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Neutron-Inelastic-Scattering Peak by Dissipationless Mechanism in the s++ -wave State in Iron-based Superconductors

We investigate the neutron scattering spectrum in iron pnictides based on the random-phase approximation in the five-orbital model with a realistic superconducting (SC) gap, Delta=5meV. In the normal state, the neutron spectrum is suppressed by large inelastic quasi-particle (QP) scattering rate gamma* ~ Delta. In the fully-gapped s-wave state without sign reversal (s++), a hump-shaped enhancement appears in the neutron spectrum just above 2Delta, since the inelastic QP scattering is prohibited by the SC gap. That is, the hump structure is produced by the dissipationless QPs for Ek<3Delta. The obtained result is more consistent with experimental spectra, compared to the results of our previous paper with Delta=50meV. On the other hand, both height and weight of the resonance peak in the fully-gapped s-wave states with sign reversal (s+-) are much larger than those observed in experiments. We conclude that experimentally observed broad spectral peak in iron pnictides is created by the present"dissipationless mechanism"in the s++ -wave state.


I. INTRODUCTION
Since the discovery of superconductivity in iron pnictides with high transition temperature (T c ) 1 , substantial experimental and theoretical works have been performed to clarify the mechanism of superconductivity. The superconducting (SC) gap in many iron pnictides is fully-gapped and band-dependent, as shown by the penetration depth measurement 2 and the angle-resolved photoemission spectroscopy (ARPES) 3,4 . The fullygapped state is also supported by the rapid suppression in 1/T 1 (∝ T n ; n ∼ 4 − 6) below T c 5-7 . On the other hand, P-doped Ba122 8 and LaFePO 9,10 show the nodal line behavior (T -linear dependence) in penetration depth measurements. In these compounds, A 1g symmetry pairing states with accidental nodes are expected theoretically. 11,12 In iron pnictides, the intra-orbital nesting of the Fermi surface (FS) between the hole-and electron-pockets is expected to induce the antiferromagnetic (AF) fluctuations. Taking this fact into account, the fully-gapped sign-reversing s-wave state (s ± -wave state) mediated by the AF fluctuation had been predicted 13,14 . On the other hand, we have demonstrated that the orbital fluctuation mediated fully-gapped s-wave state without signreversal (s ++ -wave state) is realized by the inter-orbital nesting, by taking the electron-phonon interaction into account. 15,16 In the latter scenario, the close relation between T c and the crystal structure revealed by Lee 17 , e.g., T c becomes the highest when the As 4 cluster is regular tetrahedron, is automatically explained 16 . Moreover, the latter scenario is consistent with the large SC gap on the z 2 -orbital band in Ba122 systems 16 , observed by bulk-sensitive laser ARPES measurement 18 . In addition, the orthorhombic structure transition and the corresponding shear modulus softening is well explained theoretically 19 . In newly discovered K x Fe 2 Se 2 with T c ∼ 30K, in which only electron-pockets exist, orbital-fluctuation-mediated fully-gapped s ++ -wave state 20 or spin-fluctuation-mediated nodal d x 2 −y 2 -wave state 20,21 had been predicted theoretically.
Thus, it is important to clarify the sign of the SC gap via phase-sensitive experiments. Nonmagnetic impurity effect offers us useful phase-sensitive information. In iron pnictides, the SC state survives against high substitution of Fe sites by other element (more than 10%). [22][23][24][25] These results support the s ++ -wave state since the s ± -wave state is very fragile against impurities, similar to other unconventional superconductors. 26,27 Moreover, impurity driven crossover from s ± -wave state to s ++ -wave state had been discussed in Refs. 12,21 .
Another promising method is the neutron scattering measurement: As discussed by Monthoux and Scalapino in Ref. 28 , existence of the resonance peak at a nesting wavevector Q is a strong evidence for AF fluctuation mediated superconductors with sign reversal [28][29][30][31] . The resonance occurs under the condition ω res < 2∆, where ω res is the resonance energy and ∆ is magnitude of the SC gap. The sharp and large resonance peak has been observed in many AF fluctuation mediated unconventional superconductors, like high-T c cuprates 32-34 , CeCoIn 5 35 , and UPd 2 Al 3 36 . The measurements of phonon spectral function for |ω| 2∆ would also be useful. 37 Neutron scattering measurements for iron pnictides had been performed 38-43 after the theoretical predictions 44,45 . Although clear peak structures were observed in FeSe 0.4 Te 0. 6 39 , BaFe 2−x Co x As 2 40,42 and Ca-Fe-Pt-As 43 , these weights are much smaller than those in high-T c cuprates and CeCoIn 5 . Moreover, the resonance condition ω res < 2∆ is not surely confirmed since it is difficult to determine the value of ∆ accurately. For example in BaFe 1.85 Co 0.15 As 2 , ω res is observed as 10meV in neutron scattering measurement. 40 In this material, 2∆ = ∆ h + ∆ e , where ∆ h(e) denotes magnitude of gap on the hole (electron) pocket. It was estimated as 11meV according to an earlier ARPES measurement. 46 However, ∆ h +∆ e was estimated as 7meV by a recent measurement of the specific heat 47 . We also obtain ∆ h + ∆ e = 7meV from a recent penetration depth measurement in Ref. 48 , by the linear interpolation for x = 0.14 and x = 0.17.
In our previous paper 49 , we revealed that for ∆ = 50meV a prominent hump structure free from the resonance mechanism appears in neutron scattering spectrum just above 2∆ in the s ++ -wave state. The hump structure originates from the dissipationless quasi-particles (QPs) free from the inelastic scattering in the SC state. Although the broad spectral peak observed in iron pnictides was naturally reproduced based on the s ++ -wave state, rather than the s ± -wave state, used model parameters were not realistic.
In this paper, we investigate the dynamical spin susceptibility χ s (ω, Q) based on the five-orbital model 13 for both s ++ -and s ± -wave states, by improving the method of numerical calculation. Using a realistic parameter ∆ = 5meV, the obtained results are more realistic than our previous results for ∆ = 50meV. 49 In the normal state, χ s (ω, Q) is strongly suppressed by the inelastic QP damping γ * , which is large due to the strong correlation. However, this suppression is released in the SC state since the inelastic damping γ * disappears for |ω| 3∆. This "dissipationless mechanism" induces a hump-shaped enhancement in χ s (ω, Q) in the s ++ -wave state, just above 2∆ till ∼ 3∆. In the s ± -wave state, very high and sharp resonance peak appears at ω res < 2∆ even in the case of ∆ = 5meV. We demonstrate that the broad spectral peak observed in iron pnictides is naturally reproduced based on the s ++ -wave state, rather than the s ± -wave state.
In Sec. III.C, we comment that Nagai et al. 50 fail to reproduce the spectral gap in the two-particle Green function 2∆ and are therefore unreliable. In appendix, we introduce the similar hump structure of the neutron scattering spectrum in CeNiSn much below the Kondo temperature T K . This compound is called Kondo semiconductor since the hybridization gap ∆ opens much below T K , while it is an incoherent metal with large inelastic scattering above T K . This is another example of the hump structure by the "dissipationless mechanism", since the inelastic scattering is prohibited by the singlet gap ∆.
We note that numerical results are improved from results in the first version of preprint 51 , in which the value of γ * (ǫ) for 3∆ < ǫ < 4∆ was incorrect in our previous numerical calculation.

A. Method of calculation
Now, we study the 10 × 10 Nambu BCS Hamiltonian H k composed of the five-orbital tight-binding model and the band-diagonal SC gap introduced in ref. 26 . Then, the 10 × 10 Green function is given bŷ where ω n = πT (2n + 1) is the fermion Matsubara frequency,Ĝ (F ) is the 5 × 5 normal (anomalous) Green function, andΣ k is the self-energy in the d-orbital basis.
In this paper, we assume that the magnitude of the SC gap is band-independent; |∆ ν | = ∆.
Here, we have to calculate the spin susceptibility as function of real frequency. Numerically, it is rather easy to use the Matsubara frequency method and the numerical analytic continuation (pade approximation). 44,45 In the present study, however, we perform the analytical continuation before numerical calculation in order to obtain more reliable results. The irreducible spin susceptibility in the singlet SC state is given by 31 where x + = x+ω, k + = k+q, l i = 1, · · · , 5 represents the d-orbital, and A (R) represents the advanced (retarded) ll ′ (x, k))/2πi are one particle spectral functions. Here, we divideχ 0R(A) into the "Hermite part"χ 0′ and "non-Hermite part"χ 0′′ , Then,χ 0′ andχ 0′′ are expressed aŝ We explain that the non-Hermite part satisfies the relationχ 0′′ (ω, q) = 0 for |ω| < 2∆ at T = 0. Now, we assume ω > 0. ρ G,F ll ′ (x, k) = 0 is given for |x| < ∆ since the SC gap opens. Then, in order to satisfy both In order to obtain the finite value ofχ 0′′ (ω, q) in eq. (5), all three inequalities must be satisfied for some x. Considering the third inequality, the first two inequalities are restricted to They are satisfied for some x only when ω > 2∆. There- In the present numerical study, we calculate exactlŷ χ 0′′ using eq. (5), and calculate approximatelyχ 0′ using the Hermite part of eq. (6) in Ref. 49 . Using this method, we can calculate accurately the imaginary part of the spin susceptibility as we will discuss later.
B. Inelastic QP damping rate γ * In strongly correlated systems, χ s (ω, q) is renormalized by the self-energy correction. We phenomenologically introduce a band-diagonal self-energy as z · ImΣ R k (ǫ) = iγ * (ǫ)1, where z ≡ m/m * is the renormalization factor. First, we estimate the QP damping in the normal state from the experimentally observed conductivity. From the Nakano-Kubo formula, the conductivity is given by (0), where γ(0) ≡ γ * (0)/z is the "unrenormalized" damping at zero energy, and N ν (0) and v ν are the density of states (DOS) and the Fermi velocity of the ν-th FS, respectively. Using the fiveorbital model, we obtain ρ ≈ (2.0γ(0)[meV]) µΩcm for the inter-layer spacing c = 6Å and ρ ≈ (2.8γ(0)[meV]) µΩcm for c = 8Å. 26,27 In table I, we show the Tdependence of ρ estimated by fitting the experimental data below ∼ 100K 52-54 , and the inelastic damping γ(0) is derived from the theoretical relation between ρ and γ. For example, in BaFe 1.85 Co 0.15 As 2 (c ≈ 6Å), the unrenormalized inelastic damping γ(0) is estimated as 3.7T , which is comparable to that in over-doped cuprates.
Then, we derive the (ǫ, T )-dependences of the "renormalized" inelastic scattering: In the presence of the strong spin and orbital fluctuations, the damping follows the approximate relation γ * (ǫ) ≈ b(T + |ǫ|/π) according to spin (orbital) fluctuation theories. 12,55 According to Table I, we obtain b ∼ 1.9 in BaFe 1.85 Co 0.15 As 2 if we assume z ∼ 0.5. In the present study, we use a larger value b = 2.5. Note that the result is not so sensitive to the value of b.
C. Hump structure in Imχ s due to dissipationless Here, we explain an intuitive reason why the QP is "dissipationless" for |ω| < 3∆ at zero temperatures. 30 In Fig. 1 (a), we show an inelastic scattering process, in which a QP at k is scattered to k − q, with exciting a particle-hole (p-h) pair (k ′ + q, k ′ ). Since a QP in the SC state cannot exist in the thin shell |ω| < ∆, the particlehole excitation energy E p−h is always larger than 2∆. Since the energy of the final state E k−q is also larger than ∆, the inelastic scattering is prohibited when E k ≤ 3∆. Thus, the relationship γ * (ω) = 0 for |ω| < 3∆ is obtained. Form this relation, the peak of the DOS at ω = ∆ for the isotropic SC gap remains to be sharp.
Then, the dissipationless QPs in the SC state produce the hump-shaped enhancement in the spin spectrum. In the normal state, Imχ s has no gap structure, and it is suppressed by the inelastic QP damping γ * induced by the strong correlation. In the SC state as illustrated in Fig. 1 (b), Imχ s has the p-h excitation gap 2∆. Since the QP is dissipationless for |ω| < 3∆ in the SC state, the suppression in Imχ s (ω) is released just above the excitation gap ω 2∆ so as to form a hump structure. For this reason, a prominent hump structure appears in Imχ s (ω, Q) just above 2∆ till ∼ 3∆ in strongly correlated s ++ -wave superconductors.

III. NUMERICAL RESULT
A. Spin susceptibility at the nesting vector Q = (π, π/8) Figure 2 shows the obtained Imχ s (ω, Q) at the nesting vector between the hole-and electron-pockets Q = (π, π/8): We fix T = 1meV hereafter, since the obtained results are insensitive to the temperature for T ≤ 3meV. In the normal state with γ * 0 = 20(15)meV, the Stoner factor is α St = 0.950(0.959) for U = 1.32eV. In the SC states with γ * s = 10meV, α St = 0.956 (0.982) in the s ++ -wave (s ± -wave) state for U = 1.32eV. In the s ± -wave state, α St increases due to the coherence factor. Inversely, α St in the s ++ -wave state decreases due to absence of coherence factor. As shown in Fig. 2, in the normal state with γ * 0 = 20meV, the peak position of Imχ s is about 20 − 25meV, which is consistent with experimental result in BaFe 1.85 Co 0.15 As 2 40 . Thus, the value of Imχ s in the normal state with γ * 0 = 15meV is overestimated. A broad hump structure appears in the s ++ -wave state at ω 2∆ even in the case of ∆ = 5meV, and its overall shape is consistent with experimental results 39,40 . We had neglected the energy-dependence of ∆ in the previous study 49 . However, in reality, the SC gap ∆ will be cut off when the energy of the ν-th band ǫ ν k measured from the Fermi energy exceeds the characteristic energy scale of the pairing interaction. To take this fact into account, we introduce a Gaussian cutoff ∆ ν k = ∆ ν exp{−[ǫ ν k /∆E] 2 } following Refs. 50,56 . We put ∆E = 20meV, which correspond to the Fe ion optical phonon frequency ω D ∼ 20meV employed in the orbital fluctuation theory. 12,15 .
When the cutoff is applied in the s ++ -wave state, the hump structure becomes more prominent as shown in Fig. 2 (a). We confirm that the obvious hump appears over the normal state even with γ * 0 = 15meV. The enhancement of hump structure originates from the increment of the Stoner factor by introducing the cutoff, from α St = 0.956 to 0.965 for U = 1.32eV.
On the other hand, in the s ± -wave state, very high and sharp resonance peak appears at ω res < 2∆ even in the case of |∆| = 5meV as shown in Fig. 2 (b). This result is apparently inconsistent with experimental results. In order to explain the experimental result by the s ± -wave state, large inhomogeneity would be required, although the s ± -wave state is fragile against inhomogeneity. The height of the resonance peak exceeds 100eV −1 for a(0)γ * s = 0.5meV, while it diverges for a(0) → 0 if k-meshes are fine enough. Imχ s is slightly suppressed by considering the cut off, ∆E = 20meV.
We also study the spectra for both s ++ -and s ±wave states with cutoff for U = 1.33eV: In Fig. 2(c) and 2(d), we show the results for the normal state with γ * 0 = 15meV (α St = 0.965) and γ * 0 = 20meV (α St = 0.956). We also show results for the s ++ -wave state with γ * s = 10meV (α St = 0.971), and s ± -wave state with γ * s = 10meV (α St = 0.984). We note that the effect of multiband on Imχ s , which was discussed in Ref. 40 , is automatically included in our calculation. By increasing U from 1.32eV to 1.33eV, the hump structure in the s ++ -wave state is more enhanced. Also, the resonance peak in the s ± -wave state develops, and ω res shifts to lower energy. In Figure 3, we confirm that the hump in the s ++wave with ∆ = 5meV state is enhanced as the value of U increases. Thus, the hump becomes prominent as system comes close to the AF order.
In this paper, we have calculated χ s introduced in Eq. 8. To obtain the value of spin susceptibility χ neu observed in neutron measurements, we have to take the spin magnetic moment (= 1µ B ) and the factor of spin degeneracy. Its z-component is χ neu
In the s ++ -wave state, Imχ s (ω, Q) = 0 for |ω| < 2∆ at T = 0 as we discussed in Sec. II. This relation is correctly satisfied in the present method if we put a(0) → 0 in Eq. (10). In the present method, we perform the numerical calculation ofχ 0′′ using Eq. (5) exactly. In fact, in Fig. 4(a) and (b), we verify that the spectral gap of Imχ s (ω, Q) is well reproduced in the present method with a(0)γ * s = 0.5meV, demonstrating the superiority of the present method to the previous method in Ref. 49 . In the case of s ± -wave state, we obtain Imχ s (ω, Q) ∝ δ(ω − ω res ) for |ω| < 2∆ if the numerical calculation is performed accurately.
In the present paper, we calculateχ 0′′ in eq. (5) exactly, whileχ 0′ is calculated approximately using eqs. (6) and (7) in Ref. 49 . We consider this is justified since we had verified that the present "approximated RPA" is reliable in our previous paper 49 : In Fig. 1 (b) of Ref. 49 , we had performed the "exact RPA calculation" for bothχ 0′ andχ 0′′ with ∆ = 400meV, and confirmed that overall behavior of Imχ s (ω, Q) is well reproduced by the present approximated RPA.
Here, we comment on the q dependence of Imχ s (ω, q) around q = Q. In our two-dimensional model 13 , it is difficult to discuss the q dependence of Imχ s (ω, q) because q dependence of Imχ s (ω, q) is drastic even in the normal state, which is inconsistent with the neutron scattering measurements.
Recently, Nagai et al. 50 had calculated the neutron scattering spectrum using the method proposed in Ref. 49 , and claimed that (i) hump structure in the s ++ -wave state is smeared when γ * ∼ 10meV and ∆ ∼ 5meV compared to the case of ∆ > 25meV, and (ii) resonance peak in the s ± -wave state becomes very low and broad. Moreover, they had also claimed that (iii) one can distinguish between the s ++ -wave and the s +− -wave states from the spectrum at q = (π, π).
First, we explain that (i) and (ii) are incorrect statements based on their inaccurate numerical calculation. First, their result fails to reproduce the spectral gap of Imχ s for ω < 2∆ as shown in Fig. 5 (a). (One can prove rigorously that Imχ s = 0 for ω < 2∆ at T = 0.) Second, the peak position of the result of Nagai et al. 50 is about 2∆, while it must be higher energy (∼ 3∆). In the s ± -wave state, the resonance peak should be δ functional structure when a(0) in eq. (10) is enough smaller than ∆. Thus, the low and broad resonance peak of Nagai et al. 50 is far from the exact behavior of the resonance peak. In Nagai's results, fine structures in Imχ s (ω, q) seem to be inappropriately smeared in both s ++ -and s ± -wave states.
Next, we comment on the claim (iii). They pointed out the spectrum in the s ± -wave state with ∆ = 5meV are different from that in the s ++ -wave state with ∆ = 25meV. Here, we show the results of both s ++ -and s ±wave states in Figs. 5, for (b) |∆| = 5meV and (c) |∆| = 25meV. Since both spectra are almost identical, we cannot distinguish between the s ++ -and s ± -wave states by the spectrum at the wave vector q = (π, π) for the same ∆. This result is reasonable because sign of the SC gap is preserved through the (π, π) shift for both the s ++ -and s ± -wave states. Although claim (iii) is based on their numerical result in which the hump of the s ++wave state appears only for ∆ 25meV, the prominent hump appears in the s ++ -wave state with ∆ = 5meV in our improved numerical results as shown in Fig. 2(a) and (c). Thus, we conclude it is impossible to distinguish between the s ++ -and s ± -wave states with ∆ = 5meV.

IV. CONCLUSION
We have studied the dynamical spin susceptibility χ s (ω, Q) in iron-based superconductors for both s ++and s ± -wave states, by developing more accurate numerical method and introducing the high-energy dependence of the SC gap. 49 In the s ++ -wave state, the dissipationless QPs for |ω| < 3∆ produce a prominent humpshaped enhancement in χ s (ω, Q) just above 2∆ till ∼ 3∆. This "dissipationless mechanism" is unrelated to the resonance. The peak energy of the hump will shift to lower energy if we consider the band-dependence and/or the anisotropy of the SC gap, as we discussed in Ref. 49 .
On the other hand, in the s ± -wave state, very high and sharp resonance peak appears at ω res < 2∆. In order to explain small and broad peaks observed in Refs. 39,40 as the resonance peak in the s ± -wave state, sufficient inhomogeneity or small SC volume fraction would be required. However, the s ± -wave state is fragile against inhomogeneity. We concluded that the small and broad spectral peak observed in iron pnictides is naturally reproduced based on the s ++ -wave state in the absence of inhomogeneity, rather than the s ± -wave state.
In the Comment on the present paper written by Nagai and Kuroki on arXiv 57 , the authors repeated their claim "smallness of the hump in the s ++ -wave state" based on the "old method" that was first developed in Ref. 49 . In Sec. III, however, we actually obtained large hump using the "new method", which is mathematically superior to the old method. This discrepancy originates from the calculation method as well as the numerical accuracy, not from the detail of model parameters, as we discussed in our Reply on arXiv 58 . Imχ s (ω). Readers can find more detailed explanations in the original paper 62 .  Figure 6 (b) shows the obtained single-particle spectrum A(ω). For U = 0, the hybridization gap in A(ω) is ∆ = 0.62. For U = 4, ∆ is renormalized to 0.35 at T = 1/16 (< T K ), while the gap is smeared out by thermal fluctuations above T K 62 . At T = 0, inelastic QP scattering is suppressed by the hybridization gap, such that γ * (ω) = 0 for |ω| < 3∆ 63 in analogy to Fig. 1 (a). Figure 6 (c) shows Imχ s (ω) for U = 4. In the metallic state at T = 1/4 (≫ T K ), Imχ s (ω) shows a gapless metallic behavior. In the semiconducting state at T = 1/16 (≪ T K ), in contrast, it shows a spectral gap ∆ s and the relation ∆ s ≈ 2∆ ≈ 0.7 is recognized. At the same time, large hump structure emerges around ω ∼ 3∆. Because of the absence of spin resonance mechanism, its natural explanation is the reduction in the inelastic QP scattering (γ * (ω) = 0 for |ω| < 3∆), as we discussed in Fig. 1 (b). We must stress the hump structure in Fig. 6 (c) is obtained exactly in the DMFT, by including both the self-energy and vertex corrections. Therefore, experimental and theoretical studies in CeNiSn strongly support the idea of "hump structure in the s ++ -wave state" given in Fig. 2 (a), that is obtained by the RPA by introducing the inelastic QP scattering γ * (ω) phenomenologically.