On interband pairing in multiorbital systems
Adriana Moreo, Maria Daghofer, Andrew Nicholson, Elbio Dagotto
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p On Interband Pairing in Multiorbital Systems
Adriana Moreo, Maria Daghofer, Andrew Nicholson, and Elbio Dagotto
Department of Physics and Astronomy,University of Tennessee, Knoxville, TN 37966-1200 andOak Ridge National Laboratory,Oak Ridge, TN 37831-6032,USA, (Dated: June 3, 2018)The discovery of high- T c superconductivity in the pnictides, materials with a Fermi surface deter-mined by several bands, highlights the need to understand how superconductivity arises in multibandsystems. In this effort, using symmetry considerations and mean-field approximations, we discusshow strong hybridization among orbitals may lead to both intra and interband pairing, and wepresent calculations of the spectral functions to guide the experimental search for this kind of state. PACS numbers: 74.20.-z, 74.20.De, 74.20.Rp
I. INTRODUCTION
Iron-based high- T c superconductors have acomplex Fermi surface that is determined by severalbands, an effect resulting from the hybridization of the3 d orbitals of iron. Band structure calcula-tions have shown that the bands that define the twohole pockets around the Γ point have mostly d xz and d yz character, while the two electron pockets around theM point have d xz , d yz , and a smaller amount of d xy contributions. For this reason it is importantto understand superconductivity in multiorbital systemsin general terms.Among the first to address this complex problem sev-eral years ago were Suhl et al. using a model consist-ing of two orbitals, one s and one d , that did not hy-bridize with each other. Thus, in this case each bandwas determined by one single orbital. They showedthat BCS pairing could occur in each band and, sincein the most general case the electron-phonon interac-tion would have different strengths for electrons in thedifferent bands, it was proposed that two different su-perconducting gaps could arise. Almost 50 years wereneeded to observe experimental evidence of this phe-nomenon. In 2001, superconductivity with T c = 39 Kwas observed in MgB . Despite the high- T c , it becameclear that the BCS mechanism was at play and forthe first time two different superconducting gaps wereobserved. As shown in Ref. 18, the Fermisurface (FS) is determined by two bands: the π bandformed by the p z orbitals of B, and the σ band consti-tuted by a linear combination of the p x and p y B-orbitals.Although three orbitals determine the FS, it is interestingto notice that only two different BCS gaps are observed.This occurs because two of the three orbitals hybridizewith each other and determine one single band, whichcouples strongly to the lattice phonons. This opens alarge superconducting gap on the σ FS. The other or-bital, p z , does not hybridize and forms the π band thatcouples weakly to the lattice phonons determining a sec-ond, smaller, superconducting gap at the FS of the π band. Thus, the number of different gaps that can arisein a multiorbital system is related to the degree of hy-bridization among the orbitals. Also note that in this early effort interband hopping of pairs of electrons be-longing to the same band was included but the possibilityof interband pairing, i.e., pairs formed by electrons be-longing to two different bands , was not considered.In this paper, the subject of superconductivity in mul-tiorbital systems is revisited, in particular to shed lighton the possible symmetry of the pairing operator of thepnictides superconductors. The motivation is that thepairing operators that have been discussed the most thusfar assume that only intra-band pairing should occur, namely the two electronsof the Cooper pair belong to the same band. How-ever, numerical simulations performed on a two-orbitalmodel for the pnictides favor an interorbital pair-ing operator that, when transformed to the band rep-resentation, results not only in intraband pairing butit includes interband pairing as well. For the pnic-tides, interband hopping of pairs formed between elec-trons in the same band is often denoted as “Inter-band Superconductivity”. The situation discussed inthe present paper is different and involves Cooper pairswhere the two electrons come from two different bands,which we will call “Interband pairing”. Using symmetryarguments and mean-field approximations, the plausibil-ity and physical meaning of such an interband pairing inmultiorbital systems will be discussed.Interband pairing has previously been addressed inthe context of Quantum Chromodynamics (QCD) andcold atoms, heavy fermions, cuprates , and BCSsuperconductivity. In the case of heavy fermions, it wasargued that interband pairing could occur if two Fermisurfaces arising from different bands are very close toone other, while in QCD and cold atoms it was pre-sented as a possibility for the case of sufficiently strongattractive pairing interactions, or for weaker attractionsamong particles with very different masses. As it willbe discussed for a simple model in Sec. III, three dif-ferent regimes, shown schematically in Fig. 1, can re-sult from a purely interband pairing as a function of thestrength of the pairing potential g : (1) a normal regimewhere the ground state is not superconducting (namely inpurely interband pairing an infinitesimal attraction doesnot lead to superconductivity), (2) an exotic supercon-ducting “breached” regime where gaps open at the nor-mal Fermi surfaces while new Fermi surfaces defining re-gions containing unpaired electrons are created, and (3) asuperconducting regime resembling BCS states, at largeattractive coupling. gBCSbreachednormal FIG. 1: Schematic representation of the three regimes thatcan arise as a function of the strength g of an interbandpairing attraction. The label “normal” denotes a non-superconducting state. The case “breached” is an exoticregime with superconductivity and gaps, coexisting withFermi surfaces (or several nodes) and electrons that do notpair. “BCS” is the large attraction region, where the groundstate resembles that of a BCS superconductor and all elec-trons participate in the pairing. The paper is organized as follows. In Sec. II, the gen-eral form of pairing operators in multiorbital systems willbe presented, remarking how the symmetry is determinedby the spatial and the orbital characteristics of the op-erator. The interorbital pairing operator with B g sym-metry obtained numerically in a two-orbital model forthe pnictides is discussed, emphasizing that this oper-ator presents a mixture of intra and interband pairingin the band representation. In Sec. III, a simple toymodel with pure interband pairing attraction is intro-duced. This simplified model is discussed in order toillustrate the effects of interband pairing on observables,such as the occupation number and the spectral func-tions A ( k , ω ). The stability of the interband paired stateis also discussed. The occupation number, spectral func-tions, and the stability of the B g pairing state are thesubject of Sec. IV which is directly related to the physicsof pnictides, while Sec. V is devoted to the conclusions. II. PAIRING OPERATORS IN MULTIORBITALMODELS
In single-orbital models, the symmetry of a spin-singletpairing state is completely determined by the propertiesof its spatial form factor. More specifically, the pairingoperator will have the form:∆( k ) = f ( k )( c k , ↑ c − k , ↓ − c k , ↓ c − k , ↑ ) , (1)where c k ,σ destroys an electron with momentum k andspin projection σ and f ( k ) is the form factor that trans-forms according to one of the irreducible representationsof the crystal’s symmetry group. Thus, f ( k ) determinesthe symmetry of the operator. These form factors dependon the lattice geometry and generally they may be very complex. However, in materials with short pair coherencelengths, such as the high- T c cuprates, the assumptionthat the two particles that form the pair can be very closeto one other is usually made. The Cu-oxide planes in thecuprates have the symmetry properties of the D h groupand the case f ( k ) = cos k x − cos k y , which transformsaccording to the irreducible representation B g , providesthe well-known d -wave symmetry pairing.In multiorbital systems, on the other hand, a spin sin-glet pairing operator will have both spatial and orbitaldegrees of freedom and it will be given by∆( k ) = f ( k ) τ α,β ( d k ,α, ↑ d − k ,β, ↓ − d k ,α, ↓ d − k ,β, ↑ ) , (2)where d k ,α,σ destroys an electron with momentum k , inorbital α , and with spin projection σ , f ( k ) is the spatialform factor as indicated above, and τ α,β is a matrix in thespace spanned by the orbitals involved. The dimensionof τ is equal to the number of orbitals that are consideredto be of relevance. In this case, notice that the symmetryof the pairing operator would in general be determinedby the product of the symmetry properties of f ( k ) andthe symmetry of the orbital contribution τ α,β . Only if τ α,β is the identity matrix do the orbital contributionbecome trivial, because the identity matrix transformsaccording to A g . Thus, this is the only case where f ( k )fully determines the symmetry of the pairing operator,as in the single-orbital example.The minimum model for the pnictides considersthe two orbitals d xz and d yz , which are stronglyhybridized. All the possible pairing operators, upto nearest-neighbor distance, that are allowed bythe lattice and orbital symmetries have been al-ready calculated.
Numerical simulationsperformed on the two-orbital model suggest that the fa-vored pairing operator at intermediate couplings, wherethe state is both magnetic and metallic, has symme-try B g and is given by Eq. (2) with f ( k ) = (cos k x +cos k y ), which transforms according to A g , and τ = σ which transforms according to B g (where σ i are Paulimatrices). Thus, the non-trivial symmetry under rota-tions arises from the orbital portion of the operator. Thispairing operator has been studied at the mean-field levelin Ref. 40. In the orbital representation, the Bogoliubov-de Gennes Hamiltonian matrix is given by: H MF = ξ xx ξ xy k ξ xy ξ yy ∆ k
00 ∆ k − ξ xx − ξ xy ∆ k − ξ xy − ξ yy , (3)with ξ xx = − t cos k x − t cos k y − t cos k x cos k y − µ,ξ yy = − t cos k x − t cos k y − t cos k x cos k y − µ,ξ xy = − t sin k x sin k y , (4)and ∆ k = V (cos k x + cos k y ) , (5)where V = V ∆, with V being the strength of the pair-ing interaction and ∆ the mean-field parameter obtainedby minimizing the energy. Since the two orbitals are hy-bridized via ξ xy , in the band representation the Hamil-tonian matrix becomes: H ′ MF = ǫ V B V A ǫ − V A V B V B − V A − ǫ V A V B − ǫ , (6)where V A and V B are given by V A = 2 u ( k ) v ( k )∆ k , (7) V B = ( v ( k ) − u ( k ) )∆ k , (8)and u ( k ) and v ( k ) are the elements of the change of basismatrix U given by U = u ( k ) v ( k ) 0 0 v ( k ) − u ( k ) 0 00 0 v ( k ) u ( k )0 0 − u ( k ) v ( k ) , (9)with U − = U T . Remember that V A and V B , are func-tions of the momentum k , and u ( k ) + v ( k ) = 1. Thus,it is clear that in the band representation, in additionto the intraband pairing given by V A , there is also in-terband pairing given by V B . Among pairing operatorscompatible with the symmetry of the model, the onesthat do not lead to interband pairing not only cannotmix orbitals, but have to contain τ = σ , i.e., the iden-tity matrix. One example for such an operator wouldbe the s ± pairing. The discussion above describes general properties ofhybridized multi-orbital systems. If the orbitals are hy-bridized, but not related to one another by symmetry,there is no reason to expect that the coupling betweenthe electrons in each orbital, and the interaction that pro-duces the pairing, will have the same strength for all theorbitals and lead to a unit matrix in the orbital sector.Then, it is expected that interband pairing will arise ingeneral and, thus, it is important to understand its con-sequences, providing the main motivation for the presentmanuscript.
III. INTERBAND PAIRINGA. Generic properties
1. Model and non-interacting limit
To address qualitatively the issue of interband pairing,postponing the matters of stability to the next subsec-tion (Sec. III B), let us consider the following two-bands simplified model with interband pairing: H k = X α,σ ǫ α ( k ) c † k ,α,σ c k ,α,σ + V X α = β ( c † k ,α, ↑ c †− k ,β, ↓ + h.c. ) , (10)where α, β, = 1 , σ is the spin projection, and for simplicity ǫ α ( k ) = − k m α + C, (11)which gives parabolic bands that are degenerate at k = 0with energy C , and with a chemical potential µ = 0.This can be considered as a crude representation of thetwo hole-pocket bands around the Γ point in the pnic-tides, but more importantly presents a simple toy modelwhere the effects of interband pairing can be studied. Asbefore, the parameter V = V ∆ is the product of an at-tractive potential V between electrons in the two differ-ent bands and a mean-field parameter ∆ determined byminimizing the total energy. The band dispersion with-out the interaction is presented in the inset of Fig. 2.The Bogoliubov-de Gennes matrix expressed in the basis k -4-2024 E V=0.5V=0 ε ε π ∆ ∆ E A E B -E B -E A FIG. 2: (Color online) Mean-field band dispersion for themodel defined by Eq. (10), for the indicated values of V (defined in the text) as a function of the momentum k = p k x + k y . The case shown is for m = 1, m = 2, and C = 2. Inset: Non-interacting band dispersion for the sameparameters. expanded by B = { c † k , , ↑ , c − k , , ↓ , c † k , , ↑ , c − k , , ↓ } has theform: H = ǫ V V − ǫ ǫ V V − ǫ . (12)This matrix can be diagonalized becoming H D = E A − E B E B
00 0 0 − E A , (13)in the basis expanded by B AB = { γ † k , A , ↑ , γ − k , B , ↓ , γ † k , B , ↑ , γ − k , A , ↓ } and the change-of-basis matrix is given by U = u k v k − v k u k u k v k − v k u k , (14)with u k + v k = 1.The four energy eigenvalues are E i ( k ) = ± " ( ǫ − ǫ )2 ± r ( ǫ + ǫ + V , (15)where the first positive (negative) sign corresponds tothe eigenvalues of the upper (lower) block, labeled E A and − E B ( E B and − E A ) in Eq. (13) and in Fig. 2. Thesecond sign differentiates between the two solutions ineach block. Figure 2 shows the eigenvalues for V = 0 and V = 0 .
5. When V = 0, then E A = ǫ and E B = ǫ , thetwo bands define two circular Fermi surfaces with radius k F and k F where they cross the chemical potential ( µ =0). This is illustrated schematically in Fig. 3. k’kk’ F2F2 F1 k F1 ∆∆ kk xy pairedelectronsunpairedelectrons 1 FIG. 3: (Color online) Schematic diagram of the FS deter-mined by the two parabolic bands of the simple model usedin Sec. III. k F and k F indicate the Fermi momentum of thetwo bands when V = 0, while k ′ F and k ′ F indicate the posi-tion of the Fermi momenta for the case of a finite but smallpairing potential. The shaded rings indicate the regions withwidth ∆ and ∆ in momentum space where electrons canpair. The white region in between the rings contains unpairedelectrons in band 1. Notice that number operators can be defined in the twobases that are being considered here, i.e. B and B AB ,which of course become equivalent when V =0. Thus, inthe basis B the number operator is, n α ( k ) = X σ c † k ,α,σ c k ,α,σ , (16) while in basis B AB the number operator is ( I =A,B) n I ( k ) = X σ γ † k , I ,σ γ k , I ,σ . (17)The total electronic occupation of the system is given by n ( k ) = X α n α ( k ) = X I n I ( k ) . (18)Then, for V = 0 we find that n ( k ) = 0 for | k | ≤ | k F | , n ( k ) = 2 for | k F | < | k | < | k F | since in this region n ( k ) = n A ( k ) = 2 while n ( k ) = n B ( k ) = 0, and finally n ( k ) = 4 for | k | > | k F | since both orbitals are totallyfilled with electrons. These results are represented by thedashed lines in Fig. 4. n ( k ) V=0V=0.5 n B ( k ) n A ( k ) n ( k ) k n ( k ) (a)(b)(c)(d)(e) k F1 k F2 k F1 k F1 k F2 k F2 k’ F2 k’ F2 k’ F2 k’ F2 k’ F1 k’ F1 k’ F1 k’ F1 FIG. 4: (Color online) Mean-field state population as a func-tion of momentum along the diagonal k x = k y for (a) thewhole system; (b) band B, (c) band A; (d) orbital 1; and (e)orbital 2 for the indicated values of the pairing potential V ,and m = 1, m = 2, and C = 2.
2. Weak attraction
In the nontrivial case of V different from zero, thebands ± E A and ± E B result from the hybridization ofthe ǫ and ǫ bands due to V . It is interesting to ob-serve that an internal gap opens at the crossing of bands E A with − E B above the chemical potential and between E B and − E A below the chemical potential, as indicatedwith circles in Fig. 2 where results for V = 0 . ± E A bands are separated by a gap, thebands ± E B still cross the chemical potential determin-ing two Fermi surfaces at k ′ F > k F and at k ′ F < k F ,even in this pairing state (again, this is different from theone-orbital pairing standard BCS ideas). The new Fermisurfaces are shown schematically in Fig. 3: the interiorFS has expanded and the exterior one has contracted. Infact, it will be shown below that the effect of V is to tryto equalize the two Fermi surfaces, as this pairing attrac-tion grows in magnitude. Calculating n ( k ) we observethat n ( k ) = 4 v k for k < k ′ F and k > k ′ F , where v k is theelement of the change-of-basis matrix in Eq. (14). Thisagrees with the BCS expression for n ( k ) but it jumpsdiscontinuously to n ( k ) = 2 for k ′ F < k < k ′ F . Such aresult is shown by the solid lines in Fig. 4(a). The jumpsindicate the existence of the two Fermi surfaces, whichare here present even in the paired state. Thus, someelectrons in the region in between the two Fermi surfacesmay behave like normal unpaired electrons.A better understanding of the electronic behavior canbe achieved by studying the electronic density in the twobases B AB and B . We find that n A ( k ) = 2 v k , as shownin Fig. 4(c), which is the standard BCS behavior in agree-ment with the fact that bands ± E A are separated by agap. Thus, all the electrons in this band participate in thepairing and they do not have a FS. On the other hand, itcan be shown that n B ( k ) = n A ( k ) = 2 v k for k < k ′ F and k > k ′ F , but for k ′ F < k < k ′ F there is a discontinuouschange of behavior to n B ( k ) = u k = 1 − v k [Fig. 4(b)].Thus, the Fermi surfaces are determined by electrons inthe band E B . However, note that in this region n A ( k ) isan increasing function of | k | while n B ( k ) is a decreasingfunction of | k | which satisfies n A ( k ) + n B ( k ) = 2 for all | k | .This behavior can be better understood by calculat-ing the (photoemission) spectral functions A ( k , ω ), whichallow us to obtain n ( k ) = R µ =0 −∞ A ( k , ω ) dω . In the non-interacting case, shown in Fig. 5(a), the spectral functionshows two peaks, corresponding to the two bands ǫ and ǫ , for each value of | k | . This is the expectation for freeelectrons in a non-interacting multiorbital system. Thetwo non-interacting FS’s are located where each bandpasses across the chemical potential.When the pairing interaction becomes finite, the spec-tral function develops four peaks for each value of | k | asit can be observed in Fig. 5(b) for V = 0 .
5. This is alsothe expected result since the BCS interaction generatesa “shadow” or Bogoliubov band for each band present inthe non-interacting system. For example, close to | k | = 0the bands E A and E B have almost all the spectral weight,given by u k which is very close to 1 in this region, and fol-low a dispersion similar to the non-interacting bands ǫ and ǫ , while the bands − E A and − E B appear with verysmall spectral weight given by v k = 1 − u k . The latterare the Bogoliubov or “shadow” bands. These shadowbands appear above the chemical potential for large val-ues of | k | , as expected.But what happens in the intermediate region k ′ F 0. The presence of a gap for all the momenta shown isnow clear. has appreciable shadow spectral weight indicating thatthere are also some paired electronic population as in-dicated in Figs. 4(b) and (c). As k increases, spectralweight is transferred continuously from − E B to − E A ,behavior associated with the internal gap opened by thepairing interaction, so that when k approaches k ′ F mostof the spectral weight below the chemical potential is in − E A , paired electrons, and − E B , unpaired electrons, hasshadow spectral weight as seen in Figs. 4(b) and (c). At k = k ′ F the second FS is determined by the crossing of ± E B at the chemical potential indicated by the suddenjump in n B . Thus, the unpaired electrons in this regioncoexist with paired electrons and the spectral functionsdo not resemble the non-interacting ones.It is also illuminating to analyze what happens with n ( k ) in the basis B . While n ( k ) = n ( k ) = 2 v k in theregions where there are no unpaired electrons, we finddiscontinuities associated with Fermi surfaces in the twodistributions that are given by n ( k ) = 0 and n ( k ) = 2for k ′ F < k < k ′ F [Figs. 4(d,e)]. This indicates thatthe pairing interaction V has been able to promote someelectrons from above to below k F in orbital 2. Also,electrons have been transferred from their original loca-tion in orbital 1, in the neighborhood of k F and above k F , to both orbitals 1 and 2 around k F . These arethe electrons in 1 and 2 that have become paired (thepairing is indicated by the shadowed circular regions inFig. 3). But the interaction was not strong enough toprovide pairing partners to all the extra electrons origi-nally in orbital 1 and, thus, they have been left unpairedin between the two paired regions, as indicated in Fig. 3.Thus, the interband pairing attraction creates pairs ofelectrons belonging to different orbitals within an interval∆ k Fi around each of the two original Fermi surfaces. Thewidth of the pairing region increases with V . The pair-ing partners are obtained by promoting electrons withmomentum k ≈ k F and k ≈ k F in both orbitals andby moving electrons from the more populated to the lesspopulated orbital. This creates the conditions to pairelectrons near both Fermi surfaces. The electrons in band1 that could not find promoted partners remain unpaired.Whether this state is stable or not depends, of course, onthe balance between kinetic and pairing energies, whichwill be discussed in Sec. III B. 3. Strong attraction As the interaction V increases further, the numberof unpaired electrons is reduced. This means that k ′ F and k ′ F become closer to each other making the size ofthe intermediate region with unpaired electrons in Fig. 3smaller. Eventually, the two momenta become the same k ′ F = k ′ F for V = 0 . 71, and for V > . 71 the regionwith unpaired electrons vanishes and a full gap opens inthe system whose physics now resembles BCS, except forthe fact that the pairs are constituted by electrons fromdifferent orbitals. The electronic population of the sys-tem in such a case, e.g. at V = 1 . 0, is presented in Fig. 6and the corresponding spectral functions are shown inFig. 5(c). It can be shown that now n ( k ) = n ( k ) forall values of k and all the particles around the two non-interacting Fermi surfaces now participate in the pairing.The spectral functions show four peaks, i.e. Bogoliubovbands for all values of k , as it can be observed in Fig. 5(c). n ( k ) V=0V=1 n ( k ) k n ( k ) (a)(b)(c) k F1 k F2 k F1 k F2 π FIG. 6: (Color online) Mean-field state population as a func-tion of the momentum (main diagonal) for (a) the whole sys-tem, (b) band 1, and (c) band 2 for the indicated values ofthe pairing potential V , and for m = 1, m = 2, and C = 2.The case V = 1 . Thus, notice that while band 1 contained more elec-trons than band 2 for V = 0 [see Figs. 6(b) and (c)],the interband pairing mechanism transfers electrons fromone band to the other so that both bands have the samenumber of electrons in the superconducting state. Con-sequently, the smaller FS expands and the larger oneshrinks. Then, when the pairing becomes strong enough,the two Fermi surfaces become equalized and no unpairedelectrons remain. Whether this situation can be achievedwill depend on the strength of the interaction and the en-ergy balance, as discussed in the next subsection.If the two non-interacting Fermi surfaces are very closeto each other in momentum space, even relatively weakpairing interaction could effectively be strong enough tomake the interband pairing resemble BCS pairing as inthe case of large V in the present example. B. Stability of the interband paired state 1. The case without intraband pairing As discussed in the Introduction, the possibility of in-terband pairing has been previously discussed in the con-text of QCD and cold atomic matter. Similar effects onthe FS as found in the present study were described, al-though the physics was different because in the QCDcontext each band contained different kinds of particlesand the pairing was thus not able to promote particlesfrom the majority to the minority band. The issueof stability was explored in the QCD framework, and itwas found that a purely interband paired state could bestabilized for pairing attractions above a certain cut-offvalue, which could become very small for a large differ-ence between the masses of the two paired species. In the case of our model, however, we have found (seebelow) that the purely interband-paired state only be-comes stable when the attraction is sufficiently strongthat no unpaired particles are left, i.e. when the twoshaded regions overlap and the unpaired region in Fig. 3vanishes. This means that, although the pairs wouldbe formed by electrons in different orbitals, the physicswould be analogous to BCS. A gap will be opened in thefull Fermi surface of the simple model studied here.In order to study the issue of stability, let us assumethat the interaction term responsible for the interbandattraction is given by H attr = 1 N X k , k ′ ,α V k , k ′ c † k ,α, ↑ c †− k , − α, ↓ c − k ′ , − α, ↓ c k ′ ,α, ↑ , (19)where V k , k ′ = − V and N is the number of sites. Per-forming the standard mean-field approximation: b k ′ = h c − k ′ , − α, ↓ c k ′ ,α, ↑ i and b † k = h c † k ,α, ↑ c †− k , − α, ↓ i and makingthe substitution c † k ,α, ↑ c †− k , − α, ↓ = b † k + ( c † k ,α, ↑ c †− k , − α, ↓ − b † k ) (and an analogous substitution for the product of an-nihilation operators), the mean-field results are obtained.As usual, the fluctuations around the average given by( c † k ,α, ↑ c †− k , − α, ↓ − b † k ) are assumed to be small. Defining∆ = N P k b k = N P k b † k we obtain the following mean-field Hamiltonian: H MF = X α,σ ǫ α ( k ) c † k ,α,σ c k ,α,σ − V ∆ X k ,α = β ( c † k ,α, ↑ c †− k ,β, ↓ + h.c. ) + 2 V ∆ N. (20)Equation (12) can be recovered by defining − V ∆ = V ,and disregarding the constant last term of Eq. (20). Wecan calculate the total energy E MF for Eq. (20) as a func-tion of V = V ∆. If for a given V = 0 the energy has aminimum, this indicates that the interband-paired stateis stable. Note that having a term linear in ∆ in Eq.(20)is not sufficient to conclude the appearance of a supercon-ducting state at small ∆, since the sign of the coefficientof the linear term can change sign with V . A similarsituation occurs for the magnetic state of undoped pnic-tides: a finite Hubbard U must be reached to stabilizethe “striped” state . Returning to superconductivity,there are two regions of interest: (i) 0 < V < . 71, whichcorresponds to the case in which two Fermi surfaces arepresent in the paired state; (ii) V > . 71, which corre-sponds to the fully gapped case. In Fig. 7(a), E MF /N vs. V is shown for different values of V . It can be observedthat a second minimum develops for V > V > 3. The minimum always occurs for V > . 71 which means that it corresponds to the case inwhich there are no unpaired electrons in the system. Theresults shown in the figure are robust in the sense thatchanges in the values of m , m or the chemical poten-tial were not found to stabilize the state with unpaired -4-3 E m f V =2V =3V =3.5V =4V =5 (a) V=V ∆ E m f V =2.5V =3V =3.5V =4 (b) FIG. 7: (Color online)(a) Mean-field energy Eq. (20) per sitevs. V = V ∆ for different values of V . The case V = 0 . V / V = V ∆, for different values of V . electrons. Thus, in this respect an attraction that is onlyinterband can only lead to a stable superconducting statein the strong attraction region. 2. Stability when both inter and intraband pairing coexist The results of the previous paragraphs may seem neg-ative with respect to the relevance of the “intermediate”state with simultaneous coexistence of pairing and Fermisurfaces. However, as pointed out in Sec. II, most of thepairing operators allowed by the lattice and orbital sym-metry in the pnictides are characterized by a mixture ofboth intra and interband pairing. Thus, it is importantto consider such a situation in our simple model as well.In the case of the B g pairing operator, Eqs. (6), (7), and(8) indicate that the pairing is purely interband only for k x = 0 or π and k y = 0 or π , because V A = 0 alongthese lines. Thus, let us now consider our simple modelin a Brillouin zone (BZ) defined by − π < k x , k y , ≤ π and with the addition of intraband pairing with intensity V / k x = 0 or π and k y = 0 or π , i.e., the pairing is purely interband only along thosedirections. The energy bands now behave as in Fig. 2only along (0 , − ( π, 0) and (0 , − (0 , π ), while the dis-persion along any other direction is shown in Fig. 8 for V = 0 and V = 0 . k x =k y -505 E V=0V=0.5 π FIG. 8: (Color online) Mean-field band dispersion for themodel given by Eq. (10) (along the main diagonal) for theindicated values of the pairing potential V , with the additionof an intraband pairing with strength V / 2, as described inthe text. Performing the mean-field approximation similarly asexplained above, we have found that the superconduct-ing state now becomes stable on both sides of the originalcritical value V = 0 . 71. Figure 7(b) shows that the pair-ing state is stabilized for V > . V < . x and y axes. Increasing the value of V , the minimum eventuallyoccurs for V > . 71. For these larger values of V , therewould consequently be no nodes.Then, in this section it has been shown using a simplemodel that the interorbital paired state can become sta-ble if the attraction V is sufficiently strong. In this case,it is the nodeless case that is stable even for purely inter-band attraction. In addition, the very interesting novelphase with coexisting nodes and unpaired electrons inthe majority band also requires intraband pairing to bestable, with strength similar to that of the interband, atleast in parts of the BZ. IV. THE INTERORBITAL B g PAIRINGOPERATOR In the previous section, a simple model was presented,both with exclusively interband pairing and with bothinter and intraband pairing, and it was found that theintraband pairing stabilizes the state with a mixture ofsuperconductivity and metallicity. As mentioned in theIntroduction, it is expected that in the most general cases FIG. 9: (Color online) (a) One-particle spectral function forthe two-orbital model Eq. (3), with vanishing pairing interac-tion V = 0. Parameters: t = 1 . t = − t = t = − . µ = 1 . 54. (b) Same as (a) but for V = 0 . the pairing operators allowed by the symmetry of the lat-tice and of the orbitals will, in the band representation,have both intra and interorbital pairing. Thus, now wewill present and discuss, at the mean field level, the oc-cupation number and the spectral functions for the pair-ing operator obtained from the numerical study of thetwo-orbital model for the superconducting state of thepnictides introduced in Sec. II. A. Non-interacting limit Let us start with the non-interacting case in which V = 0. The spectral functions along high-symmetry di-rections in momentum space are presented in Fig. 9(a)and, as expected, they reproduce the non-interactingband dispersion. We also show the total occupationnumber n ( k ) along the same directions [dashed lines inFig. 10(a)], as well as the occupation number for each or-bital n x ( k ) [dashed lines in panel (b)] and n y ( k ) [dashedlines in panel (c)] and the orbital occupation in thequadrant of the first BZ defined by 0 ≤ k x , k y ≤ π inFig. 11(a). It is clear that the electron-like and hole-likeFermi surfaces are determined by an admixture of the π ) ( π ,0) ( π , π ) (0,0) ( π ,0) k n ( k ) (e) n ( k ) (d) n y ( k ) (c) n x ( k ) (b) n ( k ) (a) FIG. 10: (Color online) (a) Total occupation number n ( k ) forthe two-orbital model Eq. (3) with pairing interaction V = 0 . V = 0 (dashed lines). Parameters: t = 1 . t = − t = t = − . µ = 1 . 54. (b) Same as(a) but for the orbital d xz . (c) Same as (a) but for the orbital d yz . (d) Same as (a) but for band 1. (e) Same as (a) but forband 2. two orbitals.On the other hand, in the band representation, band1 determines the electron pockets while band 2 formsthe hole pockets as it can be seen from the behavior of n ( k ) and n ( k ) (indicated by the dashed lines in panels(d) and (e) of Fig. 10) and by the light (orange) anddark (red) surfaces in Fig. 11(b) where the FS is alsoindicated. It is clear that the electronic occupation ofband 1 is smaller than the electronic population of band2, so that unpaired electrons would be expected to belongpredominantly to band 1, as in the simple model of theprevious Sec. III. It is interesting to notice that in theorbital representation, on the other hand, the electronsare equally distributed among the xz and yz d orbitals. B. Nonzero pairing Let us discuss what occurs when the pairing interac-tion becomes nonzero. To simplify the discussion, definethe following points in momentum space: X = ( π, Y = (0 , π ), Γ = (0 , M = ( π, π ). As remarkedin Ref. 40, the B g pairing operator always has nodesalong the X − Y direction because the spatial form fac-tor f ( k ) = cos k x + cos k y vanishes along that line. But,as soon as V is finite, a gap opens along the Γ − M direc-tion [notice that along this direction the pairing is purelyintraband since in Eq. (6), v = u and thus V B = 0 in FIG. 11: (Color online) (a) Occupation number n ( k ) for thetwo-orbital model Eq. (3) without pairing interaction for theorbital d xz (orange/light) and the orbital d yz (red/dark). The‘floor’ indicates the FS in red. (b) Same as (a) but for band1 (orange/light) and band 2 (red/dark) Eq. (8)]. Along Γ − X , Γ − Y , X − M , and Y − M nodesassociated to the different number of electrons in band 1and band 2 remain [notice that along these directions thepairing is purely interband since V A = 0 because Eq. (3)and Eq. (6) become identical to one other]. When thepairing interaction V becomes strong enough to make n ( k ) = n ( k ), as described in the simplified model pre-sented in the Sec. III A 3, these nodes vanish. Along anyother direction in the BZ a mixture of intra and interor-bital pairing will be present. Let us first consider a relatively small pairing V = 0 . n ( k ) is presented along high symmetry di-rections. Along Y − X there is no pairing and, thus, n ( k )is unchanged from the non-interacting case shown in thefigure with dashed lines. Along X − M , where only inter-band pairing occurs, no effects are observed at the elec-tron pocket FS but a rounding in n ( k ) indicating pairingis observed at the hole pocket FS. However, n ( k ) showsdiscontinuities at two points indicating the existence ofnodes. Along the diagonal direction M − Γ, where allthe pairing is intraband, it is found that n ( k ) exhibitsstandard BCS behavior at both hole Fermi surfaces indi-cating the opening of gaps. Finally, along Γ − X it can be0observed a rounding of n ( k ) at the hole Fermi surfaces,indicating pairing, and a sharp jump at the electron FS.We can further analyze the pairing in the orbital rep-resentation. The occupation number for the orbitals xz and yz is shown in Figs. 10(b-c). Along Y − X , wherethe pairing is zero, we observe how the FS for the elec-tron pocket at Y ( X ) is totally determined by electronsin the orbital xz ( yz ) and how the population of eachorbital varies smoothly between the two Fermi surfaces,always satisfying n x ( k ) + n y ( k ) = 2. From X to M , n x ( k ) = n y ( k ) for k F h < k < M indicating that the elec-trons at the hole pocket FS are paired in a wide regionaround it, but the pairing region is very narrow aroundthe electron pocket because the pairing is reduced bythe small value of f ( k ). Along the diagonal, i.e. from M to Γ, standard intraband pairing occurs at both holeFermi surfaces and, thus, n x ( k ) = n y ( k ) while from Γto X n x ( k ) = n y ( k ) for Γ < k < k F h indicating pairingaround the hole FS and almost no pairing occurs at theelectron pocket FS. The behavior n x ( k ) = n y ( k ) = n ( k )2 for all k for which pairing occurs and n i ( k ) unchangedfrom the non-interacting value for all other k , was ob-served for all values of V . For this reason, figures for n i ( k ) in the orbital representation will not be shown forthe additional values of V discussed below.The population of the different bands is presented inFig.10 (d-e). The Fermi surfaces along Y − X are clearlydetermined only by band 1, which is the band that formsthe electron pockets. It can also be observed that there isalmost negligible pairing at the electron FS along X − M ,but there is clear interband pairing at the hole FS. Thisis an indication that, due to the spatial variation of thepairing interaction, the attraction is much stronger at thehole pockets than at the electrons pockets. Also noticethat at the hole pocket FS the pairing is interband andthus n ( k ) = n ( k ), but this does not happen along thediagonal direction M − Γ where intraband pairing occurs,and there are more paired electrons belonging to band 2than to band 1. Along the direction X − Γ again weobserved a stronger pairing effect at the hole FS than atthe electron one. C. Spectral functions A ( k , ω ) In this section, we discuss the form of the spectral func-tions A ( k , ω ), which can be measured in angle resolvedphotoemission spectroscopy (ARPES) experiments, forweak to strong interorbital pairing. 1. Weak attractive coupling The spectral function for V = 0 . B g oper-ator leads to intra -band coupling along the Γ − M line,where one consequently clearly sees the hole pockets tobe gapped. Along Γ − X , the pairing is purely inter -band, and one finds a Fermi surface on both the hole and elec-tron pockets, indicating that V = 0 . X − M ; the signal for the node at the electron pocket FSshould be robust while the one at the hole pocket FS willbe weak. 2. Intermediate attractive coupling As the pairing interaction increases, the nodes result-ing from the interband pairing should get closer to eachother, as discussed in Sec. III. Figures 12(a-c) show n ( k ) for V = 3 along high-symmetry directions in theband representation. One can see that there are still un-paired electrons, and V = 3 consequently falls into the“breached” region schematically represented in Fig. 1.It is interesting to notice that while k ′ F h > k F h , on theother hand k ′ F e ≈ k F e indicating that the reconstructionaround the hole pockets is much larger than around theelectron pockets. This can also be observed in Fig. 12(b),where we observe unpaired electrons along X − M , butnot along Γ − X . The effect occurs in part due to thesmaller value of f ( k ) at the electron pockets but also be-cause, due to the band dispersions, the price in kineticenergy for interband pairing is much larger at the electronpockets than at the hole pockets. The spectral densityin Fig. 13(a) further illustrates that the pairing interac-tion is more effective along Γ − X than along X − M .Only shadow spectral weight crosses the chemical poten-tial along Γ − X , while strong spectral weight crosses thechemical potential twice along X − M and leads to twonodes. 3. Strong attractive coupling Finally, let us consider a much stronger value of thepairing, such as V = 6, in the BCS region of Fig. 1, forwhich the only nodes observed are those along X − Y , dueto the vanishing of the pairing operator. Figures 12(d-f),show that n ( k ) is discontinuous only along X − Y , whileit is smooth along all the other directions indicating pair-ing. Note that n ( k ) = n ( k ) along Γ − X and X − M where interband pairing occurs, while along Γ − M , n i ( k )is smooth but different for each band because the pair-ing is intraband. The behavior of the spectral functionsdisplayed in Fig. 13(b) shows that spectral weight onlycrosses the chemical potential along the X − Y direction.In this situation, in the folded BZ, nodes should occuronly at the points where the two electron pockets crosswith each other, as indicated in Ref. 40. In the rest ofthe BZ an anisotropic gap will be observed.1 π ) ( π ,0) ( π , π ) (0,0) ( π ,0) k n ( k ) (c) n ( k ) (b) n ( k ) (a) π ) ( π ,0) ( π , π ) (0,0) ( π ,0) k n ( k ) (f) n ( k ) (e) n ( k ) (d) FIG. 12: (Color online) (a) Total occupation number n ( k ) forthe two-orbital model Eq. (3) with pairing interaction V = 3(continuous lines) and V = 0 (dashed lines). Parameters: t = 1 . t = − t = t = − . µ = 1 . 54. (b) Same as(a) but for band 1. (c) Same as (b) but for band 2. (d) Totaloccupation number n ( k ) for the two-orbital model Eq. (3)with pairing interaction V = 6 (continuous lines) and V = 0(dashed lines). (e) Same as (d) but for band 1. (f) Same as(e) but for band 2. D. Stability of the B g pairing state Finally, let us discuss the important issue of the stabil-ity of the B g pairing state. It will be assumed, follow-ing the notation in Appendix A of Ref. 40, that Eq. (3)has arisen from an interorbital attractive potential of the FIG. 13: (Color online) (a)Spectral density for the two-orbital model Eq. (3) for a pairing strength V = 3. Therest of the parameters are as in Fig. 9; (b) same as (a) butfor strong pairing V = 6. form V k , k ′ = V ∗ (cos k x + cos k y )(cos k ′ x + cos k ′ y ) , (21)and that∆ † ( k ) = ∆( k ) = V ∗ ∆(cos k x + cos k y ) , (22)where ∆( k ) = − X k ′ V k , k ′ h b k ′ i , ∆ † ( k ) = − X k ′ V k , k ′ h b † k ′ i . (23) V in Eq. (5) is then given by V = V ∗ ∆ and the mean-field energy E MF can be calculated for a given V ∗ as afunction of V . The results are presented in Fig. 14. Inthis figure, a minimum for V = 0 can be seen for V ∗ ≥ V ≥ . if the attractionexists, both multinodal (breached) and states with nodesonly along X-Y (strong coupling) are possible. V=V * ∆ -8.8-8.7-8.6-8.5-8.4 E m f / N V * =3V * =4V * =5V * =6 FIG. 14: (Color online) Mean field energy per unit site for theinterorbital pairing with symmetry B g for different values ofthe attraction V ∗ as a function of V = V ∗ ∆. V. CONCLUSIONS Summarizing, in this manuscript the possibility ofintra and interband pairing in multiorbital systemshas been discussed. While interband pairing has pre-viously been studied in the context of QCD, coldatoms, heavy fermions, cuprates , and BCSsuperconductivity, most of the pairing operators pro-posed for the pnictides are based on the premise thatthe pairing has to be purely intraband, i.e., both elec-trons in the Cooper pairs belonging to the same band,compatible with the assumption that in these supercon-ductors all the action must occur at the Fermi surfaces.However, symmetry considerations show that in modelsfor the pnictides interorbital pairing is allowed with thetwo members of the Cooper pair belonging to differentbands. This is not surprising since the bands that de-termine the electron and hole Fermi surfaces consist ofhybridized orbitals. These interorbital pairing operatorsgive rise to not only intra but also interband pairing whenthe band representation is used. In addition, numericalcalculations in a minimal two orbital model for the pnic-tides favor one of these non-trivial pairing operators. As a consequence, a clear discussion of the role of inter-band pairing is necessary.The explicit calculations shown here of the electronicoccupation in the orbital and the band representations,as well as the calculation of the spectral functions show-ing the distribution of Bogoliubov bands, may offer guid-ance in the interpretation of ARPES experiments. Inparticular, most ARPES measurements determine theFS in the normal state and study the opening of the su-perconducting gap by monitoring A ( k F , ω ) as they lowerthe temperature. Notice that this approachwould miss the nodes associated with the “breached”phase since in the superconducting state a gap would beobserved in A ( k F , ω ) while the node would be detected in A ( k ′ F , ω ). Thus, experimentalists should investigate thepossibility of nodes at points in momentum space that do not belong to the normal state FS and they must keepin mind that some of the nodes may be determined byshadow bands with very small spectral weight.The most recent experimental results with thepolarization dependence of the ARPES spectra forBaFe . Co . As provided the allowed contribution ofeach of the five 3 d orbitals to the electron and hole Fermisurfaces. While discrepancies with proposed four andfive orbital models are remarked in that publication, it isinteresting to observe that the minimal two-orbital modeladdressed here does not contradict the ARPES findingsif we disregard the additional β hole pocket FS that theypresent, which is reasonable because it has d x − y char-acter, an orbital not included in the minimal model con-sidered here. In fact, along their Γ − M direction, whichcorresponds to our Γ − X , our first hole FS is purely d xz as it is their α π hole FS; our second hole FS, which arisesupon folding our extended FS along X − Y is purely d yz ,as it is their α σ hole FS, and our electron FS is purely d yz as it is their γ electron FS. Our second electron FS(obtained upon folding) has a purely d xz character, whiletheir electron FS γ ′ /α ′ appears to be mostly d yz and d xy ,but with some amounts of d xz as well. Along the diag-onal direction Γ − X , which corresponds to our Γ − M ,the two hole Fermi surfaces are a symmetric admixture of d xz and d yz , exactly as in the two-orbital model. Thus,these similarities between the experimental results andthe band composition of the simple two-orbital model of-fer encouragement towards exploring whether the mainphysics of the pnictides can be captured with such a min-imum number of degrees of freedom.We have also mentioned in the text that the symmetryof the lattice and of the orbitals introduce constraints onthe possible pairing operators. In general, a purely in-traband pairing, such as the proposed s ± state, would occur only if the coupling of the electrons with thesource of the attraction is identical for all orbitals. Con-sidering the different spatial orientations of the orbitals,it is not obvious that this should be the case, since, asdiscussed in the Introduction, phonons couple differentlyto electrons in the p z and p x boron orbitals in the caseof MgB . 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