Orbital selective pairing and gap structures of iron-based superconductors
Andreas Kreisel, Brian M. Andersen, Peter O. Sprau, Andrey Kostin, J.C. Séamus Davis, P. J. Hirschfeld
OOrbital selective pairing and gap structures of iron-based superconductors
Andreas Kreisel,
1, 2
Brian M. Andersen, P.O. Sprau,
3, 4
A. Kostin,
3, 4
J.C. S´eamus Davis,
3, 4 and P. J. Hirschfeld Niels Bohr Institute, University of Copenhagen,Juliane Maries Vej 30, DK 2100 Copenhagen, Denmark Institut f¨ur Theoretische Physik, Universit¨at Leipzig, D-04103 Leipzig, Germany LASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA CMPMS Department, Brookhaven National Laboratory, Upton, NY 11973, USA Department of Physics, University of Florida, Gainesville, FL 32611, USA
We discuss the influence on spin-fluctuation pairing theory of orbital selective strong correlationeffects in Fe-based superconductors, particularly Fe chalcogenide systems. We propose that a keyingredient for an improved itinerant pairing theory is orbital selectivity, i.e., incorporating thereduced coherence of quasiparticles occupying specific orbital states. This modifies the usual spin-fluctuation via suppression of pair scattering processes involving those less coherent states and resultsin orbital selective Cooper pairing of electrons in the remaining states. We show that this paradigmyields remarkably good agreement with the experimentally observed anisotropic gap structures inboth bulk and monolayer FeSe, as well as LiFeAs, indicating that orbital selective Cooper pairingplays a key role in the more strongly correlated iron-based superconductors.
I. INTRODUCTION
In both copper-based and iron-based high tempera-ture superconductors, fundamental issues include the de-gree of electron correlation and its consequences for en-hancing superconductivity. In both archetypes, there aremultiple active orbitals (two O p orbitals and one Cu d orbital in the former, and five Fe d orbitals in thelatter). This implies the possibility of orbital-selectivephysics, where states dominated by electrons of one or-bital type may be weakly correlated and others muchmore strongly correlated, leading to substantial differ-ences in quasiparticle spectral weights, interactions, mag-netism and orbital ordering . Cooper pairing itselfcould then become orbital-selective, with the electronsof a specific orbital character binding to form the Cooperpairs of the superconductor. The superconducting energygaps of such a material would therefore generically behighly anisotropic , i.e., large only for those Fermi sur-face regions where a specific orbital character dominates.Such phenomena, although long the focus of theoreti-cal research on higher temperature superconductivity incorrelated multi-orbital superconductors, have remainedlargely unexplored because orbital-selective Cooper pair-ing has not been experimentally accessible.Spin fluctuations are proposed as the dominant mech-anism driving Cooper pairing in a wide variety of un-conventional superconductors: heavy-fermion systems,cuprates, two-dimensional organic charge transfer salts,and iron-based superconductors (FeSC) . There iscurrently no version of spin-fluctuation based pairingtheory that enjoys either the well-controlled derivationfrom fundamental interactions or the consensual suc-cess explaining observed properties of the BCS-Migdal-Eliashberg theory of conventional superconductivity. Onthe other hand, the calculational scheme referred to asrandom phase approximation (RPA) in the case of one-band systems , or matrix-RPA in the case of multi-band systems , has achieved considerable qualitativeprogress for unconventional systems. While material-specific calculations of the critical tem-perature T c within spin-fluctuation theory appear dis-tant, considerable success has been achieved understand-ing qualitative aspects of pairing, particularly in Fe-pnictide systems . In the 122 materials, which werethe subject of the most intensive early study, itinerantspin-fluctuation theory provided convincing, material-specific understanding of the variation of gap anisotropywith doping within the dominant sign-changing s -wavechannel, particularly the existence or nonexistence ofnodes; the interplay with d -wave pairing; the rough size of T c ; and the origin of particle-hole asymmetry in the phasediagram. In retrospect, such agreement was somewhatfortuitous, possibly because the 122 systems have largeFermi surface pockets of both hole- and electron-type,and are relatively weakly correlated. In other pnictideslike 111 , and in 11 Fe-chalcogenide systems ,correlation effects are considerably more significant. InLiFeAs, for example, angle-resolved photemission spec-troscopy (ARPES) measurements show that the Γ-centered d xz /d yz hole pockets are considerably smallerthan predicted by density functional theory (DFT), whilethe d xy pocket is larger. Taking these effects into ac-count via a set of renormalized energy bands is insuffi-cient, however, to account for the accurate gap structureof LiFeAs within spin-fluctuation theory (see Ref. 15and references therein).The consequences of correlations for the band structureof FeSC are more profound than simple Fermi surfaceshifts, however. If one examines compounds where the d -bands are closer to half-filling (5 electrons/Fe), the effectof electron-electron interactions are enhanced in a waydistinctly different from one-band systems: different d orbital effective masses are enhanced by different factors.This “orbital selectivity” predicted by theory hasbeen confirmed by ARPES experiments. While most Fe-based systems have more electrons/Fe, closer to 6, theeffects are still nontrivial in the Fe-chalcogenides. Forexample, the electrons in bands with d xy orbital charac- a r X i v : . [ c ond - m a t . s up r- c on ] M a y d xy =d xz d xy d yz =d xy d yz d xz =d yz d xz CRLH
Figure 1. Fermi surfaces together with orbital character of the models considered in this work obtained from tight-bindingmodels fit to ARPES and quantum oscillation experiments. The individual sheets are labeled as indicated: (a) model forFeSe (bulk) including orbital order, (b) 2D model for FeSe monolayer derived from the previous one where maps of ARPESintensities obtained from measurements with horizontally polarized (LH) and circular polarized (CR) initial photons have beenoverlayed to show agreement to experimental results and (c) model for LiFeAs . Plots as a function of the angle ϕ aroundthe Fermi surface sheets are done with the angle measured from the k x axis as indicated in (b). ter have been claimed to exhibit single particle massesup to 10-20 times the band mass, while in d xz /d zy statesthe renormalization is closer to 3-4 .In Fermi liquid theory, excitations in a system of in-teracting fermions are described by quasiparticles thathave the same quantum numbers but deviate from thefree particles in properties such as the quasiparticle mass,which renormalizes the Fermi velocity. Generally, inter-actions in electronic systems also lead to reduced quasi-particle weights, corresponding to reduced values of theresidue at the pole of the Green’s function describingthose dressed electrons. Even in one-band systems whereorbital selectivity does not play a role, pairing in super-fluid systems with reduced Landau quasiparticle weightis an important unsolved theoretical problem. While onegenerally expects pairing interactions to be reduced asthe quasiparticle weight is suppressed as other aspects ofpairing are held fixed, pairing in completely incoherentnon-Fermi liquids is not impossible, as discussed recentlyin Ref. 33. The effect of orbital selective quasiparticleweights on pairing in FeSC has been discussed elsewherein various approximations , with differing conclusions.In this work, we implement a simple scheme to incor-porate aspects of renormalization of the electronic bandstructure, including reduced quasiparticle coherence thatis orbital selective into spin-fluctuation pairing theory,and apply it to several FeSC. This orbital selectiveapproach to pairing provides an excellent descriptionfor the superconducting gap deduced from quasiparticleinterference measurements on the nematic Fermi surfacepockets of bulk FeSe, as shown already in Ref. 10. Herewe discuss the generality of this approach, and showhow it explains the exotic gap structures of FeSe, FeSemonolayers and in the LiFeAs system as well. Thesefindings encourage us to believe that the proposedparadigm is the correct way to understand the physicsin these materials, but we cannot rule out completely that other effects affecting the gap such as spin-orbitcoupling or orbital fluctuations may contribute. Whilethe microscopic origin of the phenomenology remainsan open challenge, we believe that it provides a majorstep towards a quantitative, material-specific theory ofsuperconductivity in strongly correlated FeSC. II. MODEL
The starting point of any uncorrelated multiband sys-tem is the electronic structure described by a tight-binding model H = (cid:88) k σ(cid:96)(cid:96) (cid:48) t (cid:96)(cid:96) (cid:48) k c † (cid:96)σ ( k ) c (cid:96) (cid:48) σ ( k ) , (1)where c † (cid:96)σ ( k ) is the Fourier amplitude of an operator thatcreates an electron in Wannier orbital (cid:96) with spin σ and t (cid:96)(cid:96) (cid:48) k is the Fourier transform of the hoppings. By a unitarytransformation from orbital to band space, H becomesdiagonal H = (cid:80) k σµ ξ µ ( k ) c † µσ ( k ) c µσ ( k ), with eigenener-gies ξ µ ( k ) and c † µσ ( k ) creating an electron in Bloch state µ, k .There is no way to determine empirically the electronicstructure ξ µ ( k ) of the uncorrelated reference system cor-responding to a given real material. However, experimen-tal probes like ARPES and quantum oscillations provideinformation on the real single-particle spectrum, whichwe will call ˜ E µ ( k ). Since we do not have access to ξ µ ( k ),we will henceforth use the term “uncorrelated” to meana model for an electronic structure where the quasipar-ticles have unit weight; in this work we only work withsuch models where the eigenenergies ˜ E µ ( k ) have beenobtained by fit to experiment. In Fig. 1 we show ex-amples of Fermi surfaces derived from the eigenenergies˜ E µ ( k ). For three dimensional (3D) models considered inthis work, the zero energy surfaces, i.e. the set of k vec-tors with ˜ E µ ( k ) = 0 are corrugated tubes identified as π ,0) ( π , π ) (0, π ) (0,0)00.511.5 Figure 2. Comparison of the orbitally diagonal components ofthe susceptibility of the uncorrelated model for bulk FeSe (a)and the same quantities including the quasiparticle weightsthat suppress contributions from orbitals with small weightfactors according to Eq. (4) (b). α , δ and ε sheets in Fig. 1(a) (FeSe, bulk) or the β and γ sheets in (c) (LiFeAs), but can also be closed surfacesas the α pocket in (c). For a 2D model as shown in (b),the Fermi surface is given by elliptical lines such that itis convenient to plot quantities as a function of the angle ϕ .In the orbital basis the “uncorrelated” Green’s functionis given by G (cid:96)(cid:96) (cid:48) ( k , ω n ) = (cid:88) µ a (cid:96)µ ( k ) a (cid:96) (cid:48) ∗ µ ( k ) iω n − ˜ E µ ( k ) , (2)where a (cid:96)µ ( k ) are the matrix elements of the unitary trans-formation mentioned above. The orbital weight | a (cid:96)µ ( k ) | becomes important when discussing low-energy (Fermi-surface driven) properties and is therefore visualized colorcoded for the important Fe d orbitals (cid:96) = { d xy , d xz , d yz } in Fig. 1 as well.In order to include the full effects of correlations, wefurther make the orbital selective ansatz that the oper-ators c † (cid:96) ( k ) create quasiparticles with weight √ Z (cid:96) in or-bital (cid:96) , c † (cid:96) ( k ) → √ Z (cid:96) c † (cid:96) ( k ) . Note that (cid:96) runs over the Fe3 d orbitals ( d xy , d x − y , d xz , d yz , d z − r ). The associ-ated Green’s function becomes˜ G (cid:96)(cid:96) (cid:48) ( k , ω n ) = (cid:112) Z (cid:96) Z (cid:96) (cid:48) (cid:88) µ a (cid:96)µ ( k ) a (cid:96) (cid:48) ∗ µ ( k ) iω n − ˜ E µ ( k ) , (3) where ˜ E µ ( k ) are the renormalized band energies. A sim-ilar approach has been used recently when parametriz-ing the normal state Green’s functions in a Fermi liq-uid picture , with the formal difference that we ex-plicitly employ the renormalized quasiparticle energies˜ E µ ( k ), which include the static real part of the self-energy, and retain the quasiparticle weights in the numer-ator. Following state-of-the-art pairing calculations fromspin-fluctuation theory (see Appendix C), impor-tant effects of the √ Z (cid:96) factors enter in two places: 1) thecalculation of the susceptibility includes the renormal-ized quasiparticle Green’s function, and 2) when project-ing the pairing interaction from orbital to band space,one needs to account for the replacement of c † (cid:96) ( k ) →√ Z (cid:96) c † (cid:96) ( k ). In cases where the Hamiltonian already cor-rectly describes the quasiparticle energies of a correlatedsystem ξ µ ( k ) → ˜ E µ ( k ) (as obtained, e.g., from fits tomeasured quasiparticle energies from spectroscopic ex-periments), the bare susceptibility in orbital space needsto be simply multiplied by the quasiparticle weights˜ χ (cid:96) (cid:96) (cid:96) (cid:96) ( q ) = (cid:112) Z (cid:96) Z (cid:96) Z (cid:96) Z (cid:96) χ (cid:96) (cid:96) (cid:96) (cid:96) ( q ) , (4)in order to obtain the corresponding quantity (with tilde)in the correlated system. Our models as shown in Fig. 1already match the true quasiparticle energies ˜ E µ ( k ), suchthat we can use Eq. (4) to examine the effect of thequasiparticle weights on the susceptibility. In Fig. 2(a),the diagonal components of the orbitally resolved sus-ceptibilities where (cid:96) = (cid:96) = (cid:96) = (cid:96) are plotted asobtained from our model of FeSe (bulk). For all or-bitals, the overall magnitude is similar (except for (cid:96) = d z Figure 3. Plot of the spectral function at zero energyin the first Brillouin zone. (a) Spectral function A ( k ,
0) = − /π Im Tr G ( k ,
0) of the uncorrelated model for FeSe (bulk)at k z = 0 with the Green’s function as in Eq. (2). (b)Spectral function ˜ A ( k ,
0) of the model including quasiparticleweights inducing orbital selective reduced coherence. For thepair scattering of Cooper pairs at momenta k to k (cid:48) on theFermi surface (arrows) two quantities determine the scatter-ing strength: (i) the susceptibility ˜ χ ( q ) to which the pairingvertex Γ k , k (cid:48) is proportional and (ii) the quasiparticle weight atinitial and final momentum. In summary, some processes getlargely suppressed (thin red and blue arrows) such that otherprocesses (thick green arrow) dominate the Cooper pairing. π ,0) ( π , π ) (0, π ) (0,0) χ ( q ) Figure 4. Results for FeSe (bulk): (a) Calculated susceptibility with quasiparticle weights ( ˜ χ , thick lines) compared to thesusceptibility without quasiparticle weights ( χ , thin dashed lines), (b) gap symmetry function as obtained from conventionalspin-fluctuation pairing and (c) the same quantity when taking into account orbital dependent quasiparticle weights. For bothcalculations, the dominant pair scattering processes leading to a large order parameter are symbolized with a double arrow.The calculations are done for a fixed ratio J = U/
6, but with an overall scale U as indicated. that does not play any role for the subsequent discus-sion), but the momentum structure is distinct: The d xy component has a maximum at q = ( π, π ), whereas thecomponents for d yz ( d xz ) have maxima at q = ( π, q = (0 , π )). Introducing quasiparticle weights as in-dicated in Fig. 2(b), it is obvious that some componentsare suppressed more than others such that for the presentchoice of {√ Z l } = [0 . , . , . , . , . d yz contribution dominates . In a similar way,the pairing interaction gets modified by prefactors fromquasiparticle weights (see Appendix C). Physically, thismeans that orbital-selective pairing occurs because pair-ing from certain quasiparticle states is suppressed morethan others because the states themselves are less coher-ent.To visualize this effect, we have plotted the spectralfunction A ( k , ω ) = − /π Im Tr G ( k , ω ) for k z = 0 atzero energy in Fig. 3(a) for the uncorrelated systemand in (b) with the same choice of quasiparticle weightsas discussed above. We use the bulk FeSe Fermisurface discussed below as an illustration of the idea,but details of the bands are not important for thispurpose. The superconducting order parameter isnow determined by the strength of the pair scatteringΓ k , k (cid:48) of a Cooper pair at k to k (cid:48) which is proportionalto the susceptibility within the spin-fluctuation ap-proach. In the uncorrelated case, scattering processesinvolving three pairs of k -vectors as depicted by thearrows in Fig. 3 are comparable in magnitude (withthe process in blue involving d xy states being slightlylarger). Taking into account the quasiparticle weights,the spectral function and thus the pair scattering issuppressed on parts of the Fermi surface. Consequently,the processes involving d yz states (green, thick ar-row) dominate over those involving d xy states (blue)and d xz states (red), making the pairing orbital selective. III. BULK FeSe
Early thermodynamic and transport studies of bulk FeSe, as well as STM supported a state with gapnodes . However, more recent measurements oflow-temperature specific heat , STM , thermalconductivity and penetration depth have founda tiny spectral gap, indicating that the gap function ishighly anisotropic but may not change sign on any givensheet. The only experiments that provide information onthe location of these deep minima are an ARPES mea-surement on the related Fe(Se,S) material and a re-cent quasiparticle interference (QPI) experiment , bothof which find deep minima on the tips of the hole ellipseat the center of the Brillouin zone. The latter also distin-guishes deep minima on the tips of the ε electron pocket“ellipse”.To test the mechanism of orbital selective pairing de-termined by reduced coherence of some quasiparticles,we show first how this mechanism modifies results forthe susceptibility and the superconducting gap for bulkFeSe. Our starting point is a tight-binding model withhoppings adapted such that the spectral positions of thequasiparticle energies fit recent findings using ARPES,quantum oscillations and STM experiments . Asthe band energies are “measured” in this case, these canbe identified with the renormalized band energies ˜ E µ ( k )in the presence of correlations, yielding the Fermi surfacein Fig. 1(a).To construct a proper approximation of the quasipar-ticle Green’s function [Eq. (3)], we need to addition-ally include quasiparticle weights. Next, we fix the ra-tio J = U/ andoptimize the weights in the orbital basis. The resultis {√ Z l } = [0 . , . , . , . , . α pocket, as seenfrom Fig. 4(c). These values for Z l are in reasonableagreement with general trends in FeSC: the d xy orbitalexhibits strongest correlations (smallest weight) , whilethe d x − y orbital is the most weakly correlated . Wenote that the resulting gap structure is very different fromthe one obtained from conventional spin-fluctuation cal- − − Figure 5. Results for FeSe (bulk): Plot of the gap functionaround the Fermi surface pockets for (a) the conventional spinfluctuation calculation and (b) a calculation using the spin-fluctuation pairing in presence of quasiparticle weights. Fordirect comparison, the data from a Bogoliubov QPI analysisfrom Ref. and a ARPES investigation on a related com-pound FeSe(S) are displayed as well. culations (which also show a distortion from tetragonalsymmetry as expected) , a result of the very differentmomentum structure of the pairing interaction [compareFig. 4(b,c)]: The largest gap magnitude is on the tipelectron pocket ( ε ) centered at the X point for the con-ventional calculation, because the largest pair scatteringΓ k , k (cid:48) connects this area of the Fermi surface with thecorresponding one on the Y centered pocket [blue arrowin Figs. 3(a) and 4(b)]. It appears on the α pocketwhen using the orbital selective pairing ansatz, becausethe dressed electrons mediate the strongest Cooper pairscattering from the flat area of the α pocket to the flatarea of the ε pocket, where also the gap is maximal [greenarrow in Figs. 3(b) and 4(c)]. The physical origin of thiscan be attributed to the strong splitting of weights of the d xz and d yz orbitals where states of the d xz orbital arevery incoherent.We observe that the susceptibility ˜ χ , originallystrongly dominated by ( π, π ), now shows dominant stripefluctuations with q = ( π,
0) [see Fig. 4(a)]. This re-sult is in agreement with findings from neutron scatter-ing experiments which find strong stripe fluctuationsat low energies. Taking into account the results of a re- cent ARPES experiment with the conclusion that theelectronic structure of FeSe evolves in such a way thatit becomes less correlated as temperature increases, wecan conclude that weight of the spin-fluctuations shouldshift from ( π,
0) towards ( π, π ) as temperature increases.This can be understood directly from Eq. (4), wherethe different orbital components of the susceptibility areweighted according to the quasiparticle weights; the d xy components which are peaked at ( π, π ) get suppressed.The d xz components, peaked at (0 , π ), are suppressed aswell (see Fig. 2). On individual pockets, the gap func-tion then follows the orbital content of the orbital withstrongest contribution (in this case, the d yz orbital) [com-pare Fig. 1 (a)].Consequently, the pairing is changed by two mecha-nisms: First, it is modified directly by the quasiparticleweights as discussed earlier and, second, the peakshifts in q in the (RPA) susceptibility. Both of theseeffects make the pair scattering in the d yz orbital moreimportant [green thick arrow in Fig. 3(b)] yieldingthe gap structure as shown in Fig. 4(c). To make theagreement to experiment evident, we plot in Fig. 5 thegap function at a cut of the Fermi surface at k z = π comparing to results from two different spectroscopicmethods. While the conventional calculation [5 (a)] doesnot show any similarities, the correspondence in (b) isevident. Finally, we note that this picture is differentthan that ascribed to orbital selective physics in the“strong-coupling” t − J model approach, where the d xy pairing channel is enhanced rather than suppressed . IV. MONOLAYER FeSe ON SrTiO Despite considerable excitement over the high criticaltemperature in the FeSe/STO monolayer system, lim-ited information is available regarding the structure ofthe superconducting gap. Early ARPES measurementssuggested an isotropic gap on electron pockets . The-oretical possibilities for pairing states in the presence ofmissing Γ-centered hole band were discussed in Ref. 15.Quite recently, a new ARPES study identified significantand unusual anisotropy on a single unhybridized ellipti-cal electron pocket , whereby the gap acquired globalmaxima at the ellipse tips, and additional local max-ima at the ellipse sides. These authors showed that thestructure cannot be explained using any of the low-orderBrillouin zone harmonics expected from so-called “strongcoupling” electronic pairing theories.Within the model for the electronic structure of bulkFeSe, we perform a calculation with a few modificationsto account for differences in the monolayer from the bulk:(1) We ignore all hoppings out of the plane, yielding astrictly 2D system. (2) We neglect orbital order, whichhas never been observed in the monolayer. (3) Experi-mentally, only electron-like Fermi pockets have been de-tected, suggesting that the monolayer is actually elec-tron doped. Possible reasons for this doping are chargetransfers from the substrate or surface defects. We there- xy d xz d yz − −
90 0 90 180 (a) − −
90 0 90 18051015 (b) − −
90 0 90 18051015 (c)
Figure 6. Results for monolayer FeSe: (a) Orbital weight at the Fermi surface. (b) Superconducting gap obtained fromconventional spin-fluctuation theory, and (c) the same quantity including orbital dependent quasiparticle weights compared tomeasured gap functions in ARPES. Symmetry operations of the tetragonal system have been applied to the measured data.All calculations were done for a fixed ratio J = U/
10, with overall scale U as indicated. fore apply a rigid band shift by δµ = 60 meV, whichremoves the Γ-centered hole pocket and leaves electronpockets that have the size and shape of measured spec-tral functions in ARPES , with n = 6 .
12 electrons/Fe,see Figs. 1(b) and 6(a) for a plot of the orbital character.The quasiparticle weights in the monolayer may be dif-ferent from the bulk for two reasons: (1) The absenceof the orbital order, i.e., the tetragonal crystal struc-ture dictates that the weights for d xz and d yz orbitalsare degenerate (unlike bulk FeSe). (2) Correlations maybe different in the monolayer where a tendency towardsweaker correlations was found recently , such that we fixthe ratio J = U/
10 in this case.At this point, we note that the states on the Fermisurface have only tiny orbital weight of d z and d x − y character, and additionally there are no pair scatter-ing processes from k to k (cid:48) with q = ( π,
0) [or q =(0 , π )] such that a fit procedure with all quasiparti-cle weights will be under-determined. In the opti-mization procedure, we therefore fix the weights to (cid:112) Z x − y = 0 . > √ Z z = 0 . {√ Z l } =[0 . , . , . , . , . . This result doeschange the susceptibility slightly, but keeps the ( π, π )fluctuations dominant; for details we refer to Fig. S 1 inthe Appendix. These fluctuations drive an overall (node-less) d -symmetry ground state as expected, but with anunusual structure modified strongly by orbital correla-tions, with the result as shown in Fig. 6(b,c). Evidentlythe gap function for the standard spin-fluctuation calcu-lation [Fig. 6(b)] mostly follows the orbital content of the d xy orbital [compare Fig. 6(a) for a plot of the orbitalweights as a function of angle ϕ around the X-centeredpocket ]. For the current Fermi surface, this is expectedbecause the pairing interaction is dominated by intra-orbital processes, and the d xy orbital has large weight atpositions k and k (cid:48) on the Fermi surface which are sep-arated roughly by ( π, π ) and can take advantage of thestrong peak in the susceptibility at that q vector. Theother two orbitals play a negligible role in the pairing process. This situation is modified once the pairing in-teraction is renormalized by the quasiparticle weights andtherefore reduces the contribution of the d xy orbital. Themain effect is that a second maximum in the gap functionappears at a position in momentum space where the d xz or d yz orbital is dominant [see Fig. 6 (c)].In the pairing process, intra-orbital, inter-pocket contributions dominate, whereby one pair on the X pocket of d yz character scatters into another pair onthe Y pocket with the same orbital character, meaningthat the latter pair must be located on the tip of the Y -pocket where the gap has largest magnitude. Becausethe total weight of this orbital is smaller there, theorder parameter for k states dominated by this orbitalis enhanced. In summary, one gets a gap structure witha large maximum at the tip of the ellipse and a smallmaximum at the flat part of the ellipse, remarkablysimilar to that detected by experiment. V. LiFeAs
LiFeAs is another Fe-based superconductor thatis known to have a Fermi surface quite differentfrom that predicted from DFT. Several theoreticalattempts to understand the ARPES-determinedgap structure were reviewed recently in Ref. 15.All were based on an “engineered” tight-binding bandstructure consistent with ARPES data , i.e., contain-ing the correct spectral positions of the bands (includingthe orbital content). Despite some success in explainingcertain features of the gap structure, others were not re-produced properly in all approaches, although Ref. 34claimed a good overall fit to experiment.To reveal how and whether the standard spin-fluctuation theory result changes upon inclusionof quasiparticle weights, we use the same methodas described above for a band structure relevantto LiFeAs . The corresponding Fermi surface isshown in Fig. 1(c). First, we note that moderatechanges in the quasiparticle weights which we set to α β γ (a) g ( k ) U = − −
90 0 90 180 − − − − (b) − −
90 0 90 180 − − − − (c) Figure 7. Results for LiFeAs: (a) 3D plot of the gap function as obtained from spin-fluctuation calculation includingquasiparticle weights. (b) Cut at k z = π of the result of the s-wave gap function from conventional spin-fluctuation theory(solid lines) plotted as a function of angle ϕ (as defined in Fig. 1) around the pockets (Γ-centered hole pocket ( α , magenta),M-centered hole pocket ( γ , cyan) and X-centered electron pocket ( β , black)) together with experimental results. The measuredmagnitudes of the gap from an ARPES experiment are symmetrized and displayed as crosses, and those from a BogoliubovQPI experiment as filled dots. (c) The same quantity for the gap function as shown in (a) also compared to experimentaldata. All calculations are done for a fixed ratio J = 0 . U , but with overall scale U as indicated. {√ Z l } = [0 . , . , . , . , . s ± symmetry, evenwith small values of J . Note that the conventionalspin-fluctuation scenario, d and s wave solutions arenearly degenerate, a consequence of the poor ( π, . Secondly, orbitalselectivity enhances the gap on the small Γ-centeredhole pocket ( α pocket), see Fig. 7(a). This appearsto correct the crucial discrepancy in the calculation ofWang et al . relative to experiment [see Fig. 7 (b,c)].Finally, the procedure leads to weaker anisotropy of thegap on the large d xy dominated pocket, also in betteragreement with experiment, whereas small deviationsbetween the ARPES data and our calculation onthe electron pockets persist which could be due tohybridization of the corresponding bands. We did notinvestigate effects of spin-orbit coupling in this casesince these are supposed to be small . Note furtherthat the (angular) position of the maximum gap on theelectron pockets change from 0 degrees to slightly off90 degrees, opening the possibility of two maxima (andtwo minima). Unlike the models for FeSe (bulk) andmonolayer FeSe, all three orbitals ( d xy , d xz , d yz ) play animportant role in determining the gap anisotropy on the β pockets, making it more sensitive to changes in theelectronic structure. VI. DISCUSSION
The above results are extremely encouraging, sug-gesting that the orbital selective correlation effects areindeed required when applying spin-fluctuation pairingtheory to Fe-chalcogenide and more strongly correlated Fe-based superconductors. We caution, however, thatwe have not derived the renormalizations entering thepair vertex self-consistently from a microscopic theory.Efforts along these lines are in progress. Secondly, byconstruction the quasiparticle renormalizations describeonly the states near the Fermi level. Comparison withARPES measurements should be performed carefully, asthese analyses tend to emphasize renormalizations onmuch larger energy scales, which may be quite different.Possible imprints of the orbital selectivity could be vis-ible in the penetration depth if calculated within thesame theoretical framework, or Friedel oscillations closeto impurities in the case of bulk FeSe which are rotatingin direction as a function of energy . Calculations alongthese lines are also in progress. VII. CONCLUSIONS
In the absence of a fully controlled many-bodytreatment of electronically paired superconductivity, itmay be very valuable to have a simple phenomenologicalyet microscopic approach that includes aspects of thelow-energy quasiparticle renormalizations that affectpairing most strongly. We have presented a paradigmthat allows for suppressed quasiparticle weight withinthe framework of conventional spin-fluctuation pairingtheory, and argued that it provides accurate descriptionsfor the previously inexplicable superconducting energygap structures of the most strongly correlated FeSC. Wehave given results of explicit calculations in three caseswhere correlations are known to play an important role,bulk FeSe, monolayer FeSe on STO, and LiFeAs. Theseresults reveal an immediate challenge to determine if ourapproach can be combined with microscopic calculationsof quasiparticle weights to yield a material-specifictheory with predictive power for strongly correlatedFeSC.
ACKNOWLEDGMENTS
We would like to thank A.V. Chubukov, D.J.Scalapino, and D. D. Scherer for useful discussions.A.Kr. and B.M.A. acknowledge support from a Lund-beckfond fellowship (Grant No. A9318). P.J.H. acknowl-edges support through Department of Energy DE-FG02-05ER46236. J.C.S.D. acknowledges gratefully supportfrom the Moore Foundation’s EPiQS Initiative throughGrant GBMF4544, and the hospitality and support ofthe Tyndall National Institute, University College Cork,Cork, Ireland. P.O.S. and A.Ko. acknowledge sup-port from the Center for Emergent Superconductivity,an Energy Frontier Research Center, headquartered atBrookhaven National Laboratory and funded by the U.S.Department of Energy under DE-2009-BNL-PM015.
Appendix A: Hamiltonian and construction ofGreen’s function
Considering the tight binding Hamiltonian, Eq. (1)together with its diagonalization to band basis, onecan construct the Green’s function in the band basis G µ ( k , ω n ) = [ iω n − ξ µ ( k )] − . The unitary transforma-tion that takes one from the band basis (Greek indices)to the orbital basis (Roman indices) is c (cid:96)σ ( k ) = (cid:88) ν a (cid:96)ν ( k ) c νσ ( k ) . (A1)Unitarity implies (cid:88) (cid:96) a (cid:96)ν ( k ) a (cid:96)µ ( k ) ∗ = δ µν (A2)so we can invert (A1) to find the orbital basis Green’sfunction as stated in the main text, G (cid:96)(cid:96) (cid:48) ( k , ω n ) = (cid:88) µ a (cid:96)µ ( k ) a (cid:96) (cid:48) ∗ µ ( k ) G µ ( k , ω n )= (cid:88) µ a (cid:96)µ ( k ) a (cid:96) (cid:48) ∗ µ ( k ) iω n − ξ µ ( k ) . (A3) Appendix B: Quasiparticle description in band space
At this point, we make a short remark about the impli-cations of quasiparticles in band representation. Startingfrom Eq. (3), we can transform back to the band basisand obtain the quasiparticle Green’s function˜ G ν ( k , ω n ) = (cid:88) s,p a sν ∗ ( k ) a pν ( k ) ˜ G sp ( k , ω n )= (cid:32)(cid:88) s,p | a sν ( k ) | | a pν ( k ) | (cid:112) Z s (cid:112) Z p (cid:33) G ν ( k , ω n )= ˜ Z ν ( k ) G ν ( k , ω n ) ≡ ˜ G ν ( k , ω n ) , (B1) where ˜ Z ν ( k ) ≡ [ (cid:80) s | a sν ( k ) | √ Z s ] are the quasiparticleband weights near the Fermi surface. If the point k onthe Fermi surface sheet ν is dominated by a particularorbital weight | a sν ( k ) | , the quasiparticle weight for thatband will be given predominantly by Z s . Calculating thespectral function from such a Green’s function and plot-ting versus k at ω = 0, one directly sees that part of theFermi surface is strongly suppressed in intensity when-ever an orbital dominates that has small quasiparticleweight, i.e., is strongly correlated. In Fig. 3 we showthis effect of the spectral function on the example of ourmodel for FeSe (bulk).We stress that the approach applied in this paper isphenomenological in the sense that the band renormal-izations and the quasiparticle weights are not obtainedself-consistently from the same bare interaction param-eters. Thus we do not address the problem of how toquantitatively capture nontrivial self-energy effects andthe eventual transition to non-Fermi-liquid behavior withincreasing correlations or hole-doping , but simply relyon a wealth of previous theoretical studies showing theexistence of orbital selectivity, and study their influenceon the superconducting pairing structure. Appendix C: Spin-fluctuation pairing: uncorrelatedmodel
Here, we remind the reader of the approach to calculat-ing the gap function in the usual spin fluctuation pairingmodel . First, local interactions are included via thefive-orbital Hubbard-Hund Hamiltionan, H = H + U (cid:88) i,(cid:96) n i(cid:96) ↑ n i(cid:96) ↓ + U (cid:48) (cid:88) i,(cid:96) (cid:48) <(cid:96) n i(cid:96) n i(cid:96) (cid:48) + J (cid:88) i,(cid:96) (cid:48) <(cid:96) (cid:88) σ,σ (cid:48) c † i(cid:96)σ c † i(cid:96) (cid:48) σ (cid:48) c i(cid:96)σ (cid:48) c i(cid:96) (cid:48) σ (C1)+ J (cid:48) (cid:88) i,(cid:96) (cid:48) (cid:54) = (cid:96) c † i(cid:96) ↑ c † i(cid:96) ↓ c i(cid:96) (cid:48) ↓ c i(cid:96) (cid:48) ↑ , where the interaction parameters U , U (cid:48) , J , J (cid:48) aregiven in the notation of Kuroki et al. with the choice U (cid:48) = U − J , J = J (cid:48) , leaving only U and J/U tospecify the interactions. Here, (cid:96) is an orbital indexwith (cid:96) ∈ (1 , . . . ,
5) corresponding to the Fe 3 d orbitals( d xy , d x − y , d xz , d yz , d z − r ). The orbital susceptibilitytensor in the normal state is now given as χ (cid:96) (cid:96) (cid:96) (cid:96) ( q ) = − (cid:88) k,µν M µν(cid:96) (cid:96) (cid:96) (cid:96) ( k , q ) G µ ( k + q ) G ν ( k ) , (C2)where we have adopted the shorthand k ≡ ( k , ω n ), anddefined M µν(cid:96) (cid:96) (cid:96) (cid:96) ( k , q ) = a (cid:96) ν ( k ) a (cid:96) , ∗ ν ( k ) a (cid:96) µ ( k + q ) a (cid:96) , ∗ µ ( k + q ) . (C3)The Matsubara sum in Eq. (C2) is performed analyt-ically, and we then evaluate χ (cid:96) (cid:96) (cid:96) (cid:96) by integratingover the full Brillouin zone. As noted earlier , theFermi surface nesting condition gives significant con-tributions to the susceptibility, but finite-energy nest-ing also contributes. The spin- ( χ RPA1 ) and charge-fluctuation ( χ RPA0 ) parts of the RPA susceptibility for q = ( q , ω n = 0) are now defined within the random phaseapproximation as χ RPA1 (cid:96) (cid:96) (cid:96) (cid:96) ( q ) = (cid:110) χ ( q ) (cid:2) − ¯ U s χ ( q ) (cid:3) − (cid:111) (cid:96) (cid:96) (cid:96) (cid:96) , (C4a) χ RPA0 (cid:96) (cid:96) (cid:96) (cid:96) ( q ) = (cid:110) χ ( q ) (cid:2) U c χ ( q ) (cid:3) − (cid:111) (cid:96) (cid:96) (cid:96) (cid:96) . (C4b)The total spin susceptibility at ω = 0 is then given bythe sum χ ( q ) = 12 (cid:88) (cid:96)(cid:96) (cid:48) χ RPA1 (cid:96)(cid:96)(cid:96) (cid:48) (cid:96) (cid:48) ( q ) . (C5)The interaction matrices ¯ U s and ¯ U c in orbital spaceare composed of linear combinations of U, U (cid:48) , J, J (cid:48) andtheir forms are given, e.g., in Ref. 39. We focus here onthe spin-singlet vertex for pair scattering between bands ν and µ ,Γ νµ ( k , k (cid:48) ) = Re (cid:88) (cid:96) (cid:96) (cid:96) (cid:96) a (cid:96) , ∗ ν ( k ) a (cid:96) , ∗ ν ( − k ) (C6) × Γ (cid:96) (cid:96) (cid:96) (cid:96) ( k , k (cid:48) ) a (cid:96) µ ( k (cid:48) ) a (cid:96) µ ( − k (cid:48) ) , where k and k (cid:48) are quasiparticle momenta restricted tothe pockets k ∈ C ν and k (cid:48) ∈ C µ , and is defined in termsof the the orbital space vertex functionΓ (cid:96) (cid:96) (cid:96) (cid:96) ( k , k (cid:48) ) = (cid:20)
32 ¯ U s χ RPA1 ( k − k (cid:48) ) ¯ U s (C7)+ 12 ¯ U s −
12 ¯ U c χ RPA0 ( k − k (cid:48) ) ¯ U c + 12 ¯ U c (cid:21) (cid:96) (cid:96) (cid:96) (cid:96) . Using this approximation to the vertex, we now considerthe linearized gap equation − V G (cid:88) µ (cid:90) FS µ dS (cid:48) Γ νµ ( k , k (cid:48) ) g i ( k (cid:48) ) | v F µ ( k (cid:48) ) | = λ i g i ( k ) (C8)and solve for the leading eigenvalue λ and correspondingeigenfunction g ( k ). Here v F µ ( k (cid:48) ) is the Fermi velocityof band µ and the integration is over the Fermi surfaceFS µ . The eigenfunction g i ( k ) for the leading eigenvaluethen determines the symmetry and structure of the lead-ing pairing gap ∆( k ) ∝ g ( k ) close to T c . Finally, thearea of the Fermi surface sheets is discretized using a De-launay triangulation algorithm that transforms the inte-gral equation Eq. (C8) into an algebraic matrix equationwhich is solved numerically. Typically, we use a k -meshof 80 × ×
30 points for the k integration and totally ≈ k mesh is on the orderof 100 ×
100 and ≈
200 points on all Fermi sheets arerequired for reasonably converged results. π ,0) ( π , π ) (0, π ) (0,0) χ ( q ) Figure S 1. Susceptibility ˜ χ for our model for the mono-layer FeSe as calculated from the orbital selective ansatzusing the quasiparticle Green’s functions with {√ Z l } =[0 . , . , . , . , . χ ), where the interactions have been scaleddown. π ,0) ( π , π ) (0, π ) (0,0) χ ( q ) Figure S 2. Total susceptibility χ for LiFeAs as calculatedfrom the electronic structure using a 3D model and samequantity ˜ χ , but calculated using the quasiparticle Green’sfunctions with {√ Z l } = [0 . , . , . , . , . Appendix D: Spin-fluctuation pairing includingquasiparticle weights
In this appendix, we show the modified equations forthe pairing calculation as outlined above, but includingquasiparticle weights from dressed electrons. Taking theansatz for the dressed Green’s function, Eq. (3), it isobvious that from Eq. (C2) immediately follows Eq. (4)which is then used in Eqs. (C4) instead of χ (cid:96) (cid:96) (cid:96) (cid:96) ( q )for the dressed quantities. The total susceptibility thenreads as ˜ χ ( q ) = 12 (cid:88) (cid:96)(cid:96) (cid:48) ˜ χ RPA1 (cid:96)(cid:96)(cid:96) (cid:48) (cid:96) (cid:48) ( q ) . (D1)For the FeSe (bulk) model, the total susceptibility is dis-played and discussed in the main text, because the quasi-particle weights have a strong effect on the qualitativebehavior. At this point, it is worth mentioning that this0 − −
90 0 180 180 − Δ k − −
90 0 180 180 − Δ k − −
90 0 180 180 − Δ k − −
90 0 180 180 − Δ k − −
90 0 180 180 − Δ k − −
90 0 180 180 − Δ k − −
90 0 180 180 − − Δ k − −
90 0 180 180 − − Δ k Figure S 3. Comparison of the calculated gap function for FeSe (bulk) to experimental data from Refs. and . Calculatedgap function from the two-dimensional model at k z = 0 with conventional spin-fluctuation pairing and interaction parameters U = 0 .
33 eV, J = U/
6, (a), a calculation with the orbitally selective pairing ansatz as described in the main text (b). Sincethe quasiparticle weights reduce the susceptibility in general, a slightly larger interaction of U = 0 .
54 eV was chosen, while theratio J = U/ k z = 0 cut from theconventional spin-fluctuation calculation, (d) the same cut from the orbitally selective ansatz, cuts for k z = π are shown in Fig. 4.Variations of the fits for the 2D model, where the ratio of the quasiparticle weights of the d yz and d xz orbital is constrainedto the value as indicated on the figure (g,h). The resulting values are then {√ Z l } = [0 . , . , . , . , . {√ Z l } = [0 . , . , . , . , . is not the case for the model of monolayer FeSe, wherethe quasiparticle weights are chosen closer to unity (ac-counting for smaller correlation effects in this material).In Fig. S 1, it can be seen that the total susceptibilityis practically unchanged. Similar conclusions can alsobe drawn from the comparison of the total susceptibili-ties for LiFeAs in the uncorrelated and correlated model,see Fig. S 2. Note that the quasiparticle weights Z l are consistent with DMFT results where it is found that t g orbitals are strongly correlated with d xy strongest,and components of the susceptibility get suppressed ( d xy strongest) .The equation˜Γ (cid:96) (cid:96) (cid:96) (cid:96) ( k , k (cid:48) ) = (cid:20)
32 ¯ U s ˜ χ RPA1 ( k − k (cid:48) ) ¯ U s (D2)+ 12 ¯ U s −
12 ¯ U c ˜ χ RPA0 ( k − k (cid:48) ) ¯ U c + 12 ¯ U c (cid:21) (cid:96) (cid:96) (cid:96) (cid:96) for the orbital space vertex function is basically un-changed except for the addition of the tilde. In the con-struction of the pair scattering vertex, additional quasi-particle weights enter from the replacement c † (cid:96) ( k ) →√ Z (cid:96) c † (cid:96) ( k ) such that it reads˜Γ νµ ( k , k (cid:48) ) = Re (cid:88) (cid:96) (cid:96) (cid:96) (cid:96) (cid:112) Z (cid:96) (cid:112) Z (cid:96) a (cid:96) , ∗ ν ( k ) a (cid:96) , ∗ ν ( − k ) × ˜Γ (cid:96) (cid:96) (cid:96) (cid:96) ( k , k (cid:48) ) (cid:112) Z (cid:96) (cid:112) Z (cid:96) a (cid:96) µ ( k (cid:48) ) a (cid:96) µ ( − k (cid:48) ) (D3)and enters Eq. (C8) instead of Γ νµ ( k , k (cid:48) ). Appendix E: Comparison of 2D calculations and 3Dcalculations
In the present paper, we discuss three different phys-ical systems, two of them parametrized using a bandstructure including a k z dispersion as well. As noted al-ready earlier, the susceptibility as calculated from a 3Dmodel (with weak dispersion in k z direction) shows onlyvery small dependence on k z . Conclusions similar tothe ones in the main text can also be drawn in a two-dimensional calculation, where the initial band structureis just the one at k z = 0. Taking the same interactionparameters and quasiparticle weights, one obtains qual-itative similar results as for the 3D calculation. Thisis expected since the electronic structure is found to bequasi-two-dimensional, and especially since the suscep-tibility and thus the pairing interaction have little de-pendence on q z . Differences in the relative magnitudesof the gap functions on the individual pockets can, how-ever, arise due to the variation of the Fermi velocities asa function of k z , e.g. the weight at k z = 0 as includedin a 2D calculation is not just the average of the partialcontributions to the density of states from different k z .In the solution of the linearized gap equation, this can in-crease the gap on individual pockets or reduce the gapas seen on the α pocket for the 3D calculation in Fig. S3 (d). Overall, the variation of the results is small andmostly of quantitative nature rather than qualitative. Wenote that the Fermi surface properties can still strongly1influence the actual superconducting order parameter insuch a calculation even if the pairing interaction itself hasnegligible variation in q z . This will occur in a 2D calcu-lation for the LiFeAs model where the Fermi surface isdifferent at cuts in k z = 0 and k z = π because of theclosed α pocket. Because of this, we have not consideredany results of a 2D calculation for this model further. Finally, we present results for the gap structure obtainedfrom a fit where the relative magnitudes of the quasipar-ticle weights of the d xz and d yz orbital are kept fixed.Even when lowering the ratio between those, the agree-ment is still good [see Fig. S 3 (g-h)], but not allowinga larger quasiparticle weight in the d yz orbital does notyield an agreement (not shown). L. de Medici,
Iron-Based Superconductivity, Weak andStrong Correlations in Fe Superconductors , edited by PeterD. Johnson, Guangyong Xu, and Wei-Guo Yin, SpringerSeries in Materials Science (Springer, 2015). Elena Bascones, Bel´en Valenzuela, and Maria Jos´eCalder´on, “Magnetic interactions in iron superconductors:A review,” C. R. Phys. , 36 (2016), iron-based super-conductors / Supraconducteurs `a base de fer. Ambroise van Roekeghem, Pierre Richard, Hong Ding,and Silke Biermann, “Spectral properties of transitionmetal pnictides and chalcogenides: Angle-resolved photoe-mission spectroscopy and dynamical mean-field theory,”C. R. Phys. , 140 (2016), iron-based superconductors /Supraconducteurs `a base de fer. Z. P. Yin, K. Haule, and G. Kotliar, “Kinetic frustrationand the nature of the magnetic and paramagnetic statesin iron pnictides and iron chalcogenides,” Nat. Mater. ,932 (2011). Yu Li, Zhiping Yin, Xiancheng Wang, David W. Tam,D. L. Abernathy, A. Podlesnyak, Chenglin Zhang, MengWang, Lingyi Xing, Changqing Jin, Kristjan Haule,Gabriel Kotliar, Thomas A. Maier, and Pengcheng Dai,“Orbital selective spin excitations and their impact on su-perconductivity of LiFe − x Co x As,” Phys. Rev. Lett. ,247001 (2016). S. Mandal, P. Zhang, and K. Haule, “How correlatedis the FeSe/SrTiO system?” ArXiv e-prints (2016),arXiv:1609.06815 [cond-mat.supr-con]. Z. R. Ye, Y. Zhang, F. Chen, M. Xu, J. Jiang, X. H. Niu,C. H. P. Wen, L. Y. Xing, X. C. Wang, C. Q. Jin, B. P.Xie, and D. L. Feng, “Extraordinary doping effects onquasiparticle scattering and bandwidth in iron-based su-perconductors,” Phys. Rev. X , 031041 (2014). Naoya Arakawa and Masao Ogata, “Orbital-selective su-perconductivity and the effect of lattice distortion in iron-based superconductors,” J. Phys. Soc. Jpn. , 074704(2011). Rong Yu, Jian-Xin Zhu, and Qimiao Si, “Orbital-selectivesuperconductivity, gap anisotropy, and spin resonance ex-citations in a multiorbital t - J - J model for iron pnictides,”Phys. Rev. B , 024509 (2014). P. O. Sprau, A. Kostin, A. Kreisel, A. E. B¨ohmer, V. Tau-four, P. C. Canfield, S. Mukherjee, P. J. Hirschfeld, B. M.Andersen, and J. C. S´eamus Davis, “Discovery of Orbital-Selective Cooper Pairing in FeSe,” ArXiv e-prints (2016),arXiv:1611.02134 [cond-mat.supr-con]. Y. Zhang, J. J. Lee, R. G. Moore, W. Li, M. Yi,M. Hashimoto, D. H. Lu, T. P. Devereaux, D.-H. Lee, andZ.-X. Shen, “Superconducting gap anisotropy in monolayerFeSe thin film,” Phys. Rev. Lett. , 117001 (2016). Y. Wang, A. Kreisel, V. B. Zabolotnyy, S. V. Borisenko,B. B¨uchner, T. A. Maier, P. J. Hirschfeld, and D. J. Scalapino, “Superconducting gap in LiFeAs from three-dimensional spin-fluctuation pairing calculations,” Phys.Rev. B , 174516 (2013). D. J. Scalapino, “A common thread: The pairing inter-action for unconventional superconductors,” Rev. Mod.Phys. , 1383–1417 (2012). Arzhang Ardavan, Stuart Brown, Seiichi Kagoshima,Kazushi Kanoda, Kazuhiko Kuroki, Hatsumi Mori, MasaoOgata, Shinya Uji, and Jochen Wosnitza, “Recent topicsof organic superconductors,” J. Phys. Soc. Jpn. , 011004(2012). P. J. Hirschfeld, “Using gap symmetry and structure to re-veal the pairing mechanism in Fe-based superconductors,”C. R. Phys. , 197 (2016), iron-based superconductors /Supraconducteurs `a base de fer. A. Chubukov,
Iron-Based Superconductivity, Itinerantelectron scenario for Fe-based superconductors , edited byPeter D. Johnson, Guangyong Xu, and Wei-Guo Yin,Springer Series in Materials Science (Springer, 2015). N. F. Berk and J. R. Schrieffer, “Effect of ferromagneticspin correlations on superconductivity,” Phys. Rev. Lett. , 433–435 (1966). D. J. Scalapino, E. Loh, and J. E. Hirsch, “ d -wave pair-ing near a spin-density-wave instability,” Phys. Rev. B ,8190 (1986). Tetsuya Takimoto, Takashi Hotta, and Kazuo Ueda,“Strong-coupling theory of superconductivity in a degen-erate Hubbard model,” Phys. Rev. B , 104504 (2004). Michaela Altmeyer, Daniel Guterding, P. J. Hirschfeld,Thomas A. Maier, Roser Valent´ı, and Douglas J.Scalapino, “Role of vertex corrections in the matrix for-mulation of the random phase approximation for the mul-tiorbital hubbard model,” Phys. Rev. B , 214515 (2016). P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin,“Gap symmetry and structure of Fe-based superconduc-tors,” Rep. Prog. Phys. , 124508 (2011). C. Platt, W. Hanke, and R. Thomale, “Functional renor-malization group for multi-orbital Fermi surface instabili-ties,” Adv. Phys. , 453 (2013). Johannes Ferber, Kateryna Foyevtsova, Roser Valent´ı, andHarald O. Jeschke, “LDA + DMFT study of the effects ofcorrelation in LiFeAs,” Phys. Rev. B , 094505 (2012). Geunsik Lee, Hyo Seok Ji, Yeongkwan Kim, ChangyoungKim, Kristjan Haule, Gabriel Kotliar, Bumsung Lee, Se-unghyun Khim, Kee Hoon Kim, Kwang S. Kim, Ki-SeokKim, and Ji Hoon Shim, “Orbital selective fermi surfaceshifts and mechanism of high T c superconductivity in cor-related a FeAs ( a = Li, Na),” Phys. Rev. Lett. , 177001(2012). Rong Yu, Qimiao Si, Pallab Goswami, and Elihu Abra-hams, “Electron correlation and spin dynamics in ironpnictides and chalcogenides,” Journal of Physics: Confer- ence Series , 012025 (2013). S. V. Borisenko, V. B. Zabolotnyy, D. V. Evtushinsky,T. K. Kim, I. V. Morozov, A. N. Yaresko, A. A. Kordyuk,G. Behr, A. Vasiliev, R. Follath, and B. B¨uchner, “Super-conductivity without nesting in LiFeAs,” Phys. Rev. Lett. , 067002 (2010). Sergey V. Borisenko, Volodymyr B. Zabolotnyy, Alex-nader A. Kordyuk, Danil V. Evtushinsky, Timur K. Kim,Igor V. Morozov, Rolf Follath, and Bernd B¨uchner, “One-sign order parameter in iron based superconductor,” Sym-metry , 251 (2012). Rong Yu and Qimiao Si, “Mott transition in multiorbitalmodels for iron pnictides,” Phys. Rev. B , 235115 (2011). M. Yi, D. H. Lu, R. Yu, S. C. Riggs, J.-H. Chu, B. Lv,Z. K. Liu, M. Lu, Y.-T. Cui, M. Hashimoto, S.-K. Mo,Z. Hussain, C. W. Chu, I. R. Fisher, Q. Si, and Z.-X.Shen, “Observation of temperature-induced crossover toan orbital-selective mott phase in A x Fe − y Se ( a =K, Rb)superconductors,” Phys. Rev. Lett. , 067003 (2013). Rong Yu and Qimiao Si, “Orbital-selective mottphase in multiorbital models for alkaline iron selenidesK − x Fe − y Se ,” Phys. Rev. Lett. , 146402 (2013). M. Yi, Z.-K. Liu, Y. Zhang, R. Yu, J.-X. Zhu, J. J. Lee,R. G. Moore, F. T. Schmitt, W. Li, S. C. Riggs, J.-H.Chu, B. Lv, J. Hu, M. Hashimoto, S.-K. Mo, Z. Hussain,Z. Q. Mao, C. W. Chu, I. R. Fisher, Q. Si, Z.-X. Shen,and D. H. Lu, “Observation of universal strong orbital-dependent correlation effects in iron chalcogenides,” Nat.Comm. , 7777 (2015). Z. K. Liu, M. Yi, Y. Zhang, J. Hu, R. Yu, J.-X. Zhu, R.-H. He, Y. L. Chen, M. Hashimoto, R. G. Moore, S.-K.Mo, Z. Hussain, Q. Si, Z. Q. Mao, D. H. Lu, and Z.-X. Shen, “Experimental observation of incoherent-coherentcrossover and orbital-dependent band renormalization iniron chalcogenide superconductors,” Phys. Rev. B ,235138 (2015). Max A. Metlitski, David F. Mross, Subir Sachdev, andT. Senthil, “Cooper pairing in non-Fermi liquids,” Phys.Rev. B , 115111 (2015). Tetsuro Saito, Seiichiro Onari, Youichi Yamakawa, HiroshiKontani, Sergey V. Borisenko, and Volodymyr B. Zabolot-nyy, “Reproduction of experimental gap structure in lifeasbased on orbital-spin fluctuation theory: s ++ -wave, s ± -wave, and hole- s ± -wave states,” Phys. Rev. B , 035104(2014). Helmut Eschrig and Klaus Koepernik, “Tight-bindingmodels for the iron-based superconductors,” Phys. Rev.B , 104503 (2009). F. Ahn, I. Eremin, J. Knolle, V. B. Zabolotnyy, S. V.Borisenko, B. B¨uchner, and A. V. Chubukov, “Supercon-ductivity from repulsion in LiFeAs: Novel s -wave symme-try and potential time-reversal symmetry breaking,” Phys.Rev. B , 144513 (2014). Youichi Yamakawa, Seiichiro Onari, and Hiroshi Kontani,“Nematicity and magnetism in FeSe and other families ofFe-based superconductors,” Phys. Rev. X , 021032 (2016). S. Graser, T. A. Maier, P. J. Hirschfeld, and D. J.Scalapino, “Near-degeneracy of several pairing channels inmultiorbital models for the Fe pnictides,” New J. Phys. ,025016 (2009). A. F. Kemper, T. A. Maier, S. Graser, H.-P. Cheng, P. J.Hirschfeld, and D. J. Scalapino, “Sensitivity of the super-conducting state and magnetic susceptibility to key aspectsof electronic structure in ferropnictides,” New J. Phys. , 073030 (2010). A. Kreisel, Y. Wang, T. A. Maier, P. J. Hirschfeld, andD. J. Scalapino, “Spin fluctuations and superconductivityin K x Fe − y Se ,” Phys. Rev. B , 094522 (2013). Note that the d x − y component is still large because ofthe choice of a quasiparticle weight close to 1. It thereforecontributes to the physical susceptibility, but has little in-fluence on the supercondicting order parameter since theorbital weight for states at low energies is small, see Fig.1. Can-Li Song, Yi-Lin Wang, Peng Cheng, Ye-Ping Jiang,Wei Li, Tong Zhang, Zhi Li, Ke He, Lili Wang, Jin-Feng Jia, Hsiang-Hsuan Hung, Congjun Wu, Xucun Ma,Xi Chen, and Qi-Kun Xue, “Direct observation of nodesand twofold symmetry in FeSe superconductor,” Science , 1410 (2011). Shigeru Kasahara, Tatsuya Watashige, Tetsuo Hanaguri,Yuhki Kohsaka, Takuya Yamashita, Yusuke Shimoyama,Yuta Mizukami, Ryota Endo, Hiroaki Ikeda, KazushiAoyama, Taichi Terashima, Shinya Uji, Thomas Wolf,Hilbert von L¨ohneysen, Takasada Shibauchi, and YujiMatsuda, “Field-induced superconducting phase of FeSein the BCS-BEC cross-over,” Proc. Natl. Acad. Sci. USA , 16309 (2014). J.-Y. Lin, Y. S. Hsieh, D. A. Chareev, A. N. Vasiliev,Y. Parsons, and H. D. Yang, “Coexistence of isotropic andextended s -wave order parameters in FeSe as revealed bylow-temperature specific heat,” Phys. Rev. B , 220507(2011). Lin Jiao, Chien-Lung Huang, Sahana R¨oßler, Cevriye Koz,Ulrich K. R¨oßler, Ulrich Schwarz, and Steffen Wirth, “Su-perconducting gap structure of fese,” Scientific Reports ,44024 EP – (2017), article. J. K. Dong, T. Y. Guan, S. Y. Zhou, X. Qiu, L. Ding,C. Zhang, U. Patel, Z. L. Xiao, and S. Y. Li, “Multigapnodeless superconductivity in FeSe x : Evidence from quasi-particle heat transport,” Phys. Rev. B , 024518 (2009). P. Bourgeois-Hope, S. Chi, D. A. Bonn, R. Liang, W. N.Hardy, T. Wolf, C. Meingast, N. Doiron-Leyraud, andLouis Taillefer, “Thermal conductivity of the iron-basedsuperconductor FeSe: Nodeless gap with a strong two-bandcharacter,” Phys. Rev. Lett. , 097003 (2016). Meng Li, N. R. Lee-Hone, Shun Chi, Ruixing Liang, W. N.Hardy, D. A. Bonn, E. Girt, and D. M. Broun, “Super-fluid density and microwave conductivity of FeSe super-conductor: ultra-long-lived quasiparticles and extended s-wave energy gap,” New J. Phys. , 082001 (2016). S. Teknowijoyo, K. Cho, M. A. Tanatar, J. Gonzales, A. E.B¨ohmer, O. Cavani, V. Mishra, P. J. Hirschfeld, S. L.Bud’ko, P. C. Canfield, and R. Prozorov, “Enhancementof superconducting transition temperature by pointlike dis-order and anisotropic energy gap in FeSe single crystals,”Phys. Rev. B , 064521 (2016). H. C. Xu, X. H. Niu, D. F. Xu, J. Jiang, Q. Yao, Q. Y.Chen, Q. Song, M. Abdel-Hafiez, D. A. Chareev, A. N.Vasiliev, Q. S. Wang, H. L. Wo, J. Zhao, R. Peng, andD. L. Feng, “Highly anisotropic and twofold symmetric su-perconducting gap in nematically ordered FeSe . S . ,”Phys. Rev. Lett. , 157003 (2016). Taichi Terashima, Naoki Kikugawa, Andhika Kiswandhi,Eun-Sang Choi, James S. Brooks, Shigeru Kasahara,Tatsuya Watashige, Hiroaki Ikeda, Takasada Shibauchi,Yuji Matsuda, Thomas Wolf, Anna E. B¨ohmer, Fr´ed´ericHardy, Christoph Meingast, Hilbert v. L¨ohneysen, Michi- To Suzuki, Ryotaro Arita, and Shinya Uji, “AnomalousFermi surface in FeSe seen by Shubnikov - de Haas oscil-lation measurements,” Phys. Rev. B , 144517 (2014). Alain Audouard, Fabienne Duc, Lo¨ıc Drigo, Pierre Toule-monde, Sandra Karlsson, Pierre Strobel, and Andr´eSulpice, “Quantum oscillations and upper critical magneticfield of the iron-based superconductor FeSe,” Europhys.Lett. , 27003 (2015). M. D. Watson, T. K. Kim, A. A. Haghighirad, N. R.Davies, A. McCollam, A. Narayanan, S. F. Blake, Y. L.Chen, S. Ghannadzadeh, A. J. Schofield, M. Hoesch,C. Meingast, T. Wolf, and A. I. Coldea, “Emergence ofthe nematic electronic state in FeSe,” Phys. Rev. B ,155106 (2015). M. D. Watson, T. K. Kim, L. C. Rhodes, M. Eschrig,M. Hoesch, A. A. Haghighirad, and A. I. Coldea, “Ev-idence for unidirectional nematic bond ordering in fese,”Phys. Rev. B , 201107 (2016). Takashi Miyake, Kazuma Nakamura, Ryotaro Arita, andMasatoshi Imada, “Comparison of ab initio low-energymodels for LaFePO, LaFeAsO, BaFe As , LiFeAs, FeSe,and FeTe: Electron correlation and covalency,” Journal ofthe Physical Society of Japan , 044705 (2010). Daniel D. Scherer, A. C. Jacko, Christoph Friedrich, ErsoyS¸a¸sıo˘glu, Stefan Bl¨ugel, Roser Valent´ı, and Brian M. An-dersen, “Interplay of nematic and magnetic orders in FeSeunder pressure,” Phys. Rev. B , 094504 (2017). A. Kreisel, Shantanu Mukherjee, P. J. Hirschfeld, andBrian M. Andersen, “Spin excitations in a model of FeSewith orbital ordering,” Phys. Rev. B , 224515 (2015). Qisi Wang, Yao Shen, Bingying Pan, Xiaowen Zhang,K. Ikeuchi, K. Iida, A. D. Christianson, H. C. Walker,D. T. Adroja, M. Abdel-Hafiez, Xiaojia Chen, D. A. Cha-reev, A. N. Vasiliev, and Jun Zhao, “Magnetic groundstate of fese,” Nat. Comm. , 12182 EP – (2016). Qisi Wang, Yao Shen, Bingying Pan, Yiqing Hao, Ming-wei Ma, Fang Zhou, P. Steffens, K. Schmalzl, T. R. For-rest, M. Abdel-Hafiez, Xiaojia Chen, D. A. Chareev, A. N.Vasiliev, P. Bourges, Y. Sidis, Huibo Cao, and Jun Zhao,“Strong interplay between stripe spin fluctuations, ne-maticity and superconductivity in FeSe,” Nat. Mater. ,159–163 (2016), letter. Y. Kushnirenko, A. A. Kordyuk, A. Fedorov, E. Haubold,T. Wolf, B. B¨uchner, and S. V. Borisenko, “Anomalous temperature evolution of the electronic structure of FeSe,”ArXiv e-prints (2017), arXiv:1702.02088 [cond-mat.supr-con]. D. Liu, W. Zhang, D. Mou, J. He, Y.-B. Ou, Q.-Y. Wang,Z. Li, L. Wang, L. Zhao, S. He, Y. Peng, X. Liu, C. Chen,L Yu, G. Liu, X. Dong, J. Zhang, C. Chen, Z. Xu, J. Hu,X. Chen, X. Ma, Q. Xue, and X.J. Zhou, “Electronic originof high-temperature superconductivity in single-layer FeSesuperconductor,” Nat. Comm. , 931 (2012). Shiyong Tan, Yan Zhang, Miao Xia, Zirong Ye, Fei Chen,Xin Xie, Rui Peng, Difei Xu, Qin Fan, Haichao Xu, JuanJiang, Tong Zhang, Xinchun Lai, Tao Xiang, JiangpingHu, Binping Xie, and Donglai Feng, “Interface-inducedsuperconductivity and strain-dependent spin density wavesin FeSe/SrTiO3 thin films,” Nat. Mater. , 634–640(2013). The Y-centered pocket is symmetry related and will notbe discussed further at this point. M. P. Allan, A. W. Rost, A. P. Mackenzie, Yang Xie, J. C.Davis, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, and T.-M. Chuang, “Anisotropic energy gaps of iron-based super-conductivity from intraband quasiparticle interference inLiFeAs,” Science , 563–567 (2012). Z. P. Yin, K. Haule, and G. Kotliar, “Spin dynamics andorbital-antiphase pairing symmetry in iron-based super-conductors,” Nat. Phys. , 845 (2014). K. Umezawa, Y. Li, H. Miao, K. Nakayama, Z.-H. Liu,P. Richard, T. Sato, J. B. He, D.-M. Wang, G. F. Chen,H. Ding, T. Takahashi, and S.-C. Wang, “Unconventionalanisotropic s -wave superconducting gaps of the lifeas iron-pnictide superconductor,” Phys. Rev. Lett. , 037002(2012). Kazuhiko Kuroki, Seiichiro Onari, Ryotaro Arita,Hidetomo Usui, Yukio Tanaka, Hiroshi Kontani, andHideo Aoki, “Unconventional pairing originating fromthe disconnected Fermi surfaces of superconductingLaFeAsO − x F x ,” Phys. Rev. Lett. , 087004 (2008). R. Nourafkan, G. Kotliar, and A.-M. S. Tremblay,“Correlation-enhanced odd-parity interorbital singlet pair-ing in the iron-pnictide superconductor LiFeAs,” Phys.Rev. Lett.117