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Mottness underpins the anomalous optical response of Iron Pnictides

M. S. Laad,1 L. Craco,2,3 S. Leoni,2 and H. Rosner21Max-Planck-Institut fur Physik Komplexer Systeme. 01187 Dresden, Germany

2Max-Planck-Institut fur Chemische Physik fester Stoffe, 01187 Dresden, Germany3Leibniz-Institut fur Festkorper- und Werkstoffforschung Dresden, 01069 Dresden, Germany

(Dated: August 17, 2013)

The recent discovery of high-temperature superconductivity (HTSC) in doped Iron pnictides isthe latest example of unanticipated behavior exhibited by d- and f -band materials. The symmetryof the SC gap, along with the mechanism of its emergence from the “normal” state is a centralissue in this context. Here, motivated by a host of experimental signatures suggesting strong cor-relations in the Fe-pnictides, we undertake a detailed study of their normal state. Focussing onsymmetry-unbroken phases, we use the correlated band structure method, LDA+DMFT, to studythe one-particle responses of both LaO1−xFeAsFx and SmO1−xFeAsFx in detail. Basing ourselveson excellent quantitative agreement between LDA+DMFT and key experiments probing the one-particle responses, we extend our study, undertaking the first detailed study of their normal stateelectrodynamic response. In particular, we propose that near-total normal state incoherence, result-ing from strong, local correlations in the Fe d-shell in Fe-pnictides, underpins the incoherent normalstate transport found in these materials, and discuss the specific electronic mechanisms leading tosuch behavior. We also discuss the implications of our work for the multi-band nature of the SCby studying the pairing “glue” function, which we find to be an overdamped, electronic continuum.Similarities and differences between cuprates and Fe-pnictides are also touched upon. Our studysupports the view that SC in Fe-pnictides arises from a bad metallic, incoherent “normal” state thatis proximate to a Mott insulator.

PACS numbers: 74.25.Jb, 72.10.-d, 74.70.-b

I. INTRODUCTION

Discovery of high-Tc superconductivity (HTSC) in theFe-based pnictides1 is the latest among a host of other,ill-understood phenomena characteristic of doped d- andf -band compounds. HTSC in Fe-pnictides emerges upondoping a bad metal with spin density wave (SDW) or-der at q = (π, 0). Preliminary experiments indicate2,3

unconventional SC. Extant normal state data indicatea “bad metal” with anomalously high resistivity O(mΩ-cm) even at low temperature.1 These observations in Fe-pnictides are reminiscent of underdoped cuprate SC. Thesmall carrier density, along with Uemura scaling fromµ-SR1 similar to hole-doped cuprates strongly suggestsa SC closer to the Bose condensed, rather than a BCSlimit. A brief general review4 gives a chemist’s overviewon the subject.

Several theoretical works have addressed the issue ofthe “degree of correlated electronic” behavior in Fe-pnictides.5,6,7 This is an important issue, bearing as itdoes upon a characterization of charge and spin fluctu-ations: are they itinerant,8,9 or closer to localized?5,6,10

This itinerant-localized duality is a recurring theme incorrelated systems in general,11 and in fact is at the rootof early formulations of the Hubbard model itself.12

In Fe-pnictides, HTSC results from the Fe d bandstates hybridized with As p states: this leads to two hole,and two electron-like pockets13 in one-electron band-structure calculations. Within weak coupling HF-RPAstudies of effective, two- and four-orbital Hubbard mod-els,9,14 this gives a q = (π, 0) SDW order, in seemingagreement with inelastic neutron scattering (INS) re-

sults.15 Observation of quasi-linear (in T ) behavior in theresistivity, pseudogap in optical reflectance,16 and a spingap in NMR3 in doped Fe-pnictides, among other observa-tions, however, are benchmark features showing the rel-evance of strong, dynamical spin and charge correlationsin the pnictides. In analogy with cuprates, this suggeststhat the Fe-pnictides might be closer to a Mott insula-tor (MI) than generally thought.6 Actually, the undopedpnictides of the A1−xFeAsFx type with A = La, Sm,the so-called “111” pnictides, show an insulator-like re-sistivity without magnetic order for T ∗ > 137 K,1 de-pendent upon the specific Fe-pnictide considered. On-set of bad metallic behavior correlates with a structural(tetragonal-orthorhombic (T-O)) distortion at T ∗, belowwhich SDW order sets in. This is different from theAFe2As2 pnictides with A = Ba, Sr (the so-called “122”pnictides), where the bad metallic resistivity is observedonly above T ∗. Nevertheless, as we will discuss below,optical measurements on the 122 family also indicate anon-Fermi Liquid (nFL) metallic behavior at low T . Sodue care must be exercised when one attempts to classifyFe-pnictides into the “weakly” or “strongly” correlatedcategory. The small carrier number seemingly generatedupon the structural distortion in the 111 pnictides ac-cords with the observed high resistivity, lending furthercredence to such a view.

Optical conductivity is a time-tested probe for char-acterizing the charge dynamics in solids. Specifically, itmeasures how a particle-hole pair excitation, created byan external photon field, propagates in the system, un-covering the detailed nature of the excitation spectrum it-self. In a normal Fermi liquid (FL), low-energy scattering

2

processes leave the identity of an excited electron (hole)intact. This fact, noticed first by Landau, implies a longlifetime for excited quasiparticles: near the Fermi sur-face, their lifetime, τ−1(ω, T ) ≃ ω2, T 2, greatly exceedsω, T , their energy. In optical response, this fact mani-fests itself as the low-energy Drude part (after subtract-ing the phonon contribution), corresponding to coherentpropagation of particle-hole excitations built from suchquasiparticles. The Drude parametrization, a lorentzianwith half-width Γ = τ−1(ω, T )

σ(ω) =ne2

m∗

1

1 + iωτ(ω)(1)

describes the low-energy optical response of normal met-als. This allows one to estimate the transport relexationrate and dynamical mass from17

1

τ(ω)=

Ne2

m0

Re

[

1

σ(ω)

]

(2)

and

m∗(ω) =Ne2

ωIm

[

1

σ(ω)

]

. (3)

With σ(ω) ≃ Γ/(ω2 + Γ2) at low-energy, m∗(ω) = m0,a constant. As long as the FL survives, even in f -bandrare-earth metals with e−e interactions much larger thanthe (band) kinetic einergy, this observation holds. Obser-vation of a non-Drude optical conductivity in clean met-als with low residual resistivity is thus a diagnostic fornon-FL charge dynamics, i.e, where τ−1(ω) ≃ ω2, m∗ =const at low energy no longer hold. One can, however,continue to use the Drude parametrization, at the cost ofhaving a complicated ω-dependence of τ−1, m∗ at low en-ergy. Such non-FL optical conductivity in the symmetry-unbroken bad-metallic phases is characteristic of severalstrongly correlated systems, from quasi-one dimensionalLuttinger liquids,18 high-Tc cuprates up to optimal dop-ing,19 f -electron systems close to quantum phase transi-tions20 and MnSi,21 among others. Additionally, stronglycorrelated d- and f -band FL metals routinely exhibit anon-Drude optical response above a low-T scale, the so-called lattice coherence scale (O(1− 20) K), below whichcorrelated FL behavior obtains.11 So the material diver-sity and range of distinct ground states exhibited by theabove strongly suggests a common underlying origin ofthe anomalous charge dynamics. The known importanceof strong, short-ranged electronic correlations in d-andf -band systems then implies that Mott-Hubbard physicsmay underlie such generic, anomalous features.

On the theoretical side, observation of non-Drude, in-coherent, or power law optical response forces one to dis-card the Landau FL theory, together with perturbationtheory in interactions upon which it is based: an electron(hole) is no longer an elementary excitation of the sys-tem. The one-fermion propagator exhibits a branch cut

analytic structure, leading to power-law fall-off in optics.This reflects the fact that the action of the electronic cur-rent operator (within linear response theory)22 does notcreate well-defined elementary excitations at low energy,leading to an incoherent response.

II. EARLIER WORK

Extant LDA+DMFT (local-density approximationplus dynamical mean-field theory) works on Fe-pnictidesgive either a strongly renormalized FL5 or an orbitalselective (OS), incoherent, pseudogapped metal.7 Verygood semiquantitative agreement with key features seenin both photoemission (PES) and X-ray absorption (XAS)for SmO1−xFxFeAs,23,24 as well as with the low-energy(15 meV) kink in PES is obtained using LDA+DMFT.25

Focusing on the “normal” state of Fe-pnictides, isLDA+DMFT adequate for describing their correlatedelectronic structure, or are cluster extensions (cluster-DMFT) of DMFT needed? If the observed SDW orderhas its origin in a Mott-Hubbard, as opposed to a weak-coupling Slater-like SDW picture, one would expect thatincorporation of “Mottness”26 is adequate, at least inthe symmetry unbroken phases (T > T ∗ at x = 0 andat all T > Tc, the SC transition temperature, for dopedcases) without SDW/SC order. If two-particle spectra,e.g, the optical conductivity, could be described withinthe same picture, this would serve as strong evidence forrelevance of large D (DMFT) approaches in this regime.Specifically, given that vertex corrections in the Bethe-Salpeter equations for the conductivity identically vanishin D = ∞,27 a proper description of the optical responseof SmO1−xFxFeAs within DMFT would imply negligiblevertex corrections, justifying use of DMFT a posteriori.

An optical study on La- and Sm-oxypnictides hasalready been carried out.28,29 While detailed spectralweight analysis remains to be done, characteristic strongcorrelation features are already visible: a small “Drude”peak, weak mid-infra red feature, and a slowly de-creasing contribution up to high energy, O(2.0) eV inLa-pnictides, and a power-law like decay of the re-flectance in Sm-pnictide, all testify to this fact, and ac-cord with their bad metallic resistivity. Onset of SCin SmO1−xFeAsFx results in reflectivity changes over abroad energy range,29 a characteristic signature of under-lying “Mottness”. Apart from these common features,there are quantitative differences in results from differ-ent groups.28,29 These can be traced back to the factthat while an effective medium approximation is invokedfor SmO1−xFeAsFx,

29 no such analysis is performed forLaO1−xFeAsFx.

28 Given the intrinsic polycrystal natureof the samples used by both, as well as the differencesin analysis, this may not be surprising. Optical workon single-crystal samples is thus highly desirable; thismay be close at hand. Nevertheless, with these caveats,these observations are strongly reminiscent of cupratesup to optimal doping,11 and constrain theories to un-

3

derstand SC as an instability of an incoherent, non-FLmetallic state. More recently, Yang et al.30 have indeedperformed a detailed analysis of optics for single crys-tals of Ba0.55K0.45Fe2As2. Slow (in energy, ω) decay inreflectivity and an anomalously large τ−1(ω) with sub-linear ω dependence, implying no FL quasiparticles, re-inforce similar features found in earlier optical resultsfor La- and Sm-oxypnictides. Further, an extraction ofthe α2F (Ω) bosonic “glue”30 reveals strong coupling tothe low-energy bosonic fluctuation modes (due to inter-orbital, coupled charge-spin density modes?). Observa-tion of non-FL quasiparticle features in optics impliesthat these bosonic modes are themselves strongly over-damped. Strong inelastic scattering from short-ranged,dynamical spin, charge and orbital correlations can in-deed lead to such behavior, as has been investigated moreextensively in the cuprate context. More studies are re-quired to check whether these features are generic forFe-pnictides: in view of incoherent features already seenin all investigated cases, we believe that this will indeedturn out to be the case.

In what follows, we compute the correlated bandstructure and optical conductivity of both La- andSmO1−xFeAsFx, extending previous work, where verygood semiquantitative agreement with the one-particlespectrum was found.25 We show how “Mottness” inthe Fe d-bands underpins the charge dynamics in Fe-pnictides. In particular, we show how an excel-lent theory versus experiment comparison for the one-particle spectral function (DOS) is obtained for dopedLaO1−xFeAsFx, and build upon this agreement to obtainvery good quantitative agreement with the reflectivity aswell. Armed with this agreement, we analyze these the-oretical results in detail and predict specific non-FL fea-tures that should be visible in future experimental work.Finally, we estimate the “glue function”, α2F (ω), andpropose that it should be understood as an electronic(multiparticle) continuum that can be interpreted as anoverdamped bosonic spectrum. We conclude with a briefqualitative discussion of its implications for SC.

III. CORRELATED BAND STRUCTURE

The bare one-electron band structure of both LaOFeAsand SmOFeAs used in this work was computed using thelinear muffin-tin orbital (LMTO) scheme31 in the atomicsphere approximation (ASA).32 As evidenced in severalstudies, the overall structures are remarkably similar inboth cases, confirming that the active electronic statesinvolve the carriers in the FeAs layers. Shown in Fig. 1,the orbital-induced anisotropies in the band structure aremanifest: the xy, 3z2−r2 bands are almost gapped at theFermi energy (EF ), while the xz, yz, x2 − y2 bands haveappreciable weight at EF . The only essential differencebetween the LDA DOS for La- and Sm-pnictides is thatin the latter case, due to larger chemical pressure (causedby the smaller size of Sm relative to La), the LDA band

−4.0 −2.0 0.0 2.0ω (eV)

0.0

0.3

0.6

ρσ x2 −y2 (ω

)

LaSm

0.0

0.3

0.6

0.9

ρσ 3z2 −

r2 (ω)

−4.0 −2.0 0.0 2.0 4.0ω (eV)

0.0

0.3

0.6

ρσ

xy (ω)

0.0

0.3

0.6

0.9

ρσ

xz,yz (ω)

FIG. 1: (Color online) Orbital-resolved LDA density-of-states(DOS) for the Fe d orbitals in LaOFeAs (solid) and SmOFeAs(dashed), computed using the LMTO method. Notice theoverall similarity between the two DOSs. This shows thatthe electronic states generically relevant to Fe-pnictides areFe d-band states.

width is slightly (O(0.5 eV)) larger. The one-electronpart of the Hamiltonian for Fe-pnictides is then given by,

H0 =∑

k,a,σ

(ǫk,a − ǫa)c†k,a,σck,a,σ , (4)

where ǫk,a label the five d bands, and ǫa denote the bandenergies. The inter-orbital splitting arises from the realcrystal field (of S4 symmetry in Fe-pnictides), which liftsthe five-fold degeneracy of the atomic d-shell. This givesthe two hole- and two electron-like pockets, as apparentlyobserved by de Haas van Alphen (dHvA) studies.33

However, a direct comparison between LDA resultsand the PES/XAS experiments (which must be consid-ered together when a comparison of the theoretical spec-tral function is attempted) shows substantial mismatchbetween theory and experiment. Related discrepanciesare found in optical studies,28 where the actual plasmafrequency, ωp, is 2 − 3 times smaller than the LDA pre-diction. Neither can the high (O(mΩ-cm)) resistivitybe understood within an almost band-like (free electron)picture. Also, the dHvA study33 reveals that the LDAbands need to be shifted by 0.2 eV to get a proper fitwith experiment. Further, the effective masses are en-hanced by a factor of 2 − 3 over their LDA values (thisagrees with the renormalization of the plasma frequencyfound in optics). Taken together, these features stronglysuggest sizable electronic correlations, which, moreover,are also exhibited by a host of other, known, correlatedmetals.11

Theoretically, while LDA (LDA+U) generically ac-counts for ground state properties of weakly (strongly)

4

correlated systems, their inability to describe excitedstates (and hence the charge and spin dynamics) incorrelated systems is well-documented.34 Marrying LDAwith DMFT opens the way toward resolving this short-coming of traditional band structure approaches, andLDA+DMFT has proven successful in describing phys-ical properties of various correlated materials in terms oftheir correlated electronic structures.34

The discussion above clearly shows that incorporationof strong, multi-orbital electronic correlations is a re-quirement for a proper description of Fe pnictides. Theinteraction part is given by,

Hint = U∑

i,a

nia↑nia↓ + U ′∑

i,a6=b,σ,σ′

niaσnibσ′

− JH

∑

i,a,b

Sia · Sib , (5)

where naσ = c†aσcaσ and Sa are the fermion numberand spin density operators for an electron in orbitala. We take U ≃ U ′ + 2JH , as is commonly knownfor TMO.34 The relevance of strong correlations in Fe-pnictides has been recognized by several authors.5,6,10

While these works explicate the important role of multi-orbital (MO) correlations (in particular, the sensitivity toJH),5 other works7 conclude that correlations are weak.This is a highly relevant, and open, issue in the field ofFe-pnictides and their SC: are they weakly correlated,itinerant metals, with a conventional, BCS like instabil-ity to SC, or are they strongly correlated metals, givingway to SC via a non-BCS-like instability? A detailedcomparison with extant experimental results should go along way toward resolving this important issue.

Here, guided by good success obtained in a theory-experiment comparison of one-particle spectra in our pre-vious study,25 we use LDA+DMFT to compute the de-tailed optical response in the “normal” state of La- andSm-based Fe-pnictides. We solve H = H0 + Hint withinmulti-orbital (MO) DMFT. In this study, MO-IPT isempolyed as the “impurity solver” to solve the impu-rity model of DMFT. Though not numerically “exact”,it has been shown to be quantitatively accurate for a widerange of correlated d-band materials.35,36 If only the FeAslayer states are relevant in Fe-pnictides,13 we expect ourwork to provide a generic picture of charge dynamics inFe-pnictides. We find excellent quantitative agreementwith both, the one-particle spectral data (PES/XAS) aswell as two-particle data (reflectance) for the La-basedFe-pnictide. Based on this, we argue that Fe-pnictidesshould be viewed as strongly correlated, MO systems,with incoherent low-energy behavior and describe the op-tical response of both La- and Sm-based Fe-pnictides indetail. Implications of our work for the high-Tc SC inFe-pnictides are touched upon, and intriguing similari-ties (and differences) with cuprates are highlighted.

Starting with the five Fe d-orbitals, we use MO-DMFTto extract the correlated spectral functions for the five d-orbitals. We choose values of U = 4.0 eV, JH = 0.7 eV

−4.0 −2.0 0.0 2.0ω (eV)

0.0

0.1

0.2

0.3

ρσ x2 −y2 (ω

)

x=0.1x=0.2x=0.3LDA

0.0

0.1

0.2

0.3

0.4

ρσ 3z2 −

r2 (ω)

−4.0 −2.0 0.0 2.0 4.0ω (eV)

0.0

0.1

0.2

0.3 ρσ

xy (ω)

0.0

0.1

0.2

0.3

0.4

ρσ

xz,yz (ω)

−4.0 −2.0 0.0 2.0ω (eV)

0.0

0.1

0.2

0.3

ρσ x2 −y2 (ω

)

La (x=0.2)Sm (x=0.2)

0.0

0.1

0.2

0.3

0.4

ρσ 3z2 −

r2 (ω)

−4.0 −2.0 0.0 2.0 4.0ω (eV)

0.0

0.1

0.2

0.3 ρσ

xy (ω)

0.0

0.1

0.2

0.3

0.4

ρσ

xz,yz (ω)

FIG. 2: (Color online) Top panel: Orbital-resolvedLDA+DMFT (solid, dashed and dot-dashed) and LDA (dot-ted) DOS for electron-doped LaOFeAs for three doping val-ues. The parameters are U = 4.0 eV, U ′ = 2.6 eV andJH = 0.7 eV. The drastic modification of the LDA spectrato almost totally incoherent character by large-scale dynam-ical spectral weight transfer is clearly visible. Bottom panel:Comparison between LDA+DMFT spectra for La- and Sm-based Fe-pnictides.

and U ′ ≃ (U − 2JH) = 2.6 eV, as employed in ourearlier work.25 These are shown in Fig. 2 for La-basedFe-pnictide, for three values of electron doping, so thatntotal =

∑

a na = (6 + x), with x = 0.1, 0.2, 0.3, alongwith the respective LDA DOS. Electronic correlations areseen to lead to dramatic and interesting modifications ofthe LDA spectra:

(i) the spectra describe an incoherent, non-FL metalfor each of the d-bands, with orbital dependent low-energy pseudogap features. Correspondingly, the imag-inary parts of the self-energies (not shown) show devi-

5

ations from the −ω2 form at small ω, being consistentinstead with a sub-linear ω-dependence, along with a fi-

nite value at EF (= 0), as also seen in our earlier work.25

(ii) In striking contrast to the LDA band struc-tures, the LDA+DMFT band structure shows that multi-orbital electronic correlations “self-organize” the spectralfunctions. While the xy-orbital DOS shows the maxi-mum itinerance, and has a shape distinct from the others,the much more localized xz, yz, x2 − y2, 3z2 − r2 orbital-DOS are seen to closely resemble each other with regardto their lineshapes. Dramatic spectral weight transfer(SWT) over large energy scales O(5.0) eV is also appar-ent in the results. In our MO-DMFT calculation, strong,incoherent inter-orbital charge transfer leads to dramaticspectral weight redistribution between the different d-orbital DOS. This is a characteristic also exhibited byother, correlated, MO systems,37 and points to the rele-vance of MO correlations in the Fe-pnictides.

(iii) There is no OS metallic phase for our choice ofparameters, and all d-orbital DOS cross EF . The DMFTresults are sensitive to changes in U, U ′ for fixed JH , andour results describe a metal very close to the OS-metallicone, which occurs for U ≥ 5.0 eV.7

Very similar features have been reported by us for Smpnictides in an earlier study.25 This is shown in the lowerpanel of Fig. 2. Given that the LDA bands for Sm-based Fe-pnictide are slightly wider than for La-basedFe-pnictides, we expect localization-like features to bemore enhanced in La-based Fe-pnictides within a Mott-Hubbard picture, as is indeed seen in the comparison.These results imply that strong, MO correlations mayhave a generic consequence of self-organizing the corre-lated spectra, an observation not readily seen in the LDADOS. This may have important consequences, and, forexample, could be of aid when one seeks to constructeffective correlation based models.

In Fig. 3, we compare our LDA+DMFT results withextant photoemission (PES) and X-ray absorption (XAS)data for LaO0.93FeAsF0.07.

38,39 Since only the five d-bands have been included in the LDA+DMFT, we re-strict ourselves to the energy window (which, however,is rather wide) −0.6 ≤ ω ≤ 1.2 eV around EF (thisis the region where only the five d-bands dominate inthe LDA). Clearly, excellent quantitative agreement withboth PES and XAS results is obtained. In particular, thelow-energy pseudogap is faithfully reproduced, as is thedetailed form of the lineshapes. Taken together with ourearlier results on SmO1−xFeAsFx,

25 this strongly sug-gests that the FeAs states with sizable d-band electroniccorrelations are a universal feature of Fe-pnictides.

In particular, a noteworthy fact is that for both Fe-pnictides, a low energy kink at approximately 15.0 −25.0 meV, along with a pseudogap, and stronglyasymmetric incoherent features at higher energies (at−0.28 eV in PES and at 0.6 eV in XAS) are clearlyresolved.38,39 Remarkably, our LDA+DMFT calculationreproduces all these features in excellent agreement withboth PES and XAS results. The low-energy kink is inter-

−0.6 −0.3 0.0 0.3 0.6 0.9 1.2ω (eV)

0

Inte

nsity

(ar

b. u

nits

)

LDALDA+DMFT (x=0.1)PES (x=0.07)XAS (x=0.075)

FIG. 3: (Color online) Comparison between the LDA+DMFTresult for LaO0.9FeAsF0.1 (solid line) and angle-integratedphotoemission (PES, diamonds)38 and X-ray absorption(XAS, vertical crosses)39 for LaO0.93FeAsF0.07. As inSmOFeAs,25 very good quantitative agreement is clearly seen.In particular, the low-energy kink at −15.0 meV in PES isaccurately resolved in the DMFT spectrum. Also notice thesubstantial improvement obtained by LDA+DMFT over theLDA result (dotted line).

preted as arising from low-energy, collective inter-orbitalfluctuations, as discussed earlier25 in detail.

Additionally, two more interesting features are also vis-ible from Fig 3. First, a good “fit” between the experi-mental and LDA values of the Fermi energy is achieved byshifting the LDA spectrum downwards by 0.15 eV. Theneed for such a shift of LDA DOS in this context hasalready been noticed earlier38 and attributed to correla-tion effects. Hitherto, their quantification has not beenundertaken. Here, this already arises selfconsistently inthe MO-DMFT from the MO-Hartree shift mentionedabove, bringing the LDA+DMFT value of EF in verygood agreement with experiment. Second, as we willestimate below in optical analysis, the effective plasmafrequency is reduced by a factor of 2 − 3 over its LDAvalue. This translates into an average effective enhance-ment of the band (LDA) mass as 2−3m0, where m0 is thebare LDA mass. Both these observations are completelyconsistent with the dHvA results, where very similar esti-mates for the band shift as well as the effective mass wereextracted.33 In addition, the latter is also consistent withthe 2 − 3-fold enhancement in the γ co-efficient of thelow-T specific heat in LaO1−xFePFx.

40 Taken together,these results constitute a consistent, quantitative ratio-nalization of basic one-particle responses in both (La-and Sm-based) pnictides.

6

IV. OPTICAL CONDUCTIVITY USING

LDA+DMFT

We now study the optical conductivity of both, dopedLaOFeAs and SmOFeAs using the LDA+DMFT prop-agators for all d-orbitals. In D = ∞, the computationof the optical conductivity simplifies considerably. Thisis because the irreducible vertex functions entering theBethe-Salpeter equation for the evaluation of the current-current correlation function vanish exactly in this limit.27

Thus, the optical response is directly evaluated as convo-lution of the DMFT propagators. For MO-systems, thegeneral expression for the real part of the optical conduc-tivity is given by

σ′ab(ω) =

2πe2~

V

∑

k

∫

dω′ f(ω′) − f(ω + ω′)

ω

× Tr[Ak(ω′ + ω)vk,aAk(ω′)vk,b] , (6)

with a, b labelling the various orbitals used in the DMFTcalculation. Ak(ω) is the one-particle spectral function,a matrix in the orbital sector, and vk,a = 〈k|Pa|k〉 is thefermion velocity in orbital a. The corresponding matrixelement of the momentum, Pa, weights the different tran-sitions, and is determined by the band structure. Esti-mation of the vk,a for Fe-pnictides is especially difficult,where, in addition to the d-states (which can be writ-ten in localized, Wannier-like basis sets), one also hasthe much more delocalized As p-states to contend with.Hence, we simplify our analysis by replacing this matrixelement by a constant, vk,a = 〈k|Pa|k〉 = va. Further, werestrict ourselves to intraband transitions.41 Both theseapproximations will turn out to be justified later. Withthese simplifications, the optical conductivity is writtenas

σ′ab(ω) = δa,bv

2a

2πe2~

V

∑

k

∫

dω′ f(ω′) − f(ω + ω′)

ω

× Ak,a(ω′ + ω)Ak,a(ω′) , (7)

where

Aa(k, ω) = −1

πIm

[

1

ω − ǫk,a − Σa(ω)

]

(8)

is the fully renormalized one-particle spectral functionfor orbital a, and Σa(ω) is the corresponding one-particleself-energy. The total reflectivity, R(ω) =

∑

a Ra(ω), canbe computed using

Ra(ω) =

∣

∣

∣

∣

∣

√

ǫa(ω) − 1√

ǫa(ω) + 1

∣

∣

∣

∣

∣

2

, (9)

with

ǫa(ω) = 1 +4πiσa(ω)

ω(10)

being the complex dielectric constant.Using the Kramers-Kronig (KK) relations, a detailed

analysis of the reflectivity, R(ω), and the optical conduc-tivity can be readily carried out for the normal state.In addition, an extended Drude analysis of the resultsshines light on the nature of the strong renormalizationscaused by MO electronic correlations. In particular, esti-mation of the frequency-dependent carrier lifetime, τ(ω),and effective mass, m∗(ω), for each orbital state yieldsmicroscopic information on the mechanism of incoherentmetal formation. Finally, the electronic “glue” responsi-ble for pairing is estimated therefrom as,17

Fa(ω) ≡ α2aFa(ω) =

1

2π

d2

dω2

(

ω

τa(ω)

)

(11)

whence microscopic information concerning the detailedstructure of the pairing interaction is obtained.

V. RESULTS FOR ELECTRODYNAMIC

RESPONSE

We now describe our results. In doing so, we adopt thefollowing strategy:

(i) we use our LDA+DMFT results for dopedLaOFeAs, which give a very good description of the one-particle spectral features, Fig. 2, as discussed above. Us-ing the DMFT propagators, we compute the intraband

optical conductivity of LaO1−xFeAsFx. Excellent quan-titative agreement with extant reflectivity data as mea-sured28 is obtained, and, building upon this agreement,we describe the optical response (i.e, optical conductiv-ity, dielectric constants, plasma frequency, ac penetrationdepth) in detail.

(ii) Given the observation that the electronic structureof the Fe-pnictides is determined by the electronic statesin the FeAs layers, we use our DMFT results for both La-and Sm-based Fe-pnictides to compute the normal stateelectrodynamics in the correlated metal. We provide thefirst theoretical estimates of the anisotropic carrier life-times and effective masses as a function of frequency.These results show, in accord with the incoherent metalclassification of single layer Fe-pnictides, that their nor-mal state cannot be described within FL theory.

In Fig. 4, we show the theory-experiment comparisonfor the reflectivity of LaO0.9FeAsF0.1. The experimen-tal result was taken from earlier work.28 Quite remark-ably, excellent semiquantitative agreement is clearly vis-ible for U = 4.0 eV,U ′ = 2.6 eV, and JH = 0.7 eVin LDA+DMFT. To highlight the importance of strongelectronic correlations, we have also plotted theoreticalresults for smaller, unrealistic values of U = 2.0 eV,U ′ = 1.3 eV and U = 1.1 eV, U ′ = 0.7 eV. Clearly,

7

0.0 1.0 2.0 3.0ω (eV)

0.0

0.2

0.4

0.6

0.8

1.0R

(ω)

exp (x=0.1)x=0.0 (U=4.0eV)x=0.1 (U=4.0eV)x=0.2 (U=4.0eV)x=0.3 (U=4.0eV)x=0.0 (U=2.0eV)x=0.0 (U=1.1eV)

FIG. 4: (Color online) Comparison between the experimen-tal reflectivity for LaO0.9FeAsF0.1

28 and the LDA+DMFTresults for ntotal = (6 + x). Very good quantitative agree-ment over the whole energy range, including the kink in therelectvity around 0.6 eV is clearly seen for U = 4.0 eV and x =0.1. Also, progressive disagreement between theory and ex-periment with decreasing U is clear: for U = 1.1, 2.0 eV, sub-stantial disagreement over the whole energy range is strongevidence for the strongly correlated character of the metallicstate in Fe-pnictides.

in all respects, the agreement gets worse with decreas-ing U, U ′, strongly supporting the strong correlation-based view. This excellent quantitative agreement withboth one- and two-particle spectra in the normal state ofLaO1−xFeAsFx encourages us to make a deeper analysisof transport properties for both La- and SmO1−xFeAsFx.

Using Eq. (7), we have computed the optical conduc-tivity, which is shown in Fig. 5 for all d-orbitals for bothFe-pnictides. Anisotropic responses, dictated both byLDA band-structure, as well as by correlation effects (seebelow) are clearly visible. A very interesting aspect of theresults is the observation of strong incoherent features inσa(ω): a Drude-like peak, with very small weight, existsonly in σxy(ω), and is even smaller, almost vanishing, inthe xz, yz optical response. For all other orbitals, theoptical response is totally incoherent, with distinct nonDrude contribution (also see the carrier mass/lifetime re-sults below). Large scale SWT across huge energy scalesO(4.0) eV with doping is also explicit in the results. Inthe MO-DMFT, this SWT is driven by the dynamical cor-relations associated with large on-site interactions U andU ′: the latter causes inter-orbital SWT on the observedscale, and is intimately linked with underlying “Mot-tness” in the MO model. The orbital resolved opticalconductivity shows a distinctly non-Drude component,along with a very slow decrease with increasing ω up tohigh energy, O(3.0) eV. This is observed clearly in Fig. 5for both pnictides. At higher energies, one expects both,

0.0 1.0 2.0ω (eV)

0

σ’x2 −

y2 (a.

u.)

x=0.1x=0.2x=0.3

0

σ’3z

2 −r2 (

a.u.

)

0.0 1.0 2.0 3.0ω (eV)

0

σ’xy (a.u)

0

σ’xz,yz (a.u.)

0.0 1.0 2.0ω (eV)

0

σ’x2 −

y2 (a.

u.)

La (x=0.2)Sm (x=0.2)

0

σ’3z

2 −r2 (

a.u.

)

0.0 1.0 2.0 3.0ω (eV)

0

σ’xy (a.u.)

0

σ’xz,yz (a.u.)

FIG. 5: (Color online) Top panel: Orbital-resolved opticalconductivity of LaO1−xFeAsFx for x = 0.1, 0.2, 0.3, withinLDA+DMFT. Apart from the very small quasi-coherent com-ponent in σxy(ω), all other orbital components exhibit inco-

herent non-FL responses, with clear wipe-out of the “Drude”response at low-energy. Bottom panel: Comparison betweenthe orbital resolved optical spectra for doped La- and Sm-based Fe-pnictides, showing very similar responses in bothcases.

the inter-orbital transitions, as well as those between thebands neglected within the DMFT to start contributingto σa(ω). Thus, one should expect to find good agree-ment with experiment up to ω ≃ O(3.0) eV. It is rathersatisfying to notice that precisely the above features, i.e,non-Drude low-energy part, a slowly decaying (in ω) con-tribution at higher energies, and large-scale SWT withdoping, have indeed been observed in Fe-pnictides. Theseimply that Fe-pnictides should be considered as stronglycorrelated systems, in the proximity of a MI state.6 Fromthe total optical conductivity, σ(ω) =

∑

a σa(ω), we have

8

estimated the average plasma frequency from the sumrule,

∫ ∞

0

σ(ω)dω =ω2

p

4π, (12)

yielding ωp = 0.76 eV, in excellent agreement with theexperimental estimate of 0.68 eV for LaO1−xFeAsFx withx = 0.1. This is a renormalization of a factor of 3 rela-tive to the LDA estimate. For comparison, we have alsocomputed ωp with reduced U, U ′, whence ωp increasessmoothly toward its LDA value (this is also visible fromthe reflectivity curves, where the kink moves to higherenergy with decreasing U, U ′).

Using the extended Drude parametrization allows usto estimate the frequency-dependent transport scatter-ing rate, τ−1(ω), as well as the dynamical mass, m∗(ω).In Figs. 6 and 7 (top panels), we show these quantitiesfor each orbital, a, as a function of doping. Distinctivenon-FL features are clearly evident. We emphasize thatthese results are valid in the symmetry unbroken metal-lic state, i.e, without SDW or SC order: this is the caseabove Tc in doped Fe-pnictides for x ≥ 0.1. Both τ−1

and m∗ show distinctly orbital-selective non-FL behav-iors. With the exception of the xy orbital, m∗(ω) forthe other (x2 − y2, xz, yz, 3z2 − r2) orbitals continuesto increase in a power-law-like fashion, or in a fashionconsistent with the onset of incoherent pseudogap be-havior,42 down to lowest energy. Correspondingly, therespective scattering rates clearly exhibit a sublinear ωdependence (with anisotropic, ω = 0 values) at low en-ergy. Interestingly, the DMFT optical spectra of bothLa and Sm-based pnictides show that the dominant low-energy “metallic” contribution comes from the dxz,yz,xy

bands, while clear pseudogap response is manifest in thedx2−y2,3z2−r2 channels: the latter are almost Mott local-ized. This observation is consistent with ARPES data,43

where the correlated spectral function is measured; ex-tant results show broadened “quasiparticle” bands onxy, yz, xz character crossing EF . In our DMFT, onlythe “more metallic” xy, yz, xz bands will then show upas quasicoherent bands crossing EF , while, for the pseu-dogapped x2 − y2, 3z2 − r2 bands, the extremely largescattering rates (with a pseudogap) will obliterate thesein ARPES.

It is instructive to analyze the differences betweenthe optical spectra for La- and Sm pnictides. In thelower panels of Figs. 5, 6, and 7, we explicitly showthese. Clearly, the anisotropic pseudogap features dis-cussed above are more pronounced for the La-based Fe-pnictide. To understand this difference, we recall thatthe LDA band widths for Sm-based Fe-pnictide are aboutO(0.5) eV wider than those for La-based Fe-pnictide, anobservation that is consistent with the higher chemicalpressure induced by the smaller (Sm) ion in the formercase. Hence, in our DMFT picture, the correspondinglynarrowed d orbitals for La-based Fe-pnictide will be closerto Mott localization vis-a-vis those for the Sm-based Fe-

0.0 1.0 2.0ω (eV)

0

1/τ x2 −

y2 (a.

u.) x=0.1

x=0.2x=0.3

0

1/τ 3z

2 −r2 (

a.u.

)

0.0 1.0 2.0 3.0ω (eV)

0

1/τxy (a.u.)

0

1/τxz,yz (a.u.)

0.0 1.0 2.0ω (eV)

0

1/τ x2 −

y2 (a.

u.) La (x=0.2)

Sm (x=0.2)

0

1/τ 3z

2 −r2 (

a.u.

)

0.0 1.0 2.0 3.0ω (eV)

0

1/τxy (a.u.)

0

1/τxz,yz (a.u.)

FIG. 6: (Color online) Top panel: Orbital-resolved scat-tering rates of LaO1−xFeAsFx for x = 0.1, 0.2, 0.3, withinLDA+DMFT. All orbital components exhibit incoherent non-FL responses, but the scattering rate for the xy orbital car-riers remains most metallic. For the others, the scatteringrates exhibit large values at ω = 0, in full accord with theemergence of low-energy pseudogap-like features in their cor-responding one-particle and optical lineshapes. Bottom panel:Comparison between the orbital resolved scattering rates fordoped La- and Sm-based Fe-pnictides, showing very similarresponses in both cases. Notice the enhanced low-energy in-coherence for the La-pnictide.

pnictide; this manifests itself in the observation of pseu-dogap signatures in τ−1

a (ω), m∗a(ω) for La pnictide, while

these are weaker for Sm pnictide. Apart from these ma-terial dependent differences, the aspect of large normalstate incoherence is clearly reflected in the results for both

pnictides.Finally, we remark that, while such extended Drude

analysis for the “single layer” Fe-pnictides remains tobe done with single crystals, a non-FL scattering rate

9

0.0 1.0 2.0ω (eV)

0mx2 −

y2 (a.

u.)

x=0.1x=0.2x=0.3

0m3z

2 −r2 (

a.u.

)

0.0 1.0 2.0 3.0ω (eV)

0

mxy (a.u.)

0

mxz,yz (a.u.)

0.0 1.0 2.0ω (eV)

0mx2 −

y2 (a.

u.)

La (x=0.2)Sm (x=0.2)

0

m3z

2 −r2 (

a.u.

)

0.0 1.0 2.0 3.0ω (eV)

0

mxy (a.u.)

0m

xz,yz (a.u.)

FIG. 7: (Color online) Top panel: Orbital-resolved dy-namical masses of LaO1−xFeAsFx for x = 0.1, 0.2, 0.3,within LDA+DMFT. All orbital components exhibit strongω-dependence at low energy, in line with the incoherent opti-cal responses. Only the mass for the xy orbital carriers seemsto approach a constant, correlation-enhanced value at low en-ergy. For the others, the dynamical masses exhibit strong ω

dependence up to low-energy, in full accord with the emer-gence of low-energy incoherent, or pseudogap-like features intheir corresponding one-particle and optical lineshapes. Bot-tom panel: Comparison between the orbital resolved dynami-cal masses for doped La- and Sm-based Fe-pnictides, showingvery similar responses in both cases. Notice the enhancedlow-energy incoherence for the La-pnictide.

is indeed extracted from the optical spectra for dopedBaFe2As2.

30 Based on our results, we predict that sim-ilar incoherence will characterize the normal state opti-cal response of the single-layer Fe-pnictides as well. Thesublinear-in-ω scattering rate is also consistent with thesublinear-in-T dependence of the dc resistivity in dopedFe-pnictides6 above Tc(x).

VI. DISCUSSION

The lack of any Drude component in the low-energyoptical response implies that the symmetry-unbrokenmetallic state above Tc in the doped Fe-pnictides is not aFL, in the sense that the one-particle propagators exhibita branch cut structure, rather than a renormalized polestructure, at low energies.

What is the microscopic origin of the non-FL fea-tures found in the DMFT solution? In MO systems, theorbital-resolved hopping matrix elements (diagonalizedin the LDA) are very directional, being sensitive func-tions of orbital orientation in the real structure. Further,the d bands are shifted relative to each other because ofthe action of the crystal field (of S4 symmetry in Fe-pnictides), and the six d electrons are distributed amongall d orbitals. In this situation, strong, MO correlationscause two, intimately linked changes:

(i) the static MO-Hartree shift, which depends uponthe occupations of each orbital, as well as on the inter-orbital U ′ and JH , renormalizes the on-site energies ofeach orbital in widely different ways. In particular, itcauses inter-orbital charge transfer between the variousd-orbitals, self-consistently, modifying their energies andoccupations. This effect is also captured in LDA+U ap-proaches. Specifically, the lower-lying orbital(s) in LDAare pushed further down by the MO-Hartree shift, theamount of which is determined by their occupation(s)and by the values of U ′, JH relative to their respectiveLDA band width(s), and to the bare crystal field split-ting in LDA.

(ii) More importantly, the dynamical correlations as-sociated with U, U ′, JH results in a large-scale transferof dynamical spectral weight. Small changes in the LDAband structure induced by (i) above (or by changes inexternal perturbations in general) induce large changes

in SWT, drastically modifying LDA lineshapes.(iii) Crucially, the renormalized Fermi energy is com-

puted selfconsistently in DMFT by requiring consistencywith the Freidel-Luttinger theorem; i.e, EF is computedby demanding that the renormalized Fermi surface en-closes the total number of d-electrons in the system, aslong as no broken symmetry states are considered.

Generically, as U ′ increases (JH is usually fixed for agiven d-state), the lower-lying subset of d-orbitals gets se-lectively Mott localized; the metallic phase is then the OSmetal found recently in various contexts.35,44 Once thisselective localization occurs within the DMFT, the lowenergy physical response is governed by strong scatteringbetween the effectively Mott-localized and the renormal-ized, itinerant components of the matrix spectral func-tion. The problem is thus effectively mapped onto aFalicov-Kimball type of model, as has been noticed inearlier work.35,44 Within DMFT, the itinerant fermionspectral function then shows a low-energy pseudogappedform, while the “localized” spectral function shows apower-law fall-off as a function of energy, as long as therenormalized EF is pinned to the renormalized orbital

10

energy of the localized orbital(s). This is understoodfrom the mapping of the corresponding impurity modelto that of the “X-ray edge”,45 where the orthogonalitycatastrophe destroys FL behavior.46 The spectral func-tions then exhibit asymmetric continuua (branch cuts) atlow energy, instead of symmetric Abrikosov-Suhl Kondoresonance features, and the metallic phase is not a FL.This incoherence is mirrored in the optical responses inD = ∞, since the optics is entirely determined by the fullDMFT one-particle Green functions in this limit. Thisis entirely consistent with our results, and strongly sug-gests that effects akin to the orthogonality catastrophe,arising from strong, incoherent, inter-orbital scattering,produce the incoherent metal behavior in the “normal”state of Fe-pnictides.

This loss of one-particle coherence corresponds to dras-tic reduction of the carrier kinetic energy, and implies theirrelevance of one-particle terms in the RG sense.47 Inter-estingly, this clears the way for two-particle instabilitiesto take over as T is lowered; they pre-empt the non-FLbehavior from persisting down to T = 0. This could beeither:

(i) the q = (π, 0) SDW for x < xc. This is in factfurther reinforced by the near-nested character of theelectron- and hole Fermi pockets in the Fe-pnictides, al-ready visible in weak coupling RPA calculations.14,48 TheSDW instability at x = 0 is then interpretable as an in-stability in the particle-hole channel, aided by near FSnesting. That AF-SDW survives the lack of such nestingfeatures away from x = 0 is an observation which favorsa strong coupling scenario. An intermediate-to-strongcoupling version of a similar instability to exactly thesame state results from the LDA+DMFT work of Hauleet al.5 A strong coupling method as used here will yieldthe same AF instability, in analogy with one-band Hub-bard model studies, where the AF ordered state at half-filling remains the Neel ordered state, evolving from theSlater type to the Mott-Hubbard type as U/t is increased.Though weak-coupling RPA approaches, valid for smallU/t, can indeed give the correct magnetic ground statesfor U ≪ W (= 2zt), the normal state incoherence charac-teristic of Fe-pnictides, which are now identified to be inthe intermediate coupling limit (U ≃ W ), is beyond theirscope. This is because the incoherent part of Ga(k, ω) iscompletely neglected there, and one deals with propa-gators having only a coherent QP pole structure at lowenergy. This is valid for small U , but not for U > W , thenon-interacting bandwidth.

(ii) An unconventional SC for x > xc. This corre-sponds to an instability in the particle-particle channel.Independent of the precise order parameter symmetry, amatter of intense debate,3,6,8,9 the observation of smallsuperfluid density, short-coherence length, large 2∆/kTc

ratios, along with large energy scale changes in optical re-sponse across Tc, are all hallmarks of a SC closer to theBose condensation limit, rather than the weak couplingBCS limit.

Our findings are in accord with Si et al.6 and Haule et

al.,5 who, along with Baskaran,10 were the first to rec-ognize the strong coupling aspect of Fe-pnictides. Onthe other hand, several works address both AF and SCwithin weak coupling scenarios.14,48 In the latter, the“normal” state above Tc is a FL metal, and SC arisesvia a BCS like instability of this MO (multi-band) FL.In the strong correlation limit, however, the “normal”state is itself incoherent, and so there are no FL-like QPsto pair up into usual BCS-like cooper pairs. In otherwords, SC must then arise as an instability of an unusualmetallic state without FL QPs, or, to rephrase it, directlyfrom overdamped, collective multi-particle modes. Thefrequency-dependence of the self-energy is important inthe latter case, and consideration of the instability of suchan incoherent state to the SC state leads to a stronglyfrequency-dependent SC gap function. In such a “strongcoupling” picture, the physical response functions in theSC state are controlled by both, the symmetry of the SCorder parameter, as well as the strong, ω, T dependentdamping originating in the incoherent “normal” state.Observation of features such as the absence of the Hebel-Slichter peak in the NMR T−1

1 (T )3 and the large scalemodification of the optical spectral weight across Tc

29 inthe Fe-pnictides are strongly suggestive of such a strongcoupling scenario. These features are again reminiscentof cuprates,19,49 where the role of “normal” state (non-FL) incoherence is well documented. To make their rolein the Fe-pnictides more explicit, we use our optical re-sults to show the pairing “glue” function for Fe-pnictidesin Fig. 8. In line with the conclusions extracted fromthe analysis of the incoherent optical conductivity above,F(Ω) shows interesting features. We find

(i) a two “peak” structure, with both peaks stronglybroadened by incoherent scattering. We emphasize thatthis broad, two-peak structure arises from the incoher-ent one-particle propagators, and represents a multipar-ticle electronic continuum. This could be interpreted asoverdamped “bosonic” modes, if one associates the shortranged, strongly coupled spin-orbital modes with the in-coherent, short-distance components of the conventionalbosonic modes used in the itinerant descriptions. It is in-teresting to notice that similar features, namely, strongnon-FL signatures in optics, as well as a strongly damped,two-peaked, low-energy continuum is also characteristicof high-Tc cuprates up to optimal doping.19,49 Thus, ourresults show that SC arises from an incoherent normalstate, and that coupling carriers to an incoherent elec-tronic continuum pairs them up in the SC. Hence, we pro-pose that the underlying Mott-Hubbard physics knownto underpin the anomalous responses in cuprates is alsoat work in the Fe-pnictides.

(ii) All d-orbital components show up in Fa(ω), albeitanisotropically. This supports a MO (multi-band) originfor SC pairing in Fe-pnictides. Several interesting fea-tures are visible upon close scrutiny: the “glue” functionis larger for those (orbitals) bands which are closer toMott localization, as seen by comparing the respectivecurves in Fa(Ω) and ρa(ω), while the most “itinerant”

11

0.0 0.1 0.2ω (eV)

0

Fx2 −

y2 (a.

u.)

x=0.1x=0.2x=0.3

0

F3z

2 −r2 (

a.u.

)

0.0 0.1 0.2 0.3ω (eV)

0

Fxy (a.u.)

0

Fxz,yz (a.u.)

0.0 0.1 0.2ω (eV)

0

Fx2 −

y2 (a.

u.)

La (x=0.2)Sm (x=0.2)

0

F3z

2 −r2 (

a.u.

)

0.0 0.1 0.2 0.3ω (eV)

0

Fxy (a.u.)

0

Fxz,yz (a.u.)

FIG. 8: (Color online) Top panel: Orbital-resolved “glue”functions for LaO1−xFeAsFx for x = 0.1, 0.2, 0.3, withinLDA+DMFT. All orbital components contribute, albeitanisotropically, at low energy. This suggests that the instabil-ity to superconductivity in Fe-pnictides will involve multiplebands. The lack of any sharp feature shows that the “glue”is an overdamped electronic continuum (see text). The two-peak structure at low energy is reminiscent of what is seen inhigh-Tc cuprates, and has a multi-orbital character. Bottompanel: Comparison between the “glue” functions for dopedLa- and Sm-based Fe-pnictides, showing similar incoherentelectronic continuum features in both cases.

xy band has the smallest contribution to F . In otherwords, there is pronounced orbital-induced anisotropyin the pairing “glue” function. This should imply anorbital-dependent, multiple gap SC, which may not beinconsistent with recenttheoretical indications.13,48 Inother words, suppression of one-particle coherence (in theDMFT propagators, Ga(k, ω)) makes two-particle (col-lective) processes more relevant, by enhancing the re-spective collective susceptibilities in the charge and spin

channels. In the doped case, lack of nesting features inthe electron-like and hole-like Fermi sheets suppressesthe q = (π, 0) SDW found for x = 0, leaving multi-band SC as the only relevant two-particle instability.In such a MO-SC, opening up of an orbital-dependentgap should be linked to orbital-dependent coupling ofthe carriers to the electronic “glue” via Fa(ω): remark-ably, a very recent indeed shows exactly this feature.50

This agrees qualitatively with our expectations from astrongly orbital-dependent “glue” function, as derivedabove. It is also interesting to note that a recent ARPESstudy on (Sr/Ba)1−xKxFe2As2

51 also finds quasiparti-cle kinks in the quasiparticle dispersion in the bindingenergy range from 15 meV to 50 meV, again reminis-cent of what is observed in high-Tc cuprates,52 but alsoin three dimensional, correlated systems like SrVO3.

53

The latter fact points to its connection to underlyingMottness characteristic of a correlated system. In ourDMFT, we recall that the orbital-resolved DOS showlow-energy kinks precisely in the 15 − 50 meV range(see Fig. 2). These are attributed to low-energy, collec-tive, inter-orbital fluctuations,25 and we have shown thatthe resulting LDA+DMFT DOS gives excellent quan-titative agreement with the normal state kink-like fea-ture in angle integrated PES.54 Based on our calcula-tion, we propose that an ARPES measurement done for(La/Sm)O1−xFeAsFx should uncover kink structure(s) ina similar low-energy (20 meV) range.

Whether additional density-wave instabilities may alsointerfere with SC is an interesting issue. Two-band modelstudies48,55 do suggest additional channels involving ne-matic or current instabilities: whether they can seriouslycompete with SC or remain sub-leading in the full five-band model, is still unclear. In any case, our analysisshows that consideration of all d-orbitals is necessary fora proper microscopic description of multi-band SC in Fe-pnictides. We leave the detailed consideration of the in-stability of the MO, incoherent metal found here to a MOSC for future consideration.

A brief discussion of the similarities and differencesbetween cuprates and Fe-pnictides is in order at thispoint. The predominance of normal state incoherence incuprates is well-documented by now,56 and its link withthe strong on-site electronic correlations is well known.In particular, strong non-FL features (with a branch cutform of the one-particle propagator, G(k, ω)) in ARPES,dc and ac transport, magnetotransport, as well as in mag-netic fluctuations bears this out in a very remarkable way.Up to optimal doping, the cuprates are not describablein terms of the Landau FL picture, and the role of strongMottness and short-ranged AF spin correlations in thiscontext is recognized. In the underdoped cuprates theinstability to the d-wave SC state is quite far from theweak-coupling BCS variety: large-scale redistribution ofspectral weight across Tc as well as strong vortex liq-uid fluctuations, short SC coherence lengths, and smallsuperfluid stiffness, among other observations, put thistransition closer to the Bose condensed limit.

12

Many of these features, like normal state incoherence,along with pseudogap behavior in both charge and spinfluctuations, are also characteristic of Fe-pnictides. Fur-ther, in single-layer Fe-pnictides, the superfluid densityis small, the upper-critical fields, Hc2, is high, the SCin-plane coherence length is short (ξpair < 20 A)51 andappreciable redistribution of spectral weight across Tc

is visible. These similarities then strongly support thestrong correlation-based view for Fe-pnictides as well.

There are noticeable differences between the cupratesand the Fe-pnictides. First, SC in cuprates most likelyinvolves a single, strongly p − d hybridized band withstrong electronic correlations, and arises upon doping aMott-Hubbard insulator with AF order. In contrast, SCin Fe-pnictides is most probably of the multi-band vari-ety, and arises upon doping a very bad metal, which maybe close to a Mott-Hubbard insulating state.6 This meansthat the microscopic electronic “glue” for pairing in bothcases will be quite different. Nevertheless, in view of theunderlying relevance of Mottness in both cases, one mayexpect more similarities in the structure of the SC insta-bility in both cases. It is more likely, given the explicitMO situation in Fe-pnictides, that additional compet-ing instabilities may be at work. In particular, the ne-matic and current instabilities, which have been invokedas competitors of d-wave SC in cuprates,57 might alsoplay a role here.48,55 As remarked early by Baskaran,10

it is possible that being able to think about situationswhere these competing instabilities could be suppressedcan push up the SC Tc in Fe-pnictides to even highervalues.

Our analysis here has been carried out for thesymmetry-unbroken metallic state of the Fe-pnictides.At low T , this incoherent state becomes unstable to eitherAF-SDW order with QSDW = q = (π, 0) or to SC order,depending on x, though some studies also suggest co-existence of the two orders. As discussed in literature,6,58

the AF-SDW state involves strong geometric frustration(GF) in the inter-orbital hopping matrix elements. Char-acterization of magnetic fluctuations has indeed been car-ried out, both in the strong coupling (J1 − J2 Heisen-berg model),59 as well as within weak coupling HF-RPAwork.9 Observation of features akin to high-Tc cupratesfound experimentally, as discussed above, put the Fe-pnictides into the strongly correlated category, thoughnot so strongly correlated as cuprates, which are dopedMott insulators. While the fact that we are able to de-scribe both the one-electron responses and the reflectivityof La-pnictide strongly suggests that LDA+(MO)DMFTis adequate for a quantitative description of these cor-

relation effects, we expect geometrical frustration effectsto become relevant at low T < TN(x), at least for theSDW phase. In view of the fact that the spatial ex-tent of correlations is short in GF systems, we remarkthat the dynamical effects associated with these short-ranged magnetic correlations may not change the aboveresults substantially. They will certainly not modifythem qualitatively; indeed, one would expect that ad-ditional consideration of the dynamical effects of suchshort-ranged, strongly frustrated couplings (J1, J2 withJ2/J1 ≃ O(1))58 would push the almost totally incoher-ent normal state derived above even more toward inco-herence.

VII. CONCLUSION

In conclusion, we have studied the normal state elec-trodynamic response of two Fe-pnictides, La1−xFeAsFx

and SmO1−xFeAsFx, within the LDA+DMFT formal-ism. Armed with the very good agreement with one-electron responses (PES and XAS) between experimen-tal and LDA+DMFT spectral functions, we also find verygood quantitative agreement with published reflectivitydata on LaO1−xFeAsFx. Building upon these agree-ments, we have studied the optical response in detail.We find an incoherent optical response, with strongly ω-dependent effective masses and transport lifetimes (scat-tering rates) at low energy. These features are linkedto the incoherent, near pseudogap-like features found inthe orbital resolved, one-particle spectral functions inLDA+DMFT. The very good agreement found betweentheory and experiment for both pnictides is strong ev-idence for the relevance of “Mottness”, i.e, to proxim-ity of the normal state of Fe-pnictides to a correlation-driven Mott-Hubbard insulating state. Further, basedon an estimation of the “glue” function, we proposethat this should be understood as arising from a multi-particle electronic continuum, which could also be in-terpreted in terms of an overdamped “bosonic” glue.Finally, all d-orbitals should contribute to SC pairing,albeit anisotropically, and one should have an orbital-dependent, multiple-gap SC. It is an interesting taskto investigate the low-T instabilities of this incoherentmetallic state: we leave this for the future.

L.C. thanks S.-L. Drechsler for discussions. S.L. ac-knowledges ZIH Dresden for computational time. M.S.L.thanks the MPIPKS, Dresden for financial support.

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