Rotationally-invariant slave-bosons for Strongly Correlated Superconductors
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Rotationally-invariant slave-bosons for Strongly Correlated Superconductors
A. Isidori and M. Capone , Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, Piazzale A. Moro 2, 00185, Rome, Italy and SMC, CNR-INFM and ISC-CNR, Piazzale A. Moro 2, 00185, Rome, Italy (Dated: November 19, 2018)We extend the rotationally invariant formulation of the slave-boson method to superconductingstates. This generalization, building on the recent work by Lechermann et al. [Phys. Rev. B , 155102 (2007)], allows to study superconductivity in strongly correlated systems. We apply theformalism to a specific case of strongly correlated superconductivity, as that found in a multi-orbitalHubbard model for alkali-doped fullerides, where the superconducting pairing has phonic origin, yetit has been shown to be favored by strong correlation owing to the symmetry of the interaction. Themethod allows to treat on the same footing the strong correlation effects and the interorbital interac-tions driving superconductivity, and to capture the physics of strongly correlated superconductivity,in which the proximity to a Mott transition favors the superconducting phenomenon. PACS numbers: 71.10.Fd, 71.10.-w, 71.30.+h, 74.25.Jb
I. INTRODUCTION
The theoretical description of strongly correlated sys-tems and of the prototypical models introduced to un-derstand their behavior plays a central role in modernmany-body theory. Even if the number of materials ofinterest in which the mutual interaction between elec-trons has been identified as relevant is now countless,there is no doubt that the main trigger for the devel-opment of the correlated-electron field has been the dis-covery of high-temperature superconductivity in dopedcorrelated insulators such as the copper oxides. Yet,the link between strong correlation and high-temperaturesuperconductivity has not been established unambigu-ously, which prompts for theoretical methods able to de-scribe the superconducting phenomenon in the presenceof strong electron correlations.One of the main reasons why strongly correlated sys-tems and their properties are, at the same time, interest-ing and hard to solve is that they are intrinsically out ofweak-coupling regimes, where a perturbative expansioncan be performed. Starting from an uncorrelated systemand imagining to continuously increase the degree of cor-relation, the relevance of local repulsion gradually intro-duces constraints to the electronic motion, leading even-tually to the localization of the carriers (Mott transition).Thus a proper method for correlated electrons should beable to introduce local constraints onto an otherwise un-correlated state, that would be naturally delocalized, i.e.,spatially unconstrained.A popular strategy which formally implements thispoint of view is based on slave bosons . Within theseapproaches, the Hilbert space is enlarged to include, be-sides fermionic degrees of freedom associated to Lan-dau quasiparticles, suitable extra degrees of freedomof bosonic character which are typically related to lo-cal states. The auxiliary (slave) degrees of freedomare then treated in a mean-field approximation, lead-ing to an effective low-energy theory for the quasipar-ticles. The high-energy physics can only be recovered introducing fluctuations of the fields describing the aux-iliary particles . We can see this strategy as a way toenforce a local point of view, which is expected to becorrect in very strong coupling, starting from delocalizednon-interacting states. In its most popular version, in-troduced by Kotliar and Ruckenstein , one defines oneboson for each local configuration (namely, | i , |↑i , |↓i , |↑↓i ), and the equivalence between the physical Hilbertspace and the new extended space is enforced by impos-ing constraints which, as we shall discuss below, implythat the local configurations should be coherently labeledby the fermionic and bosonic degrees of freedom, and thatprecisely one boson should be present on each lattice site.Yet, as thoroughly discussed in Ref. 4, the standard4-boson representation for the single-band Hubbardmodel, as well as its simplest generalizations to multi-orbital models , are not suitable to handle arbitraryforms of the interaction Hamiltonian, characterized byterms which cannot be put in the form of density-densityinteractions, such as exchange interactions associated tothe Hund’s rule coupling. Furthermore, even for puredensity-density interactions, the Kotliar-Ruckenstein ap-proach still remains inadequate to handle charge sym-metry breaking order parameters such as the supercon-ducting one in the Hubbard model. Slave-boson ap-proaches to superconductivity in the Hubbard modelhave indeed mostly used the approximately equivalentstrong-coupling t-J model, and have been based on spe-cific assumptions .The first attempt in overcoming the inadequacy ofKotliar-Ruckenstein’s representation was made by Li etal. , who proposed a spin-rotation invariant slave-bosonformulation of the single band Hubbard model, whilein Ref. 8 Fr´esard and W¨olfle introduced a more gen-eral representation for single-band models, in which spinand charge degrees of freedom are treated on the samefooting and rotational invariance involves both spin andparticle-hole transformations. Although the formalismpresented by these authors refers only to a 4-state sys-tem, with the slave-boson fields labeled in correspon-dence with the specific SU (2) ⊗ SU (2) generators of spinand particle-hole rotations, it already has all the ingredi-ents required for describing systems with local supercon-ducting pairing. Such method has been indeed appliedto the single-band attractive Hubbard model in Ref. 9.Developing the ideas of these pioneering works in a moresystematic way, Lechermann et al. finally built a com-pletely basis-independent slave-boson formalism, suitableto describe, within a generic multi-orbital model, anyarbitrary form of local interaction. As it is found forKotliar-Ruckenstein’s approach, it is worth mentioningthat, at mean-field level, such formalism turns out to beequivalent to analogous extensions of the Gutzwillerapproach .While the possibility of extending the formalism tosuperconducting states is mentioned in Ref. 4, the ex-plicit derivation is limited to normal solutions, impos-ing no charge symmetry breaking. In the present work,instead, we lift this restriction and consider explicitlythe more general case of full rotational-invariance underany local transformation of the electronic degrees of free-dom, and we apply the formalism to solve a three-orbitalmodel which has been proposed to describe alkali-dopedfullerides . Besides its relevance to the fullerides, themodel has important properties that led us to choose itas an optimal benchmark for our method. The modelhas indeed been shown to present “strongly correlatedsuperconductivity” , i.e., the enhancement of phononmediated superconductivity in the proximity of a Motttransition. The key of the phenomenon is that a smallattraction which involves orbital and spin degrees of free-dom is not screened when charge fluctuations are frozenby strong correlations. This leads to an enhancement ofsuperconductivity, since the unscreened attraction nowacts on strongly renormalized quasiparticles with a largereffective density of states. This effect has been identifiedusing Dynamical Mean-Field Theory (DMFT) , whichfully takes into account local quantum fluctuations, butit has not been reproduced by ordinary slave-boson meth-ods due to the difficulties in treating interactions whichare not of the charge-charge form, such as those drivingsuperconductivity in the model we are dealing with. Inthis light, the model is an ideal test ground of the abilityof the rotationally-invariant slave boson method in ac-curately treating general forms of interactions. On theother hand, the model has only local (on-site) interac-tions, which simplifies the approach.The paper is organized as follows. In Sec. II we intro-duce the rotationally-invariant slave boson method formodels with local superconducting pairing. In Sec. IIIwe present the multi-orbital model used for the descrip-tion of fullerenes, illustrating the way it can be solved bymeans of the slave boson approach. In Sec. IV we presentthe results obtained with our method, and finally Sec. Vis dedicated to concluding remarks and perspectives. II. THE GENERAL FORMALISM
In this section we will explicitly extend therotationally-invariant slave-boson formalism introducedby Lechermann et al. to the possibility of describing su-perconducting states. To facilitate the reading and thecomparison with the formalism of Ref. 4, whenever pos-sible we shall use the same notation for correspondingquantities. A. Motivations
Without entering in details, we shall first provide abrief reminder on slave-boson formulations, in order toface the difficulties encountered with non-invariant ap-proaches such as Kotliar-Ruckenstein’s one.In a generic multi-orbital model the local Hilbert spaceof electronic states is defined as the set of all the possible“atomic” configurations at a given lattice site (for sim-plicity, we will drop site indices throughout this section);a natural choice for the basis set of this space is providedby the 2 M Fock states | n i ≡ (cid:16) d † (cid:17) n · · · (cid:16) d † M (cid:17) n M | vac i , [ n α = 0 ,
1] (1)where α = 1 , . . . , M are the local orbital species and d † α the corresponding electron-creation operators. A slave-boson representation is then constructed by mapping thelocal Hilbert space H (e.g., the Fock states | n i ) onto an“enlarged” Hilbert space H generated by the tensor prod-ucts of boson operators φ † µ and auxiliary fermion opera-tors f † α : H : {| n i} 7−→ H : nQ µ (cid:0) φ † µ (cid:1) N µ | vac i ⊗ | n i f o . In the above expression | n i f refers to the Fock states gen-erated by the auxiliary fermions f † α , which correspond toquasiparticle (QP) degrees of freedom: their presence en-sures the possibility of describing Fermi liquid propertieswithin the auxiliary-fields representation (note that theorbital basis for quasiparticle degrees of freedom may notcoincide, in general, with that of the physical electron op-erator d † α ). On the other hand, an arbitrary number ofauxiliary bosons φ µ , for each species µ , can in princi-ple be present in the enlarged Hilbert space, unless someconstraints, which characterize the specific form of theslave-boson representation, are imposed to its states. Inother words, a given representation is defined by the wayin which the auxiliary states are selected out of the en-larged space H in order to represent uniquely the originalphysical states | n i , | n i 7−→ | n i , | n i ≡ X { N µ } , m K ( n, { N µ } , m ) Y µ (cid:0) φ † µ (cid:1) N µ | vac i ⊗ | m i f . (2)Needless to say, the choice of a specific representation K , apart from being consistent with the above unique-ness assumption, must also provide some simplificationsin the (local) interaction Hamiltonian: the whole purposeof introducing auxiliary bosons is indeed the possibilityof writing local interactions as a sum of quadratic termsin the boson fields, at the expense of a larger numberof degrees of freedom and a more complex structure ofhopping terms.In multi-orbital generalization of Kotliar-Ruckenstein’sapproach, 2 M boson fields φ µ ≡ φ n are introduced incorrespondence with the original Fock states | n i , and therepresentation of such states in the enlarged Hilbert spacereads | n i ≡ φ † n | vac i ⊗ | n i f . (3)The “physical states” in H are therefore those states con-taining exactly one boson and whose quasiparticle con-tent | n i f matches the Fock configuration associated tothe boson field φ n . From Eq. (3) it should be evidentthe non-invariant nature of this representation under ro-tations of the quantization basis. Consider, in fact, an SU ( M ) rotation of the orbital indices d † α = P β U αβ ˜ d † β , f † α = P β U αβ ˜ f † β . (4)This rotation will induce a corresponding unitary trans-formation on both physical and QP Fock states, | n i = P m U ( U ) nm | e m i , so that the representation of physicalstates would now read | e m i = X n m ′ (cid:0) U † mn φ † n U nm ′ (cid:1) | vac i ⊗ | e m ′ i ˜ f = X m ′ ˜ φ † mm ′ | vac i ⊗ | e m ′ i ˜ f . (5)In the new orbital basis, therefore, the slave-boson repre-sentation do not retain its original form, and more specif-ically the definite relation between physical states andtheir quasiparticle content no longer holds. As discussedin Ref. 8, indeed, only disentangling physical and quasi-particle degrees of freedom it becomes possible to formu-late rotationally-invariant slave-boson representations.As a consequence of its basis-dependent nature,Kotliar-Ruckenstein’s approach can be applied only tosystems whose local Hamiltonian can be written, in anappropriate basis, in terms of purely orbital-density op-erators ˆ n α = d † α d α , H loc = X α ǫ α ˆ n α + X αβ W αβ ˆ n α ˆ n β , (6)i.e., when the Fock states | n i are eigenstates of H loc .In this case, the representation of H loc in the enlargedHilbert space can be easily written as a free-boson Hamil-tonian, H loc = X n E n φ † n φ n , (7) with E n = P α ǫ α n α + P αβ W αβ n α n β . We remark, how-ever, that the definite relation imposed between quasipar-ticle degrees of freedom and the (physical) Fock contentof boson fields inhibits the development of those spon-taneous symmetry-breaking order-parameters that can-not be expressed in terms of orbital-density operators(e.g., superconductivity, magnetization perpendicular tothe spin-quantization axis, etc.). B. Representation of physical states
The electron Hamiltonian for a generic multi-orbitalmodel with purely local interactions is given by H = H kin + X i H loc [ i ] , (8) H kin = X k X αβ ǫ αβ ( k ) d † k α d k β , (9)where all the local terms, including the chemical potentialand the orbital energy levels, are included in H loc , so that P k ǫ αβ ( k ) = 0.In comparison to the previous subsection, we choosehere, as the basis set for the (physical) local Hilbertspace, a generic set of states {| A i} (not necessarily Fockstates) that are eigenstates of the local particle-numberoperator ˆ n ( d ) = P Mα =1 d † α d α , with eigenvalues N A . Even-tually, among these sets, we can choose the eigenstates {| Γ i} of the local Hamiltonian, since H loc commutes withthe local number operator. The basis set for quasiparti-cle states, instead, is still given by the Fock states | n i f generated by the auxiliary fermion operators f † α .As discussed previously in pointing out the limita-tions of Kotliar-Ruckenstein’s approach, the key ingre-dient in constructing rotationally-invariant slave-bosonrepresentations is to disentangle physical and quasipar-ticle degrees of freedom . Therefore, we introduce a setof auxiliary boson fields φ µ ≡ φ An associated, in princi-ple, to each pair ( | A i , | n i f ) of physical and quasiparticlestates, without assuming any a priori relation betweenthose states in the enlarged Hilbert space representation.Depending on the phases one takes into account, how-ever, there exist some limitations in the possible H –states φ † An | vac i ⊗ | n i f which can figure in the representation ofa physical state | A i , | A i ∝ X n φ † An | vac i ⊗ | n i f . (10)Indeed, if we limit to normal phases as in Ref. 4, it issufficient to consider, for a given state | A i , only thosestates | n i f which have exactly the same number of parti-cles of | A i ; in other words, physical states with a definitenumber of electrons are represented by a superpositionof auxiliary states characterized by the same number ofquasiparticles. On the other hand, when allowing for thespontaneous breaking of particle-number conservation, asin superconducting states, we need to consider, for eachstate | A i , all the Fock states | n i f characterized by " M X α =1 n α − N A (mod 2) = 0 , i.e., all the 2 M − quasiparticle states with the samestatistics of | A i . While the former representation is in-variant only under rotations of the QP basis that areblock-diagonal in the quasiparticle occupation number P Mα =1 n α , the latter is invariant under a larger class ofQP rotations, represented by all the unitary transforma-tions that preserve the statistics of quasiparticle states:in such representation, the quasiparticle number opera-tor is no longer a conserved quantity, and its expectationvalue does not correspond to any physical observable, asparticle-hole transformations may change its value.The explicit representation of | A i will then read, in thefully-invariant formalism, | A i ≡ √ M − X n φ † An | vac i ⊗ | n i f , (11)with the sum running over the 2 M − QP Fock stateswhose particle-number parity equals that of | A i . It isworthwhile to remark that the above representation can-not be further enlarged, including, for example, in thedefinition of | A i , the remaining quasiparticle states withopposite statistics. This, in fact, would lead to unphys-ical results such as non-vanishing expectation values ofodd numbers of fermion operators. Constraints
In order to characterize uniquely the physical statesamong all the states of the enlarged Hilbert space H , itis necessary and sufficient that the selected states satisfy,as operator identities, the following constraints: X An φ † An φ An = 1 , (12) X A X nn ′ φ † An φ An ′ h n ′ | f † α f α ′ | n i = f † α f α ′ , (13) X A X nn ′ φ † An φ An ′ h n ′ | f † α f † α ′ | n i = f † α f † α ′ . (14)The first two types of constraints are already present inthe normal-phase formalism of Lechermann et al. : thefirst equation, indeed, limits the physical subspace of H to one-boson states only, while the second set of con-straints ensures the rotational invariance of (11) underQP rotations that preserve quasiparticle number. Onthe other hand, the last set of constraints promotes therotational invariance to particle-hole rotations, enablingthe non-conservation of quasiparticle number.It is worthwhile to remark that, for single-band models( M = 2), the above equations reduce to the same set of constraints characterizing the SU (2) ⊗ SU (2) spin-chargeinvariant formalism introduced in Refs. 8 and 9, as longas the appropriate changes in notation are made. Forthis purpose, Ref. 9 provides a useful link between ournotation and the one presented in Ref. 8.A more compact form of Eqs. (12-14) will be derivedin Sec. II D, where we shall relate the role of constraintsto the gauge group structure of the slave-boson represen-tation. C. Physical electron operator
The representation of the physical electron creation op-erator in the enlarged Hilbert space is defined by d † α | B i = X A h A | d † α | B i | A i . (15)When the constraints (12-14) are satisfied exactly, itsexpression in terms of bosons and quasiparticle operatorsreads d † α = 1 M X AB, nm, β h A | d † α | B i φ † An φ Bm ×× h h n | f † β | m i f † β + h n | f β | m i f β i , (16)with the normalization factor 1 /M coming from the fol-lowing relation: X p, β h h n | f † β | p i f † β | p i + h n | f β | p i f β | p i i = M | n i . (17)We can thus summarize the non-diagonal relation be-tween physical and quasiparticle degrees of freedom inthe form d † α = R ( p ) αβ [ φ ] ∗ f † β + R ( h ) αβ [ φ ] f β (18)(summation over repeated indices is implied), where wehave defined the R -matrix operators as R ( p ) αβ [ φ ] ∗ = 1 M X AB, nm h A | d † α | B i φ † An φ Bm h m | f β | n i , (19) R ( h ) αβ [ φ ] = 1 M X AB, nm h A | d † α | B i φ † An φ Bm h m | f † β | n i . (20)In the above expressions, we have taken advantage ofthe reality of the matrix elements between Fock states, h n | f β | m i = h m | f † β | n i , in order to guarantee the correcttransformation properties of the R -operators under thegauge group transformations discussed in Sec. II D.At the saddle-point level, on the other hand, when theboson fields are treated as probability amplitudes and theconstraints are satisfied only on average, the expressionof d † α must be modified , in order to recover the cor-rect normalization of transition amplitudes in the non-interacting limit. For this purpose, it is easier to definethe physical electron operator in the orbital basis { λ } inwhich the quasiparticle and quasihole density matricesare diagonal,ˆ∆ ( p ) αβ ≡ X Anm φ ∗ An φ Am h m | f † α f β | n i = X λ U αλ ξ λ U † λβ , (21)ˆ∆ ( h ) αβ ≡ X Anm φ ∗ An φ Am h m | f β f † α | n i = X λ U αλ (1 − ξ λ ) U † λβ , (22)where ξ λ = P An | Ω An | n λ is the probability to find thesystem in a state such that n λ = 1, i.e., with a quasi-particle in the orbital λ (note that P An | Ω An | = 1). Inthese expressions, the quasiparticle operators referred tothe new orbitals are related to the old ones by the unitarytransformation f † α = X λ U αλ ψ † λ , (23)while the boson fields transform with the correspondingrotation of the Fock states (summation over repeated in-dices is implied) | n i f = U ( U ) nm | m i ψ , φ An = U ( U ) nm Ω Am . (24)In such basis, the transition amplitude between stateswith n λ = 0 in the initial [final] configuration, and n λ = 1in the final [initial] one, must be normalized by the factor1 / p ξ λ (1 − ξ λ ), yielding the following expression for thephysical electron operator: d † α = X AB, nm, λ h A | d † α | B i Ω ∗ An Ω Bm p ξ λ [Ω](1 − ξ λ [Ω]) ×× h h n | ψ † λ | m i ψ † λ + h n | ψ λ | m i ψ λ i . (25)Rotating back to the original basis, we finally get, for thesaddle-point expressions of the R -matrices, R ( p ) αβ [ φ ] ∗ = X AB, nm, γ h A | d † α | B i φ ∗ An φ Bm h m | f γ | n i M γβ ,R ( h ) αβ [ φ ] = X AB, nm, γ h A | d † α | B i φ ∗ An φ Bm h m | f † γ | n i M βγ , (26)where M γβ = (cid:20) (cid:16) ˆ∆ ( p ) ˆ∆ ( h ) + ˆ∆ ( h ) ˆ∆ ( p ) (cid:17)(cid:21) − γβ (27)is the particle-hole symmetrized version of the normal-ization factor, expressed in the original basis. D. Functional integral representation
The partition function of a generic multi-orbital Hamil-tonian (8) can be formally written, in terms of auxiliaryfields, as Z = Z D [ f, f † ] D [ { φ } , {A} ] e − R β dτ L ( τ ) , (28)where we have introduced, along with slave bosons andauxiliary fermions, a set of Lagrange multiplier fields {A i ( τ ) } that allow to enforce, at each lattice site i andimaginary-time value τ , the constraints (12-14). The La-grangian functional entering the above expression reads L ( τ ) = X i (cid:16) φ † An,i ∂ τ φ An,i + f † α,i ∂ τ f α,i + H const [ i ] + H loc [ i ] (cid:17) + H kin (29)(except for lattice sites, summation over repeated indicesis always implied throughout this section), where H loc and H kin are, respectively, the representatives of the lo-cal and kinetic part of the Hamiltonian (8) in the enlargedHilbert space, while H const contains the constraint inter-actions between auxiliary fields and Lagrange multipliers.In order to derive the expressions of the Hamiltonianterms in (29), and thereby identify the underlying sym-metry group of the Lagrangian, it is convenient to collectall the local fermionic degrees of freedom (either physi-cal or auxiliary) into a 2 M -component Nambu-Gor’kovspinor: Ξ i ≡ (cid:18) { d α,i }{ d † α,i } (cid:19) , Ψ i ≡ (cid:18) { f α,i }{ f † α,i } (cid:19) . In such formalism, the representation of physical elec-trons in terms of bosons and quasiparticles (18) is simplywritten as Ξ i = R i Ψ i , where the local 2 M × M matrixoperator R i ≡ R [ φ i ] = (cid:18) R ( p ) [ φ i ] R ( h ) [ φ i ] ∗ R ( h ) [ φ i ] R ( p ) [ φ i ] ∗ (cid:19) (30)is defined in terms of the boson fields φ An,i associated tothe corresponding site i (note that, to lighten the nota-tion, site indices were omitted in previous sections). Therepresentation of the kinetic Hamiltonian is then readilyobtained as H kin = X ij t αβij d † α,i d β,j = 12 X ij Ξ † i ˜ t ij Ξ j = 12 X ij Ψ † i R † i ˜ t ij R j Ψ j , (31)˜ t ij = (cid:18) t ij − t ∗ ij (cid:19) , (32) t ij being the real-space hopping matrix, whose Fouriertransform gives the band dispersion matrix ǫ ( k ) definedin Eq. (9).On the other hand, the local terms of the model Hamil-tonian, which may include any kind of on-site interac-tion between (physical) electrons, are represented, withinthe enlarged Hilbert space, by a purely quadratic bosonHamiltonian: H loc [ i ] = h A | H loc | B i φ † An,i φ Bn,i = E Γ φ † Γ n,i φ Γ n,i , (33)where the physical states {| Γ i} denote the eigenstates of H loc . Gauge invariance
In order to discuss the symmetry structure of the La-grangian (29), we begin to notice that the auxiliary-fieldsHamiltonian H = H kin + P i H loc [ i ] is invariant under thefollowing gauge transformations:Ψ i ( τ ) → U i ( τ )Ψ i ( τ ) , (34) φ An,i ( τ ) → e iξ i ( τ ) × U [ U ] nn ′ φ An ′ ,i ( τ ) . (35)The unitary matrix U i ( τ ) in (34) represents an arbitrary SO (2 M ) rotation of quasiparticle operators acting inde-pendently on each site, and it is conveniently parameter-ized as U i ( τ ) = e iξ ai ( τ ) T a , (36)the T a matrices being a 2 M -dimensional representationof the M (2 M −
1) group generators. We note, however,that such matrices are not expressed in the usual form ofan orthogonal group generator (namely, as purely imagi-nary antisymmetric matrices), but are instead of the form T a = (cid:18) T a H T a A − ( T a A ) ∗ − ( T a H ) ∗ (cid:19) , (37)with T a H and T a A denoting, respectively, M × M Her-mitian and antisymmetric matrices. In other words, the2 M -dimensional Nambu spinors do not transform withreal orthogonal matrices under SO (2 M ) rotations, eventhough they clearly form a real representation of thegauge group: Ψ † i = Ψ Ti E , (38) U ∗ i = E U i E , (39) E = (cid:18) (cid:19) . (40)The transformation law of the boson fields, on the otherhand, is characterized by the unitary transformationof Fock states | n i → U [ U ] nn ′ | n ′ i that is associated tothe SO (2 M ) rotation of quasiparticle operators, plus anadditional U (1) factor e iξ i ( τ ) under which the Nambu spinors are neutral. Following the exponential parame-terization of U i ( τ ), we can then similarly write U [ U ] nn ′ = h e iξ ai ( τ ) J a i nn ′ , (41)where J ann ′ are the SO (2 M ) group generators expressedin the Fock space representation.While the gauge invariance of H loc [ i ] follows immedi-ately from its definition (33), in the case of H kin we stillneed to establish the transformation properties of the R i operators, defined in Eq. (30). These, however, canbe readily obtained by writing such operators directly interms of the physical and auxiliary Nambu spinors, inmatrix notation: R i = 1 M h A | Ξ | B i φ † An,i φ Bm,i h m | Ψ † | n i , (42) R i ( τ ) → R i ( τ ) U † i ( τ ) . (43)The above transformation law ensures the gauge invari-ance of the physical electron operator Ξ i = R i Ψ i , andhence of the whole kinetic Hamiltonian (31).In the discussion of the gauge invariance of the La-grangian (29), we are thus left with the time-derivativesand constraints terms, whose transformation propertiesare closely related to each other. The time-derivativeterms, in fact, are clearly not invariant under the trans-formations (34) and (35), which generate inhomogeneousterms proportional to the time derivatives of the rota-tion parameters (e.g., ∂ τ ξ ai and ∂ τ ξ i ). Such terms, how-ever, can be reabsorbed by a corresponding inhomoge-neous transformation of the Lagrange multiplier fields (which may be regarded as “gauge bosons”), making thewhole Lagrangian gauge invariant.To show how this mechanism works, we rewrite the M + M ( M −
1) local constraints (13) and (14) in thefollowing way: φ † An,i φ An ′ ,i h n ′ | Ψ † T a Ψ | n i = Ψ † i T a Ψ i , (44)where we have made use of the “orthogonality” betweenthe SO (2 M ) T a matrices. It is then straightforward(though somewhat lengthy) to verify that the Fock-spacegenerators J ann ′ , introduced in Eq. (41), are representedby J ann ′ = − h n ′ | Ψ † T a Ψ | n i (45)(in other words, the above matrices provide a faithfulrepresentation of the SO (2 M ) Lie algebra), so that wecan finally write: H const [ i ] = A ai (cid:18)
12 Ψ † i T a Ψ i + φ † An,i J ann ′ φ An ′ ,i (cid:19) + A i (cid:16) φ † An,i φ An,i − (cid:17) . (46)Together with the time-derivative terms, the above inter-actions may be arranged in “covariant derivatives” actingon the auxiliary fields (fermions and bosons),12 Ψ † i D τ Ψ i = 12 Ψ † i [ ∂ τ + A ai T a ] Ψ i , (47) φ † An,i D τ φ An,i = φ † An,i (cid:2)(cid:0) ∂ τ + A i (cid:1) δ nn ′ + A ai J ann ′ (cid:3) φ An ′ ,i , (48)where the role of gauge fields is played by the Lagrangemultipliers A ai and A i . It is then easily checked that theLagrangian L = X i (cid:20) φ † An,i D τ φ An,i + 12 Ψ † i D τ Ψ i + H loc [ i ] − A i (cid:21) + H kin (49)is indeed invariant under the SO (2 M ) ⊗ U (1) gaugegroup, provided the Lagrange multipliers transform asgauge fields in the adjoint representation : A i → A i − i∂ τ ξ i ( τ ) , (50) A ai T a → U i ( τ ) [ A ai T a + ∂ τ ] U † i ( τ ) . (51)Note that the transformation law of A ai T a induces a cor-responding transformation of A ai J ann ′ that has exactlythe same structure of (51), with the U i matrix replacedby U [ U ] nn ′ . More precisely, both transformation laws de-scend from that of the Lagrange multiplier field A ai ( τ ),which for infinitesimal rotations transforms as A ai → A ai + f abc A bi ξ ci − i∂ τ ξ ai + O ( ξ ) , (52) f abc being the structure constants of the group.Finally, it is worth mentioning that for M = 2 (single-band models) the orthogonal group SO (4) is locally iso-morphic to SU (2) ⊗ SU (2), so that the gauge group struc-ture of the present formalism reduces to that of the spin-charge invariant formalism of Ref. 8. Gauge fixing
As discussed in Ref. 8, the gauge invariance of thefunctional integral representation causes Eq. (28) to con-tain integration over spurious degrees of freedom, namelythe physically equivalent field configurations connectedto each other by gauge group trajectories. It is thus nec-essary to impose a “gauge fixing” condition that removesthe integration over the unphysical degrees of freedom,as it is usually done in gauge field theories .We choose to work in the so-called “radial gauge” , inwhich the complex boson fields are represented throughreal amplitudes and complex phase fields. In this rep-resentation, the SO (2 M ) ⊗ U (1) gauge transformationsallow to remove M (2 M −
1) + 1 phase variables, so thata corresponding number of boson fields can be reducedto purely real amplitudes, with no phase fluctuations.The boson fields that remain fully complex, on the otherhand, continue to display some phase dynamics, which isresponsible for the incoherent features of the spectrum (e.g., lower and upper Hubbard bands). It is beyond our purpose to enter into the formal detailsof the radial gauge representation, thoroughly derived inRef. 17. Nevertheless, it is worthwhile to observe, here,that such gauge fixing procedure allows to avoid Elitzur’stheorem , which would prevent the slave bosons to ac-quire a non-zero expectation value, making therefore le-gitimate the use of the saddle-point approximation. E. Saddle-point approximation
The saddle-point approximation of the functional inte-gral (28) is obtained by considering the slave bosons andLagrange multipliers as static variables, corresponding tothe time-averages of these fields. Moreover, we will as-sume a homogeneous spatial structure of the saddle-pointsolution, so that we can finally set: φ An,i ( τ ) ϕ An , A ai ( τ )
7→ A a and A i ( τ )
7→ A .In such approximation, all the bosonic amplitudes ϕ An are assumed to have a constant phase, in contrast to theradial gauge representation, where we are allowed to re-move (fix) a limited number of complex phases (namely M (2 M −
1) + 1). The phase fluctuations of those fieldsthat remain intrinsically complex are thus ruled out,precluding the possibility of describing the high-energyphysics of a given model. The low-energy features, onthe other hand, will be suitably described in terms ofcoherent Landau-Bogoliubov quasiparticles, providing aFermi-liquid description of metallic and superconductingstates.At the saddle point, the free energy per site Ω = − (1 /βN ) ln Z is obtained as the (minimum) stationaryvalue of the following free-energy functional:Ω[ { ϕ } , {A} ] = − βN ln Z f − A (53)+ X AB nm ϕ ∗ An (cid:2) h A | H loc | B i δ nm + (cid:0) A δ nm + A a J anm (cid:1) δ AB (cid:3) ϕ Bm , where N is the total number of sites and Z f representsthe Gaussian integral over the auxiliary fermions, Z f = Z D Ψ exp − Z β dτ X ij Ψ † i (cid:2) δ ij D τ + R † ˜ t ij R (cid:3) Ψ j = Z D Ψ exp " − Z β dτ X k Ψ † k [ ∂ τ + h ( k )] Ψ k . (54)In the last expression, h ( k ) denotes the momentum-spacequasiparticle energy matrix: h ( k ) = R † [ ϕ ] (cid:18) ǫ ( k ) 00 − ǫ ( − k ) ∗ (cid:19) R [ ϕ ] + A a T a . (55)To evaluate the functional integral in Eq. (54), we notethat, up to irrelevant constants, Z f = det [ ∂ τ + h ( k )], sothat we can writeln Z f = 12 ln det [ ∂ τ + h ( k )]= 12 X k tr ln h e − β h ( k ) i . (56)The saddle-point equations are then obtained by set-ting to zero all the partial derivatives of the free-energyfunctional (53) with respect to the slave boson ampli-tudes and Lagrange multipliers. For practical calcula-tions, however, it is useful to consider a different basisset for the M (2 M −
1) Lagrange multipliers A a , belong-ing the adjoint representation of SO (2 M ). In place ofthem, in fact, we can equivalently use the set of indepen-dent matrix elements of A a T a , parameterized as follows: A a T a = (cid:18) Λ Π − Π ∗ − Λ ∗ (cid:19) , (57)Λ αβ = Λ ∗ βα , Π αβ = − Π βα . With this choice, the saddle-point equations can be de-rived without knowing the explicit expressions of the T a matrices, differentiating the free-energy functional di-rectly with respect to Λ αβ and Π αβ . To this end, we notethat Ω[ { ϕ } , {A} ] can be easily expressed in terms of thenew set of Lagrange multipliers by means of the followingrelation: A a J ann ′ = − h n ′ | Ψ † ( A a T a )Ψ | n i = − h Λ αβ h n ′ | (cid:0) f † α f β − f β f † α (cid:1) | n i (58)+ Π αβ h n ′ | f † α f † β | n i + Π ∗ αβ h n ′ | f β f α | n i i . F. Green’s functions and observables
After solving the saddle-point equations, we can finallyobtain the expressions for the quasiparticle and phys-ical electron propagators, conveniently written here inNambu notation. For quasiparticles, the propagator isdefined as D f ( k , τ ) = −h T Ψ k ( τ )Ψ † k (0) i (59)= G f ( k , τ ) F † f ( k , − τ ) F f ( k , τ ) − G Tf ( − k , − τ ) ! , where G f ( k , τ ) αβ = −h T f k α ( τ ) f † k β (0) i and F f ( k , τ ) αβ = −h T f †− k α ( τ ) f † k β (0) i are the normal and anomalous quasi-particle Green’s functions. Following Eq. (54), we canthen readily write the quasiparticle inverse propagatoras D − f ( k , ω ) = ω − h ( k ) . (60) The physical electron propagator, on the other hand, isdefined by D d ( k , τ ) = −h T Ξ k ( τ )Ξ † k (0) i , (61)where Ξ † k ≡ (cid:16) { d † k α } , { d − k α } (cid:17) is the Nambu spinor con-taining the physical degrees of freedom, represented interms of slave boson amplitudes and quasiparticles byΞ k = R [ ϕ ]Ψ k . The expression for the inverse physicalpropagator is thus written as D − d ( k , ω ) = [ R † ] − [ ω − h ( k )] R − . (62)Using the corresponding expression for the “bare” phys-ical propagator, D − d ( k , ω ) = ω − [˜ ǫ + ˜ ǫ ( k )] , (63)˜ ǫ + ˜ ǫ ( k ) = (cid:18) ǫ + ǫ ( k ) 00 − [ ǫ + ǫ ( − k )] ∗ (cid:19) (64)( ǫ represents the one-body part of H loc ), we can thenfind the saddle-point approximation for the self-energy: Σ d ( ω ) = D − d − D − d = ω (cid:0) − [ RR † ] − (cid:1) − ˜ ǫ + [ R † ] − (cid:18) Λ Π − Π ∗ − Λ ∗ (cid:19) R − . (65)The cancellation of the k -dependence, in the above equa-tion, follows from the definition of the QP energy matrix,Eq. (55). Indeed, this form of the self-energy is just theone we would expect from the saddle-point (mean-field)approximation, which freezes spatial and dynamical fluc-tuations. From the linear term in ω , one readily obtainsthe matrix of quasiparticle spectral weights: Z = (cid:20) − ∂ Σ d ∂ω (cid:21) − ω =0 = RR † . (66)We conclude this formal section by writing the rep-resentation, in the enlarged Hilbert space, of the localphysical density operator, whose expectation value de-fines the average number of electrons per site:ˆ n ( d ) = X α d † α d α = X A N A X n φ † An φ An . (67)As mentioned previously, we remark that this expressiondiffers substantially from that for the local quasiparticledensity, X α f † α f α = X An X α n α φ † An φ An , (68)which is not a physical quantity and depends from thechoice of the QP basis set. More generally, the represen-tation of any (local) two-particle physical operator maybe easily obtained in terms of boson operators only: forparticle-hole operators we find d † α d β = X AB h A | d † α d β | B i X n φ † An φ Bn , (69)while for particle-particle operators we have d † α d † β = X AB h A | d † α d † β | B i X n φ † An φ Bn . (70) III. APPLICATION TO THE THREE-ORBITALMODEL FOR FULLERIDESA. The model
In this section we present an explicit application of therotationally invariant slave boson approach. To this aimwe have chosen a three-orbital Hubbard-like model thathas been used to understand the role of strong correla-tions in alkali-doped fullerides. The physics of alkali-doped fullerene systems A n C represents an optimalplayground in understanding the key role of strong cor-relations on high-temperature superconductors: indeed,these systems display a relatively high T c compared toordinary Bardeen-Cooper-Schrieffer (BCS) superconduc-tors, and, similarly to what is found in cuprate systems,the enhancement of T c seems to be closely related to theproximity of a Mott insulating phase . Althoughthe nature of the pairing mechanism is most likely differ-ent from that characterizing cuprate superconductors, infullerene systems being due to ordinary electron-phonon(vibron) interaction , and in spite of the different sym-metry of the order parameter ( s –wave in fullerides), thephase diagram as a function of the inter-molecule separa-tion in these systems presents strong similarities to thatof cuprates as a function of doping , providing an inde-pendent example of the key role of electronic correlationsin enhancing superconductivity.The local Hamiltonian describing the C − molecularion, assuming rotational invariance within the threefolddegenerate level t u hosting the valence electrons pro-vided by the alkali metals, can be written, for a genericsite i , as H loc [ i ] = U n i − J H (cid:20) S i · S i + 12 L i · L i + 56 (ˆ n i − (cid:21) + H JT , (71)where the three terms represent, respectively, the globalon-site Coulomb repulsion, Hund’s rules splitting ( J H >
0) and the Jahn-Teller coupling between electrons andthe vibrational modes (vibrons) of C . In the above ex-pression, ˆ n i = P aσ d † i,aσ d i,aσ is the local electron numberoperator ( a = 1 , , t u orbitals and σ = ↑ , ↓ the spin components), while S i and L i are, respectively, the local spin and orbital angular-momentum operators, S i = 12 X a, σσ ′ d † i,aσ ˆ σ σσ ′ d i,aσ ′ , (72) L i = X ab, σ d † i,aσ ˆ ℓ ab d i,bσ , (73)where ˆ σ σσ ′ are the Pauli matrices and ˆ ℓ ( a ) bc = iε bac the O (3) group generators characterizing orbital rotations.The Jahn-Teller Hamiltonian involves both electronand vibron field operators, but if we are interested only inthe electron dynamics we can formally integrate out thevibronic degrees of freedom, obtaining an effective actionfor the electron operators only. If performed exactly, thisprocedure would clearly generate non-instantaneous (i.e.,time-dependent) interaction terms, preventing a purelyelectronic Hamiltonian formulation of the effective ac-tion; however, if we assume the vibronic frequencies tobe much higher than the relevant electronic scales, wecan take the anti-adiabatic limit of the electron effec-tive action, neglecting retardation effects and consider-ing an instantaneous interaction term which preservesthe symmetries of the original local Hamiltonian. Whilethis approximation may be questionable for fullerides, itwill not affect our claims since the neglect of retarda-tion can only disfavor superconductivity, and analogousresults have been obtained in a similar model that takesinto account the phonon dynamics . The Jahn-Tellerinteraction can then be reabsorbed in Hund’s term, andthe resulting Hamiltonian is simply given by the first twoterms of Eq. (71) with a renormalized Hund’s coupling J H
7→ − J = J H − E JT < E JT being the char-acteristic Jahn-Teller energy gain . The net effect ofthe electron-vibron coupling is therefore that of revers-ing Hund’s rules, favoring atomic configurations with lowspin and orbital angular momentum. The inversion of theHund’s rule is experimentally confirmed by the low-spinstate of both tetravalent and trivalent fullerides andby the presence of a spin gap .Given the local Hamiltonian for the C − molecularion, the expression for a tight-binding electronic Hamil-tonian describing A n C solids will then read H latt = X ij, ab, σ t abij d † i,aσ d j,bσ + X i H loc [ i ] − µ X i ˆ n i , (74)where t abij are the inter-site hopping amplitudes (includ-ing possible inter-band hybridization terms) and µ is thechemical potential controlling the average electron den-sity. We should note, however, that inter-band hybridiza-tion can actually be avoided whenever the hopping termsare restricted to nearest-neighbors only, t abij = t ab δ h ij i ; insuch case, indeed, we can exploit the O (3) orbital sym-metry of H loc in order to diagonalize t ab , so that we canset, without loss of generality, t abij = t a δ ab δ h ij i . Through-out our analysis we will use the latter expression for thehopping matrix elements, focusing in particular on theorbitally degenerate case t a = − t (we will consider the0possibility of a crystal-field splitting of the three bandsand of different bandwidths in a future study). B. Slave-boson representation of the model
As pointed out in previous studies , the zero-temperature phase diagram of the model (74) as a func-tion of the ratios
J/W and
U/W ( W being the non-interacting bandwidth) displays several interesting fea-tures, the most striking one being undoubtedly the pres-ence of a strongly enhanced superconducting phase in theproximity of the metal-to-insulator Mott transition. Themodel represents therefore a valuable test for the rota-tionally invariant slave boson method, at the same timeproviding an analytical tool to treat strongly correlatedsuperconductors. Slave-boson amplitudes
From the rotationally-invariant slave-boson represen-tation of physical states defined by Eq. (11), it shouldbe clear that there are, in principle, 2 M − (with M = 6in our model) slave-boson fields φ An describing the sys-tem. However, we must note that when the partitionfunction is approximated at saddle-point level, i.e., whenthe slave-boson fields are replaced by their mean-field ex-pectation values ( φ An → h φ An i ≡ ϕ An ), the number ofindependent slave-boson amplitudes entering the saddle-point equations becomes much smaller, as we will showin the following, its specific value depending on whichsymmetries of the model Hamiltonian remain unbroken.As discussed at end of Sec. III A, we choose the hoppingmatrix to be diagonal and degenerate in both spin and or-bital indices, so that the atomic SU (2) ⊗ O (3) symmetrycharacterizing H loc can be promoted to a global symme-try for the full lattice Hamiltonian H latt . If we imposethis symmetry to be preserved at saddle-point level, i.e.,we do not allow for any spin and orbital ordering, wemust then set to zero all the possible order parameterswhich are not invariant under SU (2) ⊗ O (3), and thiswill strongly limit the possibility of independent slave-boson amplitudes. Indeed, considering the normal statesolution of the saddle-point equations, i.e., do not al-lowing for a superconducting order parameter, it is quitestraightforward to prove, using Wigner-Eckart’s theorem,that the non-zero slave-boson amplitudes must be of theform ϕ Γ n = h n | Γ i Φ( E Γ ) , (75)where we have taken as the basis set for the local physicalstates the eigenstates {| Γ i} of H loc , with eigenvalues E Γ .Note that in this case the quasi-particle Fock states | n i have exactly the same number of particles of the physicalstate | Γ i to which they are linked, assuring the solutionto represent a normal state; indeed, as long as the latter condition is satisfied, no superconducting order parame-ter can be ever developed, as can be easily seen takingthe expectation values of Eq. (70), h d † α d † β i = X ΓΓ ′ h Γ | d † α d † β | Γ ′ i X n ϕ ∗ Γ n ϕ Γ ′ n . (76)On the other hand, if we do allow for a superconductingsymmetry-breaking, we must add to the normal ampli-tudes defined in Eq. (75) also those amplitudes connect-ing physical states to QP states with a different num-ber of particles, so that particle number would no longerbe conserved; however, if we still want to preserve the SU (2) ⊗ O (3) symmetry as in the normal case, we shouldconsider only those amplitudes which correspond to aninvariant, with respect to spin and orbital rotations, su-perconducting order parameter. Assuming pairing to bepurely local, corresponding to an s –wave order parame-ter, the only invariant pairing amplitude is then given bythe spin and orbital singlet channel ψ sc ≡ DP a d † i,a ↑ d † i,a ↓ E . (77)At this point, it is worthwhile to observe that (77) repre-sents the most favorable pairing channel even if we do notexplicitly impose the SU (2) ⊗ O (3) symmetry, since thelocal pairing attraction is driven by the reversed Hund’sterm, which favors the formation of two-particle stateslocked in the L = S = 0 spin-orbital configuration. Ourassumption of preserving the SU (2) ⊗ O (3) symmetry isthen fully justified whenever the system turns out to besuperconducting, since any rotational symmetry break-ing pairing would be ruled out by the singlet channel;on the other hand, in our study we will only comparethe superconducting solution with a rotationally invari-ant normal state, and therefore we cannot exclude thepossibility that some other ordered phase would win thecompetition for the lowest-energy phase.We can now turn to the problem of establishing the in-dependent slave-boson amplitudes required by our modelin order to describe a superconducting solution charac-terized by the L = S = 0 order parameter defined inEq. (77). Using the Wigner-Eckart theorem as for thenormal state solution, in this case we find the followingexpression for the non-zero amplitudes: ϕ Γ n = h n | Γ i Φ( E Γ ) + X q =1 h n | ˆ ψ q | Γ i q h Γ | ( ˆ ψ † ) q ˆ ψ q | Γ i Ψ( E Γ , q )+ h n | ( ˆ ψ † ) q | Γ i q h Γ | ˆ ψ q ( ˆ ψ † ) q | Γ i Ψ( E Γ , − q ) , ˆ ψ = X a d † a ↑ d † a ↓ . (78)We have denoted with Φ( E Γ ) the “normal” slave-bosonamplitudes, which relate physical and quasi-particlestates characterized by the same number of particles,and with Ψ( E Γ , ∆ N ) the “anomalous” ones, in which thenumber of particles characterizing the QP state | n i differs1by ∆ N from that of the physical state | Γ i . We remark,however, that the presence of non-vanishing anomalousamplitudes is not sufficient, by itself, to assure a super-conducting solution, the latter requiring the supercon-ducting order parameter ψ sc , which is a specific quadraticform in the slave-boson amplitudes, to be finite. Thenormalization factors for the anomalous amplitudes, inEq. (78), are chosen in order to simplify the expressionfor the probability associated with the physical configu-ration | Γ i , P (Γ) = X n | ϕ Γ n | = | Φ( E Γ ) | + X ∆ N | Ψ( E Γ , ∆ N ) | . (79)Using Eq. (78), which relates all the slave-boson am-plitudes ϕ Γ n to the independent variables Φ( E Γ ) andΨ( E Γ , ∆ N ), we are almost ready to write the free-energyfunctional (53) in terms of only Φ’s and Ψ’s, obtainingtherefore a much smaller number of saddle-point equa-tions to be solved. The last step needed to achieve thisgoal, in fact, is to evaluate the matrix elements h n | ˆ O d | Γ i between the eigenstates of H loc , which form the basis forthe local physical Hilbert space, and the Fock states | n i d expressed in terms of the physical electron operators d † aσ .As discussed in Sec. III A, the effective local Hamilto-nian for the electron dynamics of the C − ion is given,in the anti-adiabatic limit, by H loc = U n − − µ ˆ n (80)+ J (cid:20) S · S + 12 L · L + 56 (ˆ n − (cid:21) , where we have included also the chemical potential term,as required by the general formalism described in Sec. II,and we have dropped, for simplicity, the redundant lat-tice site index. In comparison to Eq. (71), the Coulombinteraction is here written in a particle-hole symmetricform by properly redefining the chemical potentialWe can then readily identify the eigenstates of H loc among the atomic multiplets | Γ i which are simultaneouseigenstates of the density operator ˆ n and of the orbitaland spin angular momentum operators L and S , witheigenvalues E Γ = E ( n, ℓ, s ) (81)= U n − − µn + J (cid:2) s ( s + 1) + ℓ ( ℓ + 1) + ( n − (cid:3) . The degeneracy associated to each eigenvalue is given by g [ n,ℓ,s ] = (2 ℓ + 1)(2 s + 1) (82)and it is therefore natural to choose as a basis set forthe corresponding degenerate subspace the simultaneous eigenstates of both one of the components of L and S ,say L z and S z , so that we can finally set | Γ i ≡ (cid:12)(cid:12) n, ( ℓ, ℓ z ) , ( s, s z ) (cid:11) . (83)These states must then be expressed in terms of theFock states | n i d , in order to evaluate the matrix elementswhich characterize Eq. (78). We note, however, that | n i d ≡ Y a =1 Y σ = ↑ , ↓ (cid:0) d † aσ (cid:1) n aσ | vac i [ n aσ = 0 ,
1] (84)are not eigenstates of any of the orbital angular mo-mentum operators L a , making the representation of theatomic multiplets (83) in terms of such states a bit in-volved: it is more convenient, instead, using the rota-tional invariance of H loc , to choose an orbital basis forthe physical electron operators in which L z is diagonal, c † mσ = X a U ma d † aσ , (85) L z = X m, σ m c † mσ c mσ , [ m = 1 , , −
1] (86)so that the corresponding Fock states | n i c ≡ Y m, σ (cid:0) c † mσ (cid:1) n mσ | vac i (87)= U ( U ) nn ′ | n ′ i d are eigenstates of both L z and S z . Since we are assumingthe three bands to be degenerate, the rotation of theorbital basis (85) does not change the form of the kineticterm in H latt , while the expression for the singlet pair-creation operator ˆ ψ ≡ P a d † a ↑ d † a ↓ reads, in the new basis,ˆ ψ = c † ↑ c †− ↓ − c † ↓ c †− ↑ − c † ↑ c † ↓ . (88)The representation of the atomic eigenstates | Γ i interms of the Fock states | n i c is listed in Table I, wherethey are classified according to the quantum numbers( n, ℓ, s ) which determine, through Eq. (81), the corre-sponding eigenvalues; note, however, that for a givenvalue of the particle number n , the Pauli-principle pre-vents the orbital and spin angular momenta ℓ and s to take independent values, so that each degenerate-multiplet can actually be identified specifying only twoquantum numbers, n and ℓ . For each multiplet, we havewritten out explicitly only the component characterizedby the maximum value of ℓ z and s z , all the other compo-nents being easily obtainable from the former by repeat-edly acting on it with the lowering operators L − = √ X σ (cid:16) c † σ c σ + c †− σ c σ (cid:17) , (89) S − = X m c † m ↓ c m ↑ . (90)2 | Γ i n ( ℓ, ℓ z ) ( s, s z ) ϕ Γ n | , , i ,
0) (0 ,
0) Φ(0 , ,
0; 2)Ψ(0 ,
0; 4)Ψ(0 ,
0; 6) |↑ , , i ,
1) ( , ) Φ(1 , ,
1; 2)Ψ(1 ,
1; 4) |↑↓ , , i ,
2) (0 ,
0) Φ(2 , ,
2; 2) |↑ , ↑ , i ,
1) (1 ,
1) Φ(2 , ,
1; 2) √ h |↑ , , ↓i − |↓ , , ↑i − | , ↑↓ , i i ,
0) (0 ,
0) Φ(2 , , − ,
0; 2)Ψ(2 ,
0; 4) |↑↓ , ↑ , i ,
2) ( , ) Φ(3 , √ h |↑ , ↑↓ , i + |↑↓ , , ↑i i ,
1) ( , ) Φ(3 , , − ,
1; 2) |↑ , ↑ , ↑i ,
0) ( , ) Φ(3 , |↑↓ , ↑↓ , i ,
2) (0 ,
0) Φ(4 , , − |↑↓ , ↑ , ↑i ,
1) (1 ,
1) Φ(4 , , − √ h |↓ , ↑↓ , ↑i − |↑ , ↑↓ , ↓i − |↑↓ , , ↑↓i i ,
0) (0 ,
0) Φ(4 , , − , − ,
0; 2) |↑↓ , ↑↓ , ↑i ,
1) ( , ) Φ(5 , , − , − |↑↓ , ↑↓ , ↑↓i ,
0) (0 ,
0) Φ(6 , , − , − , − Table I: Electronic eigenstates of the C − molecular ion,and the corresponding slave-boson amplitudes: the latter areselected in order to preserve, at the saddle-point level, therotational invariance of the local Hamiltonian. The last column of the Table contains the independentslave-boson amplitudes Φ( n, ℓ ) and Ψ( n, ℓ ; ∆ N ) associ-ated, according to Eq. (78), to all the components ofa given degenerate-multiplet: the total number of ampli-tudes required by our model is therefore 35, if we considerthe full rotationally-invariant solution, while it reduces to13 if we force the system into the normal state, settingΨ( n, ℓ ; ∆ N ) ≡ Lagrange multipliers
The Lagrange multipliers A , Λ αβ and Π αβ , intro-duced in Secs. II D and II E to enforce the constraintequations (12-14), form, together with the slave-bosonamplitudes, the set of variables on which the free-energyfunctional (53) is defined. However, similarly to what weestablished in the case of the slave-boson amplitudes, wemust note that the symmetries of our problem greatlyreduce the number of independent Lagrange multipliersrequired for the solution of the model, and in the follow-ing we will identify the form of such variables.Denoting with α ≡ ( m, σ ) both the orbital and spin indices, the constraints (12-14) read, at the saddle-pointlevel, X Γ P (Γ) = 1 , (91) Q Nαβ = h f † α f β i = X k G f ( k , − ) βα , (92) Q Aαβ = h f † α f † β i = X k F f ( k , − ) βα , (93)where P (Γ) is the probability distribution defined inEq. (79), G f ( k , τ ) and F f ( k , τ ) are the normal andanomalous quasiparticle Green’s functions, and Q Nαβ ≡ X Γ nn ′ ϕ ∗ Γ n ϕ Γ n ′ h n ′ | f † α f β | n i , (94) Q Aαβ ≡ X Γ nn ′ ϕ ∗ Γ n ϕ Γ n ′ h n ′ | f † α f † β | n i (95)are defined as the normal and anomalous quasiparticledensity matrices. As for P (Γ), we can then make use ofEq. (78) in order to rewrite the left-hand side of Eqs. (92)and (93) directly in terms of the independent slave-bosonamplitudes, obtaining the following expressions: Q Nαβ = Q N [Φ , Ψ] δ αβ , (96) Q Aαβ = Q A [Φ , Ψ] δ α, ¯ β ( − η α , (97) η α = m + σ + , [ σ = ± ]where the spin and orbital indices of ¯ α are opposite tothose of α . The specific choice we have made for the in-dependent slave-boson amplitudes thus reflects in a verysimplified form of the quasiparticle density matrices, andit is not hard to recognize in this structure the symmetryproperties which characterize our model, i.e., the band-degeneracy and the SU (2) ⊗ O (3) rotational invariance.The quasiparticle energy matrix h ( k ) must then be ro-tationally invariant as well, in order to yield quasiparti-cle expectation values with the same structure of the QPdensity matrices: h f † α f β i ∝ δ αβ , h f † α f † β i ∝ δ α, ¯ β ( − η α . The kinetic part of h ( k ), namely R † ˜ ǫ ( k ) R , is guaranteedto be SU (2) ⊗ O (3) invariant, since it depends only onthe degenerate band dispersion ǫ αβ ( k ) = ǫ ( k ) δ αβ andon the slave-boson amplitudes ϕ Γ n [Φ , Ψ] (through the R -matrices) which have been properly selected in orderto yield rotationally invariant solutions. On the otherhand, the Lagrange multipliers matrices Λ and Π are,in principle, two generic Hermitian and antisymmetricmatrices, respectively, and we must therefore setΛ αβ = Λ δ αβ , Π αβ = Π δ α, ¯ β ( − η α , (98)3in order to guarantee the rotational invariance of thequasiparticle Hamiltonian.In the end, we are left with just three Lagrange mul-tipliers, A , Λ, and Π, to which we can eventually addthe chemical potential µ if we decide to solve the modelkeeping the physical electron density fixed: in the lattercase, in fact, the chemical potential plays the role of aLagrange multiplier for the number equation n phys ≡ X m, σ h c † mσ c mσ i = X Γ n (Γ) P (Γ) (99)rather than being an external parameter as in the grand-canonical ensemble. Spectral weight and low-energy excitations
As shown in Sec. II F, the expression for the quasipar-ticle spectral weight matrix Z , defined as Z = (cid:20) − ∂ Σ ∂ω (cid:21) − ω =0 , (100)is given, in terms of the slave-boson amplitudes, by Z = RR † , (101) R [ { ϕ } ] being the matrix which relates the physical elec-tron operators to the quasiparticle ones (see Eqs. (18)and (30) for its definition). In our model, this relationreads c † mσ = r p [Φ , Ψ] ∗ f † mσ + ( − m + σ + 12 r h [Φ , Ψ] f − m ¯ σ , (102)and it can be easily recognized, as in the case of thequasiparticle density matrices, the specific structure ofthe normal ( r p ) and anomalous ( r h ) terms, dictated bythe symmetries of the problem. Inserting this relationin the definition of the quasiparticle weight matrix, wefinally obtain Z = (cid:0) | r p | + | r h | (cid:1) , (103)which states that, for rotationally invariant solutions tothis model, all the electronic degrees of freedom are renor-malized by the same factor and do not mix each other dueto the interaction terms.Besides the quasiparticle spectral weight, we can actu-ally extract, from the saddle-point values of the slave-boson fields and Lagrange multipliers, the entire low-energy spectrum of the system, i.e., its coherent single-particle excitations. They are defined as the frequency-poles of the physical electron propagator D c ( k , ω ) = R D f ( k , ω ) R † = R [ ω − h ( k )] − R † (104)and, in terms of the saddle-point variables, they are given by the six-fold degenerate branches E ( k ) = ± q(cid:2) ǫ k (cid:0) | r p | + | r h | (cid:1) + λ (cid:3) + | ˜∆ | , (105) λ = Λ (cid:0) | r p | − | r h | (cid:1) + r p r h Π + ( r p r h Π) ∗ | r p | + | r h | , ˜∆ = r p Π − ( r h Π) ∗ − r p r ∗ h | r p | + | r h | . From Eq. (105) we can then readily establish the expres-sion for the low-energy spectral gap,∆ = q | ˜∆ | + R (cid:2) | λ | − W (cid:0) | r p | + | r h | (cid:1)(cid:3) ,R ( x ) ≡ x θ ( x ) , (106)where we have assumed, for the free-electron dispersion, ǫ k ∈ (cid:2) − W , W (cid:3) , W being the uncorrelated bandwidth. Itis important, however, to keep in mind that the onset ofsuperconductivity in the system is signaled by the pres-ence of a non-zero order parameter ψ sc rather than bythe opening of a gap in the spectrum: these two quanti-ties, in fact, are directly proportional to each other onlyin the weak-coupling regime ( J/W ≪ U . J ),where we find the solution to be BCS-like, becoming in-stead disentangled for stronger values of either the elec-tron correlation U or the pairing attraction J . IV. ILLUSTRATION OF RESULTS
As mentioned previously, the major strength of slave-boson approaches relies in the possibility of obtaining,with a relatively low computational effort, approximateanalytical solutions which describe quite well, on a quali-tative footing, the effects of electronic interactions on thelow-energy part of the spectrum. With such methods wecan thus investigate the behavior of a given model overthe entire range of variability of the parameters on whichthe model is defined.Using the slave-boson representation introduced in theprevious section for the description of superconduct-ing fullerides, we will here illustrate the behavior ofthe saddle-point solutions across the zero-temperature“phase diagram”, where the external parameters of themodel are represented by the electron density n phys andthe two ratios U/W and
J/W , which measure, respec-tively, the strength of the Coulomb and Jahn-Teller in-teractions with respect to the kinetic bandwidth W ∝ t .We will primarily focus on the half-filled case, where itcan be found the most interesting experimental feature ofthese systems, namely the relatively high superconduct-ing critical temperature in comparison to the strength ofthe pairing coupling, and then we will briefly analyze howthe superconducting behavior extends to finite values ofdoping. The solutions are obtained using, for simplicity,a flat density of states, D ( ǫ ) = W , since the qualitativebehavior of the system does not depend much on thespecific form of D ( ǫ ).4 J/W ZZ N ψ sc ∆ /J ( Ω−Ω N )/W Figure 1: (Color online) Normal and superconducting solu-tions at half-filling and U = 0: Z N and Z are the correspond-ing quasiparticle weights, ∆ and ψ sc are the superconductinggap and order parameter, (Ω − Ω N ) represents the free-energydifference between the normal and the superconducting state. A. Half-filling ( n phys = 3 ) We begin the analysis of the half-filled model illus-trating, in Fig. 1, the normal and superconducting so-lutions obtained, at U = 0, for increasing values of theJahn-Teller coupling J/W . Since we are turning off theCoulomb repulsion, this is a purely attractive model, withthe Jahn-Teller coupling playing the role of an attractivelocal interaction acting on spin and orbital degrees offreedom; the physics of this system is therefore charac-terized by the competition between singlet formation andkinetic delocalization, and we find the results of this com-petition to be remarkably different whether we are con-sidering a purely normal-state solution or we are allowingfor a superconducting order parameter. As expected fora purely attractive interaction, the superconducting so-lution is always energetically favored at finite J .As soon as the pairing interaction J is turned on, thebehavior of the normal-state solution is initially char-acterized by a slow decrease of the quasiparticle weight Z N from the non-interacting value Z N (0) = Z (0) = 1,which is then followed by a steep descent towards zerofor J/W & .
2; finally, when the coupling is further in-creased, the metallic state turns into an insulating one,where all the electrons are locked in local singlets formedby two or four electrons , the binding energy of the sin-glet configuration being much more favorable than thekinetic energy gain associated to the electron hopping.This state is analogous to the paired insulator found,at strong coupling, in the normal solution of the at-tractive Hubbard model within DMFT . On the otherhand, if we do allow for superconducting ordering, wefind the static singlet-formation mechanism characteriz-ing the normal solution to be replaced by the more fa- vorable Cooper pair formation, in which the singlet pairscan still gain some kinetic energy through their propaga-tion: the solution, in this case, is therefore characterizedby a finite quasiparticle weight Z over the entire rangeof the pairing interaction. We must however notice thatthe difference in the behavior of the normal solution be-tween the weak and strong coupling regimes (metallic vs.insulating) can still be traced in the behavior of the su-perconducting solution. In fact, for increasing values of J/W , we observe a crossover between a weak-couplingBCS-like superconductivity, where the gap ∆ /J and thesuperconducting order parameter ψ sc are proportional toeach other and exponentially small in the pairing cou-pling (Fig. 2),∆ /J = ψ sc ∼ λ e − λ , λ = ( J/W ) (107)and a strong-coupling superconductivity associated toBose-Einstein Condensation (BEC) of preformed pairs,where both the gap and the superconducting order pa-rameter are saturated . While in the former regime theformation of Cooper pairs subtracts some kinetic energyfrom the normal state in order to gain the binding en-ergy associated to the superconducting singlets, as evi-denced by the lower spectral weight Z < Z N , which cor-responds to more localized particles, in the large J/W regime, where the local singlets are already formed, theenergy gain of the superconducting state is due to thefinite kinetic energy of the Cooper pairs in comparisonto the static singlets characterizing the insulating normalstate .The most interesting aspect of the half-filling solutions,however, is represented by the behavior of the quasiparti- J/W
BCS estimate: ∆ /J = ψ sc = W/J e − λ ψ sc ∆ /J e − λ Figure 2: (Color online) Superconductivity in the weak-coupling regime: slave-boson solution (circles and crosses) vs.BCS estimate. For
J/W ≪ U = 0 the superconductingparameters satisfy the BCS relation ∆ /J = ψ sc and, apartfrom a constant prefactor, they follow the BCS functionalform ∆( J ) BCS = W e − λ ( λ = ( J/W ) is the dimensionlesssuperconducting coupling constant). U , at different values of the Jahn-Teller cou-pling J .In Fig. 3 we plot the U -dependence of the normal quasi-particle weight Z N and of the observables characterizingthe complete solution ( Z , ∆ and ψ sc ) at J/W = 0 .
04, cor-responding to a coupling strength located in the upperend of the weak-coupling regime, or, in other words, justbefore the U = 0 crossover region. The relevant featureto be noticed in this figure is the non-monotonic behaviorof the superconducting parameters for increasing valuesof the electron correlation: while at small U the net ef-fect of the Coulomb repulsion is to rapidly destroy thesuperconducting order, as expected in a weak-couplingBCS superconductor (notice the small value of the su-perconducting order parameter at U = 0 and its suddendisappearance as soon as U is turned on), at larger U values the system turns back superconducting, with anenhancement in the values of ∆ and ψ sc in comparisonto U = 0, until it undergoes a first-order Mott transitionat U = U c , just above the corresponding Mott transitionof the normal state. It is evident the huge enhancementof the superconducting amplitude ψ sc with respect to thenon-correlated regime.From a physical point of view, the reemergence of su-perconductivity at large U has been explained in termsof the “strongly correlated superconductivity” scenarioput forward using DMFT to solve the same model and a related simplified model . The key mechanismis the different effect of the correlation on the variousinteraction terms: when strong repulsion freezes chargefluctuations, the resulting strongly correlated quasipar-ticles experience a strongly reduced repulsion, while theattraction is essentially unscreened. As a result, the neteffect is that J/W → J/ ( ZW ) : when the electrons U/W ZZ N ψ sc ∆ /J ( Ω−Ω N )/W x 10 Figure 3: (Color online) U -dependence of the half-filling solu-tions at J/W = 0 .
04. The condensation energy (Ω − Ω N ) /W is multiplied by a factor of 10 for visibility reasons. become more localized, the relative strength of the Jahn-Teller interaction grows in comparison to the renormal-ized hopping. The precise nature of the interaction, in-volving orbital and spin degrees of freedom, is crucialin this effect, and proves the ability of our rotationallyinvariant slave bosons to properly treat every kind of in-teraction. The superconducting behavior in this regionis clearly non-BCS-like, as evidenced by the non propor-tionality between the gap and the order parameter: ∆ /J is indeed much larger (up to ten times) than ψ sc , andits maximum is located much closer to the Mott tran-sition than the order parameter’s one. On the otherhand, the pairing mechanism cannot be explained withina purely strong-coupling BEC-like picture, since in thiscase the pairing singlets are not already “preformed” inthe normal phase (which is either a correlated metal oran S = 1 / U/W ZZ N ψ sc ∆ /J ( Ω−Ω N )/W x 10 ψ sc at U=0 ∆ /J at U=0 U/W ZZ N ψ sc x 10 ∆ /J ( Ω−Ω N )/W x 10 [ ψ sc at U=0] x 10 [ ∆ /J at U=0] x 10 Figure 4: (Color online) U -dependence of the half-filling so-lutions for fixed ratios J/U = 0.02 (top) and 0.01 (bottom).Circles and crosses represent the superconducting parametersevaluated at U = 0 for the same values of J . U/W ZZ N ψ sc ∆ /J ( Ω−Ω N )/W U/W ZZ N ψ sc ∆ /J ( Ω−Ω N )/W Figure 5: (Color online) U -dependence of the half-filling solu-tions at intermediate-to-strong pairing couplings J/W = 0.1(top) and 0.2 (bottom). rather in the presence of a strongly correlated supercon-ductor, in which a small local pairing coupling turns outto be enhanced, rather than suppressed, by the effects ofa strong on-site repulsion.The correlation-driven enhancement of superconduc-tivity in the proximity of the Mott transition is evenmore evident in Fig. 4, where the solutions are evaluatedat a fixed ratio
J/U for increasing values of the correla-tion; together with the normal and complete solutions,we have plotted for comparison the (BCS-like) supercon-ducting parameters ∆ /J and ψ sc obtained, at U = 0,for the same values of J . Besides the different relationbetween ∆ /J and ψ sc in the correlated case compared tothe U = 0 solutions, these plots emphasize how the en-hancement of the superconducting gap becomes stronger(up to three orders of magnitude in the lower panel) atsmaller values of the pairing coupling: for J/W ≪ /J in the proximity of the Motttransition turns out to be O (1), while it is exponentiallysmall in J/W in the BCS regime (see Fig. 2). As already found in DMFT in a two-orbital model ,a completely different scenario is instead observed forlarger values of the Jahn-Teller coupling, correspondingto the strong-coupling regime of the U = 0 attractivemodel (shown in Fig. 5). In these cases, in fact, the su-perconducting order parameter decreases monotonicallywith U from the large U = 0 value, until a weakly first-order Mott transition (which becomes second-order when J/W is increased) turns the system into an insulator; asimilar behavior characterizes the superconducting gap,except for an initial rise at small values of
U/W in thecase
J/W = 0 .
1. At strong-coupling values of the pair-ing attraction the superconducting solution is thereforealways energetically-favorable compared to the metallicone, and the electronic correlation has only the effect ofreducing, throughout the non-insulating phase, the su-perconducting ordering.We conclude the analysis of the half-filling solutions
J/W ZZ N ψ sc ∆ /J ( Ω−Ω N )/W x 10 J/W ZZ N ψ sc ∆ /J ( Ω−Ω N )/W x 10 Figure 6: (Color online) J -dependence of the half-filling solu-tions in the strongly-correlated regime: the top panel corre-sponds to a fixed value of the correlation, U/W = 2 .
45, whilein the bottom panel the solution follows the Mott-transitionline U c ( J ). /J and ψ sc , as functionsof J , in the strongly-correlated region of the phase dia-gram: in the top panel the value of U/W is held fixed,while in the bottom one it follows the Mott-transitionline from below, U ( J ) = U c ( J ) − δU . Combining theseresults with the ones discussed previously, we can theninfer the existence of a second region in the J – U plane,beside the strong-coupling BEC-like region at J ≫ U ,in which superconductivity is found to be optimal: al-most surprisingly, it is located at very small values of thepairing attraction J and at correspondingly large valuesof the on-site electron repulsion U , just before the Mottlocalization transition line.The results presented in this section confirm that therotationally invariant slave boson approach is able to ac-curately treat interactions of different kinds and, partic-ularly, it is not limited to charge interactions. Indeed wehave found that the present approach is able to reproducethe relevant physics of a three-orbital Hubbard model forthe fullerenes, and, in particular, the huge enhancementof phonon-mediated superconductivity in the proximityof the Mott transition. The only qualitative aspect of theDMFT solution which is not found in the present studyis the second-order (or very weakly first-order) characterof the superconductor-insulator transition. B. Finite doping ( n phys = 3 − δ ) In this section we consider the effect of doping the half-filled three-band model. While this situation can not beexperimentally realized, at the moment, in fullerides, theeffect of doping is clearly suggestive for analogies withthe physics of cuprates.The behavior as a function of doping, in the neigh-borhood of n phys = 3, is shown in Fig. 7 for correla-tion strengths respectively below and above the criticalMott-transition value U c ( J ); in both cases, however, wehave U ≫ J , so that they both belong to the strongly-correlated region of the phase diagram, where the pres-ence of a finite superconducting order parameter is dueto the localization-driven enhancement of the effectivepairing coupling.We find that for U < U c the superconducting parame-ters decrease monotonically upon doping (eventually van-ishing at larger doping values), while the normal and su-perconducting quasiparticle weights increase from theirfinite half-filling values; as long as the superconductingorder is present, we have Z < Z N . On the other hand,for U > U c we observe a dome-shaped behavior in thesuperconducting parameters, the gap reaching its maxi-mum at a very small doping value δ opt ≈ .
1, while theorder parameter being maximum at a slightly larger opti-mal doping δ ( ψ )opt ≈ .
15. In this case, both the normal andsuperconducting quasiparticle weights grow linearly withthe doping δ , but while in the overdoped region δ > δ opt we find the standard weak-coupling behavior Z < Z N , at ZZ N n phys -0.0500.050.10.152.5 30 ψ sc ∆ /J ( Ω−Ω N )/W x 10 ZZ N n phys ψ sc x 10 ∆ /J ( Ω−Ω N )/W x 10 Figure 7: (Color online) Doping dependence of the solu-tions for correlation strengths respectively below (top panel,
U/W = 2,
J/W = 0 .
03) and above (bottom panel,
U/W = 5,
J/W = 0 .
02) the n phys = 3 Mott transition. lower dopings we have Z > Z N .The behaviors of both the normal and superconduct-ing solutions in the two correlation regions U ≷ U c aretherefore remarkably different from each other; however,at a closer sight, we find that they can be actually ex-plained through the same physical mechanism, namelythe competition between Mott-localization, which caneventually enhance the superconducting pairing as wehave seen in the discussion of the half-filling solutions,and the delocalization tendency introduced by doping.In fact, when the correlation strength at half-filling isnot large enough to completely destroy the quasiparticlecoherence (in other words, when the quasiparticle degreesof freedom are nor completely frozen), we find the super-conducting parameters to be maximum at zero-doping,8where the electrons are more localized and the enhance-ment of the effective pairing coupling is stronger. Whenthe system is in the Mott-insulating phase, on the con-trary, there are no available quasiparticles at half-fillingin order to develop a superconducting order parameter:the reintroduction of quasiparticle coherence due to a fi-nite level of doping becomes then essential in order torecover a superconducting solution. At small dopings,therefore, the superconducting ordering increases, dueto the regained coherence of quasiparticles and a stillstrong enhancement of pairing due to Mott localization;for larger values of doping, instead, the loosening of thelocalization-induced pairing enhancement disfavors thesuperconducting ordering, which turns to decrease as inthe U < U c scenario. It is interesting to note that inthe underdoped region we have Z > Z N , which meansthat the formation of superconducting pairs is energeti-cally more favorable also from the kinetic point of view,compared to the normal state. V. CONCLUSIONS
We have generalized to superconducting solutions (al-lowing for the spontaneous breaking of charge symme-try) the rotationally invariant slave boson approach in-troduced by Lechermann et al. on the basis of the workby Li and W¨olfle . The crucial ingredient of the rotation-ally invariant version of slave boson methods is that theboson fields cannot be simply seen as probability ampli-tudes for the different quasiparticle states, but they areexpressed as a non diagonal density matrix that connectsthe different quasiparticle states to the whole set of phys-ical states. This is easily generalized to include matrixelements between states with different number of parti-cles which allow to describe superconducting ordering.After a thorough description of the formalism, we ap-ply the method to solve a three-band model which hasbeen proposed and studied to understand the propertiesof alkali-doped fullerides . This model has beenpreviously solved using DMFT for integer fillings, pro-viding a striking realization of strongly correlated super-conductivity, i.e., of a situation in which strong electron-electron correlations favor superconductivity. A crucial element of the model is that the pairing attraction, whichcan be modelized as an inverted Hund’s rule term, onlyinvolves spin and orbital degrees of freedom, which arenot heavily affected when the charge degrees of freedomare frozen by the proximity to the Mott transition. Thisinterorbital nature of the pairing interaction is crucial togive rise to the correlation-driven enhancement of pair-ing. In this light, this model represents an ideal testfield for our approach, which is tailored to properly treatinterorbital interactions that cannot be expressed in adensity-density form. Indeed the method provides goodresults for this model, and it is actually the first mean-field-like approach able to reproduce the enhancement ofsuperconductivity observed in DMFT.The good performance of the method is very encour-aging in view of other applications. The most chal-lenging direction is obviously the study of the two-dimensional Hubbard model, which is believed to be thebasic model to understand high-temperature supercon-ductivity in the cuprates. While the full solution ofthe model on a lattice appears too cumbersome, a vi-able direction is the use of cluster extensions of DMFT,such as the cellular-DMFT or the dynamical clusterapproximation , where the rotationally-invariant slave-boson method can be used as an approximate analyticalimpurity solver for the cluster Hamiltonian. This ap-proach has been used, for example, in Refs. 4 and 41 fornormal solutions without superconducting ordering. Toinvestigate the superconducting properties of the Hub-bard model, on the other hand, our formalism can beapplied either to the 2 × , or to slightly larger clusters, such as smallrectangles in CDMFT or the 5-site “cross” in DCA, whichhave been proposed as ideal compromises between rea-sonable cluster size and adequate accuracy . Acknowledgments
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