! c~m21! , a ( ( a,a1 ,...,am21 a1,...,am21 m51 a1,•••,am21 We now proceed to the mean-field theory, and we inves- tigate the paramagnetic, paraorbital saddle point. The latter is aiÞa. ~3! obtained after integrating out the fermions, and setting all bosonic fields to their classical value. The Mott transition ˜ † The operators z a in Eq. ~3! 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 n11 and to recover the correct noninteracting limit at mean-field smaller than n21 ~if any!. The constraints allows for elimi- level, one has to observe that the classical probability for nating the variables c(n21) and c(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 V~D!5~122D !D ~Abn,M1Acn! e0 m ~Abn,M1Acn! Uc 2U 5z25 .