Superconductivity in Multi-orbital t-J1-J2 Model and its Implications for Iron Pnictides
aa r X i v : . [ c ond - m a t . s up r- c on ] A ug epl draft Superconductivity in multi-orbital t − J − J model and its impli-cations for iron pnictides Pallab Goswami, Predrag Nikolic and
Qimiao Si Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA
PACS – Theories and models of superconducting state
PACS – Nonconventional mechanisms
PACS – Pnictides and chalcogenides
Abstract. - Motivated by the bad metal behavior of the iron pnictides, we study a multi-orbitalt − J − J model and investigate possible singlet superconducting pairings. Magnetic frustrationby itself leads to a large degeneracy in the pairing states. The kinetic energy breaks this intoa quasi-degeneracy among a reduced set of pairing states. For small electron and hole Fermipockets, an A g state dominates over the phase diagram but a B g state has close-by energy. Inaddition to the nodeless A g s x y channel, the nodal A g s x + y and B g d x − y channels arealso competitive in the magnetically frustrated J ∼ J parameter regime. An A g + iB g state,which breaks time-reversal symmetry, occurs at low temperatures in part of the phase diagram.Implications for the experiments in the iron pnictides are discussed. Introduction. –
The discovery of high temperaturesuperconductivity in the iron pnictides [1–3] has spurredtremendous experimental and theoretical interest in thesesystems. While they are not Mott insulators, the un-doped iron pnictides are nonetheless “bad metals”. Thisfact has motivated the placement of these systems in anintermediate coupling regime close to the boundary be-tween Mott localization and itinerancy [5–7], where theCoulomb interactions bring about non-perturbative effectsin the form of incipient lower and upper Hubbard bands.Accordingly, the low-energy Hamiltonian contains quasi-localized moments with J − J superexchange interac-tions [6–12], and it corresponds to an effective multi-band t − J − J model for the carrier-doped systems. Impor-tantly, J ∼ J > J / π,
0) antiferromagnetic order, as seen ex-perimentally [4], but also exhibits strong magnetic frus-tration. The incipient Mott picture is supported by theobservations of the Drude-weight suppression [14–16], aswell as the temperature-induced spectral-weight transfer[15, 17, 18]. In addition, the quasi-localized moments aresupported by the inelastic neutron scattering experiments;Ref. [19], for instance, observed zone-boundary spin wavesand showed that J is indeed comparable to J . (Bycontrast, Fermi surface in the magnetically ordered statedoes not directly probe the strength of electron correla-tions [7].) Alternatively, perturbative treatments [20, 21] of the Coulomb interactions have been used to study themagnetism of the iron pnictides. The weak-coupling ap-proaches, mostly based on spin fluctuations, have beenextensively used to address the superconductivity [21–26].By contrast, strong-coupling studies of superconductivityhave been more limited [27–30].Experimentally, the pairing symmetry in the iron pnic-tides has remained inconclusive. The angle resolvedphotoemission [31, 32] (ARPES) and the Andreev spec-troscopy [33] results suggest a nodeless gap. In contrastthe nuclear magnetic resonance [34] (NMR) and some pen-etration depth [35] measurements suggest a nodal gap.The experimental results seem to vary among pnictidecompounds. As an example, the P-doped BaFe As ap-pears to have nodal gaps [36,37] and this is in strong con-trast with its K-doped counterpart.In this Letter we address the pairing in the pnictidesfrom the incipient Mott approach, and treat the competingpairing channels on an equal footing. We are particularlymotivated to consider the effect of the J ∼ J magneticfrustration, and show that it leads to a quasi-degeneracyamong several paring states.The Hamiltonian is given by H = H t + H J + H J . (1)The kinetic part is H t = − P i 0) collinear antiferromag-netic state [9, 19], it is expected to be isotropic in thetetragonal paramagnetic phases and this is consistent withspin dynamical measurements [38, 39]. Two-orbital model. – We will first consider a twoorbital model, retaining only the d xz ( α = 1) and d yz ( α =2) orbitals, and later we consider a five orbital model tobetter address the fermiology [21, 26, 40]. We further as-sume J α,βi = J i δ α,β , and J i > 0, and will subsequentlyaddress the role of interorbital exchange couplings.Under the tetrahedral point group symmetry trans-formations ( D h ), d xz and d yz orbitals transform re-spectively as x and y coordinates. The kinetic en-ergy part of the Hamiltonian is invariant under allpoint group symmetry operations ( A g ) and has thefollowing form in the extended Brillouin zone, H = P k ,s ψ † k s [ ξ k + τ + ξ k − τ z + ξ k xy τ x ] ψ k s , where k = ( k x , k y )and ψ † k s = ( c † k s , c † k s ). The identity and Pauli matri-ces ( τ , τ i ) operate on the orbital indices, and ξ k + = − ( t + t )(cos k x + cos k y ) − t cos k x cos k y − µ , ξ k − = − ( t − t )(cos k x − cos k y ), ξ k xy = − t sin k x sin k y arerespectively A g , B g , B g functions. The band disper-sion relations E k ± = ξ k + ± q ξ k − + ξ k xy , give rise to twoelectron pockets at k = ( π, 0) and (0 , π ), and two holepockets at k = (0 , 0) and ( π, π ). The carrier doping δ = | P α n iα − | . Mostly we use the minimal tight-bindingmodel of Ref. [41], t = − t , t = 1 . t , t = t = − . t obtained from a fitting of the LDA bands. Magnetic frustration and degeneracy of pairing states. We start from the case with a vanishing kinetic energy,in order to highlight the connection between magneticfrustration and enhanced degeneracy of pairing states.We define the intra-orbital spin-singlet pairing opera-tors ∆ e ,αα = h c iα ↑ c i + e α ↓ − c iα ↓ c i + e α ↑ i / 2, where e =ˆ x, ˆ y, ˆ x ± ˆ y . Without the kinetic term, the problem de-couples in the orbital basis and we can drop the or-bital indices. When J dominates, two degenerate pair-ing states s x + y and d x − y , respectively defined by thepairing functions g x ± y , k = cos k x ± cos k y are naturallyfavored. In real space, they respectively correspond to∆ x = ± ∆ y = ∆ , with ∆ x + y = ∆ x − y = 0. These s x + y and d x − y states are degenerate because the symmetry A + i B A J J (a) J J A A + i B A + i B B A B + i ( ) + (1) (2) (b) Fig. 1: Zero temperature phase diagrams of (a) a two orbitalmodel and (b) a five-orbital model, both for electron doping δ =0 . 14. The onset of B g and B g phases are respectively markedby black and red solid lines. The dotted line characterizes across-over between s A g x y [region (1)] and s A g x + y [region (2)] asthe dominant component in the A g pairing. operation c m ˆ x + n ˆ y → e i (2 m +1) π/ c m ˆ x + n ˆ y transforms theminto each other.When J dominates, the s x y and d xy states, re-spectively defined by the pairing functions g x y , k =cos( k x + k y ) + cos( k x − k y ) and g xy, k = cos( k x − k y ) − cos( k x + k y ), are preferred, and they are degenerate. Inreal space, they correspond to ∆ x + y = ± ∆ x − y = ∆ ,with ∆ x = ∆ y = 0. The s x y and d xy states trans-form into each other by the following symmetry opera-tion: We break the square lattice into two interpenetrat-ing sublattices;on the even sublattice ( m + n = even ), c m ˆ x + n ˆ y → e i ( m + n +1) π/ c m ˆ x + n ˆ y , and on the odd sublat-tice, c m ˆ x + n ˆ y → e i ( m − n +1) π/ c m ˆ x + n ˆ y .As we tune the ratio J /J , we expect a level crossing inthe magnetically frustrated regime, J ∼ J . We can thenanticipate that magnetic frustration promotes an enlargeddegeneracy among the s x y , s x + y , d xy , and d x − y pair-ing states. The effect of the kinetic energy. When the ki-netic term is incorporated, it lifts the exact degenera-cies of the paired states discussed above. We studythe full problem using a mean-field decoupling [42] ofthe two-band t − J − J model. To set the stage,we note that the D h point group symmetry opera-tions allow the following four classes of pairing statesfor an orbitally diagonal J − J model [43]: (i) A :[ s A g x + y g x + y , k + s A g x y g x y , k ] τ + d A g x − y g x − y , k τ z ; (ii)B : d B g x − y g x − y , k τ + [ s B g x + y g x + y , k + s B g x y g x y , k ] τ z ;(iii) A : d A g xy g xy, k τ z ; and (iv) B : d B g xy g xy, k τ . Eachpairing channel will have different symmetry depending onwhether it is associated with τ or τ z in the orbital space;this distinction is denoted by the superscripts. The eightpairing amplitudes s A g x + y etc. are linear combinations ofeight intra-orbital pairing amplitudes ∆ e ,αα . Mean field theory. We decouple the exchange interac-tions in the pairing channel. In terms of Ψ † k = ( ψ † k ↑ , ψ − k ↓ ),p-2uperconductivity in multi-orbital t − J − J model and its implications for iron pnictides Fig. 2: (Color online) The amplitudes of different pairing gapcomponents of a two orbital model for electron doping δ =0 . 14. For J ≫ J and J ≪ J , s A g x y ( s ± ), and d B g x − y arerespectively the dominant pairing channels. the Hamiltonian becomes H mf = X k Ψ † k (cid:20) h k ∆ k ∆ ∗ k − h k (cid:21) Ψ k (2)where h k = ξ k + τ + ξ k − τ z + ξ k xy τ x , and ∆ k =diag[∆ k , , ∆ k , ] is the orbitally diagonal gap matrix, and∆ k ,αα = P e J e ∆ e ,αα cos( k · e ), with J e = J for e = ˆ x, ˆ y ,and J e = J for e = ˆ x ± ˆ y . Diagonalizing H mf we obtainthe quasiparticle dispersion spectra E k , ± . We determinethe pairing gap matrix by minimizing the ground stateenergy density f = X e ,α J e | ∆ e ,αα | − X k ,j = ± ( E k ,j − E k ,j ) , (3)with respect to all ∆ e ,αα . The primary effect of the con-straints is to renormalize the kinetic energy via (both or-bitally diagonal and off-diagonal) t → tδ/ 2, where t isthe kinetic energy scale. Our results, with an implicittreatment of the constraint through a band renormaliza-tion, remain largely unchanged when the constraints areexplicitly incorporated.When kinetic energy is absent, the s x + y and d x − y states are indeed degenerate, and each has a ground stateenergy ≈ − . J . Likewise, the energy of either s x y or d xy state is ≈ − . J , and all four paired states becomedegenerate exactly at J = J , as anticipated earlier.We now turn to the results for the full problem in thepresence of the kinetic terms. We show an illustrativezero-temperature phase diagram for 0 ≤ J , J ≤ t , inFig. 1(a) corresponding to an electron doping δ = 0 . A g pairing exists in the entire J − J plane. In a sizableportion of the phase diagram, the pairing is in the pure A g class. In this region s A g x y is the dominant pairingchannel and coexists with the subdominant d A g x − y chan-nel. The onset of B g pairing state is marked by a solidline, which corresponds to a second order phase transi-tion. d B g x − y and s B g x y are respectively the dominant and Fig. 3: (Color online) The pairing amplitudes for d xz , d yz or-bitals, obtained from a five orbital model for δ = 0 . 14. For J ≫ J , the dominant pairing channel is s A g x y ( s ± ), and for J ≪ J , d B g x − y and s A g x + y pairing channels are dominantand nearly degenerate pairing channels. Compared to the two-orbital model s A g x + y channel is more competitive and becomessignificant for relatively smaller value of J . the subdominant components of B g phase. In the large J limit (to the right of the dotted line), s A g x + y becomesthe dominant component of the A g phase; the dotted linerepresents a crossover. The phase diagram for hole dopingis identical, but the onset of B g pairing occurs for larger J .The competition among different symmetry classes, andthe nature of the ground states are demonstrated in Fig. 2,which plot the pairing amplitudes as a function of thecoupling constants. The degeneracy between the s x y and d xy channels, occurs at very large J limit and isnot showed in Fig. 2. For moderate values of J , J ,quasi-degeneracy among the pairing states occurs in themagnetically-frustrated region corresponding to J ∼ J .In this region, Fig 2 shows that the weights of the s A g x y and d B g x − y components are comparable, and those of d A g x − y and s B g x y are close-by. This quasi-degeneracy also under-lies the phase diagram shown in Fig. 1(a). By contrast, formoderate values of J and with J ≫ J , s A g x y dominatesover s B g x y , d A g xy , d B g xy channels; likewise, for moderate val-ues of J and with J ≫ J , d B g x − y dominates over d A g x − y , s A g x + y , s B g x + y channels.At the T = 0 limit we study, the co-existing pairingchannels will lock into a definite phase. Consider firstthe case of coexisting s A g x y and d A g x − y . We define ∆ k = s A g x y g x y , k τ + d A g x − y g x − y , k τ z , and use φ to denote therelative phase. Inserting this into Eq. (3), we find that φ = π corresponds to the ground-state energy minimum,with ∂f /∂φ = 0 and ∂f /∂φ > φ = 0 is a ground-stateenergy maximum). The A g gap is therefore real. More-p-3allab Goswami et al. over, the relative minus sign associated with φ = π impliesthat the pairing function, in the band (as opposed to theorbital) basis, is s A g x y g x y , k τ − d A g x − y g x − y , k ( ξ k − τ z − ξ k xy τ x )( ξ k − + ξ k xy ) − / . Hence intraband pairing functionchanges sign between the hole and electron pockets near k = (0 , 0) and k = ( π, , (0 , π ) respectively. A similarargument shows that the relative phase between d B g x − y and s B g x y is π , and the gap changes sign between the twoelectron pockets near k = ( π, , (0 , π ) respectively.Consider next the case of coexisting s A g x y and d B g x − y .A similar analysis shows that, this time, φ = π/ φ =0 , π represent energy maxima). The resulting A g + iB g phase describes the state to the right of the solid line inFig. 1(a). This phase simultaneously breaks time rever-sal and four-fold rotational symmetries, but preserves thecombination of the two symmetries. Such a state also oc-curs in a phenomenological Landau-Ginzburg theory [44].The role of fermiology can be clearly illustrated in thelinearized gap approximation. For a set of small Fermipockets at (0 , π, 0) and (0 , π ), compared to g xy, k , g x y , k has larger overlap with the pairing kernel. Thus s A gx y and s B g x y gaps have higher T c ’s compared to d B g xy and d A g xy gaps; they become degenerate only in the large J limit. Similar reasoning shows that, unless a thresh-old value for J is exceeded, d A g x − y and d B g x − y gaps havehigher T c ’s compared to the s A g x + y and s B g x + y gaps. Re-lated observations were made by Seo et al. [28]. If weconsider a band structure that produces large pockets[27], d B g xy replaces s A g x y as the dominant pairing state for J > J , and we observe a competition between d B g xy and d B g x − y pairing states. Note that the magnetic oscillationand ARPES measurements suggest small sizes of the Fermipockets for the iron pnictides, and this should make our A g and A g + iB g phases more feasible. Effects of inter-orbital exchange couplings. We stressthat, while the detailed nature of the lattice symmetry andorbitals for the single-particle energy dispersion is impor-tant for a proper description of the Fermi surface, its coun-terpart for the exchange interactions is not obviously so.In the absence of a detailed knowledge about such struc-ture in the exchange interactions, we have considered thesimplest description and focused on the associated prop-erties that are qualitative and robust. For instance, wehave so far considered intra-orbital exchange interactions.Inclusion of inter-orbital super-exchange interactions doesnot change the phase diagram. Now the A g phase willhave a small inter-orbital d A g xy component and the B g phase will remain unchanged. Hund’s coupling may leadto on-site triplet pairing, but this is prevented by the on-site Coulomb repulsion that is built in our model. Inter-site triplet pairing may arise if the Hund’s coupling is com-parable to Coulomb repulsion [29], but this is unlikely on general or ab initio grounds; it is also unlikely on empir-ical grounds since experimental evidence has so far beenoverwhelming for singlet pairing. Using a multi-orbitalHubbard model as their starting points, both the strongcoupling calculations of Ref. [29] and the RPA calculationsof Ref. [21] find that a moderate J H enhances repulsiveinter-electron-pocket pair scattering and leads to stronger B g pairing. Thus J H can only enhance the A g + iB g part of the phase diagram. We have preferred to considerthe multiband t − J − J model, since J and J canbe more readily connected with the magnetic frustrationphysics. Five-orbital model. – To understand the robust-ness of our two-band results, against the inclusion of ad-ditional bands, and better address the fermiology we haveconsidered a five-band model with the kinetic terms ac-cording to Ref. [26]. To capture the important resultswithin a simple model of interaction, we again choosea J − J model with J αβi = J i δ αβ . Now the gen-eral intra-orbital pairing matrix has the form ∆ k = P a diag[∆ a k , , ∆ a k , , ∆ a k , , ∆ a k , , ∆ a k , ], where the in-dex a corresponds to s x + y , d x − y , s x y and d xy sym-metries. The results of minimizing the free energy withrespect to all the twenty complex pairing amplitudes aregiven in an illustrative phase diagram Fig. 1(b), and inFig. 3, which shows the competition among the pairingamplitudes for xz and yz orbitals. We find that the compe-tition between A g and B g pairings is a robust effect. Incontrast to the two band case d A g x − y amplitude is reducedand s A g x + y amplitude is enhanced in the entire phase di-agram. Indeed, for the magnetically-frustrated J ∼ J region, the nodal A g s x + y and B g d x − y states arecompetitive against the nodeless A g s x y state. Com-pared to the two band case the B g pairing occurs forsmaller J , but it still occurs in the limit when J is big-ger than the kinetic energy scale. The other three orbitalsalso demonstrate similar competition among s x y , s x + y , d x − y , and d xy pairings. Experimental implications of quasi-degeneratepairing channels. – Some of the weak-coupling stud-ies [21, 26] have also indicated a competition betweenvarious pairing states. However, the weak-coupling ap-proaches are typically restricted to an instability analysisof the linearized gap equations. By contrast, the strong-coupling approach used here has the advantage of readilyconsidering the non-linear gap equations. Our non-linearanalysis is important in bringing out the connection be-tween the quasi-degeneracy in the pairing channel and the J ∼ J magnetic frustration. With only one require-ment ( J ∼ J ), which is linked to magnetic frustrationand has been supported by both theoretical considerations[6,8–10,13] and inelastic magnetic experiments [19,38], ourresult provides a parameter-insensitive mechanism for thenear degeneracies of various paring states. Moreover, ournon-linear analysis is also essential in reaching the conclu-p-4uperconductivity in multi-orbital t − J − J model and its implications for iron pnictidession ( cf. Figs. 2 and 3) that, for J ∼ J , the pairing am-plitudes for several competing nodeless and nodal pairingchannels are comparable. This last result is particularlyimportant for experiments in the iron pnictides. Indeed,our result has anticipated the recent experimental find-ings [36, 37] that pnictide superconductors with nodelessor nodal gaps have a comparable maximum T c . Our s A g x y ( s ± ), s A g x + y states appear to be consistent with ARPESmeasurements that find full gap at hole pockets. How-ever in contrast to s A g x y , which is fully gapped on all theFermi pockets as long as the pockets are not too large, s A g x + y state has nodes on the electron pockets. This maybe the reason for seeing nodal behavior in P-doped 122compounds in contrast to the fully gapped behavior inK-doped 122 compounds.To summarize, we have shown that magnetic frustrationeffects lead to quasi-degeneracy among different pairingstates. With the bandstructures appropriate for the ironpnictides, an extended A g state is the dominant pairingstate, but this state contains both s x y and s x + y com-ponents. Moreover, a B g state has a close-by groundstate energy. The quasi-degeneracy makes it likely thatthe iron pnictides of different material families, or differ-ent dopings, have different superconducting states. Ourdetailed phase diagram contains a low-temperature phasewith time-reversal-symmetry breaking, which also can betested by future experiments. ∗ ∗ ∗ We thank E. Abrahams, B. A. 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