Anomalous isotope effect in iron-based superconductors

The role of electron-phonon interactions in iron-based superconductor is currently under debate with conflicting experimental reports on the isotope effect. To address this important issue, we employ the renormalization-group method to investigate the competition between electron-electron and electron-phonon interactions in these materials. The renormalization-group analysis shows that the ground state is a phonon-dressed unconventional superconductor: the dominant electronic interactions account for pairing mechanism while electron-phonon interactions are subdominant. Because of the phonon dressing, the isotope effect of the critical temperature can be normal or reversed, depending on whether the retarded intra- or inter-band interactions are altered upon isotope substitutions. The connection between the anomalous isotope effect and the unconventional pairing symmetry is discussed at the end.

c where α is the exponent for the isotope effect. If the dominant interaction is electron-phonon in nature, theoretical calculations give α = 1/2. In the extreme opposite, if the pairing is completely driven by electron-electron interactions, the critical temperature should not change with isotope substitutions and the corresponding exponent is α ≈ 0. In realistic superconductors, we expect the isotope exponent to be in-between. Note that, in unconventional superconductors, the phonon-mediated interactions are insufficient to explain the pairing mechanism and it is of crucial importance to study the interplay between electron-electron and electron-phonon interactions [24][25][26][27] . For instance, even when the pairing mechanism is electronic origin, dispersions observed in angle-resolved photoemission spectroscopy manifest distortions upon isotope substitutions [13][14][15][16][17][18]27 . The isotope effect observed in iron-based superconductor [28][29][30][31][32][33][34] seems to tell a more complicated story. For instance, a strong isotope effect by iron substitution 28 is found in SmFeAs(O, F) and (Ba, K)Fe 2 As 2 , almost as large as that in conventional superconductors. On the contrary, inverse isotope effect 29 is spotted in (Ba, K) Fe 2 As 2 with different isotope substitutions. Later, it was proposed that the isotope substitutions may give rise to structural change 35 and further complicate the story. On the theoretical side, Yanagisawa et al. 36 proposed a multi-band and multi-channel model to explain the possibility of observing the inverse isotope effect. However, Bussmann-Holder and Keller 37 commented that an inversion of the exponent cannot occur upon iron isotope substitutions. The controversies about the isotope effect of the iron-based superconductor are still on. And, it is of crucial importance to clarify the subtle role of the electron-phonon interactions in iron-based superconductors.

Results
Instantaneous and retarded interactions. Motivated by the controversy, we investigate the competition between electron-electron and electron-phonon interactions by the unbiased renormalization-group (RG) method. Due to the retarded nature of the phonon-mediated interactions, the energy dependence must be included. The minimal approach to include both simultaneous and retarded interactions can be accomplished by the step-shape approximation [38][39][40][41] as shown in Fig. 1(a), where g i and  g i represent (instantaneous) electronic interactions and (retarded) phonon-mediated ones. The energy scale for the retarded interactions is set by the Debye frequency ω D . Our RG analysis reveals that the pairing mechanism is dominated by the electronic interactions g i . But, the retarded interactions  g i also grow under RG transformation and become relevant in low-energy limit. Inclusion of these subdominant interactions leads to anomalous isotope effect. The isotope exponent α can be extracted numerically from RG flows in weak coupling. It is quite remarkable that the sign of the exponent α sensitively depends on whether the inter-and/or intra-band interactions are altered by isotope substitutions. Multi-band model. To illustrate how the RG scheme works, we start with a five-orbital tight-binding model for iron-based superconductors with generalized on-site interactions,  Fermiology is important in the multi-band superconductors. The electron doping x is related to the band filling = + n x 6 ( = n 10 for completely filled bands) here and the Fermi surface at = .
x 0 1 is illustrated in Fig. 1(b). There are five active bands: two hole pockets centered at (0, 0) and another hole pocket centered at (π, π) while two electron pockets located at (π, 0)and (0, π) points 43 . To simplify the RG analysis, we sample each pocket with one pair of Fermi points (required by time-reversal symmetry). This is equivalent to a four-leg ladder geometry with quantized momenta as shown in Fig. 1(b). In the low-energy limit, the effective Hamiltonian 44-46 is captured by five pairs of chiral fermions with different velocities. The RG equations for all couplings can be found in Methods.
Step-like interaction profile for simultaneous and retarded interactions. A sharp step is assumed at the Debye frequency ω D . (b) Fermiology of the five-band model = .
x 0 1. These Fermi surfaces are well sampled by five pairs of Fermi points, equivalent to the four-leg geometry with quantized momenta (dashed lines).
www.nature.com/scientificreports www.nature.com/scientificreports/ Pairing mechanism. By integrating the two sets of RG equations numerically, we found all couplings are well described the scaling ansatz 47 ,  Fig. 2(a). Meanwhile, by Abelian bosonization 45,46 , the signs of c ij from numerics lead to sign-revised (between electron and hole pockets) s ± -wave pairing, agreeing with the previous functional RG study 48 . Note that these exponents are rather robust within the doping range where the same Fermiology maintains. What about the phonon-mediated interactions? As clearly indicated in Fig. 2(b), the RG exponents for  c 11 ,  c 22 are roughly 0.6, much smaller than the dominant electronic interactions, showing the pairing mechanism is electronic origin. However, since the RG exponents are positive, the retarded interactions also grow under RG transformation. These subdominant phonon-mediated interactions can lead to anomalous isotope effect as explained in the following. Extracting isotope exponent. Numerical results for the two-step RG indicate the same superconducting phase as described in previous paragraphs but the isotope exponent α can be extracted numerically. Under RG transformation, the critical temperature satisfies the scaling form,

Discussion
To extract the isotope exponent, we study how the cutoff length scale l c varies with different Debye frequencies due to isotope substitutions. In weak coupling, we found that g i are much larger than  g i . Thus, Δ c has very weak dependence on ω D and the first term can be ignored. The contribution from the second term is shown in Fig. 4. , depending on both the bandwidth and the rescaled Debye frequency, is now quite complicated. The RG analysis alone is not sufficient to obtain α in a quantitative fashion. However, we recently found that the effective Hamiltonian at the cutoff length scale is well captured by mean-field theory (not yet published). In principle, one can combine RG and mean-field approaches together to compute the isotope exponent in intermediate coupling more accurately. www.nature.com/scientificreports www.nature.com/scientificreports/ In the end, we discuss the recent discovery of superconductivity in FeSe/STO systems 49,50 . We emphasize that our current approach includes fermiology, electron-electron interactions, and electron-phonon interactions within only the superconducting(SC) layers. One crucial assumption is the profile of the mediated electron-phonon interactions can be captured by the step function. The RG scheme built upon this approximation works as explained in the manuscript. However, according to the recent literatures in FeSe/STO systems [49][50][51][52][53][54][55][56] , to include the non-SC (SrTiO 3 ) layers we need to devise a new theoretical approach which is beyond our model at this point. The profile of the electron-phonon interactions arisen from non-SC layers is probably not captured by the simple step function anymore. One needs to find out the interaction profile generated by the non-SC layers first so that one can devise the RG scheme accordingly. This is going to be an interesting and challenging topic to explore in the future.

RG equations.
The interactions between these chiral fermions fall into two categories 39  www.nature.com/scientificreports www.nature.com/scientificreports/ Note that we separate the intra-band and inter-band couplings for clarity, i.e. ≠ i j in the above RG equations. In fact, the separation is necessary because we shall see later that inter-band and intra-band couplings play different roles in the low-energy limit. In addition, = f 0 ii and =  f 0 ii to avoid double counting.