Spin dynamics and an orbital-antiphase pairing symmetry in iron-based superconductors
SSpin dynamics and an orbital-antiphase pairing symmetry in iron-basedsuperconductors
Z. P. Yin, ∗ K. Haule, and G. Kotliar
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, United States. (Dated: November 1, 2013)
The symmetry of the wave function describing the Cooper pairs is one of the most fundamentalquantities in a superconductor but its measurement in the iron-based superconductors has proved tobe very difficult. The complex multi-band nature of these materials makes the interplay of supercon-ductivity with spin and orbital dynamics very intriguing, leading to very material dependent magneticexcitations, and pairing symmetries.
Here we use first-principles many-body method, including ab initio determined two-particle vertex function, to study the spin dynamics and superconductingpairing symmetry in a large number of iron-based superconductors. In iron compounds with hightransition temperature, we find both the dispersive high-energy spin excitations, and very strong lowenergy commensurate or nearly commensurate spin response, suggesting that these low energy spinexcitations play the dominate role in cooper pairing. We find three closely competing types of pairingsymmetries, which take a very simple form in the space of active iron d orbitals, and differ only in therelative quantum mechanical phase of the xz , yz and xy orbital contributions. The extensively discusseds + − symmetry appears when contributions from all orbitals have equal sign, while the opposite signin xz and yz orbitals leads to the d wave symmetry. A novel orbital antiphase s + − symmetry emergeswhen xy orbital has opposite sign to xz and yz orbitals. We propose that this orbital-antiphase pairingsymmetry explains the puzzling variation of the experimentally observed superconducting gaps on allthe Fermi surfaces of LiFeAs . This novel symmetry of the order parameter may be realized in otheriron superconductors. The spin and the multi-orbital dynamics of iron based superconductors is believed the play the essential role inthe mechanism of superconductivity , but a realistic modeling of magnetic excitations, and a clear physical picturefor their variation across different families of iron superconductors, is currently lacking. The Cooper pairs are lockedinto singlets but the orbital structure of the superconducting order parameter can be material dependent, and itsconnection to orbital and spin excitations is an open problem. To address these issues, we use ab initio theoreticalmethod for correlated electron materials, based on combination of dynamical mean field theory (DMFT) and densityfunctional theory (DFT) . This computational method improves on DFT description of electronic structure in ironsuperconductors, and predicts correct magnitude of ordered magnetic moments , improves electronic spectral functionand Fermi surfaces , and charge response such as optical conductivity . To successfully predict dynamical magneticresponse across different families of iron superconductors, and superconducting pairing, it is crucial to compute from ab initio the two particle scattering amplitude also called the two particle vertex function. Its complex calculationpreviously precluded ab initio description of trends across different families of iron superconductors, as spin responseis very sensitive to the value and the energy dependence of this screened interaction. In many methods used earlier,such as random phase approximation, it is treated as a static adjustable parameter , or it was computed bysumming very limited subset of Feynman diagrams . Within the functional renormalization group this screenedinteraction is computed rather well, but the electronic structure is greatly simplified, treating the Hubbard modelfor iron d -bands only, and usually neglecting renormalization of electronic structure due to electronic correlations.Similar simplification is made in strong-coupling approaches. All iron-based superconductors contain similar layers of tetrahedra with iron in the center and pnicogen/chalcogenat the corners, but their spin excitation spectra varies greatly among compounds. In Fig. 1 we plot the dynamic spinstructure factor S( q , ω )= χ (cid:48)(cid:48) ( q , ω )/(1-exp(- (cid:126) ω /k B T)) for several classes of iron compounds along the high symmetrymomentum path in the first Brillouin zone of the single iron unit cell. We overlay the neutron scattering data for some compounds where experiment is available, to show a good agreement between theory and experiment. Thespin-wave bandwidth, defined as the difference between the minimum ( q = (1 , q = (1 , J , which is inversely proportional to the strength of the low energyCoulomb interaction ( J ∝ t /U ), hence increased correlation strength leads to smaller bandwidth. Notice that thecorrelation strength in these compounds is dominated by the Fe-pnicogen distance, as shown in Ref. 8. The phosphoruscompounds (Figs. 1a-c) show the largest spin wave bandwidth of the order of 0 . .
45 eV, which is a consequenceof their most itinerant nature among these compounds . The mass enhancement due to correlations is increased inarsenides and even more in chalcogenides , hence the spin-wave bandwidth is progressively reduced to 0 . − . .
15 eV in Figs. 1g-h. The intensity of spin excitation is proportional to the size of the fluctuatingmoment in this energy range, which roughly anti-correlates with strength of correlations, hence phosphorus compounds a r X i v : . [ c ond - m a t . s up r- c on ] N ov FIG. 1:
Dynamic spin structure factor S ( q, ω ) in iron pnictides, chalcogenides and MgFeGe. The S ( q, ω ) isplotted along the high-symmetry path ( H, K, L = 1) in the first Brillouin zone of the single iron unit cell. The intensity variessubstantially across these compounds, hence the maximum value of the intensity was adjusted to emphasize the dispersionmost clearly. The maximum value of the intensity in each compound is shown in the top right corner. The color codingcorresponds to the theoretical calculations for (a) BaFe P ( T maxC < K ); (b) LiFeP ( T C = 6 K ); (c)LaFePO ( T C = 7 K );(d) SrFe As ( T maxC = 37 K ); (e) LaFeAsO ( T maxC = 43 K ); (f) BaFe As ( T maxC = 39 K ); (g) LiFeAs ( T C = 18 K ); (h) FeSe( T maxC = 37 K ); (i)MgFeGe ( T maxC = 0); (j)FeTe ( T maxC = 0); (k) BaFe . Ni . As ( T C < K ); (l) BaFe . Ni . As ( T C = 20 K );(m) Ba . K . Fe As ( T C = 39 K ); (n) KFe As ( T C = 3 . K ); (o) KFe Se . The experimental data are shown as black dotswith error bars in (f), (g), (l) and (m), digitized from Refs. 17–20. show the weakest ( M ax = 4) and FeTe shows the strongest (
M ax = 20) intensity.The low energy spin-excitations are much more sensitive to the details of both the band-structure and the two-particle vertex function, hence the trend across different compounds can not be guessed from either the correlationstrength or from the band structure. In Fig. 2 we show S ( q, ω ) for the same compounds as in Fig. 1, but we takea different cut. We keep the energy fixed at ω = 5 meV, and change momentum in the two dimensional momentumplane ( H, K ) (The momentum dependence in z direction is weak for most compounds). As is clear from Figs. 1a-c,and Fig. 2a-c, the low energy spin-excitations are extremely weak ( M ax ≈
1) in phosphorus compounds and the spinexcitations at the spin-density wave ordering vector (1 ,
0) is comparable to its value at the ferromagnetic orderingvector (0 , H, K ) = (1 , T C = 18 K). When doped, all compounds in Figs. 2d-f are high-temperature superconductors ( T c ≈ K − K ).Similarly chalcogenide FeSe (Fig. 1h), which becomes superconducting at T c = 37 K under modest pressure p = 3 GPa,has similar low energy spin response as the arsenides superconductors.On the other hand, MgFeGe is a compound with similar band structure as LiFeAs, hence the random-phaseapproximation gives similar spin response in the two compounds with low energy maximum intensity at (1 , Inclusion of the realistic two-particle vertex function, as done in this study, has a profound impact on the spin
FIG. 2:
Dynamic spin structure factor S ( q, ω ) in iron pnictides, chalcogenides and MgFeGe. The S ( q, ω ) isplotted in the 2D plane ( H, K ) at constant ω =5 meV for the same materials as in Fig.1. The maximum intensity scale for eachcompound is marked as a number in the top-right corner of each subplot. We take the cut at L = 1 for all compounds exceptMgFeGe and phosphorus compounds, where L = 0 plane is shown to emphasize their tendency towards ferromagnetism. excitations. A broad maximum appears at (0 ,
0) (see Fig. 2i), hence spin fluctuations are ferromagnetic in agreementwith calculation of Ref. 22 showing stable ferromagnetic ground state. Finally FeTe has also much broader spin-excitations covering a large part of the Brillouin zone (see Fig. 2j), and shows two competing excitations at q =(1,0)and q =(0.5, 0.5), the latter corresponds to the ordering wave vector of the low-temperature antiferromagnetic stateof Fe . Te. The common theme in high-temperature superconductors (Figs. d-h) is thus the existence of well defined highenergy dispersive spin excitations with bandwidth between 0 . − .
35 eV, and most importantly very well developedcommensurate (or nearly commensurate) low energy spin excitations at wave vector q = (1 , . The pnictide parent compounds SrFe As , LaFeAsO,BaFe As have strong low energy spin excitation centered exactly at q = (1 , As , i.e., BaFe − x Ni x As and Ba − x K x Fe As , respectively. The electron dopingslightly increases the bandwidth (comparing Fig. 1(f) with Fig. 1(k)), whereas the hole doping dramatically reducesthe bandwidth from ∼ . ∼ .
05 eV in overdoped KFe As (Fig. 1(n)). The low energy spin excitationsin the electron overdoped BaFe . Ni . As become very weak and strongly incommensurate with peak centered at q =(1.0, 0.35) (see Fig. 2k). Similarly, on the hole overdoped side in KFe As , the low-energy spectrum is suppressed(maximum intensity in Fig. 2n is 15 compared to 100 in the parent compound), and main excitation peak moves toincommensurate q =(0.75, 0) in agreement with experiment. The optimally doped compounds (Figs. 1l,m) havehigh energy spin excitations very similar to the parent compound, while the low energy excitations are slightly reducedand broadened in momentum space (Fig. 2l,m), to suppress long range magnetic order of the parent compound. Thisis very similar to the spectrum of LiFeAs and FeSe, which both have superconducting ground state. From these plots,we can deduce that near commensurate or commensurate spin excitations at q = (1 , x Fe − y Se compounds. Our results for KFe Se in Figs. 1&2(o)indicate strong low energy spin excitation peaked around q = (1 , . q = (1 ,
0) and favor superconductivity. On the other hand,vacancies in the Fe sites can move the peak to q =(0.6, 0.2) to induce novel magnetism in K . Fe . Se .Whereas the dynamic spin structure factor S ( q, ω ) dispersion and the strength of the low energy spin excitationscorrelate with experimental T c across many families of iron superconductors, the superconducting pairing symmetryand the variation of the superconducting gaps on the different Fermi surfaces cannot be extracted from the spindynamics alone. To make further progress on these issues, we computed the complete two particle scattering amplitudein the particle-particle channel within the dynamical mean field theory framework, and we solved Eliashberg equationsin the BCS low energy approximation (see Supplementary material). In the Eliashberg equations, the orbital degreesof freedom play the central role, rather than the bands, because the Coulomb interaction, and the two particleirreducible vertex function in the particle-hole channel, is large between the iron-3 d electrons on the same iron site. FIG. 3:
Fermi surfaces, pairing symmetries and the basic building blocks.
Top row: the original and unfoldedtwo-dimensional Fermi surfaces in the Γ plane for the representative compound LaFeAsO in paramagnetic state, shown in thefirst Brillouin zone of the single iron unit cell. On the top right, the Fermi surfaces are further decomposed into the dominatingFe- t g ( xz , yz , and xy ) characters. The next three rows, from top to bottom, show respectively the conventional s + − , theorbital-antiphase s + − and the d -wave pairing symmetries of the superconducting order parameter. The left column shows theFermi surfaces colored with strength of the order parameter ∆ j ( k ) = < c + k ↑ ,j c + − k ↓ ,j > ( j is the band index), while the rightcolumns decompose the order parameter in orbital space, i.e., ∆ α ( k ) = < c + k ↑ ,α c + − k ↓ ,α > ( α runs over Fe- t g orbitals: xz , yz ,and xy ). In all arsenide and chalcogenide compounds with strong (nearly) commensurate low energy spin excitation, we findthat the Eliashberg equations give three almost degenerate solutions with the largest eigenvalues. The correspondingthree eigenvectors, which are proportional to the superconducting order parameter ∆ α,α (cid:48) ( k ), are almost diagonal inorbital space ( αα (cid:48) ), and we denote the diagonal terms as ∆ α ( k ) = ∆ α,α ( k ). As shown in Fig. 3, these three states aresimilar in nature, since each orbital has sign-changing s + − structure in momentum space, described by the formula∆ α ( kx, ky ) = ∆ nnn,α cos ( kx ) cos ( ky ) + ∆ nn,α ( cok ( kx ) + cos ( ky )) / nn correspond to the nearest-neighbor and ∆ nnn to thenext-nearest neighbor pairing, and we find that | ∆ nn,α | << | ∆ nnn,α | , hence we have predominantly next-nearestneighbor pairing. In Fig. 3 we plot Fermi surfaces and gap function in the first Brillouin zone of the single iron unitcell. We plot gap function for t g orbitals only, because gaps on the e g orbitals are much smaller and their weight atthe Fermi level is also small. Although the symmetry of pairing expressed in orbital space is s + − in all three states,the projection to the Fermi surface leads to different global symmetries. When all three t g orbitals have the samephase, we recover the conventional s + − state . If the xz orbital has the opposite phase to the yz orbital, the globalsymmetry is of d -wave type. In this case the xy orbital shows negligible pairing. Finally, we find a novel type ofstate in which xz and yz orbitals are in-phase, but the xy orbital has the opposite phase. We call this state the orbital-antiphase s + − state . FIG. 4:
Superconducting pairing symmetry and pairing amplitude anisotropy in LiFeAs.
The superconductingorder parameter ∆ j ( k ) = < c + k ↑ ,j c + − k ↓ ,j > on the Fermi surface, with the (a) orbital-antiphase s + − , (b) conventional s + − and(c) a d-wave symmetry. (d)-(f) show the variation of the pairing amplitude for the orbital-antiphase s + − state on the outerhole Fermi surface and the two electron Fermi surfaces, respectively. The red lines correspond to theoretical results, and the +symbols denote the experimental measurements from Ref.4. The identification of these basic building blocks greatly simplifies our understanding of the superconducting pairingsymmetry in the iron-based superconductors, and explains why the superconducting pairing symmetry is so sensitiveto the details of the electronic structure. Since flipping the sign of the order parameter in a single orbital has only atiny energy cost, the first three eigenstates discussed above are very close in energy and have very similar eigenvalues,i.e., superconducting pairing strength. In Fig. 4 we show detailed results for LiFeAs, and we plot all the three leadingpairing symmetries on the two-dimensional Fermi surfaces in the Γ plane of the tetragonal crystallographic unit cell.Our calculations show that the orbital-antiphase s + − has the largest pairing strength in LiFeAs. Experimentally, it wasfound that the superconducting gaps have very unusual variations on the Fermi surfaces, where the superconductinggap is maximal (minimum) around ϕ =45 ◦ (0 ◦ ) on the outer hole Fermi surface (see Fig. 4(d)), whereas it is maximal(minimum) around ϕ =0 ◦ (45 ◦ ) on the two electron Fermi surfaces (Fig.4(e),(f)). This observation was used as anevidence against the spin fluctuation mechanism of superconductivity, as it is not consistent with the conventional s + − state. Indeed, our calculation show that conventional s + − order parameter can not account for the gap sizevariation in momentum space, however, as seen from (Fig.4(d)-(f)), the orbital-antiphase s + − can account for thevariation of this gap.The novel orbital-antiphase s + − pairing symmetry is not limited to LiFeAs but may be the pairing symmetry inmany other iron-based superconductors, such as Ba − x K x Fe As and K x Fe Se [29]. Our study suggests that it isimportant to describe the cooper-pairing in orbital-space, keeping the complexity of orbital and spin fluctuations,which arise due to electron correlations, rather than solving BCS equations for weakly correlated systems in bandspace. Acknowledgments:
We thank H. Park, P. Dai and H. Ding for stimulating discussion. This work is supportedby NSF DMR–1308141 (Z.P.Y. and G.K.) and NSF Career DMR–0746395 (K.H.). ∗ Electronic address: [email protected] Stewart, G. R. Superconductivity in iron compounds.
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Spin dynamics and an orbital-antiphase pairing symmetry in iron-based superconductors:Supplementary information
Z. P. Yin, K. Haule, and G. Kotliar
Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, United States.
I. METHOD
To carry out our first principles calculations taking into accounts strong correlation effects in the iron-based super-conductors, we used a combination of density functional theory and dynamical mean field theory (DFT+DMFT) [S1]as implemented in Ref. S2, which is based on the full-potential linear augmented plane wave method implementedin Wien2K [S3], together with the full two-particle vertex correction in the same footing [S4]. The electronic chargeis computed self-consistently on DFT+DMFT density matrix. The quantum impurity problem is solved by the con-tinuous time quantum Monte Carlo method [S5,6], using Slater form of the Coulomb repulsion in its fully rotationalinvariant form. The local two-particle vertex is then sampled by the continuous time quantum Monte Carlo methodon the fully-converged DFT+DMFT solution.We use the same experimentally determined lattice structures as in Ref. S7, including the internal positions of theatoms (see Table S1), from References [S8–17]. We use the paramagnetic tetragonal lattice structures, and neglectthe weak structural distortions. For convenience, we include the Table S1 in Ref. S7.
TABLE S1: Lattice constants and atomic positions of the compounds we studied. z z coordinate of P/As/Se/Teand z a (˚ A ) c (˚ A ) z (Fe) z z Se As As As P We studied the paramagnetic phase of all compounds at the same temperature and used the same Coulomb inter-action U and Hund’s coupling J as in our previous work.[S7, S20, S21] A. Unfolding
Since particle-hole irreducible vertices in our approach are nonzero only for electrons in the 3 d orbitals of the ironatoms, we can rewrite the perturbation theory in terms of vertices defined in this 3 d orbital subspace, and the singleparticle Green’s function G dd ( k, ω ), which starts and ends in the iron 3 d manifold. This Green’s function can becast into the form, where we consider only one iron per unit cell in larger Brillouin zone. This operation is usuallycalled unfolding . Note that here we keep all the information from the two-iron atom calculation (for example, we haveshadow bands in our Green’s function), we merely rewrite perturbation theory in terms of more convenient largerBrillouin zone and one atom unit cell. We checked that for our choice of unfolding, the off-diagonal terms of the singleparticle Green’s function in momentum G dd ( k, k + Q ) is negligible.To transform the electron structures from the two-iron/unit cell to the one-iron/unit cell, we used the followingtransformation: G α,α (cid:48) ( k ) = (cid:88) R,R (cid:48) e ik · ( R − R (cid:48) ) G Rα,R (cid:48) α (cid:48) ( k ) (S1)where R and R (cid:48) are the positions of the two different Fe atoms in the original crystal structure, and R (cid:48) is related to R by a group symmetry operation of the crystal structure. α , α (cid:48) labels orbitals. Special care has to be taken to thephases of the orbitals chosen on the two iron atoms. To make off-diagonal terms in the green’s function G dd ( k, k + Q )negligible, we need to choose different phases for the orbitals on the two atoms.For example, in the case of BaFe As , we choose R = (0 . , , .
25) and R (cid:48) = (0 . , , − . d orbitals on the second atom are opposite to the first atom. For LiFeAs and LaFeAsO, we choose R = (0 . , . , . R (cid:48) = ( − . , − . , − . d z , d x − y , d xy and equal for d xz and d yz orbitals. B. Magnetic and Charge Susceptibility
FIG. S1:
Feynman diagram for the local two-particle susceptibility.
The general two-particle susceptibility (depicted in Fig.S1) is defined by χ locpp (cid:48) ; rr (cid:48) = < Ψ + r ( τ )Ψ + p (cid:48) ( τ )Ψ r (cid:48) ( τ )Ψ p ( τ ) > (S2)where p , p (cid:48) , r , r (cid:48) are combined index of spin and orbital, Ψ + is the creation operator and Ψ the annihilation operator.The central quantity of this approach is the irreducible two particle vertex Γ, which consists of all Feynman diagrams,and can not be separated into two parts by cutting any two propagators. There are several types of irreducible vertices:i) the irreducible vertex in the particle-hole channel consists of all Feynman diagrams, which can not be separatedinto two parts by cutting any two propagators going in opposite direction, i.e., cutting a particle and a hole ii) theparticle-particle irreducible vertex correspondingly contains diagrams which are not separated into parts by cuttingany two propagators going in the same direction, iii) the fully irreducible two-particle vertex contains all diagramswhich can not be broken into separate diagrams by cutting either particle-particle or particle-hole pair of propagators.Within the Dynamical Mean Field Theory (DMFT), the particle-hole irreducible vertex Γ is local, and it is equalto the impurity vertex, which can be obtained from the solution of the quantum impurity model. This can be provenwith the same power-counting arguments as the self-energy is proven to be local in the limit of large coordinationnumber.[S1, S22] To compute Γ imp , we sample the two-particle susceptibility χ imp by the quantum Monte Carloimpurity solver, and by inverting the Bethe-Salpeter equation, we obtain Γ imp . For the the impurity model, thetwo-particle susceptibility is formally obtained by differentiating the partition function χ imppp (cid:48) ; rr (cid:48) = 1 Z ∂ Z∂ ∆ rp ∂ ∆ p (cid:48) r (cid:48) (S3)and resulting terms are sampled by the quantum Monte Carlo solver. Notice that the susceptibility depends on fourtimes, which can be translated into three frequencies. It turns out that this two-particle susceptibility can be brokenup into two terms, of which both depend on two frequencies only, which greatly reduces the complexity of the problem.We sample these terms directly in frequency during the Monte Carlo run, just like the single-particle Green’s function,to avoid the Fourier transform of a multidimensional object.Once the impurity two-particle susceptibility χ imppp (cid:48) ; rr (cid:48) is available, we compute the impurity polarization (bubble) χ ,imppp (cid:48) ; rr (cid:48) , and extract the particle-hole irreducible vertex Γ imppp (cid:48) ; rr (cid:48) by the Bethe-Salpeter equation: χ imppp (cid:48) ν ; rr (cid:48) ν (cid:48) = χ ,imppp (cid:48) ν ; rr (cid:48) ν (cid:48) + χ ,imppp (cid:48) ν ; p p (cid:48) ν Γ impp p (cid:48) ν ; r r (cid:48) ν (cid:48) χ impr r (cid:48) ν (cid:48) ; rr (cid:48) ν (cid:48) (S4)as Γ imppp (cid:48) ν ; rr (cid:48) ν (cid:48) = ( χ ,imp ) − pp (cid:48) ν ; rr (cid:48) ν (cid:48) − ( χ imp ) − pp (cid:48) ν ; rr (cid:48) ν (cid:48) (S5)where p , p (cid:48) , r , r (cid:48) are combined index of spin and orbital, and ν , ν (cid:48) are fermionic Matsubara frequencies. Theseequations depend also on the bosonic (center of mass) frequency Ω, which can be treated as an external parameter.With the knowledge of the particle-hole irreducible vertex, we can compute the non-local two particle susceptibilityby the Bethe-Salpeter equation χ qpp (cid:48) ν ; rr (cid:48) ν (cid:48) = χ ,qpp (cid:48) ν ; rr (cid:48) ν (cid:48) + χ ,qpp (cid:48) ν ; p p (cid:48) ν Γ locp p (cid:48) ν ; r r (cid:48) ν (cid:48) χ qr r (cid:48) ν (cid:48) ; rr (cid:48) ν (cid:48) (S6)as χ qpp (cid:48) ν ; rr (cid:48) ν (cid:48) = (( χ ,q ) − − Γ loc ) − pp (cid:48) ν ; rr (cid:48) ν (cid:48) (S7)where χ ,qpp (cid:48) ν ; rr (cid:48) ν (cid:48) = − T δ ( ν − ν (cid:48) ) (cid:80) k G k,rp ( iν ) G k − q,p (cid:48) r (cid:48) ( iν − i Ω) is the nonlocal one-particle bubble. This ladder sumin the particle-hole channel incorporates most important non-local spin and orbital fluctuations.
FIG. S2:
Feynman diagram for computing the local and momentum-dependent two-particle susceptibility inthe particle-hole spin (m) and charge (d) channels.
Here the particle-hole irreducible vertex Γ is assumed to be local.The momentum-dependent two-particle susceptibility is obtained by using the momentum-dependent one-particle bubble χ q .The spin and charge susceptibility is computed by closing the corresponding two-particle susceptibility with the spin or chargebare vertex, and tracing over the internal indices, such as orbital ( α , α (cid:48) , β , β (cid:48) ) and frequency ( ν , ν (cid:48) ). In the paramagnetic state (when the spin-orbit coupling is ignored), there is a symmetry between the stateswith different spin, and the equations can be block-diagonalized in spin. The following holds in paramagnetic state χ ↑↑ ; ↑↑ = χ ↓↓ ; ↓↓ and χ ↑↑ ; ↓↓ = χ ↓↓ ; ↑↑ where the orbital index is omitted for simplicity. As a result, the two-particlequantities can be expressed in terms of two independent channels (no mixing between the two channels), the magneticand the charge channel as χ m = χ ↑↑ ; ↑↑ − χ ↑↑ ; ↓↓ (S8)Γ m = Γ ↑↑ ; ↑↑ − Γ ↑↑ ; ↓↓ (S9) χ d = χ ↑↑ ; ↑↑ + χ ↑↑ ; ↓↓ (S10)Γ d = Γ ↑↑ ; ↑↑ + Γ ↑↑ ; ↓↓ (S11)Thus χ ↑↑ ; ↑↑ = ( χ d + χ m ) / χ ↑↑ ; ↓↓ = ( χ d − χ m ) /
2. In addition, we have the following symmetry < S z ( τ ) S z (0) > = < S + ( τ ) S − (0) > = < S − ( τ ) S + (0) > (S12)Accordingly we have χ ↓↑ ; ↓↑ = χ ↑↓ ; ↑↓ = χ m . Hence the spin index can be dropped in the paramagnetic state byusing the above vertex in the magnetic and charge channel. As shown in Fig.S2, the two-particle vertex in themagnetic/charge (m/d) channel can be written as χ m/dαα (cid:48) ; ββ (cid:48) ( ν, ν (cid:48) ) q, Ω = (( χ ) − q, Ω − Γ m/d ) − αα (cid:48) ; ββ (cid:48) ( ν, ν (cid:48) ) q, Ω (S13)The spin or charge susceptibility χ ( q, ω ) is obtained by closing the corresponding nonlocal two-particle susceptibility χ m/dαα (cid:48) ; ββ (cid:48) ( ν, ν (cid:48) ) q, Ω with the bare vertex µ and summation over the internal indices χ m/d ( q, Ω) = 2 (cid:88) α,α (cid:48) ,β,β (cid:48) ,ν,ν (cid:48) µ αα (cid:48) χ m/dαα (cid:48) ; ββ (cid:48) ( ν, ν (cid:48) ) q, Ω µ ββ (cid:48) (S14)Further computational details on magnetic susceptibility are available in Ref. S4.0 FIG. S3:
Particle-particle irreducible vertex Γ ppσ σ ; σ σ . It consists of fully-irreducible vertex Γ pp,firr , and vertex whichis reducible in the particle-hole channel. There are two ways to arrange particle-hole ladders, either horizontally (Γ pp (1) ) orvertically (Γ pp (2) ), hence there are two particle-hole contributions. C. Superconductivity
A divergent susceptibility in the particle-particle channel signals instability of the metallic state towards supercon-ductivity. To obtain this susceptibility, we need to compute the particle-particle irreducible vertex Γ pp , depicted inFig.S3. It consists of the fully irreducible vertex function Γ firr and the reducible vertex functions in the particle-holechannels. There are two particle-hole channels, because one can stack particle-hole ladders horizontally (particle-hole channel 1) or vertically (particle-hole channel 2), as shown in Fig.S3. Notice that this equation contain allspin-fluctuation diagrams [S23]. Indeed we recover the spin-fluctuation theory if we replace Γ ph by constant number U , which is treated as a phenomenological parameter in spin-fluctuation theory, and propagators with free-electronGreen’s function. Since results are very sensitive to the value and structure of this screened interaction Γ ph , it isimportant to determine it ab initio .In this report, we consider only the spin-singlet pairing and define the singlet vertex Γ pp,s byΓ pp,s = 12 (Γ pp ↑↓ ; ↑↓ − Γ pp ↑↓ ; ↓↑ ) (S15)For convenience, we rewrite Γ pp,s as the sum of the three terms depicted in Fig. S4:Γ pp,s = Γ pp,firr,s + Γ pp (1) ,s + Γ pp (2) ,s . (S16)It then follows that the fully irreducible particle-particle vertex in the spin singlet channel Γ pp,firr,s isΓ pp,firr,s = 12 (Γ pp,firr,s ↑↓ ; ↑↓ − Γ pp,firr,s ↑↓ ; ↓↑ ) , (S17)We can also express the rest of the objects in Fig. S4 in terms of the above calculated particle-hole susceptibility χ ph and particle-hole irreducible vertex Γ ph byΓ pp (1) ,s = −
12 ((Γ χ Γ) ph ↑↑ ; ↓↓ − (Γ χ Γ) ph ↓↑ ; ↓↑ )= 34 (Γ χ Γ) ph,m −
14 (Γ χ Γ) ph,d (S18)and Γ pp (2) ,s = 12 ((Γ χ Γ) ph ↓↑ ; ↓↑ − (Γ χ Γ) ph ↑↑ ; ↓↓ )= 34 (Γ χ Γ) ph,m −
14 (Γ χ Γ) ph,d (S19)These diagrams are depicted in Fig. S4.1 FIG. S4:
Decomposition of particle-particle irreducible vertex for the spin singlet channel:
We explicitly writethe spin components, which contribute to the vertex Γ pp,s in the singlet channel.
To simplify the notation, we then define(Γ χ Γ) ph ≡
34 (Γ χ Γ) ph,m −
14 (Γ χ Γ) ph,d . (S20)to connect the above computed magnetic and charge susceptibility/vertex with the particle-particle irreducible vertex.With this algebraic manipulation, we removed the need for the spin index, however, we do have four orbital indices( α , β , α (cid:48) , β (cid:48) ), three frequencies (fermionic ν , ν (cid:48) , and bosonic Ω) and three momenta ( k , k (cid:48) , q ) left. However, within DMFTthe particle-hole irreducible vertex Γ ph is local, therefore this quantity is independent of momenta k and k (cid:48) and canbe written as (Γ χ Γ) phq, Ω ( αβ, ν ; α (cid:48) β (cid:48) , ν (cid:48) ) (S21)Furthermore, the fully irreducible vertex within DMFT is local and hence independent of momentaΓ pp,firr,s ( αβν ; α (cid:48) β (cid:48) ν (cid:48) ).Finally, using these building blocks computed above, we explicitly write down the irreducible particle-particle vertexΓ pp,s ( αβkν ; α (cid:48) β (cid:48) k (cid:48) ν (cid:48) ) = Γ pp (1) ,s ( αβkν ; α (cid:48) β (cid:48) k (cid:48) ν (cid:48) ) + Γ pp (2) ,s ( αβkν ; α (cid:48) β (cid:48) k (cid:48) ν (cid:48) ) + Γ pp,firr,s ( αβν ; α (cid:48) β (cid:48) ν (cid:48) )= (Γ χ Γ) phk (cid:48) − k,ν (cid:48) − ν ( α (cid:48) α, ν (cid:48) ; ββ (cid:48) , − ν ) + (Γ χ Γ) ph − k (cid:48) − k, − ν (cid:48) − ν ( β (cid:48) α, − ν (cid:48) ; βα (cid:48) , − ν ) + Γ pp,firr,s ( αβν ; α (cid:48) β (cid:48) ν (cid:48) ) (S22)When the particle-particle ladder sum χ pp = (( χ ,pp ) − − Γ pp ) − is diverging, the normal state is unstable to super-conductivity. The sufficient condition is that the matrix of Γ pp χ ,pp has an eigenvalue equal to unity. The eigenvector2with the largest eigenvalue gives the symmetry of the superconducting order parameter. Explicitly, we are solvingthe eigenvalue problem of the following matrix − k B T (cid:88) k (cid:48) ν (cid:48) α (cid:48) β (cid:48) γδ Γ pp,s ( αβkν ; α (cid:48) β (cid:48) k (cid:48) ν (cid:48) ) χ ,ppα (cid:48) β (cid:48) γδ ( k (cid:48) ν (cid:48) )∆ γδ ( k (cid:48) ν (cid:48) ) = λ ∆ αβ ( kν ) (S23)where the eigenvalue λ is the pairing strength and the eigenfunction ∆ is the pairing amplitude.To solve this eigenvalue problem, we make an approximation consistent with the Bardeen-Cooper-Schrieffer (BCS)theory: since pairing occurs at very low energy scale, we take the particle-particle vertex at zero frequency as a proxyfor its steep rise with lowering temperature, i.e., Γ pp,s ( αβk, ν ; α (cid:48) β (cid:48) k (cid:48) , ν (cid:48) ) ≈ Γ pp,s ( αβk, + ; α (cid:48) β (cid:48) k (cid:48) , + ), hence we have − (cid:88) k (cid:48) α (cid:48) β (cid:48) γδ Γ pp,s ( αβk + ; α (cid:48) β (cid:48) k (cid:48) + ) (cid:32) k B T (cid:88) iν (cid:48) χ ,ppα (cid:48) β (cid:48) γδ ( k (cid:48) , ν (cid:48) ) (cid:33) ∆ γδ ( k (cid:48) + ) = λ ∆ αβ ( k + ) (S24)where k B T (cid:80) ν (cid:48) χ ,ppα (cid:48) β (cid:48) γδ ( k (cid:48) , ν (cid:48) )) = k B T (cid:80) ν (cid:48) G α (cid:48) γ ( k (cid:48) , ν (cid:48) ) G β (cid:48) δ ( − k (cid:48) , − ν (cid:48) ) is the one-particle bubble in the particle-particlechannel.To facilitate this calculation, we first transform the single-particle Green’s function to one-iron atom Brillouinzone. We then compute pairing strength in orbital basis ∆ αβ ( k ) with above formula, and we finally transform it toquasiparticle band basis by so-called embedding method. For DMFT calculation, we construct projector to iron-3 d states U k (for details see Ref. S2), which embeds the self-energy to the Kohn-Sham basis ( U k Σ U † k ) or projects Green’sfunction expressed in the Kohn-Sham basis to local basis ( U † k GU k ). We then compute the eigenvectors and eigenvaluesof the DMFT Green’s function at the Fermi level, defined by( ε k − µ + U k Σ( ω = 0) U † k ) ψ R ( k ) = (cid:15) k ψ R ( k ) (S25) ψ L ( k )( ε k − µ + U k Σ( ω = 0) U † k ) = (cid:15) k ψ L ( k ) (S26)where (cid:15) k are the DMFT eigenvalues at the Fermi level, which determine the Fermi surface, and ψ ( k ) are the eigen-vectors. We can then express the pairing strength on the Fermi surface as∆ k ≡ ψ Lk U k ∆( k ) U † k ψ Rk (S27)Notice that ∆ k contains both the diagonal and off-diagonal components in band basis. The latter are smaller, hencewe plot diagonal components in the manuscript ∆ k,ii . ∗ Electronic address: [email protected][S1] Kotliar, G., Savrasov, S. Y., Haule, K., Oudovenko, V. S., Parcollet, O. & Marianetti, C. A. Electronic structurecalculations with dynamical mean-field theory.
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