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Orbital-selective superconductivity, gap anisotropy and spin resonance excitations in a multiorbital t-J1-J2 model for iron pnictides

We study the orbital-dependent superconducting pairing in a five-orbital t-J1-J2 model for iron pnictides. Depending on the orbital selectivity of electron correlations and the orbital characters along the Fermi surface, the superconducting gap in an A_{1g} pairing state may exhibit anisotropy. This anisotropy varies with the degree of J1-J2 magnetic frustration. We have also calculated the dynamical spin susceptibility in the superconducting state. The frequency dependence of the susceptibility at the antiferromagnetic wavevector (\pi,0) shows a resonance, whose width is enhanced by the orbital dependence of the superconducting gap; when the latter is sufficiently strong, the resonance peak may be split into two. We discuss the implications of our results on the recent angle-resolved photoemission and neutron-scattering measurements in several superconducting iron pnictides.

Introduction. The mechanism and symmetry of the superconducting pairing is a central issue for iron-based superconductors [1,2]. In most of these materials, superconductivity appears when electron or hole doping is introduced into the antiferromagnetic parent compounds [3]. Two theoretical approaches have been proposed to address the link between the magnetism and superconductivity. In the weak-coupling limit, both properties arise from the nesting between electron and hole Fermi pockets [1]. In the strong-coupling approach, on the other hand, the superconductivity and antiferromagnetism are driven by the short-range exchange interactions among the correlation-induced quasi-local moments [5][6][7][8][9]. The superconducting gap function reflects the short-range electron pairs, and is explicitly orbital dependent. Among the evidences [10,11] for the strong-coupling approach is the fact that both the alkaline iron selenides K 1−x Fe 2−y Se 2 [12][13][14] and the singlelayer FeSe [15] lack the Fermi surface nesting (in the absence of hole Fermi pockets) and yet still display high-T c superconductivity.
A recent development in the strong-coupling approach to the iron-based superconductors is the proposal that the proximity to the Mott transition is orbital-dependent [16][17][18][19][20][21]. Experimental evidence for such orbital selectivity has already emerged [22,23]. It is therefore natural to ask how such orbital-dependent effects of the electron correlations influence the nature of the superconducting state.
The pairing symmetry of the iron-based superconductors has been studied via various experimental techniques. Angle resolved photoemission spectrum (ARPES) measurements find the superconducting gap to be nodeless and isotropic at both the hole and electron pockets in a number of materials [24][25][26][27][28]. Neutron scattering measurements on these compounds observe a clear spin resonance mode in the superconducting state [29][30][31][32]. This is consistent with the pairing order parameter changing sign between the hole pockets near the Brillouin zone (BZ) center and the electron pockets near the zone boundary, which arises within both weakcoupling [33,34] and strong-coupling approaches [35,36].
Recently, experiments have identified an anisotropy of the superconducting gap along the Fermi pockets in several iron pnictide materials [37][38][39][40]. These results hold the promise to shed new light on the understanding of the pairing mechanism. In particular, high resolution ARPES [37] has revealed that the superconducting gap is anisotropic along the electron Fermi pockets in the underdoped Na(Fe 1−x ,Co x )As; it becomes isotropic when the system reaches the overdoped regime. Because the orbital character of the electronic states varies along the electron Fermi pocket, this observation points to the possibility of the orbital dependent nature of superconductivity.
In this Letter, we study the superconducting pairing in a five-orbital t-J 1 -J 2 model for iron pnictides. We show that the orbital-dependent effects of electron correlations generally give rise to orbital-selective superconducting gaps. In particular, we emphasize two gaps that are respectively associated with the xz/yz and xy orbitals, which have the same A 1g symmetry but different pairing amplitudes. We show how this orbital-selective pairing naturally leads to a gap anisotropy, and discuss the implication of the results for the ARPES and neutron spectra of several superconducting iron pnictides.
Model and method. We consider a five-orbital t-J 1 -J 2 model. The Hamiltonian [5,35,36] reads as where c † iαs creates an electron at site i, in orbital α and spin projection s; µ is the chemical potential to fix the total electron number n. The orbital index α = 1, 2, 3, 4, 5 respectively correspond to the five Fe 3d orbitals xz, yz, x 2 − y 2 , xy, and 3z 2 − r 2 . The tight-binding parameters t αβ ij and ǫ α respectively refer to the hopping matrix and the onsite potential that reflects the crystal level splitting. For definiteness, we consider the case of NaFeAs, and obtain the tight-binding parameters by fitting its LDA bandstructure [42]. The nearest-neighbor (n.n., ij ) and next-nearest-neighbor (n.n.n., ij ) exchange couplings are respectively denoted by J αβ 1 and J αβ 2 . The spin operator is S iα = 1 2 s,s ′ c † iαs σ ss ′ c iαs ′ and the density operator n iα = s c † iαs c iαs , where σ represents the Pauli matrices. The double-occupancy prohibiting constraint from the fermion is implicitly incorporated by the renormalization of the band structure [10,36].
We study the superconducting pairing in the above t-J 1 -J 2 model by decomposing the exchange interactions in the spin singlet pairing channels. The gliding reflection symmetry of the Fe-As lattice allows us to consider the pairing channels with the choice of the one-Fe unit cell. For simplicity, we assume J αβ 1(2) = J 1(2) δ αβ (and take J 2 to be the energy unit); correspondingly, we consider intraorbital pairing. There are 20 different pairing channels, each with an amplitude and a phase, which are self-consistently determined.
We also calculate the dynamical spin susceptibility in the superconducting state.
At wavevector q and Matsubara frequency ω n the spin susceptibility χ(q, iω n ) = αβ χ α,β (q, iω n ), where and . . . refers to the expectation value with respect to the effective Hamiltonian. The susceptibility at real frequency ω is then obtained by an analytical continuation iω n → ω + i0 + .
Multiorbital nature of the Fermi surface and orbitalselective pairing.
The Fermi surface in the 1-Fe BZ for the tight-binding model at electron doping x = 0.02 (x = n − 6) is shown in Fig. 1(a). The Fermi surface contains multiple sheets with different orbital characters. The two hole pockets near (0, 0) are dominated by the degenerate xz/yz orbitals; the hole pocket near (π, π) has almost completely xy orbital character. The electron pocket near (π, 0) [or (0, π)] displays a hybridized xy and yz (xz) orbital character [ Fig. 1(b)]. The pairing amplitudes are also orbital dependent. For J 1 /J 2 1, the dominant pairing channel is s x 2 y 2 with an A 1g symmetry ( Fig. 1(c)). The amplitude of this pairing channel in the xy orbital is larger than that in the xz/yz orbital. Due to the double degeneracy of the xz and yz orbitals, a d x 2 −y 2 wave pairing channel can also have an A 1g symmetry. It serves as a subdominant pairing channel whose amplitude increases with J 1 /J 2 . The existence of orbital-selective multiple energy scales in pairing is a consequence of the orbital dependent electron correlation effects in the multiorbital model, with the xy orbital typically exhibiting strong correlation effects [16][17][18]. Correspondingly, the xy orbital has a sizeable ratio of J to the renormalized bandwidth, which in turn yields a sizeable pairing amplitude [10].
Anisotropic superconducting gap. We now turn to how the orbital-selective pairing amplitudes and the orbital character of the Fermi surface affect the momentum distribution of the superconducting gaps by inducing gap anisotropy, and how the gap amplitudes and the corresponding anisotropy can be tuned by the degree of magnetic frustration of the system. We discuss and compare the results in the five-orbital t − J 1 − J 2 model by taking J 1 /J 2 = 0.1 and J 1 /J 2 = 0.8, for illustrative purpose. For J 1 /J 2 = 0.1, the superconducting gaps are dominated by the s x 2 y 2 A 1g pairing channel. The amplitudes of this pairing channel in the xy and xz/yz orbitals are significantly different, resulting in two characteristic , and is isotropic [ Fig. 2(a)] since the dominant orbital character of a hole pocket is uniform: xy for the pocket near (π, π) and xz/yz for the pocket near (0, 0). On the other hand, the gaps along the electron pockets are strongly anisotropic [ Fig. 2(c)]. This is because the electron pocket has a hybridized xy and xz/yz orbital character, and the size of the gap at a particular wavevector depends on the dominant orbital character at that point. The gap anisotropy reflects these two characteristic superconducting gaps ∆ xy = ∆ xz/yz : as shown in Fig. 2(c), the gap cannot be fitted by a single gap function ∆ 0 cos k x cos k y though the dominant pairing channel is s x 2 y 2 A 1g . Interestingly, the gap anisotropy reduces with increasing J 1 /J 2 , and an essentially isotropic gap along the electron pocket is recovered at J 1 /J 2 = 0.8 ( Fig. 2(d)). To understand this, note that the pairing amplitude of the subdominant d x 2 −y 2 A 1g channel in the xz/yz orbital increases with J 1 /J 2 . With the contribution from this subdominant channel, the overall gap in the xz/yz orbital ∆ xz/yz ≈ ∆ xy . This then leads to an essentially isotropic gap along the electron pockets. Spin resonance excitation. The spin excitations in the superconducting state are also affected by the orbitalselective pairing. We have calculated the dynamical spin susceptibility in the superconducting state for the J 1 /J 2 = 0.1 and J 1 /J 2 = 0.8 cases discussed above. The imaginary part of the susceptibility χ ′′ (q, ω) at the antiferromagnetic wavevector q = (π, 0) exhibits two resonance peaks in the frequency dependence for J 1 /J 2 = 0.1 ( Fig. 3(a)). Our detailed analysis finds that the double peak structure of χ ′′ (q, ω) arises from the different scattering processes that connect two regimes near the electron and hole pockets, as indicated by the arrows in Fig. 1(a). By appearing in the coherence factor of the expression of χ ′′ (q, ω), the different orbital characters of the quasiparticle dispersion put a strong constraint to the scattering processes such that the spin response is enhanced only in certain regimes of the BZ, where the orbital characters of the associated hole and electron bands are compatible. For example, the dominant contribution to the lower frequency resonance peak at ω L (see Fig. 3(c)) is from a scattering between the yz orbital in regime A and the xy orbital in regime A ′ , as indicated by the dashed arrow in Fig. 1(a). The higher frequency resonance peak at ω H , on the other hand, is mainly associated with a scattering within the xy orbital between regimes B and B ′ (see Fig. 3(c) and Fig. 1(a)). As a rough estimate, the resonance frequency ω E h + E e , where E h and E e are respectively the excitation gaps of the corresponding hole-and electron-like quasiparticles, i.e. h = A, B, and e = A ′ , B ′ . Given the similar orbital character and the proximity to the equivalent points along the Fermi surface, E A ′ ≈ E B ′ . But the different orbital characters make E A = E B for J 1 /J 2 = 0.1. (Note that as a combined effect of the momentum dependence of the gap function and the multiorbital nature, E B is smaller than the gap along the nearby hole Fermi pocket [42].) There-fore, ω L = ω H ; when this difference is sufficiently large, two resonances appear in the frequency-dependent spectrum. As J 1 /J 2 increases, both E A and E B increase. But due to the subdominant d x 2 −y 2 channel in the xz/yz orbital, E A increases faster, and E A ≈ E B for J 1 /J 2 = 0.8. We thus obtain a single resonance peak at ω L ≈ ω H , as shown in Fig. 3(b) and (d)); the multi-orbital effect is then reflected in the broadening of the peak.
Discussions. Our results elucidate how the orbital selectivity of electron correlations influences the superconductivity. We show that the orbital-selective pairing gives rise to gap anisotropy along a Fermi surface with hybridized orbital characters. By promoting the subdominant d x 2 −y 2 A 1g pairing channel, the magnetic frustration may compete with the orbital selectivity, and tune the gap anisotropy. Our results are particularly pertinent to the anisotropic superconducting gap along the electron pockets in the underdoped Na(Fe 1−x ,Co x )As observed in recent ARPES measurements [37]. We are also able to understand the evolution from the anisotropic to isotropic gap with increasing electron doping. For illustrating purposes, in the model calculation of this paper, we fix the electron doping and show the evolution of gap anisotropy by tuning J 1 /J 2 . But the gap anisotropy as a consequence of the orbital selectivity is a general result. In a more realistic model, the exchange couplings would be orbital dependent. Whether an anisotropic gap shows up depends on how different the ratio of exchange coupling to the renormalized electronicbandwidth, J α 1(2) /D α , is among the different orbitals, and this ratio will be tuned by the strength of electron correlations.
Theoretically, the coexistence of antiferromagnetic order and superconductivity in the underdoped regime may also lead to an anisotropic superconducting gap along the reconstructed Fermi surface [41]. In this scenario, the Fermi surface is reconstructed for both the electron and hole pockets. But experimentally, the gap anisotropy was only observed along electron pockets. Moreover, the Fermi surface reconstruction for the gap anisotropy was not observed in ARPES [37]. It therefore is unlikely that the observed anisotropic gap is primarily driven by the coexistence of superconductivity with antiferromagnetic order.
Raman scattering has also implicated an anisotropic gap along the electron pocket in the nearly optimally hole-doped BaFe 2 As 2 . [40] Because hole doping tends to increase the orbital selectivity of electron correlations [45], it is natural to propose that the mechanism advanced here underlies this experimental observation as well.
We have also shown that the frequency dependence of the dynamical spin susceptibility at (π, 0) displays a resonance, whose width is enhanced by the orbitaldependence of the superconducting gap. When the latter is sufficiently strong, the resonance peak may be split into two. We propose that this mechanism underlies the recent neutron-scattering observation of double spin resonances in the electron underdoped NaFeAs system [44]. The double-resonance feature we have discussed is very different from the one reported in Ref. [43]. In that case, the wavevectors where the resonances take place are sensitive to the Fermi surface geometry, and are necessarily different for the two resonances; one is at q = (π, 0), while the other is at an incommensurate q. In our case, both resonances take place at the same wavevector q = (π, 0); this wavevector is determined by the qdependence of J(q).
Finally, the degree of electron correlations remains a central issue in the iron-based superconductors. This issue is typically probed in the normal state, through the bad-metal phenomenology in the optical spectrum [46] or the orbital selectivity in the ARPES spectrum [22,23]. Our theoretical results here suggest that this issue can also be fairly directly probed through the orbital selectivity of the gap function in the superconducting state. ARPES studies along this direction are already quite realistic [47,48], and we anticipate that considerable new insights will be derived through further studies along this direction.
Conclusions. Our calculation on the superconducting pairing in a five-orbital t-J 1 -J 2 model for iron pnictides reveals an orbital-selective gap structure due to the strong electron correlation effects. While both gaps have the s x 2 y 2 A 1g symmetry, the different orbital character gives rise to gap anisotropy along the electron pockets. The orbital selective pairing leads to a broadened neutron resonance at the antiferromagnetic ordering wavevector q = (π, 0) in the superconducting state. This resonance may even be split into two peaks. Our results have important implications for the ARPES and neutron measurements on the electron underdoped NaFeAs, as well as the Raman scattering results on the hole doped BaFe 2 As 2 . More generally, our results point to new ways of probing electron correlation effects of the iron pnictides through the single-particle and spin responses in their superconducting state.
To obtain the tight-binding parameters, we perform LDA calculations for NaFeAs, and fit the LDA bandstructure to the tight-binding Hamiltonian. We use the form of the five-orbital tight-binding Hamiltonian given in Ref. [1]. The tight-binding parameters so derived are listed in Table S1. Fig. S1 shows the bandstructure of the five-orbital tight-binding model for electron density n = 6.02, corresponding to x = 0.02 electron doping. The corresponding Fermi surface is shown in Fig. 1 of the main text. The Fermi surface consists of three hole pockets and two electron pockets. They have very different orbital compositions. We show the orbital weights of the hole and electron pockets in Fig. S2(b) and Fig. S3.

Momentum distribution of the excitation gap of the quasiparticles
In the conventional BCS theory for a single-band model with s-wave pairing symmetry, the superconducting gap ∆ is momentum independent, and the excitation gap for the BCS quasiparticles is E(k) = (ξ k − µ) 2 + ∆ 2 , where ξ k and µ are respectively the dispersion and chemical potential of the tight-binding model. For the five-orbital t-J 1 -J 2 model, considering the multiorbital nature of the model and the complicated structure of the superconducting pairing function, the excitation gap has a complicated momentum distribution, which can only be obtained numerically. We show the momentum distribution of the excitation gap for J 1 /J 2 = 0.1 in Fig. S2. Note that due to the momentum dependent pairing function and the nonzero inter-orbital hopping, the excitation gap at regime B (E B ) is smaller than that along the hole pocket centered at (π, π). But still E B > E A at J 1 /J 2 = 0.1.   TABLE S1. Tight-binding parameters of the five-orbital model for NaFeAs