Effect of Crystal-Field Splitting and Inter-Band Hybridization on the Metal-Insulator Transitions of Strongly Correlated Systems
Alexander I. Poteryaev, Michel Ferrero, Antoine Georges, Olivier Parcollet
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Effect of Crystal-Field Splitting and Inter-Band Hybridizationon the Metal-Insulator Transitions of Strongly Correlated Systems
Alexander I. Poteryaev, Michel Ferrero, Antoine Georges, and Olivier Parcollet Centre de Physique Th´eorique, UMR 7644, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France Institut de Physique Th´eorique, CEA, IPhT, CNRS,URA 2306, F-91191 Gif-sur-Yvette Cedex, France (Dated: November 12, 2018)We investigate a quarter-filled two-band Hubbard model involving a crystal-field splitting, whichlifts the orbital degeneracy as well as an inter-orbital hopping (inter-band hybridization). Both termsare relevant to the realistic description of correlated materials such as transition-metal oxides. Thenature of the Mott metal-insulator transition is clarified and is found to depend on the magnitudeof the crystal-field splitting. At large values of the splitting, a transition from a two-band to a one-band metal is first found as the on-site repulsion is increased and is followed by a Mott transitionfor the remaining band, which follows the single-band (Brinkman-Rice) scenario well documentedpreviously within dynamical mean-field theory. At small values of the crystal-field splitting, a directtransition from a two-band metal to a Mott insulator with partial orbital polarization is found,which takes place simultaneously for both orbitals. This transition is characterized by a vanishingof the quasiparticle weight for the majority orbital but has a first-order character for the minorityorbital. It is pointed out that finite-temperature effects may easily turn the metallic regime into abad metal close to the orbital polarization transition in the metallic phase.
PACS numbers: 71.27.+a,71.70.Ch,71.30.+h,71.10.Fd
I. INTRODUCTION
The Mott metal-insulator transition plays a key rolein the physics of strongly correlated electron materi-als. Over the last fifteen years, our theoretical under-standing of this phenomenon improved considerably, dueto the development of the dynamical mean-field theory(DMFT) (Refs. 3 and 4). A number of model stud-ies were performed in order to clarify the nature of thetransition in both a single-orbital and multi-orbital con-text .In the context of real materials, however, several im-portant features must be considered, which are not al-ways taken into account in model studies. This includesin particular two key aspects: (i) the breaking of orbitaldegeneracy by the crystalline environment and (ii) theexistence of hopping terms coupling different orbitals ondifferent sites of the crystal (inter-orbital hopping or hy-bridization). We note at this stage that the breakingof orbital degeneracy can correspond to a rather largeenergy scale (of order 1-2 eV) when one has in mindthe crystal-field splitting between t g and e g levels in atransition-metal oxide, but it can also correspond to asmaller energy scale (a small fraction of an electron volt)when considering, e.g., the trigonal splitting of the t g levels induced by a distortion of the cubic symmetry. Inthe former case, an effective model with fewer orbitalscan often be considered, but in the latter case, all or-bital components may still be relevant, albeit with dif-ferent occupancies, and one has to use a model involvingseveral orbitals with slightly different atomic level posi-tions. In the present paper, we shall designate the liftingof orbital degeneracy by the generic term of “crystal-fieldsplitting,” but it is mostly the case where this is a small energy scale (e.g., trigonal splitting of the t g multiplet)that we have in mind for applications.Indeed, the physical effects arising from the compe-tition of crystal-field splitting and strong correlationshave attracted a lot of attention recently, in particu-lar in LDA+DMFT electronic structure studies of manydifferent compounds. We now quote just a few exam-ples. Pavarini et al. pointed out that the lifting of cu-bic symmetry by the GdFeO -type distortion plays a keyrole in determining the metallic or insulating charactersof d transition-metal perovskites such as (Sr/Ca)VO (small distortion, metals) and (La/Y)TiO (larger distor-tion, insulators). Indeed, for a given value of the on-siteCoulomb repulsion U , the lifting of the orbital degener-acy makes the insulating state more easily accessible .Furthermore, correlation effects considerably enhance theeffective crystal-field splitting, hence favoring orbital po-larization (as also emphasized in Ref. 15 for these com-pounds). This correlation-induced enhancement of theeffective crystal-field splitting and this increased orbitalpolarization have also been shown to play a key rolein the metal-insulator transition of V O , with the e πg component of the t g level much more occupied than the a g component in the insulating phase (see also the pre-vious LDA+DMFT studies of V O in Refs. 18 and 19).Such effects were discussed, at a model level, in the pio-neering paper of Manini et al. , motivated by the physicsof fullerene compounds. In this work, a model consistingof two orbitals occupied by one electron (quarter-filling)was considered, and the combined effect of a crystal-fieldand of on-site repulsion was studied in the framework ofDMFT. This work identified several phases, most notablya two-band metallic phase (with partial orbital polariza-tion) and a one-band metallic phase (with full orbitalpolarization), as well as a fully orbitally-polarized Mottinsulating phase.However, some questions of great importance were leftunanswered by this early study. To quote just a few ofthese issues: (i) What is the nature of the metal-insulatortransition in the different ranges of crystal field? (ii)
Howexactly does the crossover between a two-orbital Motttransition to a one-orbital Mott transition takes place? (iii)
What is the effect of an inter-orbital hopping, al-ways present in real materials, and in particular does itwipe out the two-band metal to one-band metal tran-sition within the metallic phase? and, finally, (iv) , isit possible to obtain within DMFT the insulating phasewith partial orbital polarization, which is expected fromgeneral strong-coupling arguments? (As we shall see, theanswer is affirmative and this phase was overlooked inthe DMFT study of Ref. 6).All these questions are directly relevant to the under-standing of real materials (e.g., V O and Sr RuO ) andto a better qualitative interpretation of the results ofLDA+DMFT studies. The aim of the present article isto provide a detailed answer to these questions. This ismade possible, in particular, by the recent developmentof numerical techniques for solving efficiently the DMFTequations, in particular continuous-time Monte Carlo al-gorithms .Let us point out that another related model study re-cently appeared, namely that of the two-orbital modelat half-filling (i.e., two electrons in total) . In this case,the physical issues are quite different since one evolvesfrom a two-orbital Mott insulator in the absence of crys-tal field to a band insulator at large crystal field (not aone-orbital Mott insulator as in our quarter-filled case).Also, this study did not consider the effect of an inter-orbital hopping. In this respect, our work and Ref. 8 canbe considered quite complementary to one another.Finally, we emphasize that the interplay betweencrystal-field splitting and strong correlations is madeeven more complex in the presence of Hund’s couplingand exchange terms. In a study of BaVS it was pointedout that when Hund’s rule wins over crystal-field effects,one can observe a compensation between orbital popula-tions rather than an enhanced orbital polarization .The competition between Hund’s coupling and crystal-field is also relevant to the physics of cobaltites ,ruthenates, and monoxides under pressure . Inthe present work, however, we focus on the interplay ofcrystal field and strong correlations and on the nature ofthe Mott transition, in the simplest possible context andconsider only the effect of an on-site repulsion.This paper is organized as follows: In Sec. II, we in-troduce the model and some notations. In Sec. III A wepresent the phase diagram and discuss qualitatively eachphase. In Sec. III B, we discuss in details the insulatingphases, using both an analytical strong-coupling methodand complete numerical solution of the DMFT equations.In Sec. III C, we clarify the nature of the various phasetransitions: from a two-band to a one-band metal and from a metal to a Mott insulator, in the different crystal-field regimes. Finally, in Sec. IV, we consider the effectsof a finite inter-orbital hopping and also we discuss somefinite temperature effects in regimes where the two or-bitals have very different quasiparticle coherence scales. II. MODEL
We consider a minimal two-band Hubbard modelwith crystal-field splitting and inter-orbital hybridiza-tion, given by the Hamiltonian;ˆ H = ˆ H kin + ˆ H cf + ˆ H int , (1)with; ˆ H kin = X k X σmm ′ ˆ ε ( k ) mm ′ d † k σm d k σm ′ , (2a)ˆ H cf = ∆2 X iσ (ˆ n iσ − ˆ n iσ ) , (2b)ˆ H int = U X i X mσ = m ′ σ ′ ˆ n iσm ˆ n iσ ′ m ′ . (2c)In these expressions, i is a lattice-site index, k is the mo-mentum in reciprocal space, m = 1 , σ = ↑ , ↓ is a spin index. The sum in the interactionterm runs over all orbital and spin indices except the casewhen m = m ′ and σ = σ ′ and therefore all intraorbitaland inter orbital Coulomb interactions are included. ∆ isthe crystal-field splitting between the two orbitals (∆ > m = 2), and U is the density-density Coulomb interaction between thetwo orbitals.In this article, we focus on quarter-filling (i.e., one elec-tron in two orbitals, per lattice site), which is achievedby tuning appropriately the chemical potential µ . Weconsider only the density-density form of the interactionterm, and we do not include the Hund’s exchange, spin-flip, or pair-hopping terms. The motivation for neglect-ing these terms is to keep the Hamiltonian as simple aspossible. Note however that, with one electron per site,the effect of these terms is expected to be small and actsbasically as a renormalization of the on-site U (Refs. 11and 12).The kinetic term ˆ H kin is a two-band tight-bindingHamiltonian on the three-dimensional cubic lattice (wewill also use its Bethe lattice counterpart), which can bewritten (in k space) asˆ ε ( k ) = (cid:20) e ( k ) V ( k ) V ( k ) e ( k ) (cid:21) , (3)where diagonal elements correspond to the simple cubiclattice, and the off-diagonal ones have x − y symmetry; e ( k ) = 2 t (cos k x + cos k y + cos k z ) , (4a) V ( k ) = 2 √ V (cos k x − cos k y ) cos k z . (4b)This corresponds to a hopping between identical or-bitals on nearest-neighbor sites, equal to − t . The inter-orbital hopping connects orbitals m = 1 and m = 2 onnext-nearest-neighbor sites and is equal in magnitude to √ V /
2. It has a positive sign for the [ ± , , ±
1] neigh-bors and negative for the [0 , ± , ±
1] ones. This symme-try choice insures that, for all values of V , the on-site( k integrated) kinetic Hamiltonian is diagonal in orbitalspace . This is also the case of all local ( k integrated)quantities in the interacting model, as can be checked byexpanding the Green’s function in power of V . Hence,our model is such that the choice of local orbital basisset is adapted to the local crystal symmetry. Physically,the model [Eq. (4)] is a reasonable description, for exam-ple, of an e g doublet split by the breaking of the cubicsymmetry.For zero hybridization, V =0, the density of states(DOS) is reduced to the DOS of the cubic lattice forboth orbitals shifted by ± ∆ /
2. We set the energy unitby t = 1 /
6, or equivalently D = 1, where D is the half-bandwidth.We solve this model in the DMFT framework . Sinceour main aim is to elucidate the nature of the metal-insulator transitions in this model, we focus in this articleon the paramagnetic phases. The self-consistent impu-rity problem is solved with two numerical techniques: (i) Exact diagonalization (ED) as described in (Refs. 3 and36), with a “star-geometry” for the bath hybridizationfunction using five bath states per orbital degree of free-dom; (ii) the recently introduced continuous time quan-tum Monte Carlo algorithm (CT-QMC) using an expan-sion in the impurity model hybridization function .CT-QMC is more precise than ED and is necessary toestablish the existence of the partially polarized insulatorphase (see Sec. III B), as we shall discuss further below.
III. RESULTS IN THE ABSENCE OFINTER-ORBITAL HYBRIDIZATIONA. Zero-temperature phase diagram
The DMFT phase diagram of model [Eq. (1)] atquarter-filling and without inter-orbital hybridization( V = 0) is presented on Fig. 1. The effect of a non-zero V will be considered in Sec. IV. The general shape of thisphase diagram can be easily anticipated by consideringthe various limiting cases : (i) For ∆ = 0, one has a well documented two-banddegenerate model. The model undergoes a correlation-driven Mott transition at a critical U ∆=0 c ≃ .
76 whichis close to the results obtained by other authors forthe Bethe lattice (semi-circular DOS with identical half-width D = 1) . (ii) For very large ∆ ≫ D , the minority orbital (orbital m = 1) is pushed to very high energy and becomes com-pletely empty, so that it can be ignored altogether. Thequarter-filled two-band model thus reduces to a single- ∆ / D U / D MIT, V = 0PPM-FPM lineStrong coupling, U = ∆ )
PPM FPMFPIPPI
FIG. 1: (Color online) Zero-temperature phase diagram(paramagnetic phases), on the cubic lattice without hy-bridization ( V = 0), and for one electron per site. The solid(black) line separates metallic and insulating regions. Thedot-dashed (red) line separates the partially polarized metal(PPM) and the fully polarized metal (FPM) (for details seetext). The dashed (green) line is the result of the strong-coupling mean-field analysis (see Sec. III B 1 and Eq. 10):It separates the partially-polarized insulator (PPI) from thefully-polarized insulator (FPI). Arrows indicate the set of pa-rameters used in Figures 4 and 6. The ED solver was used. band model at half-filling . This situation has been thor-oughly studied within DMFT and yields a correlation-induced Mott transition at U ∆ →∞ c ≃ .
75. (see, e.g.,Ref. 3 and references therein). The metal-insulator tran-sition line (plain/black line on Fig. 1) interpolates be-tween the limiting critical couplings corresponding to∆ = 0 and ∆ = ∞ . The system is insulating abovethis line and is metallic below. (iii) The non-interacting model ( U = 0) obviously hasa transition between a two-band metal for ∆ < D anda one-band metal for ∆ > D . For ∆ = D , the minorityband crosses the Fermi level and becomes empty. Thiseffective-band transition separating a two-band situationat low energy from a non-degenerate band can actually befollowed through the phase diagram (dashed-dotted/redand dashed/green lines on Fig. 1), as we now discuss.We note that we have not attempted to precisely de-termine whether the orbital-polarization lines cross themetal-insulator transition (MIT) line at a single point,or whether the orbital polarization line in the insulatingphase and in the metallic phase hit the MIT boundaryat slightly different locations.In the absence of hybridization ( V = 0), we can use theorbital polarization as a faithful indicator of the transi-tion between the two-band and a one-band regime. Thisquantity is defined as δn = h ˆ n > i − h ˆ n < ih ˆ n > i + h ˆ n < i , (5)in which > and < stand for the majority and minorityorbitals, respectively, ( h ˆ n > i > h ˆ n < i ). At quarter fillingand for ∆ >
0, this reduces simply to δn = h ˆ n − ˆ n i .As the crystal-field splitting is increased, one reachesa critical value at which the orbital polarization reaches δn = 1, indicating a completely empty minority orbital.The line along which this happens in the (∆ , U ) plane, isindicated by the dashed-dotted (red) line in the metallicphase and by the dashed (green) line in the insulatingphase. Hence, four different phases are apparent on thephase diagram of Fig. 1: a partially polarized (two-band)metal (PPM), a fully polarized (one-band) metal (FPM),a partially polarized Mott insulator (PPI), and a fullypolarized Mott insulator (FPI).As already pointed out by Manini et al. , and as clearfrom Fig. 1, the value of the crystal-field, at which thetransition from the PPM to the FPM takes place, isstrongly reduced by interactions. While it is set by thehalf-bandwidth at U = 0, it is renormalized down by thequasiparticle weight Z > in the presence of interactions.Hence, a crystal-field splitting considerably smaller thanthe half-bandwidth can be sufficient to induce a two-bandto one-band metal transition.It is important to realize, however, that the value of ∆needed to fully polarize the system vanishes only in thelimit U = ∞ . In other words, the orbitally-degenerateMott insulator at ∆ = 0 has a finite orbital polarizabil-ity , even within the DMFT approach. Hence the PPIphase at large U and small values of ∆ exists. This pointwas incorrectly appreciated by Manini et al. , largely fornumerical reasons. Indeed, the ED algorithm is inap-propriate to correctly capture the PPI phase. In thepresent article, we establish (Sec. III B) the existence ofthe partially polarized insulating phase within DMFT us-ing both an analytical proof at strong-coupling limit anda complete numerical solution of the DMFT equationsbased on the new CT-QMC algorithm.Let us point out that in this zero-temperature phase di-agram, all the transitions are second order, except for thetransition from the PPM to the PPI – which is second or-der for the majority orbital and first order for the minor-ity orbital, as will be explained below in Sec. III C 2. Atfinite temperatures T >
0, the MIT becomes first orderthroughout the phase diagram, as in canonical DMFTsolutions, whereas the other transitions remain secondorder.In the two following subsections, we describe in moredetails the nature of these different phases and we inves-tigate the phase transitions between them.
B. Existence of the partially polarized insulator
1. Strong-coupling analysis: Kugel-Khomskii model
At strong coupling U ≫ D (or U ≫ t ), in theMott insulating phases, an effective low-energy modelcan be derived, following Kugel and Khomskii (see alsoRef. 38). The low-energy Hilbert space contains only the four states | i, m, σ i with one electron on each site( m = 1 , σ = ↑ , ↓ ). The effective Hamiltonian acting onthese states reads,ˆ H eff = − ∆ X i ˆ T zi ++ X h ij i n J s ( ~S i ~S j ) + J o ( ~T i ~T j ) + J m ( ~S i ~S j )( ~T i ~T j ) o . (6)In this expression, h ij i denotes the bonds betweennearest-neighbor sites, and the spin and pseudo-spin (i.e.,orbital isospin) operators are given by: ~S i = 12 X m d † iσm ~τ σσ ′ d iσ ′ m , (7a) ~T i = 12 X σ d † iσm ~τ mm ′ d iσm ′ , (7b)in which ~τ are the Pauli matrices. In particular, the z component of these operators (with eigenvalues ± /
2) isgiven by: ˆ S zi = 12 (ˆ n i ↑ − ˆ n i ↓ + ˆ n i ↑ − ˆ n i ↓ ) , (8a)ˆ T zi = 12 (ˆ n i ↑ + ˆ n i ↓ − ˆ n i ↑ − ˆ n i ↓ ) . (8b)The (superexchange) couplings J s , J o , and J m are givenby J s = J o = J m t U ≡ J. (9)The particular symmetry between these couplings is dueto the choice of a density-density interaction and to theneglect of the Hund’s exchange.At strong coupling, in the insulating phase, the DMFTsolution of the original model [Eq. (1)] reduces to astatic mean-field solution of Eq. (6). Focusing on thenon-magnetic phase ( h S zi i = 0), the orbital polarization δn = 2 h T z i is given by the self-consistent equation, atfinite temperature T = 1 /β : δn = tanh (cid:16) β − ∆ c δn ) (cid:17) , (10)where ∆ c = zJ z t U , (11)is a critical value of the crystal-field splitting and z is the coordination number of the lattice (number ofnearest neighbors). For the simple cubic lattice, with z = 6 and half-bandwidth D = zt , this yields, ∆ CLc = D / (6 U ), while for the large-connectivity Bethe latticewith nearest-neighbor hopping t = D/ (2 √ z ), one has:∆ BLc = D / (4 U ).At zero temperature ( β = ∞ ), the solution of Eq. (10)reads, δn = ( ∆ / ∆ c , ∆ < ∆ c , ∆ > ∆ c . (12)Hence, this shows that the orbitally degenerate insulatorhas a finite orbital susceptibility at T = 0, χ orb = 1 / ∆ c ,and that a finite crystal-field ∆ = ∆ c must be applied tofully polarize the insulating phase. The strong-couplingexpression ∆ = ∆ c = D / (6 U ) for the cubic lattice cor-responds to the dashed(green) line displayed on Fig. 1,separating the PPI from the FPI phases at T = 0.At finite temperature, a good approximation to thesolution of Eq. (10) turns out to be; δn ≃ ∆∆ c
11 + c β . (13)Finally, we would like to emphasize that, when think-ing of DMFT as an exact method in the limit of largelattice coordination z → ∞ , it is quite clear that a non-zero value of the critical ∆ c (and hence a finite extent ofthe PPI phase) is to be expected. Indeed, the orbital ex-change coupling [Eq. (9)] scales as 1 /z (since t ∝ / √ z ),hence, the critical ∆ c is of the order of the exchange fieldbetween a site and all its neighbors, i.e., of order zJ ,which remains O (1) as z → ∞ . The uniform orbital sus-ceptibility of the orbitally degenerate Mott insulator isindeed finite at T = 0 (this should not be confused withthe fact that the local susceptibility would scale as 1 /z and hence vanish in the large- z limit).
2. Numerical solution: Importance of global moves in thequantum Monte Carlo algorithm
The analytical estimate at finite temperature [Eq. (13)]provides a very useful benchmark when solving numeri-cally the DMFT equations for the original model in thestrong-coupling regime. Indeed, it is actually non-trivial,from the numerical point of view, to successfully stabi-lize the partially polarized insulating phase. To achievethis, we have used the CT-QMC method, and it provednecessary to implement global Monte Carlo moves, inaddition to the Monte Carlo moves proposed in Ref. 8.In CT-QMC, a configuration is given by a collection offermionic operators c α ( τ ) . . . c α N ( τ N ) at different imag-inary times τ i and the α i are the fermionic species of theoperators. The global moves are implemented by chang-ing all α i into a new set of α ′ i and accepting the movewith a probability satisfying the detailed balance condi-tion. In this work, we have used two global moves thatswitch the spin ( ↑↔↓ ) and the orbital (1 ↔
2) indices. Inthe absence of these global moves, the calculation can betrapped in some regions of the phase space at low tem-perature, leading to a wrong (overestimated) value of thepolarization. β D δ n CT-QMC w gmCT-QMC w/o gmStrong coupling
FIG. 2: (Color online) Temperature dependence of the orbitalpolarization, δn , for the cubic lattice in the PPI phase. Blackdots are the CT-QMC data with the use of global moves ( gm )in the spin and orbital space (see text for details). (Cyan)squares are the CT-QMC data without global moves. Thesolid (green) line is the strong coupling result given by Eq. 10.The arrow shows the zero-temperature value of the polariza-tion in the strong coupling limit, δn = 0.84. These results areobtained for U =4, ∆=0.035 and using the CT-QMC solver. This is illustrated in Fig. 2, which displays the tem-perature dependence of the polarization in the insulatingphase, at small ∆. The result of Eq. (10) is comparedto the CT-QMC results with and without global moves.One can see that without global moves, the polarizationis bigger than its strong-coupling value, whereas the con-trary is expected. This gives a clear indication that globalmoves are needed. When the correct implementation ofthe CT-QMC algorithm with global moves is used, thepolarization falls below its strong-coupling value. Notethat these results are actually obtained for an interme-diate value of U = 4, which shows that the range ofvalidity of the strong-coupling approximation is actuallyquite extended. The agreement between the DMFT datawith global moves and the strong-coupling result is seento be excellent and both indubitably show the existenceof the partially polarized insulator.We have not been able (as in Ref. 6), when using theED solver at T = 0 in the insulating phase, to stabi-lize the partially polarized insulating solution at small ∆.This is probably because this solution is too delicate andinvolves a number of competing low-energy scales ( J , ∆)to be faithfully reproduced given the simple parametriza-tion and limited number of states in the effective bath,which can be handled within ED in a two-orbital context.However, ED performs quite well in the metallic phase,and it is quite instructive to compare the iso-polarizationlines ( δn = const . ) in the (∆ , U ) plane, determined fromED, close to the metal-insulator transition, to the strong-coupling result ∆ / ∆ c = δn (i.e., U ∆ = zt δn ). Thiscomparison is made in Fig. 3 in the case of the Bethelattice (for simplicity). The ED data on the metallic side ∆ / D U / D MIT δ n = 0.1 δ n = 0.3 δ n = 0.5 δ n = 0.7 δ n = 0.9 δ n = 1.0 FIG. 3: (Color online) Metal-insulator transition (red) andiso-polarization lines for the Bethe lattice. Different colorsmark different values of the polarization (see legend). Dashedlines are ED results, while solid lines are the solutions ofEq. 12 for constant polarization. of the transition match very well to the strong-couplingform of the iso-polarization lines on the insulating side.Thus, this provides a complementary way, starting fromthe metal, to document the existence of the PPI regime.
C. Metallic phases and the nature of themetal-insulator transition
We now turn to the metallic phases. There, the self-energies can be Taylor expanded at low-frequency as ℜ Σ ≷ ( ω + i + ) = ℜ Σ ≷ (0) + (1 − /Z ≷ ) ω + · · · , (14)in which Z > and Z < are the quasiparticle weights of themajority and minority bands, respectively. The quasi-particle weights, Z ≷ = (1 − ∂ ℑ Σ ≷ ( iω ) /∂iω ) − | ω → wereextracted from the imaginary frequency data with theuse of third-order polynomials. The minority and major-ity Fermi surfaces in the metallic phase PPM are deter-mined, respectively, (for V = 0) by: e ( k ) = µ + ∆2 − ℜ Σ > (0) ≡ µ > , (15a) e ( k ) = µ − ∆2 − ℜ Σ < (0) ≡ µ < . (15b)The quantities µ > , µ < can be viewed as effective crystal-field levels renormalized by interactions (or effectivechemical potentials for each type of orbitals), and a renor-malized crystal-field splitting can also be defined as∆ eff ≡ ∆ + ℜ Σ < (0) − ℜ Σ > (0) . (16)The various transitions are conveniently described interms of µ ≷ and Z ≷ . On general ground, there are two simple mechanisms by which a given orbital can undergoa transition from a metallic behavior to an insulating one: (i) The quasiparticle weight Z may vanish at the MIT.This is the well-known Brinkman-Rice scenario, whichis realized, e.g., within the half-filled single-band Motttransition within DMFT. It is also realized for degenerateorbitals with ∆ = 0: Z > = Z < vanishes continuously at U ∆=0 c . (ii) It may also happen that either of the equations[Eq. (15)] fails to yield a solution, i.e., the “effective chem-ical potentials” µ > or µ < move out of the energy range[ − D, + D ] spanned by e ( k ). This, in turn, can happen ina continuous or in a discontinuous way.
1. Orbital polarization and metal-insulator transitions atlarge crystal field
We first consider values of the crystal-field splittinglarger than & .
1. Two successive transitions are ob-served as U is increased, from a two-band metal (PPM)to a single-band metal (FPM) – followed by a metal-insulator transition (FPM to FPI). Figure 4 (top panel)displays the quasi-particle residues Z > , Z < and orbitalpolarization δn as U is increased at a fixed ∆ = 0 . µ ≷ and ∆ eff .For U < U P ( ≈ . U is in-creased, and the orbital polarization gradually increases.At U = U P , the polarization saturates to δn = 1 andthe minority band becomes empty. This happens follow-ing the mechanism (ii) above: the minority band effectivelevel position µ < hits the bottom of the band ( µ < = − D at U = U P ) and the renormalized crystal-field splittingreaches ∆ eff = + D [as clearly seen from Fig. 4 (lowerpanel)]. Simultaneously, µ > vanishes at U P and remainszero for U > U P . This indicates that particle-hole sym-metry is restored at low-energy for the majority bandthroughout the FPM phase.For U P < U < U MIT , the minority band is empty andbecomes inactive. The remaining half-filled majority or-bital forms a single-band metal and is subject to the localCoulomb interaction. This is illustrated in Fig. 5 wherewe plot the DOS of both orbitals. Note that the major-ity orbital very quickly becomes particle-hole symmetricover its full bandwidth as U increases. The quasi-particleweight of the majority band Z > is strongly reduced inthis regime. Note that neither Z > nor Z < vanishes atthe orbital polarization transition U P . In fact, also theminority (empty) band self-energy remains linear in fre-quency at low energy in this regime, and a Z < can stillbe formally defined (as plotted on Fig. 4), although it nolonger has the physical meaning of a quasiparticle spec-tral weight since there is no Fermi surface for that band.In particular, the increase of Z < in this region should notbe interpreted as a decrease in the correlation effects.Eventually, the transition from a single-band strongly δ n and Z < , Z > U P U MIT δ n Z < Z > U / D -2-1012 ∆ e ff and µ < , µ > ∆ eff = ∆ − Re Σ > + Re Σ < µ < = µ − ∆/2 − Re Σ < µ > = µ + ∆/2 − Re Σ > FIG. 4: (Color online) PPM-FPM-FPI transitions along theconstant ∆ = 0 . V = 0).Top panel: Orbital polarization, δn and QP residues, Z > and Z < are shown by (black) dots, filled (green) diamonds andopen (red) squares, respectively.Bottom panel: Effective crystal field splitting, ∆ eff and ef-fective chemical potential for both bands, µ > , µ < are repre-sented by (black) dots, filled (green) diamonds and open (red)squares, respectively.The vertical lines show the full polarization (violet) and MITtransitions (magenta). The horizontal (brown) lines show thetop and bottom of the bare band. The ED solver was used. correlated metal to a Mott insulator with full orbital po-larization is found at U = U MIT ( ≃ U ∆= ∞ c ≃ . Z > vanishes continuously at the critical point andthe metal-insulator transition is second order (at T =0).The low-frequency majority self-energy ℜ Σ > ( ω + i + )acquires a pole on the real frequency axis in the insu-lating phase. The location of this pole depends on thechoice of the chemical potential within the insulating gap.For a specific choice (as done in Fig. 4), the pole is lo-cated at zero-frequency so that particle-hole symmetryis restored at low energy and the self-energy diverges as -4 -3 -2 -1 0 1 2 3 4 Energy / D DO S MajorityMinority
FIG. 5: (Color online) Density of states in the 1-band metallicphase. The majority (minority) orbital is shown in the upper(lower) half of the plot. The ED solver was used with U = 2 . ℜ Σ > ( ω + i + ) ∼ /ω .It should be emphasized that the very small value of Z > in the one-band (FPM) metallic phase implies thatthe quasiparticles are actually quite fragile in that phaseand can be easily destroyed by thermal effects. Hence,the orbital polarization transition at U = U P from atwo-band to a one-band metal at T = 0 may actuallyappear, at finite-temperature, as a transition between atwo-band metal and a one-band incoherent “bad metal”(or quasi-insulator). We shall come back to this point inmore details in Sec. IV B.
2. Metal-insulator transition at small crystal field
In the small crystal-field regime (∆ . . T = 0, the transition is second order with a quasiparticleweight Z > = Z < vanishing continuously at U ∆=0 c , whileat T > Z > , Z < as a function of U for β = 100and for a small value of ∆ = 0 .
05, along with the or-bital polarization δn . The MIT takes place at a criti-cal coupling U ∆ c , which is smaller than U ∆=0 c (Fig. 1).Note that the data in Fig. 6 is obtained for a finite tem-perature and, therefore, the critical U ∆ c is also smallerthan its zero-temperature counterpart (shown in Fig. 1).The orbital polarization continuously increases with theinteraction and does not approach the value δn =1 at δ n and Z < , Z > δ n Strong couplingZ < Z > U / D -2-1012 ∆ e ff and µ < , µ > ∆ eff = ∆ − Re Σ > + Re Σ < µ < = µ − ∆/2 − Re Σ < µ > = µ + ∆/2 − Re Σ > µ / D -20020 R e Σ ( ) Re Σ < Re Σ > FIG. 6: (Color online) PPM-PPI transition along the con-stant ∆=0.05 line for the cubic lattice without hybridization.Top panel: The (black) solid dots show the orbital polariza-tion, δn . The QP residues, Z > and Z < , are shown by the(green) filled diamonds and (red) open squares, respectively.The (blue) dashed line is the strong coupling result.Bottom panel: Effective crystal field splitting, ∆ eff and ef-fective chemical potential for both bands, µ > , µ < are repre-sented by (black) dots, filled (green) diamonds and open (red)squares, respectively. The inset shows the real part of the self-energies at the first Matsubara frequency, ℜ Σ ≷ ( iπ/β ), versuschemical potential within the gap for U = 3.The vertical magenta line shows the MIT. The CT-QMCsolver was used with β = 100. the transition point. The minority orbital quasiparticleweight Z < remains larger than the majority one Z > inthe metallic phase. Although it is a delicate issue numer-ically, our data appear to be consistent with a majorityorbital quasiparticle weight Z > , which vanishes continu-ously while Z < remains finite at the transition. Note thatboth the majority and minority orbital effective chemi-cal potentials [Eq. (15)] stay well within the energy band[ − D, + D ] for all couplings in the metallic phase. Thetransition into the insulating phase for the minority or-bital takes place by having µ < jumping out of the energyband in an apparently discontinuous manner, as we nowdescribe in more details. After the transition, the chemical potential µ can beplaced (at T = 0) anywhere within the charge gap, andtherefore, the effective chemical potentials [Eq. (15)] arenot longer defined in a unique manner. As documented inprevious work on the orbitally degenerate case withinDMFT, we expect the majority orbital self-energy to havea pole on the real frequency axis, at a position that de-pends on µ . For a special choice of µ , this pole is lo-cated at ω = 0, which should correspond to a divergenceof Σ > ( ω = 0) and to a divergent self-energy ∼ /ω atlow frequency. In order to document this behavior, weplot in the inset of Fig. 6 the real part of the imaginaryfrequency self-energies, ℜ Σ ≷ ( iπ/β ) at the first Matsub-ara point as a function of µ (for a given value of theinteraction U = 3). One can clearly see that the major-ity orbital self-energy, ℜ Σ > ( iπ/β ) becomes very big andchanges sign at µ ∼ .
22 while ℜ Σ < ( iπ/β ) stays con-stant within the gap. A careful scaling analysis showsthat Σ > ( ω = 0) indeed diverges at a critical value of µ . Together with the vanishing of Z > , this shows thatthe transition for the majority orbital follows the stan-dard DMFT scenario identical to the orbitally degener-ate case. Furthermore, since ℜ Σ < ( iπ/β ) does not varysignificantly when µ is varied within the gap, one canunambiguously define µ < also in the insulating phase. Incontrast, µ > depends on the choice of µ . One shouldnote here that the chemical potential, µ , defined in thisway in the insulating phase continuously connects to thechemical potential in the metallic phase.In Fig. 6 (bottom panel), we display these two quan-tities as a function of U , choosing for µ the special valueat which Σ > behaves as 1 /ω at low frequency. From thisplot, we see that the minority band becomes insulatingbecause µ < is jumping out of the energy band in a man-ner that appears as discontinuous (up to our numericalprecision). Hence, in contrast to the orbital polarizationtransition of the large ∆ case described above, the MITat small ∆ appears to occur in a discontinuous manner,as far as the minority band is concerned, while beingcontinuous (Brinkman-Rice like) for the majority band.Note also that the minority-orbital self-energy has a lin-ear behavior at low frequency throughout the insulatingphase.Note that in this finite-temperature calculation, theorbital polarization never reaches δn = 1 as U is furtherincreased. From the strong-coupling calculation, we ex-pect that it will saturate at δn ≃ .
987 when U → ∞ . Atzero temperature, however, there is a second-order tran-sition at a finite critical value of U where the polarizationreaches δn = 1. ∆ / D U / D MIT, V = 0.07 1-band - 2-band, V = 0.07MIT, V = 0 1-band - 2-band, V = 0
FIG. 7: (Color online) Zero-temperature phase diagram ofthe cubic lattice with hybridization V = 0 .
07 and for oneelectron per site. The (black) solid line separates metallicand insulating regions. The (black) dot-dashed line separatesthe 2-band and 1-band metals. For the sake of comparison,the corresponding zero-hybridization ( V = 0) lines are shown(in green). The ED solver was used. IV. EFFECT OF AN INTER-ORBITALHYBRIDIZATION V ( k ) A. Low-energy effective-band transition
In this section, we consider the effect of a finite hy-bridization (inter-orbital hopping V ( k ) = 0). At lowvalues of ∆, the metal-insulator transition is pushed tohigher values of U when turning on a small V . While atlarger values of ∆, the MIT line is less sensitive to V (asillustrated on Fig. 7). This is expected since at low ∆ theinter-orbital hopping increases the kinetic energy in bothbands while at higher ∆ the hybridization with a band,which is already empty, has a smaller effect on the crit-ical coupling. As we will discuss in more details below,in the presence of the hybridization, the fully polarizedphases (FPM and FPI) disappear. However, there is stilla transition from a two-band to a one-band metal at lowenergy. This transition line is pushed up at low valuesof the crystal-field splitting because of the increase in ki-netic energy. In non-interacting limit, the finite value of V acts as a k -dependent enhancement of the crystal field∆, and therefore, at small values of the interaction, thetwo-band to one-band transition line is below the corre-sponding V = 0 line.One should note that the majority (minority) banddoes not have a unique two (one) orbital character, andthe band index > ( < ) has to be distinguished from theorbital index two (one).On Fig. 8, we display the quasiparticle weights andorbital polarization as a function of U , for a fixed valueof V and a rather large crystal field ∆ = 0 .
3. One clearlysees that the MIT follows a similar mechanism than inthe V = 0 case: only Z vanishes continuously at the U / D δ n and Z , Z δ n, V = 0 δ n, V = 0.07 Z Z FIG. 8: (Color online) Orbital polarization δn (dots/black),quasiparticle weights Z (green/filled diamonds) and Z (red/open squares), as a function of U for a fixed value of∆ = 0 . V = 0 .
07. Forthe sake of comparison, the orbital polarization for V = 0is also displayed (cyan/dashed line). The vertical (magenta)line shows the MIT. The ED solver was used. transition, while Z is always finite.A noticeable difference with the V = 0 case is thatthe orbital polarization δn = n − n does not reachsaturation ( δn <
1) before the MIT (Fig. 8). This isexpected, because the low-energy bands in the metallicstate no longer have a unique (1 ,
2) orbital character, aswe now discuss.In order to understand more precisely the nature ofthe metallic phase, we use the low-frequency expansionof the self-energies and we obtain the expressions of thelow-energy majority and minority bands, which read,2 ω < ( k ) = (17a)= Z ǫ k + Z ǫ k + q ( Z ǫ k − Z ǫ k ) + 4 Z Z V k , ω > ( k ) = (17b)= Z ǫ k + Z ǫ k − q ( Z ǫ k − Z ǫ k ) + 4 Z Z V k . In these expressions ǫ k ≡ e k − µ + ∆ / ℜ Σ (0) and ǫ k ≡ e k − µ − ∆ / ℜ Σ (0). The Fermi surface (setby ω = 0) is determined by the following condition (inwhich the weights Z , do not appear):0 = ǫ k ǫ k − V k (18) ≡ [ e k − µ + ∆ / ℜ Σ (0)] × [ e k − µ − ∆ / ℜ Σ (0)] − V k . We recall that, when V = 0, an orbital polarization tran-sition is first encountered at U = U P , at which the Fermi-surface sheet corresponding to orbital 1 (determined by ǫ k = 0) disappears, since µ − ∆ / − ℜ Σ (0) reachesthe band-edge. In the presence of V = 0, a similar phe-nomenon occurs for the minority low-energy band ω < ( k ):0 Γ X M R X -2-1012 E n e r gy U = 2 Γ X M R X
U = 2.4 Γ X M R X
U = 2.7 Γ X M R X -2-1012
U = 2.9
FIG. 9: (Color online) Linearized band structure along symmetry lines of the cubic lattice for different values of the interaction.We used ∆=0.3, V =0.07 and the ED solver. Fatness shows a contribution of the spectral weight of the less occupied orbital( m = 1) to the majority band (see text for details). U =2.4 corresponds to the value where the effective crystal field splittingexceeds the bare bandwidth and the physical picture effectively becomes single band. The rightmost panel corresponds to thePPI solution and we used ℜ Σ ( ω + i + ) = ℜ Σ (0) + Ω /ω for the divergent orbital. one of the two sheets, which constitute the solution ofEq. (18) ceases to exist. This is expected from conti-nuity arguments in view of Eq. (18) and of the situa-tion at V = 0. This is furthermore demonstrated byFig. 9, which displays the majority and minority low-energy bands ω ≷ ( k ) along the main directions in the Bril-louin zone, as U is increased. It is clearly seen from thisfigure that for U ≃ . U ≃ . V = 0.For V = 0, the majority eigenstate | k , > i (correspond-ing to eigenvalue ω > ( k )) has a unique orbital character m = 2. In contrast, for V = 0, it has a component onboth orbital 2 and orbital 1. As a result, the orbitalpolarization does not reach δn = 1 (Fig. 8) at the effec-tive band transition between a two-band and a one-bandmetal. On Fig. 9, we have used a “fat band” represen-tation to illustrate this point: at each k point, we plot abar whose extension is proportional to the matrix element |h | k , > i| , measuring the projection of the less-occupiedorbital m = 1 onto the majority band.As U is increased beyond the effective-band transition,one is left with a single effective low-energy band, char-acterized by a quasiparticle weight, Z > ( k ) = ( ǫ k + ǫ k ) Z Z ǫ k Z + ǫ k Z , (19)where k lies on the Fermi surface of the majority band [seEq. (18)]. The subsequent Mott transition is character- ized by a vanishing quasiparticle weight for the majorityband Z > ∼ Z →
0, as clearly seen from Fig. 8 and fromthe narrowing of that band in the third panel of Fig. 9.The key conclusion of this section is that, even in thepresence of a finite inter-orbital hopping, two distincttransitions are observed as U is increased (in the largecrystal-field regime): first, a second-order transition froma metal with two active bands at low energy to a metalwith only one active band at low energy, and followed bya Mott metal-insulator transition of the one-band type. B. Orbital-selective coherence and the two-bandmetal to one-band bad-metal transition
We have seen above that, in a rather extended regionof the metallic phase, the quasiparticle weight of the ma-jority orbital is much smaller than that of the minorityone. This is especially true close to the two-band to one-band metal transition, where Z ≪ Z . This impliesthat thermal effects can easily destroy the fragile quasi-particles of the majority band. This has physical con-sequences, which may be important in practice. For ex-ample, the two-band metal to one-band metal transitionat finite temperature may appear in practice as a quasi-metal-insulator transition or more precisely as a transi-tion between a two-band metal and a bad (or incoherent)metal. This will happen when the temperature, at whichthe system is studied, is higher than the (small) quasi-particle coherence temperature of the majority band.In order to illustrate this point, we performed finite-1 I m Σ β = 20β = 40β = 50β = 100β = 200 i ω n / D -2-1 I m G T ρ ρ ρ T -2-10 Im Σ (0) Im Σ (0) Σ Σ G G FIG. 10: (Color online) Top panel: Imaginary part of the self-energies, ℑ Σ ( iω n ) and ℑ Σ ( iω n ) for different temperatures(see legend for temperature coding). The inset shows the ex-trapolation to zero of the imaginary part of the self-energies, ℑ Σ , (0) versus temperature.Bottom panel: Imaginary part of the Green’s functions, ℑ G ( iω n ) and ℑ G ( iω n ) for different temperatures (the color-coding is the same). The inset shows the density of states atthe chemical potential ρ , (0) versus temperature. We used U = 2, ∆ = 0 . V = 0 .
07 and the CT-QMC solver. temperature studies for the following parameter values: U = 2, ∆ = 0 .
3, and V = 0 .
07, which correspond tothe two-band metallic regime, not very far from the two-band to one-band metal transition. For these parameters,the two quasi particle residues are Z =0.34 and Z =0.59(see Fig. 8). In Fig. 10, we display the imaginary partof the Green’s functions (bottom) ℑ G , ( iω n ) and self-energies (top) ℑ Σ , ( iω n ) on the Matsubara axis, for dif-ferent temperatures. In the insets of this figure, we dis-play the extrapolated zero-frequency value ℑ Σ , ( i + ),which is related to inverse quasiparticle lifetime and zero-frequency density of states, ρ , (0) ≡ −ℑ G , ( i + ) /π ,respectively.It is seen from these figures that, while the minority or-bital quantities have quite little temperature dependence,the majority orbital, in contrast, displays very strong temperature dependence. For example for T & .
03 (i.e.,a rather low-energy scale as compared to the bandwidth),the majority orbital is clearly incoherent with a smallquasiparticle lifetime and much reduced value of the localdensity of states. At those temperatures, the frequencydependence of the self-energy is clearly non-metallic, ex-trapolating to a large value at zero frequency. Only ata low temperature T ∼ .
01 (200 times smaller than thebandwidth), the behavior of a coherent metal is recov-ered, with a linear Matsubara frequency dependence of ℑ Σ ( iω n ) extrapolating to a small value at low frequency(corresponding to a large quasiparticle lifetime). V. CONCLUSION
In this paper, we have investigated how a crystal-fieldsplitting, by lifting orbital degeneracy, affects the Mottmetal-insulator transition in the presence of strong on-site correlations. The study was performed on a sim-ple two-orbital model at quarter filling (one electron persite), and we have also considered the effect of an inter-orbital hopping (hybridization), which is important forapplications to real materials.Within the metallic phase, a second-order transitionfrom a two-band to a one-band metal takes place as thecrystal field is increased. The critical value of the crystal-field splitting, at which this transition takes place, is con-siderably lowered for strong on-site repulsion (i.e., theeffective crystal-field splitting is considerably enhanced).This transition has the nature of a effective band tran-sition for the renormalized low-energy bands (i.e., theminority band is pushed up in energy and does not crossthe Fermi energy anymore) and survives in the presenceof an inter-orbital hopping.The nature of the Mott metal-insulator transitions in-duced by on-site repulsion was found to depend on themagnitude of the crystal-field splitting. At high enoughvalues of this splitting, the Mott transition is betweena one-band metal and a one-band Mott insulator (con-ventional Brinkman-Rice scenario): only the majorityorbital is involved, and the transition is second orderand characterized by a vanishing quasi-particle weightfor that orbital. At low values of the crystal-field split-ting, the transition is from a two-band metal to a Mottinsulator with partial orbital polarization . It takes placesimultaneously for both orbitals: although the transi-tion is still continuous for the majority orbital, it hasa first-order character for the minority orbital. Elucidat-ing these transitions and, in particular, establishing theexistence of the partially orbitally polarized Mott insula-tor at low crystal fields was made possible by the recentdevelopment of the CT-QMC algorithm for the solutionof the DMFT equations.If a finite hybridization ( V = 0) is taken into ac-count, it is no longer possible to fully polarize the sys-tem. Therefore, the FPM and the FPI phases disappear.However, there is still a transition from a two-band to a2one-band metal at low energy so that the introductionof a finite V does not modify the overall picture of themodel.We have also studied the influence of the tempera-ture on the two-band metal just below the transition tothe one-band metal. The temperature can easily drivethe system into a regime where the quasiparticle weightof the majority band is destroyed and the system effec-tively becomes a single-band metal. Further increase ofthe temperature above the characteristic temperature ofboth bands leads the system into an incoherent (or bad)metal.Our study has direct relevance for the interpretation ofthe metal-insulator transitions of transition-metal oxides(see Sec. I), often accompanied by an enhanced orbitalpolarization. Acknowledgments
We are very grateful to O.K. Andersen, S. Bier-mann, A. Rubtsov, and A. Lichtenstein for the discus-sions related to this work. We also thank V. Anisi-mov, F. Lechermann, A. Millis, and P. Werner for theuseful conversations. We acknowledge the support fromCNRS, Ecole Polytechnique, the Agence Nationale de laRecherche (under contract
ETSF ), and the Marie CurieGrant No. MIF1-CT-2006-021820. This work was sup-ported by a supercomputing grant at IDRIS Orsay underProject No. 071393 (for the ED results) and at CEA-CCRT under Project No. p588 (for the CT-QMC calcu-lations). N. Mott,
Metal-Insulator Transitions (Taylor and FrancisLtd., London/Philadelphia, 1990). M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. , 1039 (1998). A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,Rev. Mod. Phys. , 13 (1996). G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko,O. Parcollet, and C. A. Marianetti, Reviews of ModernPhysics , 865 (2006). T. Pruschke and R. Bulla, The European Physical JournalB , 217 (2005). N. Manini, G. E. Santoro, A. Dal Corso, and E. Tosatti,Phys. Rev. B , 115107 (2002). M. J. Rozenberg, Phys. Rev. B , R4855 (1997). P. Werner and A. J. Millis, Phys. Rev. Lett. , 126405(2007). E. Koch, O. Gunnarsson, and R. M. Martin, Phys. Rev. B , 15714 (1999). S. Florens, A. Georges, G. Kotliar, and O. Parcollet, Phys.Rev. B , 205102 (2002). Y. ¯Ono, M. Potthoff, and R. Bulla, Phys. Rev. B ,035119 (2003). J. E. Han, M. Jarrell, and D. L. Cox, Phys. Rev. B ,R4199 (1998). L. Laloux, A. Georges, and W. Krauth, Phys. Rev. B ,3092 (1994). E. Pavarini, S. Biermann, A. Poteryaev, A. I. Lichtenstein,A. Georges, and O. K. Andersen, Phys. Rev. Lett. ,176403 (2004). M. Mochizuki and M. Imada, Phys. Rev. Lett. , 167203(2003). M. S. Laad, L. Craco, and E. M¨uller-Hartmann, Phys. Rev.Lett. , 156402 (2003). A. I. Poteryaev, J. M. Tomczak, S. Biermann, A. Georges,A. I. Lichtenstein, A. N. Rubtsov, T. Saha-Dasgupta, andO. K. Andersen, Phys. Rev. B , 085127 (2007). G. Keller, K. Held, V. Eyert, D. Vollhardt, and V. I. Anisi-mov, Phys. Rev. B , 205116 (2004). M. S. Laad, L. Craco, and E. Muller-Hartmann, Phys. Rev. B , 045109 (2006). A. Rubtsov and A. Lichtenstein, JETP Letters , 61(2004). A. N. Rubtsov, V. V. Savkin, and A. I. Lichtenstein, Phys.Rev. B , 035122 (2005). P. Werner, A. Comanac, L. de’ Medici, M. Troyer, andA. J. Millis, Phys. Rev. Lett. , 076405 (2006). P. Werner and A. J. Millis, Phys. Rev. B , 155107 (2006). F. Lechermann, S. Biermann, and A. Georges, Phys. Rev.Lett. , 166402 (2005). F. Lechermann, S. Biermann, and A. Georges, Progress OfTheoretical Physics Supplement , 233 (2005). F. Lechermann, S. Biermann, and A. Georges, Phys. Rev.B , 085101 (2007). C. A. Marianetti, G. Kotliar, and G. Ceder, Phys. Rev.Lett. , 196405 (2004). H. Ishida, M. D. Johannes, and A. Liebsch, Phys. Rev.Lett. , 196401 (2005). C. A. Perroni, H. Ishida, and A. Liebsch, Phys. Rev. B ,045125 (2007). C. A. Marianetti, K. Haule, and O. Parcollet, Phys. Rev.Lett. , 246404 (2007). A. Liebsch and A. Lichtenstein, Phys. Rev. Lett. , 1591(2000). V. I. Anisimov, I. A. Nekrasov, D. E. Kondakov, T. M.Rice, and M. Sigrist, European Physical Journal B ,191 (2002). A. Liebsch and H. Ishida, Phys. Rev. Lett. , 216403(2007). X. Dai, G. Kotliar, and Z. Fang (2006), URL http://arxiv.org/abs/cond-mat/0611075v1 . J. Kunes, A. V. Lukoyanov, V. I. Anisimov, R. T. Scalet-tar, and W. E. Pickett, Nature Materials , 198 (2008). M. Caffarel and W. Krauth, Phys. Rev. Lett. , 1545(1994). K. Kugel and D. Khomskii, Sov. Phys. Usp. , 231 (1982). D. P. Arovas and A. Auerbach, Phys. Rev. B52