Antiferro-quadrupole state of orbital-degenerate Kondo lattice model with f^2 configuration
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Antiferro-quadrupole state of orbital-degenerate Kondo lattice model with f configuration Hiroaki O
NISHI , and Takashi H OTTA Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A. (Received October 8, 2007)
To clarify a key role of f orbitals in the emergence of antiferro-quadrupole structure in PrPb , we inves-tigate the ground-state property of an orbital-degenerate Kondo lattice model by numerical diagonalizationtechniques. In PrPb , Pr has a f configuration and the crystalline-electric-field ground state is a non-Kramers doublet Γ . In a j - j coupling scheme, the Γ state is described by two local singlets, each ofwhich consists of two f electrons with one in Γ and another in Γ orbitals. Since in a cubic structure, Γ has localized nature, while Γ orbitals are rather itinerant, we propose the orbital-degenerate Kondolattice model for an effective Hamiltonian of PrPb . We show that an antiferro-orbital state is favored bythe so-called double-exchange mechanism which is characteristic of multi-orbital systems. KEYWORDS: PrPb , antiferro-quadrupole state, j - j coupling scheme It is currently one of the central issues in the research fieldof condensed-matter physics to unveil novel magnetic phasesof strongly correlated electron systems with active orbital de-grees of freedom. It has been a common understanding thatcompetition and interplay among spin, charge, and orbital de-grees of freedom cause diverse ordering phenomena involv-ing multiple degrees of freedom, as frequently observed in d -and f -electron systems.
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In the case of f -electron systems,spin and orbital are tightly coupled with each other due to thestrong intra-atomic spin-orbit interaction. To describe such acomplex spin-orbital state, the f -electron state is usually clas-sified in terms of multipole degrees of freedom.A rare-earth compound PrPb with a simple AuCu -typecubic structure has attracted great interest as a typical materialthat exhibits antiferro-quadrupolar (AFQ) ordering. In fact,this compound undergoes a second-order transition at 0.4 K, which has been confirmed to be a non-magnetic but an AFQtransition.
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In PrPb , Pr has a f configuration, and thecrystalline-electric-field (CEF) ground state is a non-magneticnon-Kramers doublet Γ with a magnetic triplet Γ lying 19 Kabove the ground state.
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The Γ state carries O and O quadrupole moments. Thus, the low-temperature property isgoverned by quadrupole degrees of freedom.Regarding the H - T phase diagram of PrPb , the so-calledreentrant phase diagram has been obtained, in which the tran-sition temperature goes up with increasing the field but turnsto decrease and the ordered phase closes at a low field. Thisreentrant behavior has been well reproduced phenomeno-logically based on a mean-field theory assuming a simpletwo-sublattice ordered structure. However, recent neutrondiffraction measurements have revealed that the quadrupoleordered structure is modulated in space, instead of a sim-ple two-sublattice structure. In principle, such a long-periodordered structure could emerge because of significant long-range quadrupole interactions, although the origin of the long-range interactions is not clear.So far, there have been no theoretical efforts to understandAFQ structure of PrPb from a microscopic viewpoint. In thispaper, we propose an orbital-degenerate Kondo lattice model,which is obtained on the basis of a j - j coupling scheme, asan effective model for PrPb . We investigate the ground-state Fig. 1. Schematic view of Γ and Γ orbitals. property of the model by using exact-diagonalization tech-niques. It is found that an antiferro-orbital state emerges dueto the so-called double-exchange mechanism which is in gen-eral relevant to multi-orbital systems.First we explain the construction of an effective model forPrPb . In the j - j coupling scheme, we first include the strongspin-orbit interaction, and we accommodate f electrons in thelower sextet with the total angular momentum j = / . Underthe cubic CEF effect, the sextet is split into a Γ doublet anda Γ quartet. To distinguish two Kramers doublets in the Γ quartet, it is useful to introduce two orbitals, while spin is alsointroduced to represent two states in each Kramers doublet.Note that the Γ doublet gives another orbital. The schematicviews of Γ and Γ orbitals are depicted in Fig. 1. Since weaccommodate f electrons in the level scheme of the one f -electron state, we refer to the level scheme of CePb , whichis a f compound with the same lattice structure with that ofPrPb . In CePb , it has been found that Γ is the ground stateand Γ is the excited state. Thus, for PrPb , we accommo-date two f electrons in this level scheme.In PrPb , the CEF ground state is the non-Kramers doublet Γ . In the j - j coupling scheme, the Γ state is described bytwo local singlets, each of which is composed of two electronswith one in Γ and another in Γ orbitals. Here we note that Γ orbitals carry quadrupole degrees of freedom. Taking ac-count of the formation of local singlets, an antiferromagnetic(AFM) coupling should be effective between electrons in Γ and Γ orbitals, although the Hund’s rule coupling causes aferromagnetic (FM) coupling. Thus, here we introduce theAFM coupling as an effective interaction to involve a high-order CEF effect B which can not be included in the j = / Hilbert space in the j - j coupling scheme. ULL P APER
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Concerning the itinerancy and localized nature of orbitals,we consider an f -electron hopping through the sigma bond.In a cubic structure, due to the spatially anisotropic shape oforbital, Γ orbital is localized, while Γ a orbital is itinerant inthe xy plane and Γ b orbital is itinerant in all three directions.Thus, to consider an effective model, we assume that electronin Γ orbital is localized, leading to a localized spin. Note that Γ electron is itinerant and couples with localized Γ spin dueto the effective AFM interaction, which can be regarded as ananalog of the Kondo coupling.Taking into account these situations, we obtain an orbital-degenerate Kondo lattice model as an effective Hamiltonianfor PrPb , given by H = X h i , j i ,τ,τ ′ σ t i − j ττ ′ f † i τσ f j τ ′ σ + J K X i S i Γ · S i Γ + U X i ,τ ρ i τ ↑ ρ i τ ↓ + U ′ X i ρ i α ρ i β + J X i ,σ,σ ′ ,τ = τ ′ f † i τσ f † i τ ′ σ ′ f i τσ ′ f i τ ′ σ + J ′ X i ,σ = σ ′ ,τ = τ ′ f † i τσ f † i τσ ′ f i τ ′ σ ′ f i τ ′ σ (1)where f i τσ is the annihilation operator for Γ electron withspin σ (= ↑ , ↓ ) in orbital τ (= α, β ) at site i , ρ i τσ = f † i τσ f i τσ , ρ i τ = P σ ρ i τσ , S i Γ = (1 / P σσ ′ τ f † i τσ σ σσ ′ f i τσ ′ , where σ σσ ′ are Pauli matrices, and S i Γ is the spin-1/2 operator for Γ spin. The summation of h i , j i is taken for nearest neigh-bor sites in the cubic lattice. The hopping amplitudes areevaluated from the overlap integral between f -orbital wave-functions in adjacent sites, which are given by t x αα = t/ , t x αβ = t x βα = −√ t/ , t x ββ = t/ for the x direction, t y αα = t/ , t y αβ = t y βα = √ t/ , t y ββ = t/ for the y direction, and t z ββ = t , t z αα = t z αβ = t z βα = for the z direction, where t = (3 / f f σ ) .Hereafter, t is taken as the energy unit. In the second term, J K is the Kondo coupling between Γ spin and Γ electron.The rest terms are interactions among Γ electrons: U , U ′ , J , and J ′ denote intra-orbital, inter-orbital, exchange, andpair-hopping interactions, respectively. Note that the relation U = U ′ + J + J ′ holds, which originates from the rotational in-variance in the orbital space, and J = J ′ is assumed. We analyze the model (1) by numerical diagonalization.Since the size of the Hilbert space becomes so large as N due to orbital degree of freedom, where N is the number ofsites, it is rather difficult to enlarge the system size. However,the method is advantageous to grasp the ground-state prop-erty, such as orbital structure, in an unbiased manner. In thepresent work, first we study a × square four-site systemin the xy plane. Then, taking account of the characteristicsgrasped within the four-site system, we proceed to a × × cubic eight-site system. In this paper, we set U ′ /W = , where W is the band width, and investigate the dependence on J K and J . Note that the band width is W = for the square systemand W = for the cubic system.First, we show the results for the four-site system. The mainresult is summarized in Fig. 2(a), which is the ground-statephase diagram in the ( J K , J ) plane. There three types of com-peting spin-orbital configurations are observed. When J K and J are small, we find a ferro-orbital (FO) state with an AFM Fig. 2. Four-site results. (a) Ground-state phase diagram in the ( J K , J ) plane. Inset denotes schematic view of electron configuration in eachphase. (b) T ( q ) as a function of J at J K = . . (c) FO and AFO struc-tures. (d) S ( q ) as a function of J at J K = . . (e) C s as a function of J K at J = . Note that C s (i) takes the equivalent value at every site due to thetranslational symmetry. configuration in each of Γ and Γ orbitals, while spins in Γ and Γ orbitals are antiparallel at each site due to J K . Withincreasing J , an antiferro-orbital (AFO) state occurs, and thespin state turns to be FM at a larger J than the transition pointfrom FO to AFO. It should be noted that even though the spinstate is characterized by AFM or FM in each of Γ and Γ orbitals, a local singlet is formed due to J K and the groundstate is totally non-magnetic.Let us here discuss the orbital state. In order to determinethe orbital structure, it is useful to introduce new operatorsfor ξ and η orbitals, which are given by linear combinationsof the original operators, such as ˜ f i ξσ = cos( θ i / f i βσ + sin( θ i / f i ασ , ˜ f i ησ = − sin( θ i / f i βσ + cos( θ i / f i ασ , (2)where θ i characterizes the orbital shape at each site. The opti-mal { θ i } is determined so as to maximize the orbital structurefactor, defined by T ( q ) = X j , k h ˜ T z j ˜ T z k i e i q · ( j − k ) /N, (3)where ˜ T z i = P σ ( ˜ f † i ξσ ˜ f i ξσ − ˜ f † i ησ ˜ f i ησ ) / and h· · · i denotes the . Phys. Soc. Jpn. F ULL P APER
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OTTA expectation value. In Fig. 2(b), we show T ( q ) as a function of J at J K = . with θ i = θ . For small J , the dominant componentis T (0 , with θ = π , indicating FO state. In the FO state of thepresent square system in the xy plane, Γ a orbitals are favor-ably occupied to gain kinetic energy, since Γ a orbital extendsto adjacent sites, as shown in Fig. 2(c). With increasing J , thedominant component changes to T ( π, π ) with θ = π/ , indi-cating AFO state in Fig. 2(c). We note that the present squarelattice structure is reflected in the orbital shape. In this con-text, it is crucial to proceed to the eight-site system to discussthe orbital structure in the cubic lattice.Concerning the spin state, we measure the spin structurefactor of Γ electrons, defined by S ( q ) = X j , k h S z j Γ S z k Γ i e i q · ( j − k ) /N. (4)As shown in Fig. 2(d), with increasing J , S (0 , is increased,while S ( π, π ) is reduced, so that the dominant spin correla-tion changes from AFM to FM. Here we note that the motionof Γ electrons leads to a FM spin arrangement. Namely, thedouble-exchange mechanism is effective, which is character-istics of multi-orbital systems.We also measure the on-site spin correlation, defined by C s ( i ) = h S z i Γ S z i Γ i . (5)In Fig. 2(e), the J K dependence of C s at J = is shown. Itis obvious that C s = at J K = , since there is no correlationbetween Γ spin and Γ electron. With increasing J K , C s de-creases and gradually approaches − / , indicating the stabi-lization of the local singlet. It is found that even when J isincreased, C s keeps a value near − / (not shown), indicat-ing the robust formation of the local singlet. Here it is worthnoting that as J K increases and the local singlet is stabilized,the AFO phase tends to extend to the region of small J , asshown in Fig. 2(a). Namely, the double-exchange mechanismbecomes significant due to J K .Now we move on to the results for the eight-site system. InFig. 3(a), we show the θ dependence of T ( q ) at J K = and J = . We find that T ( π, π, π ) is dominant, indicating AFOstate. Concerning the orbital shape, it is observed that themagnitude of T ( π, π, π ) does not depend on θ . We can notdetermine the actual orbital shape, but the AFO structure withany θ is possible to realize. The AFO structure with θ = isshown in Fig. 3(b). As for the spin state, S ( π, π, π ) is foundto be dominant. We note that the present parameter set ( J K = and J = ) is corresponding to the FO phase in the four-sitesystem, as shown in Fig. 2(a). It is naively expected that theAFO phase extends to a broad area in the phase diagram evenwhen we consider the cubic lattice. However, at J K = and J = , T ( q ) has dominant components T ( π, π, with θ = π/ , T ( π, , π ) with θ = π/ , and T (0 , π, π ) with θ = π/ , as shownin Fig. 3(c), while S (0 , , is dominant. The ( π, π, orbitalstructure is depicted in Fig. 3(d). When J is further increased,the orbital structure is considered to turn to be AFO due tothe double-exchange mechanism, suggesting a rich phase di-agram including competing orbital states.Finally, we briefly discuss possible relevance of the presentresults to the AFQ structure of PrPb . We have shown thatfor a one-dimensional j - j coupling model with an f con-figuration, an incommensurate orbital state appears due to Fig. 3. Eight-site results. (a) T ( q ) as a function of θ and (b) orbital struc-ture at J K = and J = . (c) T ( q ) as a function of θ and (d) orbital structureat J K = and J = . the competition between itinerant and localized orbitals. Byanalogy, we expect that the competition among plural orbitalstates with different nature could cause a modulated orbitalstructure as observed in PrPb .In summary, we have investigated the ground-state prop-erty of the orbital-degenerate Kondo lattice model to under-stand the quadrupole structure in PrPb from a microscopicviewpoint. We have observed several types of competing spin-orbital states. In particular, it has been emphasized that theAFO state emerges due to the double-exchange mechanism.The authors have been supported by a Grant-in-Aid for Sci-entific Research in Priority Area “Skutterudites” under thecontract No. 18027016 from the Ministry of Education, Cul-ture, Sports, Science, and Technology of Japan. T.H. has beenalso supported by a Grant-in-Aid for Scientific Research (C)under the contract No. 18540361 from Japan Society for thePromotion of Science.
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