Three orbital model for the iron-based superconductors
Maria Daghofer, Andrew Nicholson, Adriana Moreo, Elbio Dagotto
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Three Orbital Model for the iron-based superconductors
Maria Daghofer, ∗ Andrew Nicholson, Adriana Moreo, and Elbio Dagotto
Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996 andMaterials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 32831 (Dated: October 31, 2018)The theoretical need to study the properties of the Fe-based high- T c superconductors using reliablemany-body techniques has highlighted the importance of determining what is the minimum numberof orbital degrees of freedom that will capture the physics of these materials. While the shape ofthe Fermi surface (FS) obtained with the local density approximation (LDA) can be reproduced bya two-orbital model, it has been argued that the bands that cross the chemical potential result fromthe strong hybridization of three of the Fe 3 d orbitals. For this reason, a three-orbital Hamiltonianfor LaOFeAs obtained with the Slater-Koster formalism by considering the hybridization of the As p orbitals with the Fe d xz , d yz , and d xy orbitals is discussed here. This model reproduces qualitativelythe FS shape and orbital composition obtained by LDA calculations for undoped LaOFeAs whenfour electrons per Fe are considered. Within a mean-field approximation, its magnetic and orbitalproperties in the undoped case are here described for intermediate values of J/U . Increasing theCoulomb repulsion U at zero temperature, four different regimes are obtained: (1) paramagnetic, (2) magnetic ( π,
0) spin order, (3) the same ( π,
0) spin order but now including orbital order, andfinally a (4) magnetic and orbital ordered insulator. The spin-singlet pairing operators allowedby the lattice and orbital symmetries are also constructed. It is found that for pairs of electronsinvolving up to diagonal nearest-neighbors sites, the only fully gapped and purely intraband spin-singlet pairing operator is given by ∆( k ) = f ( k ) P α d k ,α, ↑ d − k ,α, ↓ with f ( k ) = 1 or cos k x cos k y which would arise only if the electrons in all different orbitals couple with equal strength to thesource of pairing. PACS numbers: 71.10.-w, 71.10.Fd, 74.20.Rp
I. INTRODUCTION
The discovery of high- T c superconductivity in a fam-ily of iron-based compounds is offering a new concep-tual framework to study the non-standard pairing mech-anism that seems to induce these exotic superconductingstates. The magnetism present in several of the parentcompounds and the high critical temperatures arereminiscent of properties observed in the cuprates. Butthere are also clear differences, since the parent com-pounds are metallic and the Fermi surface (FS) isdetermined by more than one orbital.
The multior-bital nature of the problem poses a challenge to the de-sign of minimal models that can be studied with powerfultechniques, such as numerical methods. Ab-initio calcu-lations making use of the local density approximation(LDA) indicate that the five 3 d orbitals of Fe stronglyhybridize to form the bands that are close to the chem-ical potential. However, the FS appears to be de-termined by bands that have mostly d xz and d yz char-acter, and this observation has been confirmed by polar-ized angle resolved photoemission spectroscopy (ARPES)experiments. This supports the notion that the mini-mum number of orbitals to be considered to study thepnictides could be two. In fact, a two-orbital mini-mal model has been proposed and it has been stud-ied with numerical techniques on small clusters, aswell as with several other approximations.
Bothnumerical and mean-field calculations indicate thatthe magnetic metallic regime observed experimentally inthe undoped compounds is stabilized for intermedi- ate values of the Coulomb repulsion U and the numericalcalculations suggest that, upon doping in this regime,the most favored pairing operator is interorbital and hassymmetry B g . On the other hand, several authors have claimed thata two-orbital model may miss important features of thereal system.
It has been argued that a mini-mal model for the pnictides should contain at least threeorbitals for mainly two reasons: (i)
A relatively smallportion of the electron-pocket FS of LaOFeAs is deter-mined by a band of mostly d xy character and (ii) thebands that produce the two hole pockets should be de-generate at the center of the Brillouin zone (BZ), whichis not the case when only two orbitals are considered.The important question is how much these shortcomingsof the model impact the most relevant properties of thepnictides. The aim of this paper is to construct a three-orbital model that addresses these concerns and compareits properties with the two-orbital case. This is importantbecause in other areas of condensed matter physics, suchas the manganites, we have learned that a simple single-orbital model is often sufficient to capture qualitativelythe phenomenon of colossal magnetoresistance, whileclearly a two-orbital model is still necessary to properlydescribe additional properties such as the magnetic andorbital order observed in these materials. Similarly theminimal two-orbital model for the pnictides appears toreproduce the experimentally observed magnetic order,and it is interesting to investigate to what extent theinclusion of the third orbital xy modifies these results.While the pnictides exhibit a structural phase tran-sition at similar or slightly higher temperature as theonset of antiferromagnetic (AF) order, experiments havenot yet addressed the issue of orbital order and also havenot provided consensus regarding the symmetry of thepairing operator. An investigation of the magneticand orbital orders, as well as pairing symmetries, thatare allowed in a three-orbital model compared to thosethat are possible in the two-orbital case will shed light onthe importance of the role that the additional d xy orbitalshould play in theoretical discussions.This paper is organized as follows. In Sec. II, the three-orbital model is introduced. Section III contains a mean-field analysis of the magnetic and orbital properties ofthe undoped system. Section IV is devoted to the classi-fication and analysis of the spin-singlet pairing operatorsallowed by the orbital and lattice symmetries. A sum-mary and conclusions are presented in Sec. V. II. THREE ORBITAL MODEL FOR THEPNICTIDES
As explained in the Introduction, it has been suggestedthat at least three orbitals may be needed to describe thesuperconducting pnictides because the bands that deter-mine the Fermi surface of LaOFeAs are mostly composedby orbitals d xz , d yz , and d xy . The notation xz , yz , and xy will be used for these orbitals, respec-tively, for better readability. The need to include at leastthree orbitals in a realistic model was first pointed outin Ref. 31 where a three-orbital model was constructedusing the symmetry properties of the Fe-As planes andLDA results. A shortcoming of that proposed model wasthat it contained an spurious hole-pocket FS around the M point in the extended BZ notation. It was argued that a fourth orbital should be added to remove the extrapocket. However, it will be shown in Sec. II B that thisspurious pocket can actually be removed entirely withina three-orbital model formalism, i.e. without adding afourth orbital.One important issue that needs to be addressed is theelectronic filling to be used in a three-orbital model.Band calculations have determined that the undopedpnictides contain six electrons distributed among the five3 d orbitals of each Fe atom. One procedure to determinethe filling for a model with a reduced number of orbitalsis to start from the crystal field splitting and fill the levelsaccordingly from the lowest energy up. In the two-orbitalmodel that considers only the xz and yz orbitals, such aconsideration would suggest that half-filling is the correctelectronic density, because the x − y and 3 z − r orbitals are assumed to be fully occupied with four of thesix electrons and the xy orbital is assumed empty, leav-ing two electrons to populate the xz and yz orbitals. Inaddition, this filling assignment is the only one that al-lows to reproduce the LDA calculated FS. Applying thecrystal-field splitting rationale to the three-orbital modelwith xz , yz , and xy orbitals, this argument leads to a filling of one third (i.e. two electrons in three orbitals). However, for such a filling we have not been able to re-produce the LDA shape of the FS. Thus, the filling mustbe adjusted to approximately reproduce the FS and theorbital occupation numbers obtained with LDA. In fact,band structure calculations suggest that the three-orbitalsystem should be more than half-filled and actually havea filling of roughly two thirds (i.e. four electrons in thethree orbitals).
Our analysis shows that a FS with ap-proximately a similar size for the hole and electron pock-ets can be obtained both at fillings around one and twothirds (i.e. two and four electrons in the three orbitals),but the two almost degenerate hole-pockets around Γ de-mand a filling larger than half-filling. Thus, the focus ofour effort will be on a filling of 2 /
3, as in Ref. 52. Asit will be discussed below, non-half-filled orbitals allowthe orbital degree of freedom to be active and actuallyit has been argued that orbital ordering phenomena mayplay a role in these materials. This orbital order is un-likely in the half-filled case which is the natural filling forthe two and four orbital models for the pnictides.Once again, note that some authors have considered half-filling in the three-orbital case, but this leads to the“unwanted” hole pocket around M . A. Real Space
To construct the tight-binding portion of the three-orbital Hamiltonian for the pnictides, the Slater-Kosterprocedure described in Ref. 24 will be followed. Nearest-neighbor and diagonal next-nearest-neighbor hoppingswill be considered for all the orbitals. It is clear that thehopping terms for the xz and yz orbitals are the same asin the previously discussed two-orbital model, H xz,yz = − t X i ,σ ( d † i ,xz,σ d i +ˆ y,xz,σ + d † i ,yz,σ d i +ˆ x,yz,σ + h.c. ) − t X i ,σ ( d † i ,xz,σ d i +ˆ x,xz,σ + d † i ,yz,σ d i +ˆ y,yz,σ + h.c. ) − t X i , ˆ µ, ˆ ν,σ ( d † i ,xz,σ d i +ˆ µ +ˆ ν,xz,σ + d † i ,yz,σ d i +ˆ µ +ˆ ν,yz,σ + h.c. )+ t X i ,σ ( d † i ,xz,σ d i +ˆ x +ˆ y,yz,σ + d † i ,yz,σ d i +ˆ x +ˆ y,xz,σ + h.c. ) − t X i ,σ ( d † i ,xz,σ d i +ˆ x − ˆ y,yz,σ + d † i ,yz,σ d i +ˆ x − ˆ y,xz,σ + h.c. ) − µ X i ( n i ,xz + n i ,yz ) , (1)while the intra-orbital hoppings for the xy orbital aregiven by H xy = t X i , ˆ µ,σ ( d † i ,xy,σ d i +ˆ µ,xy,σ + h.c. ) − t X i , ˆ µ, ˆ ν,σ ( d † i ,xy,σ d i +ˆ µ +ˆ ν,xy,σ + h.c. )+ ∆ xy X i n i ,xy − µ X i n i ,xy , (2)where ∆ xy is the energy difference between the xy andthe degenerate xz / yz orbitals. The hybridization be-tween the xz / yz and the xy orbitals is given by H xz,yz ; xy = − t X i ,σ [( − | i | d † i ,xz,σ d i +ˆ x,xy,σ + h.c. ] − t X i ,σ [( − | i | d † i ,xy,σ d i +ˆ x,xz,σ + h.c. ] − t X i ,σ [( − | i | d † i ,yz,σ d i +ˆ y,xy,σ + h.c. ] − t X i ,σ [( − | i | d † i ,xy,σ d i +ˆ y,yz,σ + h.c. ] − t X i ,σ [( − | i | d † i ,xz,σ d i +ˆ x +ˆ y,xy,σ + h.c. ]+ t X i ,σ [( − | i | d † i ,xy,σ d i +ˆ x +ˆ y,xz,σ + h.c. ] − t X i ,σ [( − | i | d † i ,xz,σ d i +ˆ x − ˆ y,xy,σ + h.c. ]+ t X i ,σ [( − | i | d † i ,xy,σ d i +ˆ x − ˆ y,xz,yσ + h.c. ] − t X i ,σ [( − | i | d † i ,yz,σ d i +ˆ x +ˆ y,xy,σ + h.c. ]+ t X i ,σ [( − | i | d † i ,xy,σ d i +ˆ x +ˆ y,yz,σ + h.c. ]+ t X i ,σ [( − | i | d † i ,yz,σ d i +ˆ x − ˆ y,xy,σ + h.c. ] − t X i ,σ [( − | i | d † i ,xy,σ d i +ˆ x − ˆ y,yz,σ + h.c. ] . (3)The hopping parameters t i in Eqs.(1-3) will be deter-mined by fitting the band dispersion to band structurecalculation results. The chemical potential µ is set toa two-thirds filling, as already discussed. The opera-tor d † i ,α,σ ( d i ,α,σ ) creates (annihilates) an electron at site i , orbital α = xz, yz, xy , and with spin projection σ . n i ,α = n i ,α, ↑ + n i ,α, ↓ are the corresponding density op-erators. Previously proposed three-orbital models only contained the nearest-neighbor (NN) hybridization t , but since next-nearest neighbor (NNN) terms are in-cluded for the intra-orbital component, as well as for thehybridization between xz and yz , they should also be in-cluded in the hybridization with xy . In Sec. II B, it is TABLE I: Parameters for the tight-binding portion of thethree-orbital model Eqs.(5) to (10). The overall energy unitis electron volts. t t t t t t t t ∆ xy − .
01 0 . − . − t / shown that these NNN terms with hopping t turn outto be crucial to provide the proper orbital character forthe electron pockets when compared with LDA results.Finally note that the hybridization Eq. (3) contains fac-tors ( − | i | that arise from the two-iron unit cell of theoriginal FeAs planes. B. Momentum Space
The Hamiltonian Eqs. (1-3) can be transformed to mo-mentum space using d † k ,α,σ = √ N P i e − i k . i d † i ,α,σ , where k is the wavevector and N the number of sites. Note thatthe Fourier transformed Hamiltonian is defined in the ex-tended or unfolded BZ. As pointed out in Ref. 31,the real-space Hamiltonian presented in Eqs. (1-3) is in-variant under a translation along the x or y directionsfollowed by a reflexion about the x - z plane. When theeigenstates of the Hamiltonian are labeled in terms ofthe eigenvalues of these symmetry operations, then themomentum-space Hamiltonian can be expressed in termsof a pseudocrystal momentum that will be called k thatexpands the unfolded BZ that corresponds to a singleFe unit-cell in real space. Note that folding the threebands used here into the reduced diamond-like cell de-fined in the first quadrant by k x + k y ≤ π , and symmetricpoints in the other three quadrants, doubles the numberof bands to six as expected.Equations (1-3) become in momentum space H TB ( k ) = X k ,σ,µ,ν T µ,ν ( k ) d † k ,µ,σ d k ,ν,σ , (4)with T = 2 t cos k x + 2 t cos k y + 4 t cos k x cos k y − µ, (5) T = 2 t cos k x + 2 t cos k y + 4 t cos k x cos k y − µ, (6) T = 2 t (cos k x + cos k y )+4 t cos k x cos k y − µ + ∆ xy , (7) T = T = 4 t sin k x sin k y , (8) T = ¯ T = 2 it sin k x + 4 it sin k x cos k y , (9) T = ¯ T it sin k y + 4 it sin k y cos k x , (10)where a bar on top of a matrix element denotes the com-plex conjugate. Since the Hamiltonian for a one-iron unitcell has been considered, then k runs within the corre-sponding extended BZ − π < k x , k y ≤ π .The hopping parameters t i and ∆ xy were chosen toreproduce the shape of the LaOFeAs FS obtained using −1012 Γ X M Γ ene r g y − µ PSfrag replacements(a) x / π k y / π α α β (b) % o f w e i gh t α xzyzxy00.51 % o f w e i gh t α % o f w e i gh t θ / πβ PSfrag replacements(c)(d)(e)
FIG. 1: (Color online) (a) Band structure and (b) Fermi sur-face of the tight-binding (i.e. non-interacting) three-orbitalmodel, with parameters from Tab. I and in the unfolded BZ.The diagonal thin solid line in (b) indicates the boundary ofthe folded BZ. In panels (c-e), the orbital contributions tothe two hole and one of the electron pockets are given. Thewinding angle θ is measured with respect to the k y -axis. Thesecond electron pocket is analogous to the one shown simplyreplacing xz by yz . In all panels, the dashed lines refer to the xz orbital, the solid to yz , and the dotted to xy . LDA calculations. These parameters are given in Tab. I.The chemical potential µ = 0 .
212 was fixed to this valueto ensure a filling of two thirds, as discussed in Sec. II.The resulting band dispersion is presented in Fig. 1(a)along high symmetry directions in the extended BZ andthe corresponding FS is shown in Fig. 1(b). The twohole-pocket FSs, labeled α and α , are formed by twobands that are degenerate at Γ in agreement with LDA,and both of them are found around the Γ-point instead of M in the extended zone. Thus, one of the shortcomingsof the two-orbital model has been corrected. It can alsobe observed that there is no hole-pocket FS around M which was a problem encountered in Refs. 31,52.The orbital composition of the hole pockets is displayedin Figs. 1 (c) and (d). They are given by a linear com-bination of the xz and yz orbitals, in agreement withLDA and ARPES results. The electron pockets at X ( Y ) arise from a linear combination of the yz ( xz )and xy orbitals. The orbital composition for the electronpocket around X , labeled β in Fig. 1(b), is displayedin Fig. 1(e): here it can be observed that the orbitalcharacter changes from purely xy along Γ- X to predom- inantly yz along X - M , as predicted by LDA and alsofound with ARPES techniques. As can be deduced fromEqs. (9) and (10), setting t = − t / xy character along the Γ- X and Γ- Y directions. A large yz ( xz ) contribution along X - M ( Y - M ) requires the hybridizations t and t to bequite robust. In summary, a tight-binding Hamiltonian has beenconstructed that captures the generic shape and orbitalcomposition of the FS for undoped LaOFeAs by consider-ing only the three xz , yz , and xy orbitals, and assumingthat they share four of the six electrons that populatethe five Fe 3 d levels.The Coulombic interacting portion of the Hamiltonianis given by: H int = U X i ,α n i ,α, ↑ n i ,α, ↓ + ( U ′ − J/ X i ,α<β n i ,α n i ,β − J X i ,α<β S i ,α · S i ,β + J X i ,α<β ( d † i ,α, ↑ d † i ,α, ↓ d i ,β, ↓ d i ,β, ↑ + h.c. ) , (11)where α, β = xz, yz, xy denote the orbital, S i ,α ( n i ,α ) isthe spin (electronic density) in orbital α at site i , andthe relation U ′ = U − J between these Kanamori pa-rameters has been used (for a discussion in the manganitecontext see Ref. 32 and references therein). The first lineterms give the energy cost of two electrons located inthe same orbital or in different orbitals on the same site,respectively. The second line contains the Hund’s rulecoupling that favors the ferromagnetic (FM) alignmentof the spins in different orbitals at the same lattice site.The “pair-hopping” term is in the third line and its cou-pling is equal to J by symmetry. The values used for U and J can be substantially smaller than the atomic ones,because the interactions may be screened by bands notincluded in our Hamiltonian. These Coulombic interac-tion terms have been used and discussed in several pre-vious publications where more details can be foundby the readers. All the energies in this paper are givenin electron volts. III. NUMERICAL RESULTSA. Mean-field Approximation and Ordered Phases
To investigate the magnetic and orbital order prop-erties of the three-orbital model in the presence of theonsite Coulomb repulsion, here a mean-field approxima-tion will be used. As it was carried out for the two- andfour-orbital models in Ref. 25, here only the mean-fieldvalues for diagonal operators will be considered (for de-tails see App. A). In the present case, the degenerate xz and yz orbitals make up most of the weight close to the TABLE II: Magnetic and orbital ordering wavevectors for thepossible ordered phases discussed in the text. The third col-umn indicates the panel of Figs. 2 and 3, where a schematicsketch for the corresponding ordered phase can be found.spin: q orbital: q panel in Figs. 2 or 3( π,
0) (0 ,
0) 2(a), 2(b)( π, π ) (0 ,
0) 2(c)( π,
0) ( π,
0) 2(d), 2(e)( π, π ) ( π, π ) 2(f)( π,
0) (0 , π ) 2(g)( π,
0) ( π, π ) 2(h)( π, π ) ( π,
0) 2(i)(0 ,
0) (0 ,
0) 3(a)(0 ,
0) ( π,
0) 3(b)(0 ,
0) ( π, π ) 3(c) (a) (b) (c)(d) (e) (f)(g) (h) (i)
FIG. 2: (Color online) Cartoons representing the spin- andorbital-order configurations considered in this effort. Sincefour electrons occupy three orbitals, perfectly ordered stateshave one doubly occupied orbital and the unpaired electronsare located in the other two orbitals, each singly occupied,forming a net spin S = 1. In this cartoon, the orbital drawnindicates the doubly occupied one and the arrows indicate theorientation of the total spin at that site. (a) ( π, yz orbital, (b) ( π, xz + yz ) / √
2, (c) ( π, π )-spin and FOorder. (d)-(f): states with the same ordering vector for thespins and the orbitals: ( π,
0) for (d) and (e), ( π, π ) in (f). (g)( π,
0) for spins and (0 , π ) for orbitals; (h) ( π,
0) for spins and( π, π ) for orbitals; (i) ( π, π ) for spins and ( π,
0) for orbitals;(b) and (e) illustrate phases where the orbital order does notfeature alternating xz and yz orbitals, but the combinations( xz + yz ) / √ xz − yz ) / √ (a) (b) (c) FIG. 3: Color online) Cartoons for the spin ferromagneticphases taken into consideration in this effort. Results aredepicted with the convention of Fig. 2, but for ferromagneticspins combined with (a) FO, (b) ( π, π, π ) orbitalorder. chemical potential. In previous investigations of the two-orbital xz - yz model, very weak orbital order has beenfound within a mean-field approximation at intermedi-ate Coulomb repulsion in the half-filled case, and none inthe large- U limit. Numerical simulations did not pro-vide indications of orbital order at half-filling either. However, the xz and yz orbitals are not expected to behalf filled in the present case: In the noninteracting U = 0limit, the xy orbital is approximately half filled, so thatthe subsystem consisting of the two degenerate xz and yz orbitals turns out to be approximately 3 / xz / yz -subspace couldnow occur, analogously to the case of quarter filling inthe xz / yz subsystem. Three possible orbital-order patterns will be consid-ered: (i)
Ferro-orbital (FO) order which corresponds tothe orbitals xz and yz having different electronic densi-ties, (ii) alternating orbital (AO) order, and (iii) stripeorbital (SO) order. Combined with the magnetic spinorder, these orbital orders lead to a large variety of pos-sible combinations of polarized or alternating spin andorbital order. Here, phases that can be expressed using(at most) two ordering vectors have been considered, i.e. q for magnetic order and q for orbital order. The ex-pectation values for the mean-field proposed states canthen be expressed as h n r ,xy,σ i = n xy + σ i q . r m xy (12) h n r ,α,σ i = n + σ i q . r m + α i q . r p + σα i ( q + q . ) r q , (13)where the first equation with the mean-field parameters n xy , m xy describes the xy orbital and the second equa-tion with parameters n, m, p , and q applies to the xz / yz subsystem, with α = ± xz / yz orbitals.The fact that the xz / yz space is not SU (2) sym-metric introduces another degree of freedom in additionto the ordering vector: Ferro-orbital order (i.e., site-independent orbital densities) can favor either the xz or yz orbitals [as shown in Fig. 2(a)], or any linear combina-tion | φ i = cos φ | xz i + sin φ | yz i . As an example, Fig. 2(b)illustrates a state with φ = π/ | xz i + | yz i ) / √
2. The same holds foralternating orbital order: alternating order correspond-ing to φ = π/ φ for each phase. The xy orbital might be similarly in-volved in such a linear combination, because the crystal-field splitting separating it from the xz and yz orbitals isnot very large. However, Exact Diagonalization of 2 × xy orbitals. Moreover, the xy orbital is al-most singly occupied at U = 0 and has low weight at theFermi energy, so that ordering phenomena in the mostrelevant intermediate- U regime may be expected to in-volve mostly xz and yz orbitals. The states consideredare shown in Figs. 2 and 3 and the corresponding valuesof q i are given in Tab. II. Which spin-orbital pattern isstabilized depends on the interaction parameters U and J , as described in Sec. III B. The phase with the largeststability range, and which is stable for the most realis-tic parameter choices, is the ( π, B. Magnetic and Orbital Orders in the UndopedRegime
Magnetic order with wavevector q = ( π,
0) (or (0 , π ))and OD was found to be stable in a broad - and especiallythe most realistic - range of interaction parameters. How-ever, the phase diagram in the
J/U vs. U plane turns outto contain a large variety of metallic disordered phases,metallic phases with different kinds of magnetic and/ororbital order, and insulating magnetically and orbitallyordered regions. Figure 4 shows a qualitative rendition ofthe resulting phase diagram. A more accurate quantita-tive determination of the boundaries as well as a detaileddescription of all phases will be presented in a future pub-lication. For a realistic Hund’s rule coupling J = U/
4, wediscuss the properties of the ground state in the differentregimes encountered varying U in Sec. III C below.It has been found that at large interaction strengths U ,after the magnetic order with q = ( π,
0) is established,alternating- [ q = ( π, π )] and ferro-orbital [ q = (0 , / ≤ J/U ≤ /
3. For very small val-ues of
J/U , ferromagnetic order becomes stable insteadof ( π, J → q = ( π, π ) antiferromagnetism.At J = 0 and for small 3 / . U . .
5, both the FMand the ( π, π )-AF state have pronounced orbital ordercorresponding to φ = π/
4: One of the orbitals given bythe linear combinations ( | xz i ± | yz i ) / √ ≈ . xy the remaining0 .
5. In the the FM state, ( | xz i + | yz i ) / √ | xz i−| yz i ) / √ π, π )-AF state with only slightly higher energy.At larger U > .
5, both xz and yz are almost filled for J / U UPM FM FM( π ,0), FO( π ,0)OD AO SOFO FIG. 4: (Color online) Qualitative phase diagram in the
J/U vs. U plane. The shaded area denotes the stability region ofthe realistic ( π,
0) magnetic ordering. The lines are guides tothe eye, the dashed line approximately indicates the metal-insulator transition. The data points were obtained by com-paring the energies of the various phases within the mean-field approximation described in App. A. The meaning ofthe symbols is the following: +: ( π, × : ( π, π, π ) alternating orbital (AO) order with φ = 0;empty squares: ( π, , π ) orbital stripes (SO)( φ = 0); ∗ : ( π, φ = 0); filled squares: FM with FO ordering tendencies, φ = π/
4. This FO order is weaker for small U and larger J ≈ .
2. The filled circles denote parameters that do not sup-port magnetically ordered states. For small J , some FO orderwith φ = π/ U and small J denote similar states withoutmagnetic ordering, but with extreme orbital order, where the xy orbital is (almost) empty, while xz and yz are (almost)filled. J → xy is nearly empty, and there are hardly anyunpaired spins that could support magnetically orderedphases. Notice that such small ratios of J/U are not ex-pected to be realistic for the pnictides, where the onsiteCoulomb repulsion U is strongly screened. With grow-ing
J/U , the magnetic ordering vector switches to ( π, J/U ≈ / U , and even smaller J/U forlarger U .For U .
2, the ( π,
0) magnetic order remains stablefor all values of
J/U & / J/U = 1 /
3. At thisratio, the Hund’s coupling is such that the inter-orbitalrepulsion felt by two electrons in different orbitals, but onthe same site, vanishes, and we therefore did not consider
J/U > / C. Evolution of the ground state as a function of U for J = U/ In this section, the ground state properties at a fixedratio
J/U = 1 / J/U is where we have found the realistic AF order U m s t agg / µ B (a) m tot m xz m yz m xy U n o r b (b) n xz n yz n xy FIG. 5: (Color online) (a) Orbital magnetization and (b)occupation number as a function of the Coulomb repulsionstrength U , obtained with a mean-field approximation. Thecolors indicate the different phases (for increasing U ): uncor-related metal, itinerant ( π,
0) antiferromagnet without orbitalorder, itinerant ( π,
0) antiferromagnet with alternating orbitalorder [small white window, spin-orbital order as in Fig. 2(h)],and a ferro-orbitally-ordered ( π,
0) antiferromagnetic insula-tor [spin-orbital order as in Fig. 2(a)]. Hopping parametersare from Tab. I, and J = U/
4. For the phase with alternatingorbital order, the thin lines show (a) m ± q and (b) 2 n ± p . with ordering momentum q = ( π,
0) for all values of
U > U c . Figure 5(a) shows how the staggered magneti-zation with ordering momentum ( π,
0) increases with theCoulomb repulsion U . As previously found for two- andfour-orbital models, intermediate U leads to an antifer-romagnetic metal. The system remains nonmagnetic forsmall U up to U c ≈ .
6. For
U > U c , the spin ( π, π,
0) AF order triggered by U introduces gaps and magnetically-induced “shadow”bands; for comparison, the uncorrelated A ( k , ω ) is in-cluded in Fig. 6. Since several bands are involved, thegaps are not necessarily located at the chemical poten-tial, as it has been discussed for the two- and four-orbitalmodels in Ref. 25, and the system remains metallic. Asit can be seen in Fig. 6, the overall features of the spec-tral density in this regime remain similar to those of thenoninteracting limit, with the bandwidth being slightlyreduced with increasing U . However, the onset of AForder does affect some details of A ( k , ω ), especially low- FIG. 6: (Color online) Spectral density A ( k , ω ) for the anti-ferromagnet orbital-disordered metallic phase at (a) U = 0 . U = 0 .
9, and (c) U = 1 .
1. The BZ is for the one-Fe unitcell, and J = U/ A ( k , ω ) for U = 0 is included as solid lines for comparison. energy features at the chemical potential, where one ofthe hole pockets disappears and additional pockets arise.The Fermi surface for U = 0 .
7, where the Coulomb re-pulsion is just barely strong enough to induce ( π,
0) an-tiferromagnetism, is shown in Figs. 7(a,b). More specif-ically, Fig. 7(a) shows the Fermi surface in the extendedBZ for spin stripes running along the y -direction, i.e.,for the ordering vector ( π, , π ) is hardly affected, the pocket at ( π,
0) has al-most disappeared. Of the two hole pockets, the innerone has also disappeared for momenta (0 , k y ), because agap has here developed at the chemical potential µ , seeFig. 6(a). For momenta ( k x , outer pocket lies below µ , and the band consequentlyforms a very small electron pocket. This result is in qual-itative agreement with the unconventional electronic re-construction observed with ARPES in (Ba , Sr)Fe As . Figure 7(b) shows the superposition of the Fermi surfacesobtained for the two equivalent ordering vectors ( π, , π ) in the reduced BZ corresponding to the two-Fe unit cell. If U is increased to U = 0 .
9, the gap in theouter hole pocket along ( k x ,
0) increases and pushes theouter band above the chemical potential; the small elec-tron pockets seen for U = 0 . , π ) electron pocket remains un-affected, but at ( π, U = 0 has been deformed strongly enough tocreate a small hole -like pocket at ≈ ( π/ , π,
0) elec-tron pocket once the results for ordering vectors ( π, , π ) are combined. As U continues to increasewithin the magnetic metallic phase no further qualita-tive changes are observed, as it can be seen in Fig. 6(c)and Fig. 7(e) and 7(f) where the spectral functions andthe FS are shown for U = 1 . π,
0) stripes. Note thatthe difference between m xz and m yz in Fig. 5(a) is largerthan the difference between n xz and n yz in Fig. 5(b) in-dicating that q is more important than p in Eq. (13). Inaddition, the behavior of the spectral functions appearsto be dominated by the magnetization in this phase.When a second critical coupling U c ≈ .
23 is reached,the system develops orbital order with an ordering mo-mentum ( π, π ), different from the magnetic orderingvector ( π, xz and yz orbitals, i.e., φ = 0.) The system re-mains a metal through this second transition as well, butthe spectral density is profoundly affected, see Fig. 8(a).The original hole and electron pockets around Γ and M completely disappear and only correlation-induced pock-ets remain: the hole pocket around k = ( π/ ,
0) is mir-rored at k ≈ ( π/ , π ), and four more small pockets can beseen in the FS (not shown) away from the high-symmetrydirections plotted in Fig. 8(a). Different states with dif-ferent orbital ordering patterns have only slightly higherenergies in this regime. In contrast, phases with different magnetic ordering have significantly higher energy, sug-gesting that the ( π,
0) AF “stripes” may be more robustthan the alternating orbital order.If U is further increased, a metal-insulator transition FIG. 7: (Color online) Fermi surface in the orbital-disorderedspin-antiferromagnetic metallic phase with (a,b) U = 0 . U = 0 .
9, and (e,f) U = 1 .
1. (a,c,e) show the unfoldedBZ containing one Fe, for the antiferromagnetic ordering vec-tor q = ( π, q = ( π,
0) and q = (0 , π ) in the (rotated) folded BZ corre-sponding to two Fe atoms. The ratio J = U/ finally occurs at a third critical U c ≈ .
43. At this point,the orbital order changes: as can be concluded from theorbital densities shown in Fig. 5(b), the system developsferro-orbital order. The spin-( π,
0) antiferromagnetismpersists, and the spectral density in Fig. 8(b) has a fullgap. The ferro-orbital spin-( π,
0) order in this insula-tor is the one depicted schematically in Fig. 2(a). Withgrowing U , the staggered magnetization converges to itsmaximal possible value 2 µ Bohr , as shown in Fig. 5(a).The direction of the magnetic stripes determines whichof the two degenerate xz and yz orbitals is (almost) dou-bly occupied in the FO order realized at large U . Forordering vector ( π,
0) ((0 , π )) it is the yz ( xz ) orbital.This can be understood by considering the interorbitalhopping t between the xy orbital and the yz ( xz ) orbitalalong the y -( x -)direction, which has to be large in or-der to ensure xz / yz character for the hole pockets, seeSec. II B. For spin stripes along the y -direction, i.e. or-dering vector ( π, x -direction are anti-ferromagnetic and the electrons can better take advan- FIG. 8: (Color online) Spectral density A ( k , ω ) for (a) theorbitally-ordered spin-( π,
0) antiferromagnetic metallic phaseat U = 1 .
36, and (b) the orbitally polarized spin-( π,
0) anti-ferromagnetic insulator at U = 1 .
6. The ratio J = U/ PSfrag replacements yz xzxyt t t t FIG. 9: (Color online) Magnetic order and orbital occupa-tion at large U shown for four sites. The magnetic orderingvector is ( π, y direc-tion. For each site, the xy , xz , and yz orbitals are shown:the xy is the one below the other two, and the yz is dou-bly occupied. Dashed (continuous) lines indicate inter-orbitalhopping t connecting the xy orbital to xz ( yz ) along the x -( y -) direction. tage of the AF superexchange if the two orbitals con-nected by t are both singly occupied (see the schematicillustration in Fig. 9). The remaining yz orbital thenhas to be doubly occupied. This does not cost any AFsuperexchange energy, because its connection to the xy orbital via t lies along the spin-aligned y -direction wheresuch AF superexchange would in any case not occur. Infact, the additional electron in the yz orbital has the op-posite spin from the majority spin of the stripes and can,thus, gain some kinetic energy by hopping via t to the xy orbital along the FM y -direction. Thus, in this regimeof large U the ground state corresponds to the cartoonshown in Fig. 2(a) if the magnetic order is ( π, U : (i) a disordered, param-agnetic phase for U < U c , (ii) a metallic phase with( π,
0) or (0 , π ) magnetic order for U c < U < U c , (iii)a metallic magnetic phase for U c < U < U c with alter-nating orbital order with ordering vector ( π, π ), and (iv)a ferro-orbitally ordered insulator with spin-( π,
0) mag-netic order for
U > U c , where the yz [ xz ] orbital haslarger electronic occupation for magnetic ordering vector( π,
0) [(0 , π )].
IV. PAIRING OPERATORS IN ATHREE-ORBITAL MODEL FOR PNICTIDES
In this section, the spin-singlet pairing operators thatare allowed by the lattice and orbital symmetries in thethree-orbital model for LaOFeAs will be constructed.This classification of operators has previously been madefor the two-orbital model.
An approach similar toRef. 67 will be followed. To achieve this goal, the three-orbital tight-binding portion of the Hamiltonian, H TB ,presented in Eq. (4) will be rewritten in terms of the3 × λ i which correspond to the eight Gell’mannmatrices for the cases i = 1 to 8, while λ is the 3 × H TB becomes H TB ( k ) = X k ,σ Φ † k ,σ ξ k Φ k ,σ , (14)where Φ † k ,σ = ( d † xz ( k ) , d † yz ( k ) , d † xy ( k )) σ and ξ k = ǫ k λ + δ k λ + γ k λ + α (1) k λ + α (2) k λ + h k λ , (15)with ǫ k = ( T + T + T ) / t + t + t )(cos k x + cos k y )+ 43 (2 t + t ) cos k x cos k y − µ + ∆ xy , (17) δ k = ( T − T ) / − ( t − t )(cos k x − cos k y ) , (18) γ k = T = 4 t sin k x sin k y , (19)0 TABLE III: Symmetry properties of the several terms in the H TB Hamiltonian of the three-orbital model.Term IR ǫ k A g δ k B g γ k B g ( α (1) k , α (2) k ) E g h k A g TABLE IV: Symmetry properties of Gell’mann matrices forthe orbital assignment that defines the proposed three-orbitalmodel. Matrix IR λ A g λ B g λ A g λ B g ( λ , λ ) E g ( λ , λ ) E g λ A g α (1) k = T /i = − t sin k x − t sin k x cos k y , (20) α (2) k = T /i = − t sin k y − t sin k y cos k x , (21)and h k = T + T √ − T √
3= 1 √ t + t − t )(cos k x + cos k y )+ 4 √ t − t ) cos k x cos k y − ∆ xy √ . (22)It can be shown that each element in Eqs. (15-22)transforms according to one irreducible representation ofthe D h group corresponding to the Fe lattice. The clas-sification is given in Tab. III.Since the Hamiltonian has to transform according to A g , the Gell’mann matrices in the orbital basis here cho-sen transform as indicated in Tab. IV.In multiorbital systems the general form of a spin-singlet pairing operator is given by ∆ † ( k ) = f ( k )( λ i ) α,β ( d † k ,α, ↑ d †− k ,β, ↓ − d † k ,β, ↑ d †− k ,α, ↓ ) , (23)where a sum over repeated indices is implied; the oper-ators d † k ,α,σ have been defined in the previous sectionsand f ( k ) is the form factor that transforms accordingto one of the irreducible representations of the crystal’ssymmetry group. Although f ( k ) may, in general, havea very complicated form, a short pair-coherence lengthrequires the two electrons that form the pair to be veryclose to each other. Consequently, for simplicity we focuson nearest and diagonal next-nearest neighbors, and formfactors that are allowed in a lattice with D h symmetry.The momentum dependent expression, as well as the ir-reducible representation according to which each form TABLE V: Form factors f ( k ) for pairs up to distance (1,1)classified according to their symmetry under D h operations. f ( k ) IR1 1 A g k x + cos k y A g k x cos k y A g k x − cos k y B g k x sin k y B g k x , sin k y ) E g k x cos k y , sin k y cos k x ) E g TABLE VI: Product table for the irreducible representationsof the group D h relevant to this work. A g A g B g B g E g A g A g A g B g B g E g A g A g A g B g B g E g B g B g B g A g A g E g B g B g B g A g A g E g E g E g E g E g E g A g + A g + B g + B g factor transforms, are given in Tab. V. Note that if thepairing mechanism is non-BCS and if the Coulomb repul-sion is strong on-site pairing, then f(k)=1 correspondingto onsite pairing is an unlikely factor. A. Intraorbital Pairing
The previous discussion shows that the symmetry ofthe pairing operator will be exclusively determined by thesymmetry of f ( k ) only if λ i transforms according to A g .Table IV indicates that this is the case for pairing oper-ators constructed by using λ or λ in Eq. (23). Thesetwo matrices are diagonal, which means that such pair-ing operators define intraorbital pairings. For intraor-bital pairing, with a symmetry fully determined by thespatial form factor, the basis functions are then given by: f ( k ) λ or f ( k ) λ .For xy to be different than for xz and yz which need, bysymmetry, to have OPs that can only differ by a relativesign. Thus, the addition of a third orbital may allow thepossibility of different superconducting gaps in the bandrepresentation, reminiscent of the two gaps in MgB . When any of the remaining seven matrices λ i appear inEq. (23), the symmetry of the pairing operator is givenby the irreducible representation of D h resulting fromthe product of the symmetry of the form factor and thesymmetry of the orbital component, according to theproduct table given in Tab. VI. For λ i = λ , the basisfunction is given by f ( k ) λ . The pairing is stillintraorbital but since λ transforms according to B g thesymmetry of the operator will be B g if f ( k ) transformsaccording to A g , etc. Note that this pairing operatordoes not involve the xy orbital and it has already been1presented in the context of the two-orbital model. How-ever, since the orbital composition of the bands is not thesame as for the two-orbital model, it will be importantto determine whether the gap structure of this pairingoperator has changed.
B. Interorbital Pairing
The remaining six λ i matrices lead to interorbital pair-ing. Note that λ and λ do not involve the orbital xy and the pairing operators that they generate havealready been discussed in the two-orbital model. Weare interested in the spin-singlet pairing operator for or-bitals xz/yz that has a basis f ( k ) λ . This opera-tor, with f ( k ) = cos k x + cos k y , has been found to befavored for intermediate values of the Coulomb repul-sion U in numerical calculations of the two-orbital modelfor pnictides. The addition of the xy orbital leadsto the possibility of new interorbital pairing operators,i.e., pairing between electrons in the orbitals xz and yz with electrons in the xy orbital. Thus, now the focus willbe on the interorbital spin-singlet pairing operators thatresult from the addition of xy .The interorbital case becomes very interesting becausewe need to combine ( xz , yz ) that transform as the two-dimensional representation E g with xy that transforms as B g ; thus, the product transforms as E g . The λ i matricesthat can appear in this intraorbital pairing are ( λ , λ )or ( λ , λ ). Since the focus here is on pairing operatorsthat are spin singlets, it will be required that the oper-ator is even under orbital exchange. Thus, only ( λ , λ )will be considered since the other two matrices ( λ , λ )that transform according to E g will produce operatorsodd under orbital exchange. Let us further restrict theanalysis to the case of pairing operators that transformaccording to one dimensional representations of the pointgroup because we assume that the ground state is non de-generate. Thus, the only spatial form factors f ( k ) thatwe should consider must transform according to E g . Thisleaves us with f ( k ) = (sin k x , sin k y ) , (24)for nearest neighbor pairs and f ( k ) = (sin k x cos k y , cos k x sin k y ) , (25)for diagonal pairs. Since the direct product of two E g representations is E g × E g = A g + A g + B g + B g , pair-ing operators transforming according to four irreduciblerepresentations will be obtained. The basis for the newpairing operators, labeled V i , are presented in Tab. VII. C. Band Representation
To obtain the gap structure of the pairing operators
TABLE VII: Properties of pairing operators in the three-orbital model. f indicates the symmetry of f ( k ).No. IR Basis GapI f f ( k ) λ Full or NodalII f f ( k ) λ Full or NodalIII fB g f ( k ) λ NodalIV fB g f ( k ) λ NodalV a A g sin k x λ + sin k y λ NodalV b A g λ sin k x cos k y + λ cos k x sin k y NodalV c B g sin k x λ − sin k y λ NodalV d B g λ sin k x cos k y − λ cos k x sin k y NodalV e A g sin k x λ + sin k y λ NodalV f A g λ cos k x sin k y + λ sin k x cos k y NodalV g B g sin k x λ − sin k y λ NodalV h B g λ cos k x sin k y − λ sin k x cos k y Nodal
Hamiltonian (BdG) is constructed and it is given by H BdG = X k Ψ † k H MF k Ψ k , (26)with the definitionsΨ † k = ( d † k ,xz, ↑ , d † k ,yz, ↑ , d † k ,xy, ↑ ,d − k ,xz, ↓ , d − k ,yz, ↓ , d − k ,xy, ↓ ) , (27)and H MF k = (cid:18) H TB ( k ) P ( k ) P † ( k ) − H TB ( k ) (cid:19) , (28)where each element represents a 3 × H TB ( k )given by Eq. (4) and P ( k ) α,β = V f ( k )( λ i ) α,β , (29)with i = 0, 8, 3, and 1 for pairing V is the magnitude of the OP given bythe product of the pairing attraction V and a mean-fieldparameter ∆ that should be obtained from minimizationof the total energy. For pairing i the basis listed inTable VII should be used instead of f ( k ) λ i .Up to this point we have worked using the orbitalrepresentation because this basis renders it straightfor-ward to obtain the form of the Hamiltonian, as well asthe pairing operators allowed by the symmetry of thelattice and orbitals. However, the experimentally ob-served superconducting gaps occur at the FS determinedby the bands that result from the hybridization of theorbitals. For this reason, it is convenient to expressEq. (28) in the band representation. H TB ( k ) can be ex-pressed in the band representation via the transformation H Band ( k ) = U † ( k ) H TB ( k ) U ( k ), where U ( k ) is the uni-tary change of basis matrix and U † ( k ) is the transposeconjugate of U ( k ). Since U is unitary it is known thatfor each value of k , P i ( U i,j ) ∗ U i,k = P i ( U j,i ) ∗ U k,i = δ j,k .Then, H ′ MF = G † H MF G where G is the 6 × × U . Then, H ′ MF k = (cid:18) H Band ( k ) P B ( k ) P † B ( k ) − H Band ( k ) (cid:19) , (30)2with P B ( k ) = U − ( k ) P ( k ) U ( k ) . (31)A standard assumption in superconducting multibandsystems is that the pairing interaction should be purelyintraband, meaning that P B ( k ) is diagonal. Thus, letus explore what kind of purely intraband pairing opera-tors are allowed by the symmetry properties of the three-orbital model for LaOFeAs. In the band representation,the most general BdG matrix with purely intraband pair-ing is given by H ′ MF = ǫ ( k ) 0 0 ∆ ( k ) 0 00 ǫ ( k ) 0 0 ∆ ( k ) 00 0 ǫ ( k ) 0 0 ∆ ( k )∆ ∗ ( k ) 0 0 − ǫ ( k ) 0 00 ∆ ∗ ( k ) 0 0 − ǫ ( k ) 00 0 ∆ ∗ ( k ) 0 0 − ǫ ( k ) , (32)where ǫ i ( k ) are the eigenvalues of H TB ( k ) and ∆ i ( k )denotes the band and momentum dependent pairinginteractions. As it can be deduced from the propertiesof the unitary change of basis matrix U , if all three bandshave the same pairing interaction, i.e. ∆ ( k ) = ∆ ( k ) =∆ ( k ) = ∆( k ), then the matrix P ( k ) in the orbital rep-resentation will also be diagonal. In this case, the pairingoperator is given by Eq. (23) with an arbitrary f ( k ) and λ i = λ , i.e., the pairing operator is intraorbital and theOP is the same for the three orbitals. This correspondsto pairing operator same for each of the three orbitals. How-ever, symmetry only requires that the orbitals xz and yz must have the same OP, while xy can have a differentone. Thus, there does not seem to be a reason to assumethat electrons in the many bands that determine the FSshould be affected by the same pairing interactions. Infact, in MgB the electron-phonon interaction that pro-vides the pairing is stronger on the σ -bands than on the π -bands giving, as a result, two different superconductinggaps. Thus, it can be asked whether the symmetry of thethree-orbital model allows for the possibility of two dif-ferent OPs with a pure intraband pairing interaction. Ifit is assumed that in Eq.(32) ∆ = ∆ = ∆ = f ( k ) C and∆ = ∆ ′ = f ( k ) C ′ , then in the orbital representation P ( k ) α,β = U α, ( k ) U ∗ β, ( k ) f ( k )( C ′ − C ) , (33)for the off-diagonal elements and P ( k ) α,α = ∆ + | U α, ( k ) | f ( k )( C ′ − C ) , (34)for the diagonal ones.Now let us concentrate on the diagonal part. This hasto arise from a linear combination of intraorbital pairingoperators with compatible symmetries. There are twopossibilities: P ( k ) α,α = f ( k )[ A ( λ ) α,α + B ( λ ) α,α ] , (35) or P ( k ) α,α = Df ( k )( λ ) α,α , (36)where A , B , and D are independent of momentum. Itcan be shown that Eq. (35) requires | U | = | U | whileEq. (36) requires | U | = −| U | , which are not satisfiedby the elements of the matrix U determining the changeof basis. This means that any purely intraband pairinginteraction allowed by the symmetry of the three-orbitalmodel should be the same for the three bands. On theother hand, if we had a case in which | U | = | U | =0, which means that one of the three orbitals does nothybridize with the other two, it would be possible to havea system with two different gaps. Note that this is thesituation for MgB in which the z orbital that forms the π band does not hybridize with the x and y orbitals thatconstitute the σ band.Summarizing, it has been found that independent gapsin different Fermi surfaces cannot arise if the hybridiza-tion among all the orbitals is strong and the pairing in-teraction is purely intraband. D. The s ± Pairing Operator
The next issue to be considered is whether the orbitaland lattice symmetries allow for the possibility of the of-ten discussed s ± pairing scenario. ARPES experimentsindicate the existence of two hole-pockets around Γ. Theinterior pocket, which is almost nested with the electronpockets with a nesting vector q = ( π,
0) or (0 , π ), devel-ops a constant gap ∆ h which has the same magnitudethan the gap on the electron pockets ∆ e . In addition,they find a smaller gap ∆ ′ h ≈ ∆ h / The ARPES results can be interpreted in twodifferent ways in the context of a three-orbital model: (i)
Assume that the inner hole pocket observed inARPES corresponds to two almost degenerate FS thatcannot be resolved, and assign the external hole pocketto a band that arises when extra orbitals are added. Thisis the same assumption made in the two-orbital model forwhich it was shown in Ref. 24 that the s ± pairing stateis compatible with the lattice and orbital symmetries.Under this assumption, in the three-orbital model the s ± pairing state corresponds to our pairing operator f ( k ) = cos k x cos k y , which in the band representa-tion leads to a purely intraband pairing attraction givenby ∆ i ( k ) = V cos k x cos k y for each of the three-bands.For hole pockets almost degenerate with each other, thegap in both bands will be the same and there will be asign difference with the gap at the electron pockets whoseFermi momentum differs from those of the hole pocketsby (0 , π ) or ( π, (ii) Assume that the inner and outer hole pockets ob-served by ARPES are described by the two hole pock-ets in the three-orbital model. This would force us torequest that, for example, ∆ ( k ) = − ∆ ( k + q ) where q = ( π,
0) or (0 , π ) and ∆ ( k ) is independent. Then, let3us assume that ∆ ( k ) = ∆ ( k ) = cos k x cos k y ∆ and∆ ( k ) = ∆( k ). Let us concentrate on the Γ- X direc-tion. Along this direction, γ k and α ( i ) k vanish, meaningthat there is no hybridization among the three orbitals.From Fig. 1 we observe that each of the two hole FS re-sults from the crossing of xz and yz , while the electronFS has pure xy character. From the orbital symmetry,then, it is deduced that the only reason for having dif-ferent gaps at the two hole-like FS would be a strongmomentum dependence of the gap since symmetry en-forces | ∆ xz ( k ) | = | ∆ yz ( k ) | ; in addition, the gap at theelectron pockets does not need to be related to the gap inthe hole pockets, unless the pairing operator contains λ .Thus, we observe that the s ± pairing operator could besupported under this assumption if it is given by Eq. (23)with f ( k ) = cos k x cos k y with λ i = λ and the additionalcondition that if k F S h represents the Fermi momentumof the internal hole pocket and k F S h ′ = k F S h + δ is theFermi momentum of the external hole pocket it is nec-essary that f ( k F S h ) /f ( k F S h ′ ) ≈ h ′ / ∆ h should not be 1/2for all materials.Thus, it is concluded that the s ± pairing could be sup-ported by a three-orbital model. It corresponds to pairsof electrons in the same orbital at distance one alongthe diagonals of the square lattice, i.e., on next-nearestneighbor sites, with the same pairing potential for allthree orbitals. Then, if experiments show that s ± is in-deed the correct pairing operator, it will remain to beunderstood why the pairing interaction does not appearto depend on the symmetry of each different orbital or,equivalently, why it is the same for electrons in differentbands. This should be contrasted with the case of MgB in which the strength of the electron-phonon couplingthat leads to pairing is stronger on the σ -band FS thanin the π -band FS.Since the pairing mechanism for the pnictides is notknown and the s ± pairing state is just one of many pro-posed states, our discussion will continue by analyzingthe other new pairing states that are allowed by symme-try when the xy orbital is considered. E. Properties of the Pairing Operators
In the previous subsection, it was shown that only pair-ing s ± pairing state indeed belongs to theclass represented by pairing λ , that allows a differentpairing strength for the xy orbital. This pairing op-erator is not purely intraband. This fact can be eas-ily deduced from the properties of the unitary changeof basis matrices. In this case, P ( k ) α,β = C α δ αβ with C = C = V f ( k ) / √ xz and yz , and C = − V f ( k ) / √ xy . Then, P B ( k ) α,β = X i C i U ∗ i,α ( k ) U i,β ( k ) , (37)which does not vanish for all values of k for α = β , thusindicating the existence of interband pairing terms. Sim-ilar calculations for all the pairing operators presented inTable VII show nonvanishing interband pairing terms.Then, it is concluded that starting from the orbitalrepresentation, the only way to obtain pure intrabandpairing in the band representation is by considering apairing interaction that affects equally all the orbitals in-volved, producing equal gaps in all the orbitals and/orbands with symmetry determined by the spatial formfactor. This shows that the requirement of purely intra-band pairing induces a strong constraint regarding thecoupling of the electrons in the different orbitals withthe source of the pairing attraction. On the other hand,if the requirement is relaxed, interband pairing occurs atleast in some regions of the BZ. It was verified that thisis the case for the remaining operators i . It has also been observed, by monitoring theeigenvalues of H BdG for operators f ( k ) shown in Tab. V, while operator f ( k ) = cos k x cos k y or 1 at afinite value of V . In addition, some linear combinationsof pairing f ( k ) = cos k x cos k y or 1 arealso nodeless for all finite values of V but they lead to in-terband pairing interactions. These nodeless states thatwe call s IB will be discussed in Sec.IV E 2.
1. Pairing with Pseudocrystal Momentum Q = ( π, π ) In Sec. II B, it was explained that although the three-orbital Hamiltonian retains the two-iron unit cell of theoriginal FeAs planes, it is possible to express it in termsof three orbitals in the space of pseudocrystal momen-tum k defined in the extended Brillouin zone corre-sponding to a base with one single Fe atom per unitcell. In terms of the real momentum, the Hamilto-nian consists of two 3 × H ( k ) and H ( k ) with H ( k ) = H TB ( k ) = H ( k + Q ); thus, these two blocksprovide the same eigenvalues in the unfolded Brillouinzone, but both blocks need to be considered if the actualreduced BZ is used. This means that in the reduced BZthe bands arise as combinations of six orbitals labeled bythe orbital index α = 1, 2, or 3, and the Hamiltonianblock index i = 1 or 2. Then, to consider all the possibleinterorbital pairing operators it is important to include4pairs formed by electrons in orbitals in the two differentblocks. In the extended BZ, this is equivalent to con-sidering pairs with both pseudocrystal momentum 0 and Q . Note that Cooper pairs with pseudocrystal momen-tum Q still have zero center-of-mass momentum. Exactdiagonalization studies of the two-orbital model didnot favor such operators, and it is possible that this kindof pairing does not occur in the three-orbital model ei-ther. However, since symmetry allows such a possibility,pairing operators with nonzero pseudocrystal momentumwill here be discussed for completeness.The generalized Bogoliubov-de Gennes matrix H MF k that allows us to consider interorbital pairs with pseu-docrystal momentum Q is given by: H MF k = H TB ( k ) 0 0 P ( k )0 − H TB ( k ) P ( k + Q ) 00 P † ( k + Q ) H TB ( k + Q ) 0 P † ( k ) 0 0 − H TB ( k + Q ) , (38)where P ( k ) has the form given in Eq. (29).By finding the eigenvalues of H MF k , the structure of thegap of the possible pairing operators with pseudocrystalmomentum Q can be obtained. Our analysis shows thatall the pairing operators with pseudocrystal momentum Q lead to inter and intraband pairing in the band rep-resentation and nodes on the FS for small V . We haveobserved that a nodeless gap for pairs with pseudocrystalmomentum Q develops at a finite value of V for operators f ( k ) = 1 or cos k x cos k y in a mannercharacteristic of systems with interband pairing.
2. Spectral Functions
It is straightforward to calculate the spectral func-tions A ( k , ω ) for all the pairing operators presented inthis manuscript. However, due to their large number,we will concentrate on (i) pairing operator f ( k ) = cos k x cos k y , i.e., the s ± pairing operator, (ii) a linear combination of pairing operators f ( k ) = cos k x cos k y , that we will call the s IB pair-ing operator; (iii) pairing operator f ( k ) =cos k x + cos k y , i.e., the B g pairing operator, favored bynumerical calculations in the magnetic metallic regimeof the two-orbital model, which will be called B g , and (iv) a linear combination of pairing operator f ( k ) = cos k x + cos k y and pairing operator V g withpseudocrystal momentum Q , which is the natural exten-sion to three orbitals of B g and will be called B ext2 g .In Fig. 10(a) the spectral functions A ( k , ω ) along highsymmetry directions in the reduced Brillouin zone areshown for the three-orbital Hamiltonian with V = 0, i.e.,without pairing, in order to illustrate the changes inducedby the various pairing interactions considered here. Notethat these results correspond to the system with two Featoms per unit cell, which leads to the six bands seenin Fig. 10(a). The results for the s ± pairing operator FIG. 10: (Color online) The intensity of the points representsthe values of the spectral function A ( k , ω ) for the three-orbitalmodel with pairing interaction (a) V = 0; (b) V = 0 .
2, forthe s ± pairing operator given in the text. with intensity V = 0 . f ( k ) = cos k x cos k y .This means that the ratio between the gaps at the FSis determined by | f ( k i ) /f ( k j ) | , where i and j can takethe values 1, 2, 3 corresponding to the three bands thatdetermine the FS, i.e., 1 (2) for the interior (exterior)hole pocket, and 3 for the electron pockets. This createsa constraint on how different these gaps can be if s ± represents the actual pairing symmetry of the pnictides.As previously mentioned, we can define a pairing op-erator s IB by combining operators f ( k ) = cos k x cos k y so that the orbital part of the ba-sis is given by Aλ + Bλ , where A and B are constants.This pairing operator is diagonal in the orbital represen-tation, and transforms according to A g thus, it has S symmetry; it also has intra and interband terms in theband representation. This is why this operator is called s IB . For a robust range of values of A and B , a nodelessgap opens on all FSs for any finite value of V . For ex-ample, we can choose the parameters in such a way that5 FIG. 11: (Color online) (a) The intensity of the points repre-sents the values of the spectral function A ( k , ω ) for the three-orbital model with pairing interaction V = 0 . s IB along the indicated high symmetry directions in thefolded BZ. (b) Ratio R between the gaps for pairing s IB andpairing s ± for V = 0 .
05 in the unfolded BZ. for any given k , the pairings for the three orbitals havethe same sign. The spectral functions for V = 0 . A = 3 / B = −√ /
2. The major difference with the results forthe s ± state [see Fig. 10(b)] is that the interband pair-ing present in s IB opens gaps between the bands awayfrom the FS. This is a feature that should be observed inARPES experiments. Also the band spectral functionsin both cases are very different close to (0 , π ) and ( π, s ± and s IB . In Fig. 11(b) weshow the ratio R between the two gaps in the unfoldedBZ for V = 0 .
05. It can be seen that on the hole pockets R = 1, but an appreciable difference is observed on theelectron pockets where R = 2 at the point where the elec-tron pocket is entirely formed by xy , and diminishes asthe hybridization of xy with xz or yz becomes stronger.The maximum value of R is a function of the values of A and B in the linear combination that defines s IB . Thus, while s ± is characterized by gaps with a weak momen-tum dependence and with similar magnitudes on the holeand electron pockets, the s IB state is characterized by adifferent gap on the electron pockets with stronger mo-mentum dependence due to the hybridization.Now we focus on the spectral function for operator B g presented in Fig. 12(a). This is the pairing operator thatwas favored by numerical calculations in the intermedi-ate U regime of the two-orbital model and it only pairselectrons in orbital xz with electrons in orbital yz . Al-though neither this pairing operator nor the Fermi sur-faces defining the two hole pockets involve the xy orbital,the results around the hole pockets differ in the two- andthree-orbital models. In the latter, much lower valuesof the pairing attraction V are sufficient to remove theextra nodes found close to the hole-pocket FS along theΓ- X ( Y ) directions in the two-orbital model. This hap-pens because the bands forming the two hole pockets arenow degenerate at Γ and the pockets are consequently atalmost the same momenta of the extended BZ, while theywere separated by ( π, π ) in the two-band model. As a re-sult, a small interorbital pairing can now overcome theseparation between the two FSs and induce a full gap atthe hole pockets. At the electron pockets a third node,in addition to the two of the two-orbital model, is foundalong the Γ- X ( Y ) direction, where the pocket has purely xy character and is, thus, not affected by the operator B g . Thus, this pairing operator would show full gaps onthe hole pocket FS and nodal gaps on the electron pocketFS.Figure 12(b) shows the spectral functions for pairingoperator B ext2 g , which is a linear combination of B g withthe pairing operator V g (with crystal momentum Q ), i.e.next-nearest-neighbor interorbital pairing among elec-trons in all three orbitals is allowed. We find that nodesoccur only at the electron pockets. As V increases, nodesat the electron pockets remain only along the Γ- X ( Y )directions because one of the electron pockets is formedby a non-hybridized orbital xy along this direction, andthe relevant pairing interaction is zero for the Fermi mo-mentum. We also investigated A ( k , ω ) for a similarly ex-tended B g pairing B ext1 g = (cos k x +cos k y ) λ − a (cos k x − cos k y )( λ − √ λ ) / V , with nodes in the Γ- M direction on thehole pockets, but where the nodes are already lifted forfinite but small V & .
01 for many non-zero values of a .Summarizing, we have found that among the nearestand next-nearest neighbor pairing operators allowed bysymmetry only the s ± pairing operator is purely intra-band and produces nodeless gaps for all values of the pair-ing attraction V . Thus, purely intraband pairing interac-tions occur only if electrons in each of the three orbitalsare subjected to the same pairing attraction, i.e. whenthe identity matrix λ characterizes the orbital portionof the pairing operator. We also found that some linearcombinations of pairing operators V if f ( k ) = cos k x cos k y , butinterband attraction appears in parts of the BZ. Finally,6 FIG. 12: (Color online) The intensity of the points representthe values of the spectral function A ( k , ω ) for the three-orbitalmodel with the pairing interaction V = 0 .
2, for the pairingoperators (a) B g and (b) B ext2 g discussed in the text. the interorbital pairing operators B g and B ext2 g favoredby numerical studies in a two-orbital model, presentnodeless gaps on the hole pockets but nodes appear onthe electron pockets. V. CONCLUSIONS
In this work, a simple three-orbital Hamiltonian hasbeen constructed involving the 3 d orbitals xz , yz , and xy . These orbitals have the largest weight at the FS ofthe pnictide LaOFeAs, according to LDA calculations. Itwas shown that it is possible to qualitatively reproducethe shape of the LDA-FS by fixing the electron fillingto 4 electrons per Fe. Moreover, two features that havebeen criticized in the two-orbital model have now beencorrected: both hole pockets now arise from bands degen-erate at the Γ-point, and there is no pocket around M inthe extended BZ. In addition, the xy character of a smallpiece of the electron pockets is now properly reproduced.Numerical calculations using a small 2 × π, , π )stripes when Coulombic interactions are added, resultconsistent with experimental observations. A mean-fieldanalysis confirms this tendency for physically relevant values of J/U . As in the case of the two-orbital model,an antiferromagnetic metallic phase occurs only at in-termediate values of the Coulomb repulsion. At large U , the ground state is magnetic, but it is an insulatorthat is also orbitally ordered. Additionally, a metallic,magnetic and orbitally ordered phase is encountered justbefore the metal-insulator transition. In the most inter-esting regime with a spin-( π,
0) antiferromagnetic metalwithout pronounced orbital order, the bands are similarto the uncorrelated ones, but their bandwidth is reducedwith increasing U . The Fermi surface is also very simi-lar to the uncorrelated one but, depending on U , we findsmall additional electron-like pockets near the originalhole pockets around Γ (small U ) or hole-like pockets be-tween the electron- and hole-pockets (at slightly larger U ).The possible pairing operators that are allowed bythe symmetry of the lattice and the orbitals have beenconstructed for pairs made of electrons separated by adistance up to one diagonal lattice spacing. If on-sitepairing is disregarded due to the large Coulomb repul-sion, it was found that the only purely intraband pair-ing operator that has a full gap on the FS is f ( k ) = cos k x cos k y which corresponds to the s ± pairingoperator with a momentum dependent OP that has op-posite signs on the hole and electron FSs. This operatorarises from a purely intraband pairing attraction equalfor each of the three bands. Note that the pairing op-erator only one that leads to purely intrabandpairing interactions. Since this pairing operator is pro-portional to the identity matrix λ both in the orbitaland the band representations we found that the ratio | ∆ i / ∆ j | between the gaps in two different FSs can differonly by the ratios | f ( k i ) /f ( k j ) | ; then, any experimentalindication of a different kind of ratio would indicate somedegree of interband pairing. Thus, order parameter ra-tios predicted by several authors with calculationsbased on purely intraband pairing (they allow interbandhopping of intraband pairs) are not allowed by the sym-metry of the lattice and the orbitals. In this regard, ourcalculations seem to indicate that unrelated gaps in dif-ferent FSs can occur only in systems in which at least oneorbital (or a group of orbitals) is not strongly hybridizedwith the remaining ones.We found that all the other pairing operators, exceptfor V with the exception of pairing operator s IB . In this case, the gap on the electron pockets is ex-pected to have a stronger variation at different points inthe BZ that the gap at the hole pockets. Thus, a strongindication that s ± is the appropriate pairing symmetrywould be provided by experiments in the pnictides show-ing a nodeless gap in all FSs, relatively independent ofmomentum, and with similar values on all FSs.Summarizing, we have shown that the addition of athird orbital corrects the shortcomings pointed out in7the two-orbital model: the two hole pockets now arisefrom bands degenerate at the Γ point while the elec-tron pockets contain a small piece with xy character.However, the dependence of the magnetic phases with U for the undoped case appears to be similar for threeand two orbitals except for a magnetic, orbital ordered,metallic phase that appears in the three-orbital case.In both models it is found that the only pairing oper-ator allowed by symmetry with next or diagonal nearest-neighbor interactions which is purely intraband and pro-duces a nodeless gap is the s ± state. In addition, the onlychange observed in the interorbital B g pairing state, fa-vored by numerical simulations in the two-orbital model,is that, at the mean-field level, the addition of the xy orbital renders the gap on the hole pockets nodeless formuch smaller values of the pairing attraction. VI. ACKNOWLEDGMENTS
This research was sponsored by the National ScienceFoundation grant DMR-0706020 (M.D., A.N., A.M., andE.D.) and the Division of Materials Science and Engi-neering, Office of Basic Energy Sciences, U.S. Depart-ment of Energy (A.M. and E.D.).
Appendix A: Mean-Field Equations
In this appendix, we discuss the mean-field approachused here to study the Hamiltonian given by the ki-netic energy Eq. (4) and the onsite Coulomb interactionEq. (11). Depending on the ordering vectors q and q ,listed in Table II, for magnetic and orbital order, thereal-space unit cell contains one, two or four sites: Onefor the ferro-orbital and ferromagnetic case, four if bothordering vectors are different from each other and from(0 , n xy , m xy , n , m , p , and q in Eqs. (12) and (13) for variouscombinations of ordering momenta, see Tab. II, whichcorresponds to minimizing the total energy. This is donefor all considered phases and the one with the lowest en-ergy is taken to be the stable solution. Depending on thesize of the unit cell, one to four momenta are coupled bythe Coulomb interaction. In the following, we will pro-vide the Hamiltonians for several ordering patterns withdifferent unit cells. In all cases, the sums run over thewhole extended BZ corresponding to the one-iron unitcell. The calculations were carried out in momentumspace for up to 400 × k -points. We did not observeany pronounced dependence on the number of momenta,except for very small lattice sizes.
1. Ferromagnetic and Ferro-Orbital Order:One-Site Unit Cell
In this case q = q = (0 ,
0) and H MF ( k ) = H TB ( k ) + U X k ,µ,σ n µ d † k ,µ,σ d k ,µ,σ + (2 U ′ − J ) X k ,µ = ν,σ n ν d † k ,µ,σ d k ,µ,σ − U X k ,µ,σ σ m µ d † k ,µ,σ d k ,µ,σ (A1) − J X k ,µ = ν,σ σ m ν d † k ,µ,σ d k ,µ,σ + N C , where n µ = n xy , m µ = m xy for the xy orbital, and n µ = n ± p/ m µ = m ± q for xz and yz . The sum over k runsthrough the whole BZ, N is the number of lattice sites,and the constant C is given by C = − U X µ n µ + U/ X µ m µ (A2) − (2 U ′ − J ) X µ = ν n µ n ν + J/ X µ = ν m µ m ν .
2. Antiferromagnetic and Ferro-Orbital: Two-SiteUnit Cell
For AF order with q = ( π, π ), (0 , π ) or ( π, k and k + q are coupled by the interaction. H MF ( k ) = H TB ( k ) + U X k ,µ,σ n µ d † k ,µ,σ d k ,µ,σ + (2 U ′ − J ) X k ,µ = ν,σ n ν d † k ,µ,σ d k ,µ,σ − U X k ,µ,σ σ m µ d † k + q ,µ,σ d k ,µ,σ (A3) − J X k ,µ = ν,σ σ m ν d † k + q ,µ,σ d k ,µ,σ + N C .
Again, n µ = n xy , m µ = m xy for the xy orbital, and n µ = n ± p/ m µ = m ± q for xz and yz ; and the sameconstant Eq. (A2) as above. The case of ferromagneticorder and alternating orbitals is treated in an analogousmanner.
3. Antiferromagnetic and Alternating OrbitalOrder with the Same Ordering Vector: Two-SiteUnit Cell
In some phases, both the orbital and the magnetic or-der alternate with the same ordering vector q = q =8 q = ( π, π ), (0 , π ) or ( π, H MF ( k ) = H TB ( k )+ [(4 U ′ − J ) n + U n xy ] X k ,σ d † k ,xy,σ d k ,xy,σ + [ U n + (2 U ′ − J )( n xy + n )] X k ,σµ = xz,yz d † k ,µ,σ d k ,µ,σ − ( U + J ) q X k ,σµ = xz,yz σα d † k ,µ,σ d k ,µ,σ (A4) − [( U + J ) m + Jm xy ] X k ,σµ = xz,yz σ d † k + q ,µ,σ d k ,µ,σ − ( U m xy + 2 Jm ) X k ,σ σ d † k + q ,xy,σ d k ,xy,σ + ( U − U ′ − J ) p X k ,σµ = xz,yz α d † k + q ,µ,σ d k ,µ,σ + N C.
Here, α = ± xz and yz orbitalflavors as σ does for the spin. The constant C reads C = − U ( n xy − m xy / − U ( n + p / − m / − q / − (8 U ′ − J ) n xy n − (4 U ′ − J )( n − p / Jm xy m + J ( m − q ) / . (A5)
4. Antiferromagnetic and Alternating Orbitals withDifferent Ordering Momenta: Four-Site Unit Cell
If both orbital occupation and magnetic order alter-nate with different ordering momenta, so that q i =( π, π ), (0 , π ) or ( π,
0) with q = q , the real-space unitcell contains four sites, and consequently all four mo-menta k , k + q , k + q , k + q + q are coupled, butapart from this, the Hamiltonian is very similar to theprevious case: H MF ( k ) = H TB ( k ) (A6)+ [(4 U ′ − J ) n + U n xy ] X k ,σ d † k ,xy,σ d k ,xy,σ + [ U n + (2 U ′ − J )( n xy + n )] X k ,σµ = xz,yz d † k ,µ,σ d k ,µ,σ − [( U + J ) m + Jm xy ] X k ,σµ = xz,yz σ d † k + q ,µ,σ d k ,µ,σ − ( U m xy + 2 Jm ) X k ,σ σ d † k + q ,xy,σ d k ,xy,σ + ( U − U ′ − J ) p X k ,σµ = xz,yz α d † k + q ,µ,σ d k ,µ,σ − ( U − J ) q X k ,σµ = xz,yz σα d † k + q + q ,µ,σ d k ,µ,σ + N C.
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