Superconductivity in a Two-Orbital Hubbard Model with Electron and Hole Fermi Pockets: Application in Iron Oxypnictide Superconductors
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Typeset with jpsj2.cls < ver.1.1.1 > Superconductivity in a Two-Orbital Hubbard Model with Electron and Hole FermiPockets: Application in Iron Oxypnictide Superconductors
Kazuhiro
Sano ∗ and Yoshiaki ¯Ono Department of Physics Engineering, Mie University, Tsu, Mie 514-85071Department of Physics, Niigata University, Ikarashi, Nishi-ku, Niigata 950-21812Center for Transdisciplinary Research, Niigata University, Ikarashi, Nishi-ku, Niigata 950-2181 (Received October 29, 2018)
We investigate the electronic states of a one-dimensional two-orbital Hubbard model withband splitting by the exact diagonalization method. The Luttinger liquid parameter K ρ iscalculated to obtain superconducting (SC) phase diagram as a function of on-site interactions:the intra- and inter-orbital Coulomb U and U ′ , the Hund coupling J , and the pair transfer J ′ . In this model, electron and hole Fermi pockets are produced when the Fermi level crossesboth the upper and lower orbital bands. We find that the system shows two types of SCphases, the SC I for U > U ′ and the SC II for U < U ′ , in the wide parameter region includingboth weak and strong correlation regimes. Pairing correlation functions indicate that the mostdominant pairing for the SC I (SC II) is the intersite (on-site) intraorbital spin-singlet with(without) sign reversal of the order parameters between two Fermi pockets. The result of theSC I is consistent with the sign-reversing s -wave pairing that has recently been proposed foriron oxypnictide superconductors. KEYWORDS: iron oxypnictide superconductors, two-orbital Hubbard model, pairing symmetry, exact diagonaliza-tion
1. Introduction
The recent discovery of iron oxypnictide superconduc-tors with transition temperatures of up to T c ∼ K has stimulated much interest in the relationship betweenthe mechanism of the superconductivity and the orbitaldegrees of freedom. First-principles calculations havepredicted the band structure with hole Fermi pocketsaround the Γ point and electron Fermi pockets aroundthe M point. By using the weak-coupling approachesbased on multiorbital models, the spin-singlet s -wavepairing is predicted, where the order parameter of thispairing changes its sign between hole and electron Fermipockets (sign-reversing s -wave pairing). This uncon-ventional s -wave pairing is expected to emerge owing tothe effect of antiferromagnetic spin fluctuations. Sincethe strong correlation between electrons is considered toplay an important role in the superconductivity of ironoxypnictides as well as in that of high- T c cuprates, non-perturbative and reliable approaches are required.As a nonperturbative approach, the exact diagonal-ization (ED) method has been extensively applied in theHubbard, d - p , and t - J models.
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Although these mod-els are much simplified and mostly limited to one di-mension, it has elucidated some important effects of astrong correlation on superconductivity. Using the EDmethod, we have studied the one-dimensional (1D) two-orbital Hubbard model in the presence of the band split-ting ∆. It is found that the superconducting (SC) phaseappears in the vicinity of the partially polarized ferro-magnetism when the exchange (Hund’s rule) coupling J is larger than its critical value on the order of ∆. Theresult suggests that spin triplet pairing emerges owingto the effect of ferromagnetic spin fluctuation. In thecase of ∆ = 0, spin triplet superconductivity has also ∗ E-mail address: [email protected] been discussed on the basis of bosonization and nu-merical approaches. Previous works, however, wererestricted to the case of a single Fermi surface, and theeffects of electron and hole Fermi pockets on supercon-ductivity have not been discussed therein.In this study, we investigate the 1D two-orbital Hub-bard model with electron and hole Fermi pockets, wherethe Fermi level crosses both the upper and lower bandsin the presence of a finite band splitting ∆. Using theED method, the Luttinger liquid parameter K ρ is calcu-lated to obtain the SC phase diagram as a function ofon-site Coulomb interactions in a wide parameter regionincluding both weak- and strong-correlation regimes. Itwould clarify the effects of a strong correlation on su-perconductivity in iron oxypnictides. We also calculatevarious pairing correlation functions and discuss a pos-sible pairing symmetry. Although our model is muchsimplified and limited to one dimension, we expect thatthe essence of the superconducting mechanism of ironoxypnictides can be discussed.
2. Model and Formulation
We consider the one-dimensional two-orbital Hubbardmodel given by the following Hamiltonian: H = t X i,m,σ ( c † i,m,σ c i +1 ,m,σ + h.c. )+ ∆2 X i,σ ( n i,u,σ − n i,l,σ ) + U X i,m n i,m, ↑ n i,m, ↓ + U ′ X i,σ n i,u,σ n i,l, − σ + ( U ′ − J ) X i,σ n i,u,σ n i,l,σ − J X i ( c † i,u, ↑ c i,u, ↓ c † i,l, ↓ c i,l, ↑ + h.c. ) Sano and Yoshiaki ¯Ono
U’J U ∆ tt J’ E k F ∆ (a)(b)(c) ε u (k) ε l (k) m=lm=um=u electron pocketholepocket Fig. 1. Schematic diagrams of (a) the model Hamiltonian, (b) theband structure in the noninteracting case, and (c) a correspond-ing two-dimensional Fermi surface related to our 1D model. − J ′ X i ( c † i,u, ↑ c † i,u, ↓ c i,l, ↑ c i,l, ↓ + h.c. ) , (1)where c † i,m,σ stands for the creation operator of an elec-tron with a spin σ (= ↑ , ↓ ) and an orbital m (= u, l ) atsite i and n i,m,σ = c † i,m,σ c i,m,σ . Here, t represents thehopping integral between the same orbitals and we set t = 1 in this study. The interaction parameters U , U ′ , J , and J ′ stand for the intra- and inter-orbital directCoulomb interactions, exchange (Hund’s rule) coupling,and pair-transfer, respectively. ∆ denotes the energy dif-ference between the two atomic orbitals. For simplicity,we impose the relation J = J ′ .The model in eq. (1) is schematically shown in Fig.1(a). In a noninteracting case ( U = U ′ = J = 0),the Hamiltonian eq. (1) yields dispersion relations rep-resenting the upper and lower band energies: ǫ u ( k ) =2 t cos( k ) + ∆2 and ǫ l ( k ) = 2 t cos( k ) − ∆2 , where k is thewave vector. This band structure is schematically shownin Fig. 1(b). When the Fermi level E k F crosses boththe upper and lower bands, the system is metallic withelectron and hole Fermi pockets corresponding to a char-acteristic band structure of the FeAs plane in iron oxyp-nictides, as shown in Fig. 1(c).We numerically diagonalize the model Hamiltonian upto 6 sites (12 orbitals) and estimate the Luttinger liquidparameter K ρ from the ground-state energy of finite-sizesystems using the standard Lanczos algorithm. To re-duce the finite size effect, we impose the boundary con-dition (periodic or antiperiodic) on upper and lower or-bitals independently and chose both boundary conditions Φ /2 π ∆ E U’U=U’+2J K ρ ∆ =0.8 2.8J=U’/4n=5/3 S=11.9 2.31.6 S=1 U’=1.5 ∆=1.6
Fig. 2. K ρ as a function of U ′ (= 4 J ) for n = 5 / . , . , . , .
3, and 2.8. The singlet ground statechanges into a partially polarized ferromagnetic ( S =1) state at U ′ ≃ E ( φ ) − E (0) as a function ofthe external flux φ for n = 2 / . to minimize | K ρ − | , where K ρ represents the K ρ of thefinite-size system in a noninteracting case. The typicaldeviation of K ρ from unity becomes about ∼ . K ρ as arenormalize value calculated using K ρ /K ρ , hereafter.On the basis of the Tomonaga-Luttinger liquid the-ory, various types of correlation functions are deter-mined by the single parameter K ρ in the model whichis isotropic in spin space. For a single-band model withtwo Fermi points, ± k F , the SC correlation function de-cays as ∼ r − (1+ Kρ ) , while the CDW and SDW corre-lation functions decay as ∼ cos(2 k F r ) r − (1+ K ρ ) . Thus,the SC correlation is dominant for K ρ >
1, while theCDW or SDW correlation is dominant for K ρ <
1. Onthe other hand, for a two-band model with four Fermipoints, i.e., ± k F and ± k F , low-energy excitations aregiven by a single gapless charge mode with a gappedspin mode. In this case, the SC and CDW correla-tions decay as ∼ r − Kρ and ∼ cos[2( k F − k F ) r ] r − K ρ ,respectively, while the SDW correlation decays expo-nentially. Hence, the SC correlation is dominant for K ρ > .
5, while the CDW correlation is dominant for K ρ < .
5. In either case, the SC correlation increaseswith the exponent K ρ , and then K ρ is regarded as agood indicator of superconductivity. As the noninter-acting K ρ is always unity, we assume that the conditionof K ρ >
3. Phase Diagram
Figure 2 shows K ρ as a function of U ′ for several val-ues of ∆ at an electron density n = 5 / J = U ′ / U = U ′ + 2 J . When U ′ increases, K ρ decreases for a small U ′ , while it in-creases for a large U ′ in the case of ∆ ≥ .
9, and then uperconductivity in a Two-Orbital Hubbard Model 3 J S=1K ρ >1 K ρ <1 ∆ =1.9n=5/3U=U’+2JS=2 (SCI) Fig. 3. Phase diagram of the ground state with K ρ on the U ′ − J parameter plane with U = U ′ + 2 J for n = 5 / . U ’ ∆ =1.9n=5/3J=U’/4K ρ <1K ρ >1 K ρ >1 S=1S=1(SCI)(SCII) Fig. 4. Phase diagram of the ground state with K ρ on the U − U ′ parameter plane with J = U ′ / n = 5 / . becomes larger than unity for U ′ > . .
9. When J (= U ′ /
4) is larger than a certain crit-ical value, the ground state changes into the partiallypolarized ferromagnetic state with a total spin S = 1from the singlet state with S = 0. We find that su-perconductivity is most enhanced in the vicinity of thepartially polarized ferromagnetic state. To confirm su-perconductivity, we calculate the energy difference of theground state, E ( φ ) − E (0), as a function of the externalflux φ . As shown in the inset of Fig. 2, anomalous fluxquantization is clearly observed for ∆ = 1 . . U ′ − J parameter plane under the conditionof U = U ′ + 2 J for n = 5 / .
9. It contains the singlet state with S = 0 to-gether with partially polarized ferromagnetic states with S = 1 and S = 2. The singlet state with K ρ >
1, where S luon , T luon S unn , T unn S lnn , T lnn S lon S uon ∆∆∆∆∆∆∆∆ Fig. 5. Schematic diagrams of various types of superconductingpairing symmetries, S lon , S uon , S lnn , S unn , and S luon with spin sin-glet pairings and T lnn , T unn , and T luon with spin triplet pairings. we call it the SC phase, appears near the partially po-larized ferromagnetic region at J > ∼ U ′ . It extends fromthe attractive region ( U ′ <
0) to the realistic parameterregion with J ∼ U ′ / >
0, which is expected to corre-spond to that in the case of iron oxypnictides. We haveconfirmed that similar phase diagrams are also obtainedfor ∆ = 2 . U − U ′ plane under the condition of J = U ′ / n = 5 / .
9. We ob-serve two types of SC phases with K ρ >
1, the SC I for
U > U ′ and the SC II for U < U ′ , in the wide param-eter region including both weak- and strong-correlationregimes. Note that the SC I corresponds to the SC phaseshown in Fig. 3 and belongs to the realistic parameterregion mentioned before.
4. Pairing Correlation
To examine the nature of these SC phases, we calculateSC pairing correlation functions for the various typesof pairing symmetries schematically shown in Fig. 5.Explicit forms of the SC pairing correlation functions C ( r ) are given by S lon ( r ) = 1 N X i h c † i,l, ↑ c † i,l, ↓ c i + r,l, ↓ c i + r,l, ↑ i ,S uon ( r ) = 1 N X i h c † i,u, ↑ c † i,u, ↓ c i + r,u, ↓ c i + r,u, ↑ i ,S lnn ( r ) = 12 N X i h ( c † i,l, ↑ c † i +1 ,l, ↓ − c † i,l, ↓ c † i +1 ,l, ↑ ) × ( c i + r +1 ↓ c i + r,l, ↑ − c i + r +1 ,l, ↑ c i + r,l, ↓ ) i ,S unn ( r ) = 12 N X i h ( c † i,u, ↑ c † i +1 ,u, ↓ − c † i,u, ↓ c † i +1 ,u, ↑ ) × ( c i + r +1 ,u, ↓ c i + r,u, ↑ − c i + r +1 ,u, ↑ c i + r,u, ↓ ) i , Kazuhiro
Sano and Yoshiaki ¯Ono S luon ( r ) = 12 N u X i h ( c † i,l, ↑ c † i,u, ↓ − c † i,l, ↓ c † i,u, ↑ ) × ( c i + r,u, ↓ c i + r,l, ↑ − c i + r,u, ↑ c i + r,l, ↓ ) i ,T lnn ( r ) = 12 N u X i h ( c † i,l, ↑ c † i +1 ,l, ↓ + c † i,l, ↓ c † i +1 ,l, ↑ ) × ( c i + r +1 ↓ c i + r,l, ↑ + c i + r +1 ,l, ↑ c i + r,l, ↓ ) i ,T unn ( r ) = 12 N u X i h ( c † i,u, ↑ c † i +1 ,u, ↓ + c † i,u, ↓ c † i +1 ,u, ↑ ) × ( c i + r +1 ,u, ↓ c i + r,u, ↑ + c i + r +1 ,u, ↑ c i + r,u, ↓ ) i ,T luon ( r ) = 12 N u X i h ( c † i,l, ↑ c † i,u, ↓ + c † i,l, ↓ c † i,u, ↑ ) × ( c i + r,u, ↓ c i + r,l, ↑ + c i + r,u, ↑ c i + r,l, ↓ ) i , where S lon ( r ), S uon ( r ), S lnn ( r ), S unn ( r ), and S luon ( r ) denotethe singlet pairing correlation functions on the same sitein the lower orbital, on the same site in the upper or-bital, between nearest-neighbor sites in the lower or-bital, between nearest-neighbor sites in the upper or-bital, and between the lower and upper orbitals on thesame site, respectively. Furthermore, T lnn ( r ), T unn ( r ), and T luon ( r ) are the triplet pairing correlation functions be-tween nearest-neighbor sites in the lower orbital, betweennearest-neighbor sites in the upper orbital, and betweenthe lower and upper orbitals on the same site, respec-tively.In Fig. 6, we show the absolute values of various typesof SC pairing correlation functions | C ( r ) | for n = 5 / . U ′ = 4 J = 1 .
0, and U = − .
4. Here, the electronic state of the system belongs tothe SC II phase, although the phase diagram for
U < | T unn ( r ) | < − and S luon ( r = 3) = T luon ( r = 3) = 0, which are notshown in Fig. 6. We find that S uon ( r ) and S unn ( r ) decayvery slowly as functions of r , and | S uon ( r = 3) | is thelargest among the various | C ( r = 3) | values. Therefore,a relevant pairing symmetry for the SC II phase seemsto be the spin singlet pairing in the upper orbital band,mainly consist of ’on-site’ pairing. It is considered thatsuch a pairing in attractive region with U < U . On the other hand,in repulsive region with U ′ > U >
0, the pairing maybe due to charge fluctuation, which is enhanced by alarge inter-orbital repulsion U ′ similarly to that in thecase of the d - p model in the presence of the inter-orbitalrepulsion U pd . Next, we discuss the superconductivity in the SC Iphase including the realistic parameter region mentionedbefore. Figure. 7 shows the absolute values of vari-ous types of SC pairing correlation functions | C ( r ) | for n = 5 / . U ′ = 4 J = 1 . U = 2 .
4, where the system belongs to the SC I phase,as shown in Fig. 4. Here, | T unn ( r ) | , | S luon ( r = 3) | , and | T luon ( r = 3) | are not shown, because these correlationfunctions are very small or zero. We find that | S uon ( r ) | isconsiderably suppressed compared with | S unn ( r ) | in con-trast to that in the case of the SC II phase. Furthermore, | S lnn ( r ) | increases with increasing r except at r = 2. −2 −1 r | C (r) | S lon S uon S unn S lnn S luon T luon T lnn U=−0.4 U’=4J=1.0 ∆ =1.9 Fig. 6. Absolute values of various types of SC pairing correlationfunctions | C ( r ) | as functions of r for n =5/3 (10 electrons/6 sites)at ∆ = 1 . U ′ (= 4 J ) = 1 .
0, and U = − .
4, corresponding tothe SC II phase.
Therefore, the relevant pairing symmetry for the SC Iphase seems to be an extended spin singlet pairing, andmainly consist of nearest-neighbor site pairing. −2 −1 r | C (r) | S lon S uon S unn S lnn S luon T luon T lnn U=2.4 U’=4J=1.0 ∆ =1.9 Fig. 7. Absolute values of various types of SC pairing correlationfunctions | C ( r ) | as functions of r for n =5/3 (10 electrons/6 sites)at ∆ = 1 . U ′ (= 4 J ) = 1 .
0, and U = 2 .
4, corresponding to theSC I phase.
Recently, weak-coupling approaches such as RPAand perturbation expansions have shown that the sign-reversing s -wave ( s ± -wave) pairing is realized in ironoxypnictide superconductors. The order parameterof such a pairing is considered to change its sign be-tween hole and the electron Fermi pockets. To compareour result with the result obtained by weak-coupling ap-proaches, we examine the SC pairing correlation functionbetween the lower and upper orbitals, such as S l − unn ( r ) = 12 N X i < ∆ l nn ( i ) † ∆ u nn ( i + r ) > uperconductivity in a Two-Orbital Hubbard Model 5 with∆ m nn ( i ) † = c † i,m, ↑ c † i +1 ,m, ↓ − c † i,m, ↓ c † i +1 ,m, ↑ ( m = l, u ) . We also define T l − unn ( r ) as well as S l − unn ( r ) in the aboveequation.When the s ± -wave pairing is dominant, the values ofthe interorbital SC pairing correlation function are ex-pected to be negative, since the Fermi surface of thelower (upper) orbital band in our model corresponds toa hole (electron) Fermi pocket, as shown in Fig. 1(b). InFig. 8, we show the interorbital pairing correlation func-tions S l − unn ( r ) and T l − unn ( r )(see also inset) for the sameparameters in Fig. 4 corresponding to the SC I phase.We see that the values of T l − unn ( r ) are positive and verysmall, while those of S l − unn ( r ) are negative except at r = 3and not so small. This result suggests that the rele-vant pairing symmetry of the SC I phase is the spin-singlet s ± -wave pairing, which agrees with the result ofthe weak coupling approaches. Therefore, we expect thatthe s ± -wave pairing proposed on the basis of the result ofweak-coupling approaches is realized in the wide param-eter region including both weak- and strong-correlationregimes. S l−unn , T l−unn r C (r) S l−unn T l−unn U’=4J=1.0 ∆ =1.9U=2.4 Fig. 8. Pairing correlation functions S l − unn ( r ) and T l − unn ( r ). Here,we show the absolute value of the correlation functions at U =2 .
4, ∆ = 1 .
9, and U ′ (= 4 J ) = 1 . n =5/3 (10 electrons/6sites). The inset shows a schematic diagram of the pairing sym-metries, S l − unn ( r ) and T l − unn ( r ). Finally, we discuss the mechanism of superconductiv-ity in the SC I phase. The s ± -wave pairing is consid-ered to be mediated by the antiferromagnetic fluctuationdue to the nesting effect between electron and hole pock-ets. At first glance, the SC I phase is located adjacentto the partial ferromagnetic phase (S=1), and then, theferromagnetic fluctuation seems to be related to super-conductivity. To examine the relationship between spinfluctuation and superconductivity, we also calculate thespin correlation function for finite-size systems, where ashort-range spin correlation is considered to be crucialfor the superconductivity. We obtain the ferromagneticand antiferromagnetic components of the spin correla-tion as a function of U ′ (= 4 J ) for a fixed U = 1 . U ′ together with increasing K ρ (not shown). Therefore,we conclude that the antiferromagnetic spin fluctuationis responsible for the s ± -wave pairing in the SC I phase.
5. Summary and Discussion
We have investigated the superconductivity of the one-dimensional two-orbital Hubbard model in the case ofelectron and hole Fermi pockets corresponding to a char-acteristic band structure of iron oxypnictide supercon-ductors. To obtain reliable results including those in thestrong-correlation regime, we used the exact diagonal-ization method and calculated the critical exponent K ρ on the basis of the Luttinger liquid theory. It has beenfound that the system shows two types of SC phases,the SC I for U > U ′ and the SC II for U < U ′ , in thewide parameter region including both weak- and strong-correlation regimes.We have also calculated various types of SC pairingcorrelation functions in the realistic parameter region ofthe iron oxypnictides. The result indicates that the mostdominant pairing for the SC I phase is the intersite in-traorbital spin-singlet with sign reversal of the order pa-rameters between two Fermi pockets. The result is con-sistent with the sign-reversing s ± -wave pairing that hasrecently been proposed on the basis of the result obtainedby weak-coupling approaches for iron oxypnictide super-conductors. This indicates that the s ± -wave pairing isrealized not only in a weak-correlation regime but alsoin a strong-correlation regime. We have also calculatedthe spin correlation function and found that antiferro-magnetic spin fluctuation is responsible for the s ± -wavepairing in the SC I phase.As for the SC II phase, the most dominant pairingis found to be the on-site intraorbital spin-singlet pair-ing, which is consistent with the ordinary s -wave pairingof BCS superconductors. However, the superconductingmechanism of this phase is due to the charge fluctuationenhanced by the interorbital Coulomb interaction and isdifferent from the conventional BCS superconductivitydue to the electron-phonon interaction. Although theSC II phase seems to be realized only for the unrealis-tic parameter region in our model, it might be realizedfor a realistic parameter region in the d - p model, whichis closer to iron oxypnictides.
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We will address sucha problem by applying the present method in the d - p model in the future.
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